BRS: A very common misunderstanding of how science works
In "Are the foundations stable"
Code:
physicsforums.com/showthread.php?t=429435
EnergyLoop asks
We are constantly finding new information that does not fit the current model, but by adding a new constant or variable into the equations it repairs the problem, but gives us new things to look for such as dark matter and dark energy. I can’t help thinking about Aristotle’s crystalline spheres and the Earth centered universe, this was a simple concept initially, until the motion of the planets was realized, then it became a complex mathematical model to try to explain this motion, Copernicus then simplified the problem, and removed the complexity.
I hope we are not heading in that direction again, is the foundation sound, equations are build on previously established equations does anyone re-examine these?
This seems to have become a very common refrain, particularly since the recent discovery of an accelerating expansion. (Cosmologists quickly adjusted; the general public did not!) I'd actually like to see a PF FAQ somewhere which addresses this, mentioning some of the same points I've made so often in the past, including:
All scientific knowledge is provisional. Constantly finding new information and constantly trying to fit new data or theoretical speculations into the body existing well-established theories is precisely what scientists do on a day-to-day basis. The fact that scientific theories can be tested by comparing quantitative predictions with quantitative data, and the fact that scientific theories are constantly up for revision is what makes science the most powerful tool known to man for the discovery and organization of knowledge about the natural world.
Scientific knowledge consists of a vast body of experimental/observational data plus the terminology, notation, and theories we use to interpret the data. We make theories and we make predictions from theories using mathematical reasoning, and you need to know the appropriate mathematical background to understand the theories.
When new data cannot be fit into existing theories, scientists look for explanations. First and foremost, a careful examination of the possibility that the data collection or analysis contained some subtle systematic bias or other flaw. If that fails, then one searches for the change to the theories which is "the least possible".
The public seems to generally misunderstand the nature of scientific advances: they should be astonished not by how much has changed, but by how little, even in such an extreme revelation as the "accelerating Hubble expansion". That is, one benefit of knowing the data and the theories is that you can appreciate how "introducing nonzero Lambda" is actually the smallest possible change to the theories. Also, the data hasn't changed, only our interpretation. Making a minimal change means that only a very small portion of our interpretation/understanding of the universe changed as the result of that particular "cosmological revolution".
To repeat: the public seems to generally misunderstand the nature of scientific revolutions. Newtonian gravitation was never "discarded", it is still used, and more often than gtr, which is a bit harder to work with. If and when a successful theory of quantum gravity appears, gtr will still be used (most likely) because the new theory will be (most likely) a bit harder to work with. Similar remarks hold for non-relativistic physics, and various specific theories which are known to be "wrong" but are still useful for limited purposes.
In "Dark Matter Distribution Around Galaxies"
Code:
www.physicsforums.com/showthread.php?t=429521
RLutz asks
Is there any reason why galactic black holes might have something to do with dark matter creation? The squashed beachball of dark matter sort of looks like what I would expect say field lines coming out of a pulsar to look like or something.
Ditto Chalnoth, plus a reminder that a black hole of mass M gravitates just like any star of mass M; unless you are very close to it you won't encounter the strong portions of the exterior field which result from the fact that the hole is so much more compact than the star.
In "Black Holes are Tears in Space"
Code:
www.physicsforums.com/showthread.php?t=429376
I`m not a scientist, i`m actually a 3D Artist. I just have a lot of faith in science unfortunately didn`t have the attention span to pay attention enough in high school and even more tragic is that my university didn`t offer any science courses!
Actually, that is tragic. Due to the global financial crisis, at least one state university in the U.S. has just eliminated its science majors.
Could it be possible that a black hole is a tear in space? It seems like it could be a way of explaining why some say you could travel through a black hole or worm hole and wind up somewhere else. If space itself was really in a shape we couldn't comprehend then maybe a tear in one place could wind up opening in an entirely different place. Does that make sense?
It's vaguely reminiscent of various possibilities discussed in serious physics like wormholes or curvature singularities, but much too vague to make much sense in a scientific discussion. So the best short answer is probably: a black hole is a region of spacetime characterized by the presence of an event horizon, which you can think of as an imaginary surface which you can fall through, but once you do, you can't ever emerge from behind the horizon, at least not into the same external region of spacetime in which you started. A very good book for poets which IMO can enable persons with only a high school science education to actually understand this, sort of, from (good!) pictures, is Geroch, General Relativity from A to B, University of Chicago Press.
JDoolin, whose "knowledge" of cosmological models is apparently based upon Wikipedia, not textbooks plus graduate level schoolwork, is arguing with Chalnoth and others about the Milne chart for the Minkowski vacuum, claiming that this chart is "inequivalent" to the cartesian chart. Of course that hinges on the meaning of "inequivalent"; gtr is however based upon Lorentzian geometry, and in Lorentzian geometry, any chart covering a region U in a spacetime (M,g) is just as good as any other.
In "velocity of gravity wave"
Code:
physicsforums.com/showthread.php?t=429166
spikenigma seems to think the European Gravity Field and Steady-State Ocean Circulation Explorer project has something to do with comparing the speed of gravitational and EM radiation! Needless to say, the investigators say something very different: from
Code:
www.esa.int/esaLP/ESAJJL1VMOC_LPgoce_0.html
GOCE will be gathering data or around 20 months to map the Earth's gravity field with unprecedented accuracy and spatial resolution. The final gravity map and model of the geoid will provide users worldwide with a well-defined data product that will lead to:
A better understanding of the physics of the Earth's interior to gain new insights into the geodynamics associated with the lithosphere, mantle composition and rheology, uplift and subduction processes.
A better understanding of the ocean currents and heat transport.
A global height-reference system, to serve as a reference surface to study topographic processes and sea-level change.
Better estimates of the thickness of polar ice-sheets and their movement.
One can use the next section to compute whether or not GOCE will be able to track ballistic missile submarines but in any case, it will be clear to anyone here, I think, that GOCE has nothing to do with "speed of gravity".
A-wal expresses in a memorable way one key element of Simple Physics crackpottery:
A-wal said:
You think I don’t understand the concepts just because I don’t know how to speak your language.
...
Would you watch your favorite dvd in binary code? The code is necessary but no one cares what it looks like.
... I really don’t want to think like physicists do. I want to get this straight in my head and still think the way I do. I don’t know how you can do it like that.
Translation:
satire said:
I refuse to learn to reason scientifically about scientific issues. But I insist upon exercising my constitutional right to make incontestable statements concerning scientific issues any time, any place. My way of muddled "thought" [sic] has just as much "validity" [sic] as all your hi-falutin equations. I don't need no stinkin' math!
He also offers a memorable anti black hole rant (based on willful ignorance, one might say):
A-wal said:
I don’t mind if I’m wrong. My ego isn’t tied up in this and I have nothing to prove. I find it difficult to accept what I don’t understand and I’m not convinced by what I’ve been told. How the hell can an object that can never reach the horizon from any external perspective ever cross the horizon from its own perspective? Is not just the light from those objects that’s frozen. How could it be if they could always escape? They’re moving slower and slower through time relative to you because time in that region is moving slower and slower relative to you. If the time dilation/length contraction go up to infinity then no given time can ever long enough and no distance can ever be short enough locally if it’s infinitely length contracted from a distance! Are those inertial coordinates you use to describe an object crossing the event horizon even relative? Does it take into account the fact that you’re constantly heading into an ever increasingly sharpening curve?
Multiple misunderstandings of analysis of Hagihara observers
In
Code:
www.physicsforums.com/showthread.php?t=431407
several posters seem to be trying answer this question:
How does the tidal tensor (electroriemann tensor) distort a small sphere of test particles in circular orbit around an isolated massive object?
The easiest way to analyze this is to study an appropriate frame field associated with the Hagihara observers, who move in circular orbits in planes parallel to the equatorial plane, where we are of course in the exterior region of the Schwarzschild vacuum. (We can ask the same question and proceed the same way for suitable observers in the Kerr vacuum, but that situation is more complicated in several ways.)
The good news is that some participants appear to be trying to learn about Hagihara observers. The bad news is that they are reading a Wikipedia article (I myself wrote an early version of that particular article, but we should presume that the current version has been trashed by years of edits by persons who didn't know what they were doing, or didn't care) instead of good textbooks. Thus, everyone participating in that thread (as of 23 September 2010) is badly confused on many many points, e.g.
"frame": in the WP article, frame means frame field, not "local coordinate chart" or whatever these posters are trying to do to apply the EP on a curved spacetime over a local neighborhood rather than in the tangent space to a single event,
none of the participants appear to understand the distinction between components wrt a frame field and components wrt a coordinate basis,
the WP article Lut Mentz cites gives frame field components, not coordinate basis components,
acceleration vector, expansion scalar, shear tensor, and vorticity tensor all refer to a timelike congruence (not neccessarily geodesic, although many books make that assumption for simplicity, and the Hagihara observers are geodesic observers in the equatorial plane),
the timelike unit vector in a frame field defines a timelike congruence, but the frame field also involves three unit spacelike vector fields, with all four being mutually orthogonal at each event,
the posters appear to be confusing a congruence having nonzero vorticity tensor with a spinning frame field,
the Hagihara observers are only geodesic observers in the equatorial plane; their world lines correspond to circular orbits parallel to the equatorial plane so obviously most of these require some acceleration to exist,
to quote the components of a tensor, it helps to adopt a frame and to quote the components wrt the frame, but note that one naturally derives first a spinning frame in which \vec{e}_2 points radially outward, and only later finds the nonspinning frame,
small spheres of Haghihara observers in the equatorial plane shear as they orbit, but don't change volume (vanishing expansion scalar!),
if one adopts the nonspinning frame, then Lense-Thirring precession shows up (in fact you can derive an exact version of the Lense-Thirring precession formula using this approach!); the nonspinning frame of the Hagihara observers slowly spins wrt the frame of a distant star, so to speak.
For what it is worth, you can derive a frame field adapted to Hagihara observers by starting with the frame of static observers
<br />
\begin{array}{rcl}<br />
\vec{e}_1 & = & \frac{1}{\sqrt{1-2m/r}} \; \partial_t \\<br />
\vec{e}_2 & = & \sqrt{1-2m/r} \; \partial_r \\<br />
\vec{e}_3 & = & \frac{1}{r} \; \partial_\theta \\<br />
\vec{4}_4 & = & \frac{1}{r \sin(\theta)} \; \phi<br />
\end{array}<br />
in the \vec{e}_4 direction, at each event, by an undetermined boost, with the boost parameter depending only on r. Then require that the acceleration of the new frame vanish in the equatorial plane. This gives an ODE for the boost parameter as a function of r which can easily be solved. The result is
<br />
\begin{array}{rcl}<br />
\vec{f}_1 & = & \frac{1}{\sqrt{1-3m/r}} \; <br />
\left( \partial_t + \sqrt{\frac{m}{r}} \; <br />
\frac{1}{r \, \sin(\theta)} \; \partial_\phi \right) \\<br />
\vec{f}_2 & = & \sqrt{1-2m/r} \; \partial_r \\<br />
\vec{f}_3 & = & \frac{1}{r} \; \partial_\theta \\<br />
\vec{f}_4 & = & \frac{\sqrt{1-2m/r}}{\sqrt{1-3m/r}} \;<br />
\left( \sqrt{\frac{m}{r}} \; \partial_t +<br />
\frac{1}{r \, \sin(\theta)} \; \partial_\phi \right)<br />
\end{array}<br />
(Recall that the static frame and the chart are only defined on r > 2m; notice that the Hagihara frame is only defined on r>3m, at best!) Then by construction
<br />
\nabla_{\vec{f}_1} \vec{f}_1 = 0<br /> in the equatorial plane (off this plane, the acceleration is nonzero!), the expansion scalar vanishes, the only independent nonzero component of the shear tensor is
<br />
\sigma_{24} = \frac{-3}{4} \sqrt{\frac{m}{r^3}} \;<br />
\frac{1-2m/r}{1-3m/r}<br />
while the only independent nonzero component of the vorticity tensor is
<br />
\omega_{24} = \frac{-1}{4} \sqrt{\frac{m}{r^3}} \;<br />
\frac{1-6m/r}{1-3m/r}<br />
The tidal tensor is
<br />
\begin{array}{rcl}<br />
E_{22} & = & \frac{-2m}{r^3} \; \frac{1-3m/2/r}{1-3m/r} <br />
\approx \frac{-2m}{r^3} \; \left( 1-\frac{3m}{2r} \right) \\<br />
E_{33} & = & \frac{m/r^3}{1-3m/r} <br />
\approx \frac{m}{r^3} \; \left( 1 + \frac{3m}{r} \right) \\<br />
E_{44} & = & \frac{m}{r^3}<br />
\end{array}<br />
All these expressions are refer to the frame \vec{f}_1, \ldots \vec{f}_4, and are only valid in the equatorial plane. And the acceleration vector, expansion scalar, shear tensor, and vorticity tensor all refer to the timelike unit vector field \vec{f}_1, whose integral curves are the world lines of the Hagihara observers.
So small spheres of inertial observers orbiting very nearly in the equatorial plane are sheared parallel to that plane, but maintain constant volume.
To study precession, you should introduce an undetermined secular rotation about \vec{f}_3, with the rate of rotation depending only on r, and demand that the Fermi derivatives of \vec{f}_2, \ldots \vec{f}_4 with respect to \vec{f}_1 should vanish in the equatorial plane. This gives an ODE for the rotation rate which you can solve, finding that the cumulative angle of rotation (at each event along the world line of a equatorial Hagihara observer) is
<br />
\psi = t \, \sqrt{\frac{m}{r^3}} \; \sqrt{1- 3 m/r}<br />
(The Lense-Thirring precession formula given in many textbooks is only an approximation of the exact result.)
With respect to the new, nonspinning frame, all the tensors just mentioned have different components from what we found for the first frame, in which the "principle axes" are seen to very slowly spin wrt the spatial vectors of the nonspinning frame. But the tensors are defined in terms of \vec{f}_1 which is shared by the two frames, and they are three dimensional tensors, so their traces and quadratic invariants (and higher order invariants) will neccessarily agree regardless of which frame you compute the components in!
You can compare all this with the Lemaitre frame appropriate for studying the physical experience of observers falling in freely and radially. Their tidal tensor wrt the Lemaitre frame is m/r^3 \; \operatorname{diag}(-2,1,1). So:
A small sphere of test particles released from rest inside Fr. Lemaitre's spaceship elongates radially and compresses orthogonally, while keeping constant volume, so forming a prolate spheroid; if Fr. Lemaitre looks out the window, he can see that the long axis of the prolate spheroid is aligned parallel to the direction toward the massive object.
A small sphere of test particles released from rest inside Dr. Hagihara's spaceship behaves similarly, but compresses slightly more than expected orthogonally to the equatorial plane, and slightly less radially, so forming a triaxial ellipsoid. If he looks out the window, he can see that the long axis of this triaxial ellipsoid is parallel with the direction of the massive object. In addition, assuming Dr. Hagihara's spaceship is gyrostabilized, as he keeps returning to "1 January" in his orbit, over time he notices a very gradual precession wrt the distant stars of where he is in his orbit on 1 January by his clock, and he also observes his spaceship to be very very slowly spinning as it orbits.
The tidal and precession effects have very different characteristic time scales, however--- the participants in the thread in question have not yet recognized this.
If Dr. Hagihara looks out the window, he can see that his spaceship, which is gyrostabilized, is nevertheless very slowly spinning wrt the massive object, and also wrt "the distant fixed stars". Looking at neighboring spaceships (also inertial and initially motionless wrt him), he sees that they appear to be very slowly shearing orthogonal to the equatorial plane (because the ships further out are moving more slowly in their orbits, as per Kepler) and that their trajectories are slowly swirling about his ship in the sense of nonzero vorticity.
All of these things can be studied in various limits and are consistent with Newtonian expectation in a suitable slow motion weak gravity limit.
If I am somehow making this sound hard, that is not my intent. This is not hard. It does however involve learning several new concepts (new for most people, or at least not yet understood by most people) and one needs to keep all these things straight: coordinate charts, frame fields, congruences, tensors defined on spacetime independent of any congruence, effectively three-dimensional tensors defined wrt a specific congruence, kinematic decomposition of a congruence, components of a tensor, invariants of a tensor, when orthogonal hyperslices do and do not exist, etc.
In
Code:
www.physicsforums.com/showthread.php?t=431572
the OP appears to be enthused by the recent widely publicized claims of Nikodem J. Poplawski which are based on elementary misunderstandings as I explained elsewhere in the BRS, so the short answer there is that Poplawski doesn't yet understand the theory of Lorentzian manifolds sufficiently well to avoid mistakes, and his claims are based upon such mistakes. It is possible that the OP might also be vaguely referring to notions of "baby universes" being born inside black holes, which is an intriguing speculative suggestion, but not one which is currently very well supported even by theoretical arguments, and of course not at all by observation!
In "e.p. implies no gravitational shielding?; Feynman?"
Code:
www.physicsforums.com/showthread.php?t=432327
Ben Crowell asks how/why gtr forbids gravitational shielding. I haven't yet had a chance to read that thread, but it reminds me of an amusing observation I made a decade or so ago in a UseNet post: there is a large class of static minimally coupled massless scalar field (mcmsf) solutions, in which the spacetime is a curved Lorentzian manifold, but which exhibit "zero gravity", e.g. static observers can hover over a region where the energy of the mcmsf is concentrated without any need to fire their rocket engines!
See "BRS: Static Axisymmetric "Gravitationless" Massless Scalar Field Solutions"
Code:
www.physicsforums.com/showthread.php?t=433793
(Mcmsf solutions are exact solutions in which the only contribution to the stress tensor comes from the field energy of a massless scalar field, which is minimally coupled to curvature.)
Exercise: read Post #1 in the above cited BRS. Find the static spherically symmetric solution in this class--- note that it is not given by choosing
w = \frac{1}{\sqrt{z^2+r^2}}
Compute its Komar mass. Think about matching across nested spherical shells to a Minkowski region inside the inner shell, and to a Schwarzschild or Minkowski region (depending upon what you found for the Komar mass) outside the outer shell. (Note that the scalar field "wants" to model a long range interaction, so any such matching will require some explanation: what is the physical reason why the scalar field is nonzero only between the two nested spherical shells?)
the OP is badly confused and I am not up to the task of trying to suggest what to say to him, but its good that several SA/Ms are trying to help. One minor clarification:
pervect said:
The Kerr solution, like the Schwarzschild solution, assumes a perfectly symmetrical collapse.
AFAIK no rotating analog of the Oppenheimer-Snyder collapsing dust ball, which produces a non-rotating hole, is yet known. However, if you look at geodesic congruences of world lines of freely falling test particles, the Doran congruence in the Kerr vacuum is generalizes the Lemaitre congruence from the nonrotating case (Schwarzschild vacuum), and the Doran observers in the equatorial plane do exhibit planar motion as they spiral in.
pervect said:
The Penrose diagram of the interior an actual rotating black hole caused by gravitational collapse is still a matter of some speculation and debate.
AFAIK, the situation remains the same as ten years ago: despite plausible analogies suggesting what to expect, nothing rigorous is known about the interiors of generic rotating holes (e.g. formed by generic collapse of ordinary matter, with some matter or radiation falling into the hole from the exterior region). The very nice paper on mass-inflation cited by JesseM discusses generic charged holes by way of pursuing the analogy.
relativityfan said:
After having read [the cited arXiv eprint] I do believe that even without [even with?] the mass inflation, there is no white hole.
the structure of the "white hole" if we look at the metric should be exactly the same as the structure of the black hole, and matter could excape because it can be accelerated faster than the speed of light (like matter inside the event horizon) . So this would not be a white hole but a black hole, because the metric would be the same.
Does anyone disagree with this?
I think the appropriate response is: "no-one has any idea what you are trying to say". I bolded some obviously problematic claims (although I can't tell whether he is trying to make a claim or to deny one!).
FWIW, the eternal Schwarzschild and Kerr spacetimes contain both "white hole" and "black hole" event horizons, or more properly, horizons from which particles must emerge from a past interior region into an exterior region, and horizons in which particles which fall from an exterior region into a future interior region cannot re-emerge into the original exterior region. In addition, the eternal Kerr vacuum (and the RN non-null electrovacuum) also contain Cauchy horizons, which must not be confused with event horizons. The eternal Kerr vacuum also contains both and interior asymptotically flat regions (negative Komar mass) as well as exterior asympotically flat regions (positive Komar mass).
BTW, "relativityfan" is making my troll antennae twitch uncomfortably I sense a whiff of Chip on Shoulder, which raises the question of whether this user is using an ironically chosen handle at PF. It's always worth remembering that one reason to "write defensively" when posting in the public areas is that you never know when some putative newbie is planning to cherry pick responses to some "naive question" in order to, say, quote them in some anti-science website. A large number of religiously committed creationists do this quite a bit with regard to anything related to the standard hot Big Bang theory.
In "Geodesics doubts"
Code:
www.physicsforums.com/showthread.php?t=424278
the OP is again rather confused (as the poster later admitted). FWIW,
Hurkyl said:
As I'm familiar with the phrase, "space is expanding" is not a coordinate-dependent phenomenon.
Depending on context, it might be reasonable to interpret "space is expanding" to refer, in an exact fluid solution, to the congruence of world lines of the fluid particles, and then the acceleration vector, expansion scalar, shear tensor, and vorticity tensor of this congruence are all geometric, coordinate-free notions. In particular, the expansion scalar gives a coordinate-free notion of whether or not a small ball of fluid particles has expanding or contracting volume. (Oops, just saw that in Post #14, George Jones already said something similar. And yay!, in Post #13, Ben Crowell pointed out that as soon as you start talking about "velocity in the large", things get tricky, and you need to be more precise about what operationally significant notion for defining "distance in the large" you have in mind.)
Ich said:
Expansion of space is purely coordinate dependent.
I think I know why Ich said this, and depending upon how you think of things, that's not wrong either, but wow, this certainly shows why it is so important to introduce some math, or at least to say that without using math, people are likely to wind up talking about different things and thus making apparently mutually contradictory statements.
Ich said:
Another more general set of fundamental observers, independent of FRW symmetries, is defined to be at rest in normal coordinates, maybe one could call them "Einstein observers", as they reproduce the inertial frames of SR if curvature is negligible, on which most people base their intuition.
I see that Ich is thinking of Riemann normal coordinates (on a spatial hyperslice? in a cosmological model?) but I don't understand how he intends to define his observers. If the congruence of their world lines is nonexpanding, however, it will generally be nongeodesic. Also, a rigid congruence is one with vanishing expansion tensor, but not all spacetimes admit such congruences.
Mentz114 said:
From what I've been learning from this thread, we have grounds for selecting a 'canonical' coordinate system - i.e. a point of view where the universe appears homogenous and isotropic.
I think at least two posters are thinking about "preferred timelike congruences" in cosmological models. There are at least two possible interpretations of what this might mean:
vorticity-free congruence whose spatial hyperslices are homogeneous and isotropic (this puts strong constraints on the spacetime)
congruence of the world lines of the fluid or dust particles which model galaxies; in a fluid this might not be a geodesic congruence but in a dust solution, it will be a timelike geodesic congruence; however if the vorticity tensor is nonvanishing these world lines will not admit a family of orthogonal hyperslices.
The OP asked Ben
TrickyDicky said:
bcrowell, this is what I call gaslight, are you saying that considering expansion as physical fact (as I and many others do) or as a coordinate artifact is just a matter of taste?
The problem is that he still doesn't realize that various notions of "expansion" have been referred to without definition in the thread. If one understand this to refer to the expansion scalar of a timelike congruence having some agreed upon physical significance in a given gtr model, then yes, this is coordinate-free. If one understands "expansion" to mean something else, then it could well be coordinate-dependent. Re what Ben said in his Post #36, if we consider "expansion" to refer to galaxies in a cosmological model, it is reasonable to stipulate that we will use the expansion tensor of the congruence to describe how nearby pairs of galaxies are moving wrt each other. But this only tells us about nearby galaxies, which is probably not what the OP wanted!
Hmm... velocity in the large, visual appearances... as the number of valid but conceptually subtle concepts mentioned in this thread increases, someone who doesn't already understand all of them will probably gain the (false) impression that the signal to noise ratio is increasing. Certainly the OP seems to be becoming increasingly confused, not less so. But maybe that's part of the learning experience.
User:TrickyDicky asks about the tidal tensor of the FRW models. He is correct that the Weyl tensor of these models vanishes identically; all their curvature is Ricci curvature, due to the immediate presence of matter (usually taken to be radiation fluid or dust) as per Einstein's field equation. However, the tidal tensor is the electroriemann tensor, not the electroweyl tensor! Thus, a small sphere of initially comoving test particles in an FRW model will contain nonzero mass, so it will contract; by symmetry, the gravitational attraction of nonzero mass outside the sphere cancels out. The tidal tensor shows this isotropic tidal compression.
For example, consider the FRW dust with E^3 hyperslices:
<br />
ds^2 = -dt^2 + t^{4/3} (dx^2 + dy^2 + dz^2), \; \;<br />
0 < t < \infty, \; -\infty < x, \, y, \, z < \infty<br />
Take the frame of the dust particles
<br />
\begin{array}{rcl}<br />
\vec{e}_1 & = & \partial_t \\<br />
\vec{e}_2 & = & t^{-2/3} \; \partial_x \\<br />
\vec{e}_3 & = & t^{-2/3} \; \partial_y \\<br />
\vec{e}_4 & = & t^{-2/3} \; \partial_z <br />
\end{array}<br />
Then the tidal tensor is
<br />
{E\left[\vec{e}_1\right]}_{ab} = <br />
\frac{2}{9 t^2} \; \operatorname{diag} (1,1,1)<br />
indicating isotropic tidal compression.
For comparison, for the Schwarzschild vacuum in the frame of static observers or Lemaitre observers,
<br />
{E\left[\vec{e}_1\right]}_{ab} = <br />
\frac{m}{r^3} \; \operatorname{diag} (-2,1,1)<br />
(traceless, as must happen for a vacuum solution!), which indicates radial tidal tension and orthogonal tidal compression.
The Jacobi geodesic formula states
<br />
\ddot{\vec{\xi}}^a = -{E^a}_b \, \xi^b<br />
where \vec{\xi} is a connecting vector (spacelike, short, points from fidudical geodesic to a neighboring geodesic as you let parameter run in both proper time parameterized geodesic curves), and where overdot denotes differentiation wrt proper time,
<br />
\dot{(\cdot)} = \nabla_{\vec{e}_1} (\cdot)<br />
Or in terms of matrix algebra, if we think of vectors as column matrices,
<br />
\ddot{\vec{\xi}} = -{\cal E} \, \vec{\xi}<br />
IOW, at each event we have a linear operator which acts on spacelike vectors in the projection of the tangent space to the normal hyperplane element, and this takes each vector to its second derivative wrt proper time, assuming it is connecting our fiducial geodesic to a very nearby one. This only makes sense because the tidal tensor is defined in terms of the Bel decomposition of the Riemann tensor with respect to a particular timelike congruence--- assumed here to be a geodesic congruence.
TrickyDicky said:
I say it because I've just read that the Weyl curvature only happens in in empty spacetime, without any gravitational source nearby.
That's not true, assuming "empty spacetime" means a vacuum or maybe electrovacum region. Weyl curvature can happen anywhere, even in a region containing matter. If that weren't true, we could shield against gravitational waves using cardboard boxes.
For those of you who use Maxima, here is a Ctensor file you can run in batch mode under wxmaxima, which allows you to easily verify that the Weyl tensor vanishes and that the electroriemann tensor (tidal) tensor has the frame field components I just mentioned:
Code:
/*
FRW dust with E^3 hyperslices; comoving cartesian chart; nsi coframe
All FRW dusts can be embedded with codimension one.
*/
load(ctensor);
cframe_flag: true;
ratchristof: true;
ctrgsimp: true;
/* define the dimension */
dim: 4;
/* list the coordinates */
ct_coords: [t,x,y,y];
/* define background metric */
lfg: ident(4);
lfg[1,1]: -1;
/* define the coframe */
fri: zeromatrix(4,4);
fri[1,1]: -1;
fri[2,2]: t^(2/3);
fri[3,3]: t^(2/3);
fri[4,4]: t^(2/3);
/* setup the spacetime definition */
cmetric();
/* display matrix whose rows give coframe covectors */
fri;
/* compute a matrix whose rows give frame vectors */
fr;
/* metric tensor g_(ab) */
lg;
/* compute g^(ab) */
ug: invert(lg);
christof(false);
/* Compute fully covariant Riemann components R_(mijk) = riem[i,k,j,m] */
lriemann(true);
/* Compute R^(mijk) */
uriemann(false);
/* Compute Ricci componets R_(jk) */
ricci(true);
/* Compute trace of Ricci tensor */
tracer;
/* Compute R^(jk) */
uricci(false);
/* Compute and display MIXED Einstein tensor G^a_b */
/* For (-1,1,1,1) sig Flip sign of top row to get G^(ab) */
einstein(false);
/* WARNING! leinstein(false) only works for metric basis! */
/* Compute Kretschmann scalar */
rinvariant();
expand(factor(%));
/* Covariant Einstein tensor as matrix */
matrix([-ein[1,1],-ein[1,2],-ein[1,3],-ein[1,4]],
[ein[2,1],ein[2,2],ein[2,3],ein[2,4]],
[ein[3,1],ein[3,2],ein[3,3],ein[3,4]],
[ein[4,1],ein[4,2],ein[4,3],ein[4,4]]);
/* electroriemann tensor */
matrix([lriem[2,2,1,1], lriem[2,3,1,1],lriem[2,4,1,1]],
[lriem[3,2,1,1],lriem[3,3,1,1],lriem[3,4,1,1]],
[lriem[4,2,1,1],lriem[4,3,1,1],lriem[4,4,1,1]]);
/* magnetoriemann tensor */
matrix([lriem[2,4,3,1],lriem[2,2,4,1],lriem[2,3,2,1]],
[lriem[3,4,3,1],lriem[3,2,4,1],lriem[3,3,2,1]],
[lriem[4,4,3,1],lriem[4,2,4,1],lriem[4,3,2,1]]);
/* Weyl tensor vanishes */
weyl(true);
/* Construct NP tetrad for our frame, compute Weyl spinors and Petrov type */
nptetrad(true);
weyl(false);
psi(true);
factor(psi[0]);
factor(psi[1]);
factor(psi[2]);
factor(psi[3]);
factor(psi[4]);
petrov();
For comparison, here is the Schwarzschild vacuum in the frame of the Painleve observers:
Code:
/*
FRW dust with E^3 hyperslices; comoving cartesian chart; nsi coframe
All FRW dusts can be embedded with codimension one.
*/
load(ctensor);
cframe_flag: true;
ratchristof: true;
ctrgsimp: true;
/* define the dimension */
dim: 4;
/* list the coordinates */
ct_coords: [t,x,y,y];
/* define background metric */
lfg: ident(4);
lfg[1,1]: -1;
/* define the coframe */
fri: zeromatrix(4,4);
fri[1,1]: -1;
fri[2,2]: t^(2/3);
fri[3,3]: t^(2/3);
fri[4,4]: t^(2/3);
/* setup the spacetime definition */
cmetric();
/* display matrix whose rows give coframe covectors */
fri;
/* compute a matrix whose rows give frame vectors */
fr;
/* metric tensor g_(ab) */
lg;
/* compute g^(ab) */
ug: invert(lg);
christof(false);
/* Compute fully covariant Riemann components R_(mijk) = riem[i,k,j,m] */
lriemann(true);
/* Compute R^(mijk) */
uriemann(false);
/* Compute Ricci componets R_(jk) */
ricci(true);
/* Compute trace of Ricci tensor */
tracer;
/* Compute R^(jk) */
uricci(false);
/* Compute and display MIXED Einstein tensor G^a_b */
/* For (-1,1,1,1) sig Flip sign of top row to get G^(ab) */
einstein(false);
/* WARNING! leinstein(false) only works for metric basis! */
/* Compute Kretschmann scalar */
rinvariant();
expand(factor(%));
/* Covariant Einstein tensor as matrix */
matrix([-ein[1,1],-ein[1,2],-ein[1,3],-ein[1,4]],
[ein[2,1],ein[2,2],ein[2,3],ein[2,4]],
[ein[3,1],ein[3,2],ein[3,3],ein[3,4]],
[ein[4,1],ein[4,2],ein[4,3],ein[4,4]]);
/* electroriemann tensor */
matrix([lriem[2,2,1,1], lriem[2,3,1,1],lriem[2,4,1,1]],
[lriem[3,2,1,1],lriem[3,3,1,1],lriem[3,4,1,1]],
[lriem[4,2,1,1],lriem[4,3,1,1],lriem[4,4,1,1]]);
/* magnetoriemann tensor */
matrix([lriem[2,4,3,1],lriem[2,2,4,1],lriem[2,3,2,1]],
[lriem[3,4,3,1],lriem[3,2,4,1],lriem[3,3,2,1]],
[lriem[4,4,3,1],lriem[4,2,4,1],lriem[4,3,2,1]]);
/* Weyl tensor vanishes */
weyl(true);
/* Construct NP tetrad for our frame, compute Weyl spinors and Petrov type */
nptetrad(true);
weyl(false);
psi(true);
factor(psi[0]);
factor(psi[1]);
factor(psi[2]);
factor(psi[3]);
factor(psi[4]);
petrov();
it looks like the Baez pages cleared up the main confusion, but TrickyDicky introduced a new one!
TrickyDicky said:
the spacetimes of constant sectional curvature like Schwartzschild
I don't know how Tricky got that confusion, but its not true.
Take any two-dimensional submanifold S of any Loretzian or Riemannian manifold M. It inherits a metric from the parent and thus has a Gaussian curvature at each point. Choose a point P and consider the tangent plane to S at P. Associated with this is a sectional curvature obtained from the action of the Riemann tensor on bivectors. Computing this wrt a frame field you obtain the Gaussian curvature of S at P.
The metric tensor on M at T_P M induces a metric tensor on the space of bivectors at P. If you use a coordinate basis to compute sectional curvature, you have to divide by a normalization factor which arises from this metric on bivectors. If you use a frame field, you don't need to do that.
Schwarzschild vacuum certainly does not have constant sectional curvature! That is a very rare property for a Riemannian/Lorentzian manifold to have!
My initial guess was that Tricky misread something somewhere concerning a venerable approach to finding initial data with which to start a numerical simulation using an initial value approach to gtr, such as ADM formalism. In this approach, one looks for a conformally flat three manifold to start the evolution; for example, "space at a time" in a time symmetric evolution in which dust is thrown out, hovers momentarily, and falls back---- the "space at a time" where the dust momentarily hovers may be conformally flat. But most spatial hyperslices certainly are not.
BTW, in a Lorentzian manifold, the Bach tensor measures the departure of a given hyperslice from being conformally flat, similarly to the fact that the Weyl tensor measures the departure of the spacetime itself from being conformally flat.
But then Tricky said this:
Tricky said:
Schwarschild solution has constant sectional curvature (that of a parabole)
It seems that Tricky misread a WP article (or some idgit munged a WP article). He must be referring to the Flamm paraboloid, an embedding of a t=t_0 hyperslice in the exterior region of Schwarzschild vacuum written in the Schwarzschild chart (only valid on the exterior). But that certainly does not have "constant sectional curvature".
I think this illustrates that people simply cannot hope to understand gtr from reading WP articles or popular books without sufficient mathematical background and ability to understand the math. But maybe I am too pessimistic--- Tricky did seem to learn something valuable from Baez's pages in the end.
atyy said:
the interior Schwarzschild solution (ie. one which contains matter) is conformally flat.
The bolded phrase is ambiguous: newbies typically don't know how to guess correctly whether an author means "Schwarzschild incompressible perfect fluid ball" or "future or past interior region of the Schwarzschild vacuum". Here, atyy means the first.
atyy said:
Penrose is talking about the vacuum Schwarzschild solution, which as far as I understand has non-zero Weyl curvature.
In a vacuum, the Ricci tensor vanishes, so the Weyl tensor agrees with the Riemann tensor in a vacuum region!
atyy said:
The "exact" solution describing our non-uniform universe must somehow contain corrections to all these approximations which must be joined up to each other somehow.
Correct. Just like in any field theory.
Tricky said:
is deflection of light outward the sun's rim produced by Weyl curvature or Ricci's?
Weyl. The deflection is an effect of the curvature in the vacuum region outside the Sun (in an idealized model formed by matching a static spherically symmetric perfect fluid (ssspf) ball such as the Schwarzschild perfect fluid across the zero pressure surface to an exterior Schwarzschild vacuum region. Since the curvature there is entirely Weyl curvature, this lensing is a Weyl effect.
In a conformally flat region, by definition there can be no lensing. Thus, no lensing in FRW models, or in the interior of the Schwarzschild incompressible fluid ball (pretending for the sake of argument that light can propagate freely at the speed of light there, which of course isn't true!). Note: most ssspf solutions are not conformally flat, nor are many cosmological models conformally flat.
Tricky said:
I was confusing the spatial curvature(paraboloid) of the Schwarzschild metric that is indeed of constan sectional curvature
Not if he is talking about the hyperslice which corresponds to the Flamm paraboloid, and I don't know what other paraboloid he could be talking about.
an interesting point is that a certain increasingly populous class of satellites, generally launched with little fanfare, are increasingly using various kinds of stealth technology (e.g. nonreflective surfaces), and increasingly engage on a timescale of minutes in various evasive behaviors (e.g.furl sails at certain times, change their geometry in other ways when passing over certain areas, frequently change orbits).
Interestingly, many evasive techniques appear to be aimed specifically at foiling casual observation by amateur astronomers (through telescopes, but certainly these things don't wish to needlessly draw the attention of naked eye observers either). Rather astonishing, considering the expense involved, but apparently true.
Be this as it may, the net result is an increasing frequency with which amateurs notice odd flashes in the night sky, which appear rather unlike flashes from conventional communication satellites in long term, known, stable orbits.
What if anything to say about this in the public areas of PF is, I guess, a matter for discussion in the SA forum.
Precession, Komar integrals, plus Kleinian geometry
Re "Perihelion advance expressed differently"
Code:
www.physicsforums.com/showthread.php?t=434694
As Lut Mentz hinted, one of the nice things about linear field equations is that you can separate out various effects and treat them seperately. In the case of the precession of Mercury, there is a much larger effect due to the influence of the major planets. We can get away with computing the gtr residual or extraNewtonian precession using a simplified model which ignores the outer planets entirely (and the quadrupole moment of the Sun, and...) only because we are using the linearized approximation, which is good enough to obtain the gtr residual to the desired accuracy for Mercury and other cases in our solar system.
To second order (in an appropriate perturbation expansion) you can think of the motion as Kepler motion on an ellipse in a tranparent piece of plastic which you slowly rotate with constant angular velocity. But higher order terms make the actual motion more complicated, and nonlinear corrections make it even more complicated (in a peturbative approach).
The perturbation expansion just mentioned refers to the Einstein-Binet equation for timelike geodesics in the exact Schwarzschild vacuum, but Einstein 1915 used a weak field approximate solution instead, which turns out to introduce an unwanted problem. So much better to follow the textbook approach.
Re "Newtonian generalization of Komar mass"
Code:
www.physicsforums.com/showthread.php?t=434722
Jonathan Scott (uh-oh!) asks a murky question about Komar mass. I expect Pervect will chime in there, to say (at least) that Komar mass is not defined the way JS suggested. Carroll's textbook has a very nice discussion, but I don't care for his notation! Here's how I'd describe the definition:
Suppose you have an asympototically flat spacetime (Lorentzian 4-manifold) which admits a timelike vorticity free Killing vector \vec{\xi}, i.e. we have an asymptotically flat static spacetime. Although this isn't needed, to simply the discussion let's also suppose we have introduced a Schwarzschild type chart so that our line element has the form
<br />
ds^2 = -f^2 dt^2 + g^2 dr^2 + r^2 \; d\Omega^2<br />
where f,g are metric functions of r only, where we suppose that our Killing vector field \vec{\xi} = \partial_t. Let \vec{N} be the outward pointing unit normal to the spheres r=r0, and let the timelike unit vector field \vec{U} be the normalization of the Killing vector field \vec{\xi}. Then
Average over the sphere at r = r_0 (Schwarzschild radial coordinate) the quantity
<br />
\vec{N} \cdot \nabla_{\vec{U}} \vec{\xi}<br />
(don't forget to use the appropriate Jacobian factor in the integrand!)
Let r0 tend to infinity
With more thought you can see that this does not actually depend upon adopting a particular coordinate chart, but only upon the assumptions that our spacetime manifold is asymptotically flat and static. Strictly speaking, we don't even need gtr to define any of these notions! It is true that in practice, we need to adopt a chart in which it is mathematically convenient to average over spheres, or at least approximate spheres, provided they become round as r0 -> infty, and provided that our radial coordinate has the required asymptotic properties.
If you apply this to a simple model of an isolated nonrotating star, consisting of a static spherically symmetric perfect fluid ball matched across the zero pressure surface to a portion of a Schwarzschild vacuum exterior region, then you obtain the mass of the spacetime, i.e. the mass of the star. Since the whole point is to take a limit, the Komar mass really only cares about the "shape" of the vacuum exterior, in fact only about the "shape at spatial infinity"! Scott may be thinking of a perfect fluid which has pressure decreasing only asymptotically to zero, and which is asymptotically flat, but if so, not everything which "looks" AF really is, so this needs to be checked. Or he may simply be confused about the definition of Komar mass.
The fun thing about Komar mass is that you can also apply this setup to stationary axisymmetric spacetimes, such as an asymptotically flat sheet (exterior or interior) of the Kerr vacuum, and then you can define and compute Komar angular momentum (about the symmetry axis) in addition to Komar mass. In the case of the Kerr vacuum you obtain the usual mass and angular momentum parameters for the Kerr vacuum solution as usually written (e.g. in the Boyer-Lindquist chart). This requires averaging over approximately round spheres, which turns out to be good enough, as long as they become round in the limit r0 -> infty, and as long as r is an appropriate radial coordinate wrt asymptotic flatness.
These Komar integrals only apply to spacetime models in which we have appropriate Killing vector fields. A more general definition, ADM mass, agrees with Komar mass when the Komar integrals are defined.
In "Is Minkowski space the only Poincare invariant space?"
Code:
www.physicsforums.com/showthread.php?t=434555
Arkadiusz Jadczyk (uh-oh!) wrote
arkajad said:
If you mean just "a manifold", then you can take any homogeneous space P/H, where P is the Poincare group and H its closed subgroup.
He is correct (that's the basic idea of Kleinian geometry, in fact, one of my fav topics). The OP was probably thinking of four dimensional manifolds. More generally, G/H where G is any Lie group and H any closed subgroup.
Actually, this only gives the smooth manifold portion of a much wider notion of Kleinian geometry. We can in fact take G to be any group, as in "BRS:Exploring the Rubik Group". For example, in order to study things like finite projective spaces over some finite field, we might take G=PGL(d+1,p^n). Then we get finite analogues of various familiar geometries. This turns out to be closely related to the study of finite simple groups--- in This Week, John Baez often discussed various aspects of Kleinian geometry, including these connections.
Minkowski spacetime arises from the case where G is the ten dimensional Poincare group and H is the six dimensional Lorentz group (notice that 10-6=4, as per my posts in "BRS:Exploring the Rubik Group"), and you can obtain discrete quotient spaces of Minkowski spacetime using Klein's approach. It matters very much whether or not we include "improper motions" in G or H!
It is easier to explain the simpler case of the round sphere S^2 and its discrete quotient round RP^3. Let's take G = SO(3), which will give Riemannian isometries on "round manifolds". In particular, S^2, as a homogeneous Riemannian two-manifold, arises as SO(3)/SO(2), while RP^2, as a homogeneous Riemannian two-manifold, arises as SO(3)/O(2) (quotient by a larger one dimensional closed subgroup, a discrete supergroup of SO(2), which gives a discrete two-fold quotient of S^2, by identifying antipodal points). In both cases, each element of SO(3) acts on S^2 or RP^2 as a proper isometry, in the sense of Riemannian geometry.
Other ways of representing "the sphere" give or remove various structural features, as appropriate depending on context, e.g. we might be interested in removing some of the Riemannian manifold structure. In particular we can remove some of the metric space structure, while retaining just enough structure to define conformal geometry on the sphere. To do this we should take G to be the Moebius group, and then we can find a closed subgroup H such that G/H is the sphere, but this is the sphere endowed with conformal geometry rather than Riemannian metric geometry. Now each element of the Moebious group (recall their classification into elliptic, parabolic, hyperbolic, loxodromic elements!) acts on S^2 as a conformal motion.
Twistor theory starts by exploiting the Lie group isomorphism between the Lorentz group and the Moebius group.
So it really matters here whether one thinks of something like the rotation group as SO(3) or O(3), and similarly for how you think of the "euclidean group" (the semi-direct product of the group of translations with the rotation group, which makes the group of translations a normal abelian subgroup of the euclidean group). Similarly for the Poincare group and the Lorentz group.
A great deal is known about the possible isometry groups for various kinds of exact solutions in gtr; see Stephani et al., Exact Solutions of the Einstein Field Equations, Cambridge University Press.
my main confusion comes from the fact that if the Stress-energy tensor for electromagnetic radiation is traceless, that would imply the pressure components of the tensor equal zero, and yet it's obvious radiation exerts pressure when absorbed or reflected and radiation pressure plays an important role in star dynamics.
(the bolded phrase is of course incorrect). Replies included:
phyzguy said:
What nicksauce is saying is that the stress-energy tensor of radiation looks like:
<br />
\begin{bmatrix}<br />
\rho & 0 &0 & 0 \\<br />
0 & -p & 0 & 0 \\<br />
0 & 0 &-p & 0 \\<br />
0 & 0 & 0 & -p<br />
\end{bmatrix}<br />
With p = rho/3, the trace is zero.
Components wrt a frame field, of course!
I think there might be some confusion here. For radiation pressure to accelerate a bit of matter, the radiation cannot be impinging in completely isotropic fashion. But if an electron is moving wrt the CMBR (for example), it does experience a drag force from radiation pressure which scales like the fourth power of the temperature; see Peebles, 5.6.
Also, nicksauce/phyzguy gave the contribution to the stress tensor of a radiation fluid (as in an early epoch in cosmology) rather than the contribution of an EM wave, which might be closer to what Tricky wanted. For a plane wave (components wrt a suitably "aligned" frame field!)
<br />
\begin{bmatrix}<br />
\varepsilon & \varepsilon &0 & 0 \\<br />
\varepsilon & \varepsilon & 0 & 0 \\<br />
0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0<br />
\end{bmatrix}<br />
Which is also traceless, and shows a directional pressure term. The off-diagonal terms shows the momentum. A spherical wave looks like a plane wave in a very small region far from the source of the radiation.
BRS: proper distance, underdensities, warp drives, &c., ad nauseum
In ""Proper distance" in GR"
Code:
www.physicsforums.com/showthread.php?t=437895
pervect said:
I am aware of two meanings of the term "proper distance" in GR. The first is when you have points in flat space-time, or space-time that's locally "flat enough", in which case it is defined as it is in SR, as the Lorentz interval between the two points. This usage of the term implies that one is considering short distances, or is working in a flat space-time.
Right, and in this case one can describe an unambiguous "proper distance in the large", because of a remarkable property of Minkowski spacetime: through each pair of events there is precisely one geodesic. There are a handful of other homogeneous (transitive isometry group) spacetimes with the same property --- they are often are often used as cosmological models, but that's not really relevant here--- such as de Sitter lambda vacuum.
A good exercise is to identify these spacetimes and to work out the "proper distance in the large" formula, analogous to the Pythagoras-Minkowski formula
<br />
d( (t,x,y,z), (t',x',y',z')) = -(t-t1)^2 + (x-x1)^2 + (y-y1)^2 + (z-z1)^2<br />
In a more general spacetime, there will be multiple geodesics between two "non-nearby" events, so there is no hope of a "proper-distance in the large" formula. However, if one clearly has in mind a specific geodesic curve, one can integrate ds along the curve and call that proper distance. Actually, one normally says "proper time" if the curve is everywhere timelike and "proper distance" if the curve is everywhere spacelike. Curves which are timelike here and spacelike there are not often considered! And for null curves, of course, "proper distance" makes no sense.
In the case of something like an FRW model, this is the idea behind integrating along a spacelike geodesic which lies entirely in some "constant time slice". (Note that in a generic hyperslice will bend away from a generic spacelike geodesic which is tangent to the slice at some event on the slice.)
I don't think there's anything to be gained by arguing over whether some definition of "spatial distance in the large" is the "right" definition. There are multiple distinct operationally significant definitions possible, and that's all there is to it. OTH, if you want to discuss in coordinate-free, geometrically meaningful terms the relative motion of a family of observers whose world lines are given by some timelike congruence, then the decomposition of the associated timelike unit vector field into acceleration vector, expansion tensor, and vorticity vector, is just what you want.
George Jones said:
Suppose a congruence of timelike worldlines of "fundamental" observers is picked out by phyics, symmetry, etc. Consider spacelike curve that intersects each worldlne in the timelike congruence orthogonally, and that has unit length tangent vector. Proper distance for the congruence is given by the curve parameter along such a spacelike curve. Sometimes these spacelike curve are geodesics, and sometimes they are not.
But there are many ways of continuing a curve given just one tangent vector, so there is too much multiplicity here to be really useful, I think. OTH, the decomposition is unique, but of course dependent upon choice of congruence! Also, don't forget that only an irrotational congruence admits a family of hyperslices everywhere orthogonal to the world lines--- for a congruence with vorticity, no "constant time slices" exist--- oh, I see now, George already said this:
George Jones said:
A congruence is hypersurface orthogonal if and only if the vorticity of the congruence vanishes.
pervect said:
I believe there is a general consensus, then, that the term "proper length" in GR needs additional specification besides two points: the curve along which the length may be specified, or the hypersurface of "constant time" in which the curve lies might be specified as an alternative, or indirect means might be used to specify the hypersurface (for instance it being orthogonal to a particular preferred family of observers).
Wish I'd being paying attention, because I demur: in a generic situation, there will be no "hypersurface of proper time" because the congruence has vorticity. However, the decomposition always makes sense and is always informative. But in the literature, I'd say that the general usage is that "proper distance" (or "proper time" for timelike curves) is the integral of ds along a everywhere spacelike (everywhere timelike) geodesic, bearing in mind that there may be more than one such geodesic between two events, and that most spatial hyperslices will bend away from a spacelike geodesic tangent to some event on the hyperslice, and that such a slice need not have any nice relation to any timelike congruence which may be physically interesting.
pervect said:
the curve is specified implicitly as (informally) "the shortest curve connecting the two points" or more formally the distance is specified as the greatest lower bound of all curves connecting the two points.
I think pervect was thinking of spacelike geodesics between two nearby events, but in general there will be multiple geodesics between two events (e.g. on an ordinary two-sphere) giving different lengths between the two points. Also, in the sequel of the thread, some posters appear to be confused about the variational principle behind the notion of geodesics: it says that when we have a geodesic curve between two events, and make a small variation (small to first order), the integral of ds is consant to second order. It doesn't say whether this integral increases or decreases. In flat spacetime, it is true that for a timelike geodesic, it will decrease, and for a spacelike geodesic, it will increase. But the point is that for a non-geodesic curve, a first order variation will result in a first order change in the integral of ds. So the variational principle says that the integral of ds is stationary, not that it is an extremum.
TrickyDicky said:
I thought for a specific spacetime with a specific curvature, there could only be one geodesic that precisely defines the shortest path. And that would also be the most "natural" definition of proper distance.
In Riemannian geometry, there is quite a bit of theory on inferring properties of geodesics from properties of the curvature tensor, and vice versa. Most of this depends on positive definiteness. There is also considerable theory relating properties of geodesics to properties of curvature tensor in Lorentzian geometry, but it has a different flavor since no positive definiteness. See respectively Berger, A Panorama of Riemannian Geometry and Stephani et al., Exact Solutions of Einstein's Field Equations.
In "Opposite side" of GR"
Code:
www.physicsforums.com/showthread.php?t=437613
the OP is struggling to discuss something like this: a dust solution which has a region of underdensity (in a spatial hyperslice, not neccessarily related to the world lines of the dust particles, this region should be compact), possibly spherically symmetric although that is not generic, and outside agrees locally with some FRW dust solution. This exactly the situation discussed at length elsewhere in the BRS!
the OP is caught up in the issue of multiple distinct operatationally distinct notions of "distance in the large", hence "velocity in the large". This is why warp drive metrics do not contradict the principle that at each event the tangent space is Lorentzian. But a large body of work since Alcubierre's papers shows that warp drives are almost certainly not physically realizable, and moreoever, if they were, so would be "time machines" and other outlandish devices. There is no completely solid disproof, and you can never tell what the future might bring, but right now it seems that there is no point on say spending large sums on looking for ways to make warp drives, because right now theory suggests strongly that it simply cannot be done.
the OP asks about the surface of last scattering. He wants to know why the CMBR appears to come from every direction and from a certain epoch (in FRW cosmology, a certain constant "cosmological time" hyperslice). I wonder whether putting up a figure might help? See below for a suggestion.
If the figure isn't clear, make a model!
buy a box of paper drinking straws (paper straws will buckle more readily than plastic ones),
say they are 20 cm long; bend all the straws at 15 cm,
get a piece of cardboard, and draw two equal intersecting circles each of radius 15/\sqrt{2} cm on the cardboard,
pierce a piece of cardboard at some points each circle,
push the straws through the cardboard, and gather the ones piercing each circle to make two cone shaped configuration (glue them at the apex with Elmer's glue or tie them at the apex with laundry ties, or something like that, but make sure that all the straws should everywhere make slope 1 wrt the cardboard),
from the apex of each "cone", you can hang a differently colored straw, cut so that it doesn't reach the surface of last scattering, or hang a short strand of colored yarn; these represent the world lines of the two galaxies (these world lines should be orthogonal to the surface of last scattering, or nearly so, and since the galaxies formed long after the epoch of last scattering, they shouldn't reach to the cardboard),
randomly bend each straw at a few points below the cardboard in random directions but still keeping slope 1 wrt the cardboard (that is, below the cardboard, each straw should each be bent several times in random directions, but making slope 1 wrt the cardboard, while above the cardboard, all the straws should be straight and make slope 1 wrt the cardboard).
This is very crude schematic model of an FRW model, represented in a conformal chart; the cardboard models the surface of last scattering, and the straw bending below this surface suggests the scattering. The model shows how two observers each measure radiation coming from all directions from events at the same epoch (the hypersurface corresponding to the surface of last scattering).
In "BTZ black hole"
Code:
www.physicsforums.com/showthread.php?t=441565
the OP asks about an exact solution modeling a Kerr analog in 2+1 gtr. But the analogy is not very close since in 2+1 dimensions, the Weyl tensor vanishes identically, so gravitation is not a long range force. That means that in model "2-stars", the matter filled interior (think of a 2-hemispherical cap) is curved but the exterior is locally flat (think of a conical frustrum).
Oh, noooo! In "Bode's Law"
Code:
www.physicsforums.com/showthread.php?t=441275
the OP inquires whether the notorious so-called "Bode's Law" (an infamous item of approximate numerology) is somehow validated by data on extrasolar planets. The answer is that Bode proposed a small integer relationship between the mean radii of the major planets, which is certainly not true (except very very approximately) in our solar system and even less so in others. So that would be "no, but hardly worth dignifying with an answer".
But there is a kernel of truth in the implicit observation that there is much which no one yet knows about the formation of solar systems. In particular, the above theory starts with a more or less fully formed solar system in which some near integer approximate relations happen to exist at least momentarily, and (with good sucess) tries to say whether these coincidental relations will be quickly destroyed by various perturbations, or will be preserved and even refined.
Someone mentioned tidal locking of the rotational periods of certain moons of Jupiter. The point that wasn't brought out clearly is that the theory of dynamical systems shows why looking at periods is preferable to looking at lengths if you want to look for small integer approximate relations in specific solar systems. As DH didn't quite say, many aspects of dynamical systems relevant to solar system dynamics are now well understood; in particular, there is good understanding of why some near integer relations are unstable (so that the orbits evolve to disrupt these near-relations) while others are stable (so that the system preserves them and may even make them better approximations over time). A good key phrase is KAM theory; I can provide citations to expositions aimed at mathematicians who are not specialists.
Coming back to looking for small integer ratios, the Greeks invented the very nice theory of simple continued fractions precisely to efficiently find good approximations by small integer ratios when they exist. This applies to anything, so it is in effect a machinery for doing numerology. Of course, in "applications" this is, in general, a mathematical analog of a parlor trick with cards: an artfully constructed illusion. Entertainment, not science.
In "Rod shortening of General Relativity"
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the OP asked for a "formula for rod shortening", apparently thinking of some alleged spatial analog of gravitational red shift for the gravitational field (possibly) nonrotating isolated objects. Ben Crowell pointed out that the OP failed to specify what spacetime he had in mind--- my guess is that the OP may have been trying to ask for a general "rod-shortening" formula for asymptotically flat Weyl vacuum solutions in the weak-field approximation (since the OP mentioned "potential", which makes sense, sort of, for weak-field approximations to Weyl vacuums)
User:yuiop assumed the OP was asking only about the Schwarzschild vacuum, and replied:
the simplest equation is:
<br />
dL = \frac{dr}{\sqrt{1-2m/r}}<br />
in units of G = c =1, where dL is the length of a short rod according to a local observer and dr is the length of the rod according to the Schwarzschild observer at infinity.
Because yuiop has not specified a measurement procedure to be used by a (static?) observer near spatial infinity, this doesn't make sense as stated. His claim can be fixed up, but the required procedure seems rather artificial to me:
The metric tensor, represented in Schwarzschild exterior chart, is
<br />
ds^2 = -(1-2m/r) \, dt^2 + \frac{dr^2}{1-2m/r} + r^2 \, d\Omega^2,<br />
\; \; r > 2m<br />
If t increases but radius and angles remain constant, r=r0, r0>2m, this specializes to
<br />
ds = \sqrt{1-2m/r_0} \, dt<br />
which gives the redshift for a signal sent by a static observer at r=r0, r0> 2m, to an observer at r=infty. If we hold t and angles constant but increase radius from r=r0 to r=r0+dr, we obtain instead
<br />
ds = \frac{dr}{\sqrt{1-2m/r}}<br />
On the face of things, this expression simply gives the radial scale factor near r=r0 for the Schwarzschild radial coordinate, i.e. it describes a characteristic of the coordinate chart, not the geometry.
It is true that (unlike most alternative radial coordinates), the Schwarzschild radial coordinate has some geometric significance. In particular
the area of the nested two-spheres implied by the spherical symmetry of the spacetime is proportional to r^2 as r varies, i.e. r is the "areal radius" of these spheres,
1/r0^2 is the constant Gaussian curvature of the sphere at r=r0,
1/r is the optical expansion of the principle outgoing null congruence (which has spherically expanding wavefronts),
However, one must do more work to explain a physical measurement procedure which explains what yuiop means by "measures at infinity".
The metric tensor for Weyl vacuums, in the Weyl canonical chart, is
<br />
ds^2 = -\exp(2u) \, dt^2 + \exp(2v-2u) \, (dz^2+\rho^2) + \exp(-2u) \, \rho^2 \, d\phi^2<br />
where u,v depend only on z,rho, where u is axisymmetric harmonic, u_{zz}+u_{\rho \rho}+u_{\rho}/\rho=0, and where v is determined from u by quadrature. To first order in u, v must be a constant. For an isolated object, u,v must tend to zero as r = \sqrt{z^2+\rho^2} grows, so v must be zero and then
<br />
ds^2 \approx -(1+2u) \, dt^2 + (1-2u) \, (dz^2 + d\rho^2 + \rho^2 \, d\phi^2) = -(1+2u) \, dt^2 + (1-2u) \, (dr^2 + r^2 \, d\Omega^2 )<br />
Because u is an asymptotically vanishing axisymmetric harmonic function which does not depend on t, it may be identified with the Newtonian gravitational potential of an appropriate isolated object. Thus, if t increases but the other coordinates remain constant, the line element specializes to
<br />
ds \approx (1+u) \, dt<br />
which gives the redshift for a signal sent by a static observer at r=r0 to an observer at r=infty. Redshift, since dt > ds.
If we hold t, \phi constant but increase r = \sqrt{z^2+\rho^2} from r=r_0 to r=r_0+dr, the line element specializes to
<br />
ds \approx (1-u) \, dr<br />
which is probably what the OP was trying to ask for. But does it make sense to call dr < ds "rod-shortening"? I can think of an interpretation, but it's not very straightforward.
For the weak-field approximation to the Schwarzschild vacuum, we have of course u = -m/\sqrt{z^2+\rho^2} = -m/r, the Newtonian potential for a spherically symmetric isolated massive object.
Figures:
"Surface" of last scattering (schematic)
Red-shift vs "rod-shortening" (?) in AF Weyl vacuums
Tantolos asks (duh!) "do photons create gravity?" I've seen so many threads with this title that it seems fair to call it a FAQ.
Here's my stab at a suggested stock answer:
Photons are a concept belonging to a QFT, whereas gtr is a classical field theory. So it makes little sense to ask about photons in gtr, because gtr don't know nuthin about quantum concepts. But it does make sense to ask: "according to gtr, does EM radiation contribute to the gravitational field?" The answer is "yes". In fact, according to gtr, all forms of mass-energy contribute to the gravitational field. (See "BRS: Massless Scalar Field Gravitationless Solutions" for an example showing that this does not imply that all forms of energy neccessarily "gravitate" in an intuitive sense, however.)
A particularly simple example are the plane waves associated with a null Killing vector field (the wave vector field) \vec{k}, in which wrt a suitable frame field the energy-momentum-stress tensor takes the form
<br />
T^{ab} = \varepsilon \, k^a \, k^b<br />
Both the Ricci and Weyl tensors are nonzero in such examples; the Riemann curvature tensor is built from these pieces, and in gtr it models the gravitational field. Since it is nonzero in an EM plane wave, plane waves are associated with a nonzero gravitational field. However, the gravitational field of EM waves we can create in the lab are much too small to measure.
In principle, when two laser beams pass nearby each other, the combined gravitational field of the EM field energy and momentum contained in the two waves should lense each laser beam. Again, this effect is much too small to measure.
BRS: Random Comments. Charts, singularities, lazy posters -> confusion
In "Eddington Finkelstein coordinates in the Schwarzschild spacetime",
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vitaniarain asks
do the Eddington-Finkelstein coordinates allow to cover the maximal analytic extension of the Schwarzschild spacetime? if not what region do they cover?
No. The ingoing chart covers a certain region (right exterior and future interior) of the full spacetime, and the outgoing chart covers another partially overlapping region (right exterior and past interior); see the BRS on Penrose-Carter conformal diagrams for details.
In
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jinbaw asks about some coordinate transformations of the Schwarzschild vacuum metric written in the usual chart on the exterior (except that he sets m=1/2). He transforms to two new charts:
ingoing Eddington,
"pre-Kruskal-Szekeres"
He observes that the second chart still has a coordinate singularity at r=2m. This is correct, and introducing the K-S factor removes it. See any gtr textbook which discusses the Kruskal-Szekeres chart.
The OP and DaleSpam use "singularity" to refer to the "coordinate singularity" at r=2m. In his response, bcrowell confusingly uses "singularity" to refer to "curvature singularity". This is sure to cause confusion, and the issue comes up constantly. The only solution, IMO, is to write out "coordinate singularity" and "curvature singularity". Even better, one can refer to "strong scalar spacelike singularity", meaning a curvature scalar (e.g. Kretschamann scalar) blows up and the singularity is strong in the sense that essentially any observer approaching it will experience destructive spaghettification in finite proper time. Some "weak null singularities" which occur in certain exact gravitational plane-wave solutions do not have this property; some observers will experience curvatures which diverge too rapidly, as it were, to tear/crush their bodies (think expansion tensor).
BTW, to amplify what Ben said, Penrose pointed out that pp-waves have the property that all their curvature invariants (even ones formed using scalar invariants built using covariant derivatives of the curvature tensor) vanish identically. However, their curvature tensors almost never vanish. This is roughly analogous to the fact that in Lorentzian geometry, a nonzero vector may have zero "squared length" (namely, the null vectors have this property).
In "Shear stress in Energy-momentum Stress Tensor"
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Q-reeus asks about the significance of off-diagonal terms in the stress tensor.
In fact, his question really does not refer to spacetime or gtr at all, but to the 3x3 stress tensor in continuum field theory in ordinary euclidean space. In fact, to any symmetric tensor field.
As usual, it helps greatly to use an orthogonal frame field (placa a frame---three mutually orthogonal unit vectors--- at each point, with this frame field smoothly varying from point to point). The principal axis theorem states that any symmetric matrix can be brought to diagonal form by an orthogonal transformation, i.e. by simply rotating the frame. So, in the stress tensor, can any off-diagonal terms always be transformed away?
The answer is that by rotating the original frame field appropriately at each point, one can obtain a new frame field which is aligned with the principle axes, so that no off-diagonal terms appear when the symmetric tensor is expressed in this adapted frame field. However, in elasticity problems, the adapted frame will in general not align with the surface of an object, so we cannot eliminate surface shear stresses, in general, by changing to a frame field adapted to the principle axes at each point of our stress tensor field.
See my old PF thread, "What is the Theory of Elasticity?", for more about the stress tensor, shear stresses, etc.
Pervect's Post#2 mentions the Komar integrals, but IMO this is not what is confusing the OP.
Baez said:
So one can argue that "gravitational energy" does NOT act as a source of gravity. On the other hand, the Einstein field equations are non-linear; this implies that gravitational waves interact with each other (unlike light waves in Maxwell's (linear) theory). So one can argue that "gravitational energy" IS a source of gravity."
Well it can't be both at the same time surely.
It is merely the imprecision of natural language which leads to the incorrect perception of a "paradox" here. Roughly speaking, in gtr, the gravitational field (Riemann curvature) certainly carries energy and momentum, and this certainly gravitates (affects the Riemann curvature), but the effects are not "ultralocal" in the terminology of Visser, so the contribution of the gravitational field itself is not represented in the stress-momentum-energy tensor.
why should there be any such contested topics after 95 years of GR?
There are hotly contested topics in non-relativistic field theory!
"Why haven't mathematicians found the general solution of the Navier-Stokes equations, after more than a century?" Also, "why can't physicists decide whether Mach principles make any sense?" Et cetera.
Some questions are just plain hard to definitely resolve, that's why.
I haven't read all of "Einstein Field Equations?"
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but I get the impression that the OP is making the mistake of worrying about philosophical problems before understanding the geometric intuition. Also, this thread illustrates the other two among the three most common sources of perennial but avoidable confusion due entirely to posters adopting lazy bad writing habits!
To repeat:
"flat" can mean "(locally) Ricci flat", "(locally) Riemann flat", "(locally) conformally flat", "(locally) flat spatial hyperslices", etc.,
"Schwarzschild interior solution" can mean "interior region of the Schwarzschild vacuum solution" or "Schwarzschild "incompressible" perfect fluid solution",
"singularity" can mean "curvature singularity" or "coordinate singularity" (or some other things like "shell-crossing singularity")
and if you don't write out which you mean, you'll confuse others and probably yourself.
Peter Donis says
Peter Donis said:
For instance in the Schwarzschild solution, which is one of the simplest solutions, we could 'push' all the curvature into the 'time' dimension.
Presumably referring to the fact that in the ingoing Painleve chart, the constant Painleve time charts are locally isometric to E^3. However, "curved time dimension" makes no sense and I recommend against trying to think of things like this!
Peter Donis said:
I don't know for sure if, mathematically, we can always find an embedding in a higher-dimensional *Euclidean* manifold.
If you allow indefinite signatures, any Lorentzian manifold can be locally embedded in a fairly small flat space and globally in a really huge one. If you search for embeddings in Ricci flat spaces (with appropriate signature), you need even fewer extra dimensions than for euclidean dimensions. There are known results concerning how many extra dimensions you need for various families of exact solutions.
Peter Donis said:
can we only in case of static curved spacetimes push all the curvature into the time dimension? I think the answer is yes, could someone confirm this is the case for the Schwarzschild interior solution?
That doesn't really make sense, but FWIW,
there are plenty of dynamical solutions, e.g. FRW models with E^3 slices, which allow spatially flat hyperslices; there are also static solutions which allow such hyperslices, such as as Schwarzschild vacuum (in the exterior; inside, we still admit such slices but the spacetime is dynamical there),
the Schwarzschild fluid admits spatial hyperslices orthogonal to the world lines of the fluid elements, which have constant positive curvature (locally isometric to S^3).
Figure:
It is impossible to eliminate shear stresses by using a frame adapted to principle axes at each point
Does Relativity estimate or predict the frequency of the gravitational radiation?
Can we also estimate the amount of gravitational radiation being emitted and thus the amount present at a gravity wave detector here on earth?
Are there any other causes which might explain the loss of orbital energy?
and got an excellent reply from Janus.
To elaborate slightly: a useful rule of thumb is that a inertial observer distant from an isolated system measures gravitational radiation whose properties are determined by the motion of the system projected onto a 2-plane orthogonal to the line of sight. Thus, in the case of an isolated binary system, a distant observer aligned with the axis measures circularly polarized radiation at frequency \omega, because the projected motion looks like a rotating barbell, while a distant observer in the plane of the orbit measures linearly polarized radiation at frequency 2 \, \omega because the projected motion looks like a barbell extending and compressing, as it were. (See the figure below.) At intermediate angles, a mixture of the two frequencies will be observed.
Here, only the two endpoints of the barbell are massive, and the amplitude of the radiation is determined from the second time derivative of the traceless quadrupole moment of the system. Even though the barbell is not changing shape (ignoring the very slow decay of the orbit!), because it is rotating the distribution of mass-energy and momentum is changing wrt an nonrotating inertial frame, and in particular, the quadrupole moment of the stress-monentum-energy tensor is changing wrt time. The details can be found in Schutz, A First Course in General Relativity.
Then Tanelorn asked a followup question
Tanelorn said:
So gravity waves are nothing more than variation with time of the static gravitational field?
This sounds like a variation of the issue worrying Ben Crowell. I urge responders to choose their words carefully to avoid "exciting the Van Flandern kookmode" I'd say something like this:
In the linearized approximation to the EFE, gravitational radiation emitted by an isolated gravitating system is identified as time variations in the Riemann tensor field (fourth rank) which
propagates as a wave,
is transverse to the direction of propagation,
propagates through vacuum at the speed of light,
when the gravitational field (Riemann tensor) is decomposed into a rapidly varying radiative and a slowly varying portion (the Coulomb tidal field), the radiative component decays like 1/r whereas the Coulomb component decays like 1/r^3,
when the gravitational field (Riemann tensor) is decomposed into three pieces (second rank 3-dimensional) wrt an observers world line (Bel decomposition), the electroriemann and magnetoriemann pieces of the radiative part of the gravitational field have comparable magnitude (when expressed relativistic geometric units); this is usually not true for the Coulomb component ( in a vacuum region, we can forget about the third piece, the toporiemann piece),
at large r, the radiation dominates, and then the principal Lorentz invariants of the field both vanish
<br />
R_{abcd} \, R^{abcd} = R_{abcd} {{}^\ast\!R}^{abcd} = 0<br />
heuristically, the radiative component would correspond in a QFT to spin-two massless exchange particle, the "graviton", but this turns out to be naive and no complete quantum theory of gravitation is yet known.
For comparison, in Maxwell's theory of EM, EM radiation emitted by an isolated charged system is identified as variations in the EM field tensor field (second rank) which
propagates as a wave,
is transverse to the direction of propagation,
propagates through vacuum at the speed of light,
when the EM field tensor is decomposed into a rapidly varying radiative and a slowly varying portion, the radiative component decays like 1/r whereas the Coulomb component decays like 1/r^2,
when the EM tensor is decomposed into two vector fields (first rank 3-dimensional) wrt an observer's world line into electric and magnetic vectors, the radiative portions have comparable magnitude and properties; this is usually not so for the Coulomb component,
at large r, the radiation dominates, and then the principal Lorentz invariants of the field both vanish
<br />
F_{ab} \, F^{ab} = F_{ab} {{}^\ast\!F}^{ab} = 0<br />
heuristically, the radiative component would correspond in a QFT to spin-one massless exchange particle, the "photon", and this is fully realized in QED.
MTW offer a very clear discussion of the effect of linearly and circularly polarized gravitational radiation on a cloud of intially comoving test particles.
In classical gravitation theories other than gtr, gravitational radiation is usually predicted, but may have properties and effects on test particles which differ significantly from the gtr predictions, e.g. might include longitudinal components.
The description of radiation in gtr is somewhat oversimplified: in the full, nonlinear EFE, radiation is a bit more complicated than just described, but this isn't expected to be relevant to understanding gravitational wave detectors near Earth, or even to change current expectations about sources of radiation.
Tanelorn said:
I understand that gravity wave detectors have been built deep underground to prove gravity waves exist. Would we expect to be able to detect this level of gravitational radiation here on Earth with the sensitivity of our detectors and with the level of noise and interference here and elsewhere?
Not deep underground. The on-line resources in the BRS thread "Resources for SA/Ms" will answer the second question.
In "Google Street View Camera Vehicles Collected WI FI data"
Google claims it was totally inadvertent that they collected wi fi data using their street view camera vehicles. Personally I can't bring myself to believe that...This was first revealed May but for some reason is just now hitting the fan again. It is totally unbelievable that they could have mistakenly done this world wide...But I still wonder why they did it. This was a world wide venture, and that means a lot of unintentional data was collected. It must have cost them a lot of money to collect information that they claim that they will now delete.
What, you thought I was actually going to comment?! I advise one and all to avoid using "friendship" features at social networking sites, but any curious SA/M knows what to do if they want the answer to Edward's question: shoot me an encrypted PM. (See the BRS thread "PKI Cryptosystems for SA/Ms: A Tutorial", which should have been titled "Personal Cryptography: a Tutorial for SA/Ms", but too late to change it now.)
Tanelorn got some fine resposes from marcus and twofish-quant, but remarked
it would be very good indeed to understand our universe as well as possible before we ourselves are forced to leave it.
I wonder: what precisely is his plan for leaving the universe? As regular readers know, I am unhappy with The State of Things, so I think I'd like to hitch a ride if he's offering
Figure:
Gravitational radiation from a binary system (schematic)
BRS: energy-momentum complexes, rotating star models, symmetries
Re "How does empty space curve?"
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Dalespam said:
unless I am missing some subtelty here spacetime still doesn't curve except in the presence of some stress-energy.
That's the right spirit, I think, but remember that in gtr, gravitational energy/momentum is not represented in the stress-energy tensor. So in a vacuum region (no nongravitational energy/momentum) we have zero Ricci curvature but nonzero Weyl curvature (associated with the gravitational field itself)
samalkhaiat said:
Einstein's equations can be written in the equivalent (Maxwell-Like) form
Any equation involving (one of many distinct) "gravitational-energy complexes" (which are pseudotensors) is not a true tensorial equation, which means for example that in some charts the "gravitational-energy" will vanish in a given small region while in others it will not. That kind of thing can be problematic, and despite a resurgence of interest in the past five years or so, IMO it is still fair to say that energy-momentum pseudotensors don't really provide much help in resolving the tricky issue of how to represent gravitational energy/momentum in gtr.
Dalepspam said:
Is there any reason to believe that the resulting boundary conditions would represent any "massless" physical situation? In other words, is there any reason to believe that you could physically have a Schwarzschild spacetime without the presence of a mass?
I think robphy was describing the idea that in a stellar model consisting of the world-tube of a ball filled with perfect fluid (say) matched across the surface of the "star" to a vacuum solution (if the ball is static this will be a portion of the Schwarzschild vacuum exterior region), we can isolate any vacuum neighborhood and consider that a "local vacuum solution". (Local in the sense of local neighborhood, as is standard in mathematics and hibrow physics.) In such a local vacuum solution, the presense of a source somewhere outside the domain covered is often implicit in some more or less murky fashion (but inferring the presence and location of a static spherically symmetric source is about as easy as such inferences get in gtr, I think). I think robphy was also referring to an elaboration in which one tries to match various local solutions to create solutions with strange global properties. There are quite a few examples in the arXiv (not easy to find, perhaps).
Regarding the quote by Geroch, I'd need to see more context to say more, but it is good to know that there are general theorems concerning asymptotically flat vacuum solutions and notions such as ADM and Bondi energy/momenta which give well-defined, general, and useful notions of the mass-energy and momentum of "isolated systems". But these don't apply if we toss in a bit of Lambda, so there is much work yet to be done.
Many of the newbie comments, e.g. by "Feullieton", are useless unless the posters clarify what they mean by "exist" (in Nature? in theoretical models? models in what theory?) and so forth. Basically, I think some of the newbies want to discuss the philosophy of physics (and the philosophy of manifold theory and differential geometry) but haven't yet recognized this.
In "empirical test of Einstein's famous goof"
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Passionflower said:
can we have the equation for the clock at the pole and the equator using the Kerr metric?
Note: if we can't do it please say so, but please no cop outs like "we don't need to", "we ignore rotation because the rotation is slow", we want to do GR here.
No doubt exact solutions representing vacuums outside rotating fluid balls exist, and there is good reason to think that they closely resemble the Kerr vacuum, but such have not been found. The only perfect fluid solution with symmetries resmembling Kerr is the Wahlquist fluid, which is physically unacceptable. In the mathematical sense, models obtainined by matching a realistic rotating perfect fluid ball to an exterior Kerr-like (but not quite Kerr) vacuum region certainly exist, but have not been written down (yet), and may well be impossible to write down in closed form, although nothing definitive appears to have been proven yet.
However, for the vacuum outside the Earth, existing and well-understood approximation methods are perfectly adequate; look for papers by Neil Ashby and others in the arXiv and Living Reviews offering brief descriptions of Post-Newtonian formalism models.
On the more elementary side, there are separate weak-field and slow-rotation-axisymmetric-but-possibly-strong-field approximations which can apply.
As a rule of thumb, introducing rotation often seems to make everything much more difficult in gtr. Experts have some insight into the reasons why but I don't know how to explain these insights in simple terms. But I'll offer this: in general, gravitational fields produced by isolated rotating sources have the property that physically interesting timelike geodesic congruences (world lines of a family of inertial observers) typically have vorticity and thus lack orthogonal spatial hyperslices. This effectively forces more nonzero metric functions depending on more variables, which makes exact solutions much harder to find by elementary means.
Almost all methods of finding exact solutions of (nonlinear) (systems of) PDEs involve exploiting symmetry in the sense of Lie's theory of the symmetries of differential equations (which is the historical origin of Lie theory as in Lie groups and Lie algebras). Relevant pairs of buzzwords include
independent and dependent variables,
external and internal symmetries,
base and fiber spaces
In Lie's theory, a transformation which "preserves the form" of a PDE gives a symmetry; the symmetries to be exploited (if possible) are not limited to metrical symmetries of the underlying spacetime (usually, the base space) but may also include symmetries involving the field variables. For example, when we say that Maxwell's theory of EM is "conformally invariant", we are referring to certain symmetries which have the nature of conformal transformations which when written out very concretely involve both the EM field components and the spacetime coordinates.
I never seem to have the energy to try to provide a painless introduction to Lie's theory, but there are a number of excellent textbooks by authors such as Brian J. Cantwell, Hans Stephani, and Peter W. Olver. Of these, the shortest may be the one by Stephani. Note that Lie's ideas work out differently for systems of ODEs and for systems of PDEs, so the theory falls into two complementary halves. For example, as a rule, to solve a system of PDEs one exploits symmetry to reduce the number of variables; to solve a system of ODEs, one exploits symmetry to reduce the order of the equations. A special case of the "point symmetries" studied by Lie, the "variational symmetries" studied by Noether, is also very helpful, since these involve symmetries of a Lagrangian formulation and lead directly to a "canonical" energy-momentum tensor and to "conserved" fluxes. A generalization leads to connections with methods used in the theory of solitons to construct "solitonic" exact solutions to certain types of wave equations. Interestingly enough, following Chern and others, these ideas can be expressed using curvature (in the abstract mathematical sense).
Coming back to the original topic, if a problem offers insufficiently many external and internal symmetries, closed form exact solutions may be impossible to find. Then one turns to Sobolev spaces and general theory which can establish existence, uniqueness, and some generic properties of solutions not known in closed form. See Robinsion, Introduction to Infinite-Dimensional Dynamical Systems, Cambridge University Press (maybe not the most apt textbook in this context, but it should certainly help convey the flavor).
In "Group of rigid rotations of cube"
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PhDorBust said:
Clearly the group has 24 elements by argument any of 6 faces can be up, and then cube can assume 4 different positions for each upwards face.
the group of proper symmetries of the cube is isomorphic to S_4 (order 24); the full symmetry group is the order 48 supergroup C_2 \wr S_3; see for example Coxeter, Regular Polytopes, Dover reprint. The subgroup lattice of S_4 is given in a previous BRS post; GAP painlessly computes the subgroup lattice of C_2 \wr S_3 which I can give in similar format if desired (31 conjugacy classes of proper nonidentity subgroups).
This is related to the unfinished BRS on the Rubik cube, in a rather general way, via the information theory which in some sense unifies Shannon's information theory and classical Galois theory. Recall that in that theory, the fundamental objects of study are an action by a group and the corresponding lattice of pointwise stabilizer subgroups, where considering various induced actions, combining actions in various ways, &c., ultimately blurs the distinction between pointwise and setwise stabilizers. This is important because most elementary discussions concern "setwise stabilizers" in some action, here the action by S_4 on the faces (apparently).
My book describes the rotations as follows:
3 subgroups of order 4 created by rotation about line passing through center of two faces.
4 subgroups of order 3 created by "taking hold of a pair of diagonally opposite vertices and rotating through the three possible positions, corresponding to the three edges emanating from each vertex."
My trouble lies with the second description, that is, I haven't the slightest idea of what it is saying. Any help?
(What book?...grrr...)
One can consider the action by S_4 (or the supergroup) on faces, edges, vertices, the four vertex-vertex diagonals, the three face-face axes, &c, and on sets of these. Considering various actions by a group is often an easy and efficient way to begin enumerating conjugacy classes of subgroups. In particular:
In the action on the three face-face axes, the stabilizer subgroup of an axis is C_4, and the action is transitive (any of the three axes can be moved to any other) so these must give a conjugacy class of three subgroups isomorphic to C_4.
In the action on the four vertex-vertex diagonals, the stabilizer of each diagonal is C_3, and the action is transitive (any of the four diagonals can be moved to any other) so theses must give a conjugacy class of four subgrups isomorphic to C_3.
In the action by S_4 on faces, pairs of opposite faces move in lockstep--- that is, the kinematic closure of one face is the pair consisting of that face and the opposite face, i.e. these two faces share the same stabilizer subgroup. Similarly for pairs of opposite vertices.
Also, any general comments on visualizing symmetry groups would be appreciated, I trouble going beyond dihedral group of order 4.
Maybe the problem is nonabelian groups? S/he can try a book by (I think) R. P. Burns which offers an excellent "workbook" type approach to learning about finite groups. The Schaum Outlines book on group theory by Baumslag and Chandler is also quite readable.
In "quaternions and metric of the 3-sphere"
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www.physicsforums.com/showthread.php?t=448298
("well known" ) the angle psi parameterizing the Hopf tori in the Hopf chart or toral chart on S^3 has nothing to do with the latitude angle in the polar spherical chart, so it should not be surprising that these angles have distinct ranges. One way to understand the range is to consider what happens as you slowly increase psi from 0 to pi/2. Another is to draw a tetrahedron in which a pair of opposite edges represent the two degenerate Hopf tori (two Hopf circles). We can imagine deforming the 3-sphere so that all the curvature is concentrated in these two great circles; just identify faces of the tetrahedron appropriately. Then consider how slices "parallel" to the pair of opposite edges (they form a family of parallel rectangles degenerating to the two edges) evolve as you move from one edge to the opposite edge.
Penrose's book The Road to Reality has a very nice picture of the Hopf circles which may suggest how to form the one parameter family of Hopf tori; the parameter is the angle psi the OP is worried about.
It may also help to observe that in a tubular neighborhood of one of the degenerate Hopf tori, the metric closely resembles ordinary cylindrical coordinates, locally in sense of open neighborhood; globally one shoulld imagine cutting the nested cylinders and identifying the circular ends to form nested tori. This gives a metric which is locally exactly the usual cylindrical chart on E^3; the Hopf chart on S^3 is similar, but in the sense of Riemannian geometry, locally (and even globally) gives 3-sphere geometry.
HTH.
Re "What is a Regular Transition Matrix?"
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www.physicsforums.com/showthread.php?t=352587
some useful buzzwords are "Frobenius-Perron theory", "ergodic decomposition", "invariant measure", and a very nice undergraduate level introduction to the application to finite Markov chains is in the textbook by Snell et al., Finite Mathematical Structures. There also plenty of other books which give readable introductions.
BTW, "transition matrix" is used in several fields of mathematics to mean several rather distinct notions; the OP should have stated s/he was asking about the usage in the theory of finite Markov chains.
Basically, there seems to be an assumption that after the Big Bang, the initial expansion of the universe was slowed by gravity. Many sources appear to feel that this is so self-evident that no further explanation is usually given other than a possible passing reference to GR.
I guess s/he is reading mostly popular sources, although s/he does cite one arXiv eprint, because of course this is a serious mischaracterization.
the basic GR premise that supports this conclusion.
I think pervect already had the same idea: s/he should learn about the Raychaudhuri equation, by preference by reading good textbooks rather than asking in PF! The Baez & Dolan expository paper is a good place to start, but a serious student should also study textbooks, IMO.
I think mysearch is struggling to express his desire to formulate and explore some competing models in the framework of gtr, based upon his verbal formulations. Unfortunately s/he runs into serious trouble immediately
The gravitational effects are assumed to align to the logic of Newton’s Shells... The force on an object m at radius=r>R, i.e. outside this volume, is subject to the normal inverse square law 1/r2 based on its distance r from the centre of the homogeneous volume.
I don't see much hope of making sense of this in the context of gtr!
S/he sketches his first model as envisioning:
a large spherical volume of homogeneous density, radius=R, exists within an infinite and absolute vacuum. This homogeneous volume has an effective mass and a centre of gravity.
This appears to mean something like a homogeneous dust ball surrounded by an asymptotically flat vacuum region. But then of course according to either gtr or Newton's theory, such a dust ball cannot remain static but must collapse. In context, presumably s/he meant that the dust ball is initially expanding, and then the expansion will be slowed by the gravitational self-attraction of the ball, in either gtr or Newton's theory. Assuming this interpretation, a nice simple gtr model meeting his verbal requirements would be the time reversal of the Oppenheimer-Snyder dust ball. This consists of an expanding dust ball (homogeneous time-varying density, zero pressure perfect fluid) matched across an expanding spherical surface to an exterior vacuum region, which is a portion of the Schwarzschild vacuum exterior with the appropriate mass parameter. The world lines of the dust particles form a uniformly expanding geodesic congruence (the ones near the surface of the ball are comoving with the surface), and this congruence has zero vorticity, so spatial hyperslices exist and turn out to be locally isometric to E^3 (vanishing three dimensional Riemann tensor).
S/he sketches his second model as envisioning:
Also assumes a homogeneous density, but now its volume conceptually extends to infinity.
This is could be compatible with an even larger array of gtr models; one fairly simple possibility would be a vacuum bubble inside an expanding FRW dust with E^3 hyperslices.
Since s/he says s/he is interested in better understanding gtr rather than shooting down modern cosmology, an even better model might be a hybrid consisting of an expanding shell of dust matched inside to an expanding vacuum region (portion of Minkowski vacuum) and outside to an asympotitically flat vacuum region (portion of Schwarzschild vacuum exterior region).
S/he asks about "a centre of gravity" but I don't think this really makes sense in gtr, because of the mathematics of curved manifolds leads to
the notorious difficulty of defining sensible notions of "distance in the large"
the notorious difficulty in averaging almost anything in any coherent sensible way
Because of the homogeneity s/he requires, the local versus global distinction is also relevant. Clearly there is no local (in sense of local neighborhood) "center of gravity" inside any FRW dust region! I suppose one could try to argue from the obvious nested spherical shells in our "dust ball at a time" that there is a global center. Note that one could choose any point inside the dust ball and make another family of distinct nested spherical shells; this would not include the surface of the ball, but this would need to be pointed out in order to identify a unique family of nested spherical shells.
John232 said:
An accurate big bang theory would have to explain why these burst started and stopped for several periods without matter.
I hope that won't pass without correction because this poster could learn something here: the standard hot Big Bang theory does not attempt to explain what happened before a time certain; its success consists of success in explaining (much) of what happened after that time.
Does GR radically disagree with this simplistic assessment of the underlying physics?
The conceptual foundation of gtr is radically from that of Newtonian gravitation, so the only possible short answer, I think, is "yes". The conceptual differences have technical consequences which cannot be ignored: the treatment of "conservation of energy", thermodynamics, and other core topics is significantly different in crucial ways.
To forestall the question: "why then does Newtonian gravity agree so closely with gtr?", one answer would be "in general, it agrees closely only in the weak-field slow motion limit; in addition, certain very simple and highly simplistic models may bear some points of naive agreement because there are only so many simple formulas".
Example: sometimes people ask why both Newtonian gravity and gtr give m/r^3 for the tidal accelerations outside a nonrotating isolated object of mass m. Well, that "r" is problematic; in Newtonian gravity there is no doubt what it means but the meaning has to carefully explained in gtr. So to some extent points of agreement can be illusory, or at least, there is almost always more to the story than simply stating an alleged "unambiguous agreement betweeen the predictions of N.g. and gtr".
Suddently s/he brings up dark energy, which looks rather like an ambush. If s/he really wants to understand gtr better, s/he should begin by studying a gtr textbook and in particular the best understood models. Leave speculations about dark energy out of it until one knows enough to begin to understand how such ideas fit into the big picture.
Re
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there has been a lot of study of what one might mean by "total energy of the universe". It turns out to be far from easy to define a useful general notion although there are obvious choices in certain restricted examples, including the situations Hawking has in mind (I guess, since the OP didn't cite or quote enough to be sure). It would be fair to say that the majority of researchers do not accept Hawking's occasional insistence on limiting oneself to the restricted situations he has in mind. If my guess is correct about what Hawking was referring to, I don't think the question of whether dark energy exists and if so what is properties are is relevant to the issue of whether the restrictions Hawking needs should be imposed. In fact, if I guessed right about what Hawking was discussing, the issue long predates the discoveries which have led to the inferences that
dark matter possibly exists,
dark energy possibly exists.
Re "Is mathematics a science?"
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I would say pretty much what HallsofIvy said, maybe a bit more:
the defining characteristic of science is the scientific method,
the defining characteristic of the scientific method is the comparision of experimental/observational data with quantitative theoretical predictions,
mathematics can be defined as "the art of precise reasoning about simple phenomena without getting confused" (by the ambiguities of nonquantitative language, for example); as such, it provides the necessary foundation for everything in science,
to be sure, mathematics is ultimately more about precisely defining and understanding abstract structures than about simple arithmetic, but ultimately, the mathematical ideas most likely to be used in science will involve real or complex numbers in some way,
to be sure, mathematics is not only the most powerful and practical tool in the intellectual's arsenal, but is possessed of great beauty, at least as perceived by those sufficiently capable of abstraction,
in mathematics as in any highly intellectual field, there is ample opportunity for individuals to stamp their personal style on a body of work; in particular, some arguments are widely agreed to be more beautiful than others,
in particular, statistics (a highly quantitative field, even when dealing with "categorical data") is clearly mathematical in language and content, and all experimenters must and do use statistics to interpret the meaning of their results,
in the philosophy of statistics, the question "what is a probability, that we should be mindful of it?" has been characterized as the greatest unsolved problem in math/stat/sci, a point which was emphasized by one of the very greatest mathematicians of the last century, Andrei Kolmogorov (who early in the century put the theory of probability on a theoretically sound foundation, measure theory, but this is not the same thing as answering the question just described!).
Shayan said:
MATHEMATICS IS NOT ABOUT REALITY!
Is Shayan quoting this alleged viewpoint in order to repudiate it? I can't tell.
micromass said:
Mathematicians only care about their axioms. As long as everything is consistent, then it's good.
I don't know what "Shayan" means by "reality" but I do know that since Newton, many, even most, of the mathematical problems which have been regarded by mathematicians as the most important problems, have been inspired by scientific, engineering or otherwise practical problems involving "the real world" in various ways. Further, many mathematicians, even most, are inspired by the prospect of comparing theory with experiment. Certainly that is true of everyone who functions as an applied mathematician (physicists, computer scientists, economists, even political scientists).
micromass said:
Mathematicians don't care about the realistic applications of complex numbers...
Really? S/he should search John Baez's UCR website. (JB is a mathematician by training.)
micromass said:
And people solved that polynomials because they were like games,
Really? There is a kernel of truth in that regarding Cardano and his contemporaries, but micromass should compare Newton's writings about polynomials (including solving them--- much more than just "Newton's method"). Newton himself emphasized applications of his work, and this is typical of the modern viewpoint in mathematics.
Also, ditto Gerald Edgar and Tom Gilroy, I think.
While no-one has yet mentioned the views of Hardy, I think they are implicit in several of the more passionate but less well-informed comments. The fact is, I think it is fair to say that the vast majority of mathematicians believe that Hardy's views have been discredited. Indeed, one remarkable phenomenon in the last 50 years has been the virtual fusion of some rather "pure" mathematics with some of the most popular lines of investigation in mathematical physics. It is particularly ironic that number theory has turned out to be so practical, even critical for our modern technoworld. This refusion of mathematics and physics has compelled a return to the mainstream view in Newton's day, which did not clearly distinguish between these subjects. (But "natural science" in the Newtonian sense has not been what modern practioners mean by "science" for two centuries at least.)
In "lie subgroup"
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the pullback of a closed set under a continuous map (its components are the given real-valued functions) is topologically closed, so the subgroup is a closed subgroup of a Lie group, and thus a Lie subgroup.
Re "Tikekar superdense stars and interior metrics"
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FunkyDwarf asks about a particular static spherically symmetric solution (ssspf) in gtr, the Vaidya-Tikekar ssspf (1982), referring to the original paper (which I have read) and two more recent ones (which I have not read).
Some general remarks might help:
all ssspf solutions are known in more or less closed form,
they can be nicely expressed in various "canonical" forms including
Wyman-Lake form (by far the oldest)
BVW form
Martin-Visser form
Rahman-Visser form (Shahinur Rahman, not Sabbir);
some of this nice work is quite recent; see eprints coauthored by Visser in the arXiv,
the Martin-Visser form admits a remarkable internal symmetry which enables one, given a central density and pressure, to choose a new central pressure while keeping the old density; this "pressure change transformation" (a point symmetry in the sense of Lie) gives new (geometrically and physically distinct) solution, so from anyone solution in MV form one immediately obtains infinitely many others in this way,
regardless of canonical forms, all ssspfs can be written using Schwarzschild like coordinates, or "isotropic" coordinates, or in various other ways, but those are the two most popular and you can probably guess what they are!; below I'll write r for the Schwarzschild radial coordinate,
all good ssspf solutions can be matched across the surface (r value where pressure vanishes) to an exterior vacuum which is a portion of the Schwarzschild vacuum with the appropriate mass parameter,
all ssspf solutions in gtr must obey the Buchdahl limit: the surface must be at r_s > 9/8 2m,
all ssspf solutions have the property that the congruence of world lines of fluid elements are nongeodesic but form a vorticity-free congruence, and the timelike unit vectors are parallel to a timelike Killing vector (hence static spacetime, since there exists a vorticity-free timelike Killing vector field)
however you write your ssspf Ansatz, physically there are two variables (density and pressure) so geometrically there are two metric functions which depend only on r; one will generally be completely determined from the other by solving a linear ODE, and the "master" function will generally be determined by solving a second order ODE, so we should expect two free parameters,
wrt a frame comoving with the fluid particles, the Einstein tensor is automatically diagonal and G^(33) = G^(44) is automatic, so the only condition is G^(22)=G^(33); this gives a second order ODE for your metric function,
the orthogonal hyperslices always resemble S^3 at the center r=0; graphically, if you plot the components of the three-dimensional Riemann tensor wrt the obvious frame field, r_(2323) = r_(2424) falls off more rapidly than r_(3434) as you head outwards from r=r0,
graphically, good ssspf solutions generally have positive pressure and density, pressure falling to zero at the surface where the density will usually be positive, i.e. G^(22) = G^(33) = G^(44) is everywhere non-negative and less than G^(11) and falls to zero at the surface,
most closed form ssspf solutions will be written using two parameters, as already noted; in principle the metric functions can be rewritten so that the parameters are central density and central pressure, and the solution may look much simpler when written this way, or much more complicated; typically the surface radius r_s will be a function of central density and pressure; sometimes a third parameter appears which simply sets a "standard radius" for computing the relative gravitational time dilation of observers riding with fluid elements,
plugging in arbitrary values for the parameters may very well lead to negative density and/or negative pressure, which must usually be rejected if we are making a stellar model!,
most ssspf solutions are not relativistic polytropes and most are not consistent with thermodynamical expectations (don't have an obvious notion of surface temperature, don't have a physically sensible equation of state)
"Good" rules out cases which violate energy conditions and some exceptional cases in which there is not surface at any finite r value. Even after restricting to "good" solutions, there is the problem of finding parameter values which give reasonable values of central density/pressure and surface radius for a given type of star; if there is no equation of state in view (which is usually the case), there is generally no reason to think that any values will give impressive models, so serious models are generally constructed numerically as described in MTW.
The Vaidya-Tikehar ssspf conforms to these expectations but is quite a bit more complicated than many other ssspf solutions such as the Tolman IV ssspf.
(For values of the central density and pressure corresponding to reasonable crude guesses for neutron stars, the Tolman IV solution actually does give values for surface radius which are reasonable for a nuetron star, and it has an equation of state, but a rather wacky one, and AFAIK the Tolman IV model is mainly of pedagogical value, and the other exact ssspf solutions known in simple closed form are AFAIK no better than Tolman IV, although some may give better approximations of other types of stars--- IIRC, the Tolman IV solution doesn't seem to work very well for ordinary stars, which is rather interesting in itself.)
In the original paper, the authors
propose the desideratum that the three-dimensional Riemann tensor of the orthogonal hyperslices should have a particular form; in this form, the parameter satisfies K < 1 and the case K=1 gives E^3 slices while K=0 gives S^3 slices, with the other cases deformed three-spheres as described above (including the fact that the geometry approaches S^3 geometry as r->0+),
write down the Schwarzschild chart for their ssspf, with one undetermined function of r,
transform variables and solve the condition G^(22) = G^(33) for their unknown metric function \nu,
specialize to the case K=-2, in which the slices have particularly simple three-dimensional Riemann tensor
FunkyDwarf thinks he spotted an error but I don't see the problem and the K=-2 solution given by the authors is certainly an exact ssspf.
By choosing values for their parameters one can find solutions which obey the energy conditions, e.g. R=1, A=3, B=5 works with r_s ~ 0.504. Plugging in numbers might show these values are physically out of range for a neutron star.
There is nothing that I see in Vaidya and Tikekar 1982 which implies that their ssspf neccesarily models "superdense stars", whatever that means (neutron stars?); they simply produced yet another exact solution of this kind. Some of the more recent arXiv eprints give this solution in the BVW form, as I recall.
Can't resist touting the many virtues of the new canonical forms mentioned above. Just as an example, the Martin-Visser form is (in Schwarzschild chart)
<br />
ds^2 = <br />
-\exp \left( -2 \int_r^\infty g(\bar{r}) d\bar{r} \right) \; dt^2<br />
\; + \; \frac{dr^2}{1-2m \, r^2} <br />
\; + \; r^2 \, d\Omega^2<br />
where m is a function of r satisfying
<br />
m^\prime =<br />
\frac{-2r}{1+r \, g} \; (g^\prime + g^2) \; m <br />
+ \frac{(g/r)^\prime/r + g^2/r}{1+r \, g}<br />
Here, the ingenious idea of Martin and Visser was to use as metric function
<br />
m = M/r^3<br />
where M is also a function of r, giving the total mass inside that r, so that as r approaches r_s, M approaches the mass parameter used in the Schwarzschild exterior. It turns out that this metric function m is a very clever choice.
We can plug in
<br />
G^{11} = 8 \pi \, \epsilon, \; \;<br />
G^{22} = G^{33} = G^{44} = 8\pi \, p<br />
and eliminate g and m. We find
<br />
g = \frac{-p^\prime}{p + \epsilon}, \; \;<br />
m = \frac{-p^\prime/r - 4 \pi \, p \,( \epsilon + p)}<br />
{\epsilon + p - 2 r \, p^\prime}<br />
where the mass-energy density \epsilon satisfies an Abel equation
<br />
\epsilon^\prime = <br />
A \, \epsilon^3 + B \, \epsilon^2 + C \, \epsilon + D<br />
where the coefficients A,B,C,D depend upon the pressure. Different choices of solution for epsilon in terms of p, up to two constants, gives different families of ssspf solutions, e.g. the Vaidya-Tikekar family. Finally, choosing central values for \epsilon and p determines a specific solution which can used as a model of a nonrotating isolated static fluid ball.
Here, a "good" solution should satisfy
<br />
\epsilon > 0, \; \;<br />
p > 0, \; \; <br />
p^\prime < 0, \; \;<br />
p + \epsilon > -2 r \, p^\prime > 0<br />
You can plug in the polytrope conditions (oddly, haven't yet seen this done in the literature)
<br />
\mu = \mu_0 \, (T/T_0)^n, \; \;<br />
p = p_0 \, (T/T_0)^{n+1}, \; \;<br />
\epsilon = \mu + n \, p<br />
where \mu is the mass-density, p is the pressure, and \epsilon is the mass-energy density (including the mass-energy due to nonzero temperature of the matter). Here, n is a constant, the adiabatic index. Notice that
<br />
M^\prime = 4 \pi \, r^2 \, \mu<br />
Then we have
<br />
g = \frac{r \, \left( m + 4 \pi \, p_0 \, (T/T_0)^{n+1} \right) }<br />
{1-2m \,r^2}<br />
This is a little more elaborate than a just plain ssspf because we have another variable, the temperature, and we determine everything in terms of the central density, pressure, and temperature.
Now you obtain some interesting and amusing expressions for various quantities. For example:
the gravitational acceleration of the fluid element at a given r value is:
<br />
\nabla_{\vec{e}_1} \vec{e}_1 = <br />
\frac{m r \, (1 + 4 \pi \, p)}{\sqrt{1-2m \, r^2}} \; \vec{e}_2<br />
the tidal tensor is given by
<br />
\begin{array}{rcl}<br />
E_{22} & = & -2m + 4 \pi \, (\epsilon + p) <br />
= -2m + 4 \pi \, (T/T_0)^n \; <br />
\large( \mu_0 + (n+1) \, p_0 \, (T/T_0) \large) \\<br />
E_{33} = E_{44} & = & m + 4 \pi \, p =<br />
m + 4 \pi \, p_o (T/T_0)^{n+1}<br />
\end{array}<br />
which shows how the stresses inside the fluid depend on the temperature at each point,
the three-dimensional Riemann tensor of the "constant time" spatial hyperslices is given by
<br />
r_{2323} = r_{2424} = -m + 4 \pi \, \epsilon, \; \;<br />
r_{3434} = 2m<br />
As we approach the center r=0, the acceleration vanishes and
the Riemann tensor of the hyperslices approaches
<br />
r_{2323} = r_{2424} = r_{3434} = \frac{8\pi}{3} \, \epsilon_0<br />
Just one problem, as you may have already noticed: the polytrope assumptions ensure that either there is no surface (where sphere r=r_s where p(r_s) =0)! So the polytrope assumption isn't neccessarily what we want.
You know how the U.S. Congress opens each session? Even though most of the members are probably stone-cold atheists in their innermost cynical hearts? (And as any Freudian can see, have apparently not quite separated from either the God Father or the Mother Country.) In similar faux-pious spirit, I should probably open each BRS post with this incantation.
Someone recently advised me to develop the habit of enumerating my potentially offensive remarks, and I see I have already Potentially Offended:
the faithful
Freudian psychiatrists
the Congress
the mob
the English royal family
Frenchmen who think Frenchmen should sing in French
the NSA
God Himself, assuming a notoriously lacking existence theorem.
And I haven't even gotten started in this post!
Fear of politically-motivated retribution aside, what prompts me to such sentiment? Well, all the truly useful math and physics books come from CUP. Well, not quite all, but all the ones anyone but Croesus can afford. Their London Mathematical Society Student Texts (LMSST) series is particularly excellent in terms of contemporary value. And fans of differential geometry will be particularly delighted by their recent reprint of William L. Burke's Applied Differential Geometry, which I urge every SA/M to cite in PF whenever someone asks "what the bleep is a (one-form) (tensor), anyway?"
Some Dover reprints, including some of their new Phoenix series, are also really great books, but alas I have to say that on balance they still tend more toward books which are not as relevant today, although there are some brilliant exceptions such as Flanders's classic on differential forms. I plan to mention some more below.
It's rather staggering that I began making less than laudatory remarks about Dover back when none but the odd visionary envisioned the possible Death of the Book in our lifetimes, but sadly, I now find that I must reverse myself and beg SA/Ms to impoverish themselves by buying Dover, simply in a last ditch attempt to save the book itself. One of the oldest, and still the best, form of information storage and retrieval.
Don't even get me started about such troubling issues with e-books and browsing-in-the-cloud as the question of who owns an e-book (hint: not you!) and a variety of (you guessed it) privacy and computer security issues. Let's just say that I think SA/Ms should avoid reflexively citing Google books at every opportunity, because its possible to give a coherent argument that this is contrary to the medium term interests of scholarship itself, and thus, science itself.
And now, courtesy of the Department of the Awkward Segue:
Discuss: the sudden uptick in sophisticated questions may be due to the physorg prize recently awarded to PF.
Re "Stress-energy tensor"
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wrt a frame field adapted to the EM or KG field, and denoting by \epsilon the energy density:
the stress-energy tensor of a non-null EM field takes the form:
<br />
T^{ab} = <br />
\epsilon \;<br />
\left[ \begin{array}{cccc}<br />
1 & 0 & 0 & 0 \\<br />
0 & -1 & 0 & 0 \\<br />
0 & 0 & 1 & 0 \\<br />
0 & 0 & 0 & 1 \\<br />
\end{array} \right]<br />
this applies to the magnetic field of a bar magnet, and if you like the "field-lines" picture, then intuitively speaking, "the magnetic field lines repel each other but also try to contract along their own length"; the field seeks an equilibrium configuration balancing these desiderata; the desire of each field line to contract along the spatial direction aligned with the field line, while repelling all neighboring field lines, is clearly visible in the (-1,1,1) structure seen in the spatial components on the diagonal,
the stress energy tensor of a null EM field (vanishing principle Lorentz invariants; EM radiation) takes the form:
<br />
T^{ab} = <br />
\epsilon \;<br />
\left[ \begin{array}{cccc}<br />
1 & \pm 1 & 0 & 0 \\<br />
\pm 1 & 1 & 0 & 0 \\<br />
0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 \\<br />
\end{array} \right]<br />
a massless Klein-Gordon scalar field typically has spacelike gradient in some "static region" and timelike gradient in some "dynamic region", and
in a region where the scalar field has timelike gradient, the stress-energy tensor takes the "stiff-fluid" form:
<br />
T^{ab} = <br />
\epsilon \;<br />
\left[ \begin{array}{cccc}<br />
1 & 0 & 0 & 0 \\<br />
0 & 1 & 0 & 0 \\<br />
0 & 0 & 1 & 0 \\<br />
0 & 0 & 0 & 1 \\<br />
\end{array} \right]<br />
in a region where the scalar field has spacelike gradient, the stress-energy tensor takes the form:
<br />
T^{ab} = <br />
\epsilon \;<br />
\left[ \begin{array}{cccc}<br />
1 & 0 & 0 & 0 \\<br />
0 & 1 & 0 & 0 \\<br />
0 & 0 & -1 & 0 \\<br />
0 & 0 & 0 & -1 \\<br />
\end{array} \right]<br />
on the boundary between two such regions, where the scalar field has null gradient, the stress-energy tensor takes the same form as a null EM field.
Well-known static examples of mcmsf solutions include the Janis-Newman-Winacour mcsmf and the Ellis-Bronnikov (Morris-Thorne) mcsmf, in which the scalar field has a spacelike gradient. The Roberts mcmsf has a static exterior region where the scalar field has spacelike gradient and past interior and future interior regions which are dynamic, where the scalar field has timelike gradient.
Recalling that to take the trace, we first form {T^a}_b and then contract, the EM field always makes a traceless contribution to the stress-energy tensor, while the massless KG field makes a contribution with nonzero trace except where the gradient is null.
To avoid possible misunderstanding: choose any event and boost/rotate the frame there. In the new frame, the energy-momentum tensor will most likely not assume the above forms, which are only valid for suitably adapted frame fields. But scalar invariants of the EM tensor will of course be the same no matter what frame field you use.
If you use a coordinate basis rather than a frame field (ONB in the language of MTW), then you are unlikely to find any chart in which the stress-energy tensor looks as simple as above, unless you are working with Minkowski spacetime (i.e. str, not gtr).
FWIW, the field equations are
for EM, the Maxwell equations written in formalism suitable for curved spacetimes, e.g. using differential forms (plus the Hodge star):
<br />
dF = 0, \; \; d{{}^\ast\!F} = 4 \pi \, J<br />
for massless KG, the curved spacetime wave equation
<br />
\Box \phi = 0<br />
In "Regge Wheeler Equation"
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When we perturb schwarzschild metric with linear perturbation we get Regge-Wheeler equation. Which is schrodinger equation for spin 2 particles. Gravitational waves also have spin 2. Is there a connection?
Yes, the dynamic perturbations will describe gravitational radiation propagating near an otherwise static spherically symmetric gravitational field in a vacuum region (i.e. Schwarzschild vacuum exterior), and the QFT inspired shorthand slogan for the tensorial nature of gravitational radiation in gtr (strictly: weak-field gtr) is "spin-two". For more detail the OP should see chapter 4 in Chandrasekhar, Mathematical Theory of Black Holes.
Re "Dirac brackets and gauge in special relativity"
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the discussion in Lawrie, Unified Grand Tour of Theoretical Physics, should be perfect for the OP.
Re "Solving for g_φφ=0 in charged/rotating BHs"
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stevebd wants to solve for r the equation
<br />
\frac{R^4}{\Delta}=a^2 \sin^2 \theta, \; \;<br />
\hbox{where}<br />
R=\sqrt(r^2+a^2), \; \;<br />
\Delta=R^2-2Mr+Q^2 <br />
He didn't say that he is looking at the Kerr metric written in some chart (presumably ingoing Eddington or ingoing Painleve--- if he wants a good answer s/he should write out the line element intended so that we at least know what "r" might be!). Fortunately, I can confidently guess that he is trying to find the condition that the coordinate vector \partial_\phi be null. Then the obvious circles are closed null curves, and such do indeed exist in the "deep interior" of the Kerr vacuum.
(But not to worry because there are independent good reasons to think realistic models of black hole interiors as treated in gtr do not look that the Kerr vacuum in the "deep interior", even though the exterior should closely resemble Kerr vacuum!).
tiny-tim said:
isn't it a straightforward quadratic equation in R^2?
I think I know what Tim was thinking, and for a moment I made the same mistake, but no, because plugging
<br />
R=\sqrt(r^2+a^2), \; \;<br />
\Delta=R^2-2Mr+Q^2<br />
into
<br />
R^4 = \Delta \, a^2 \sin^2 \theta<br />
gives
<br />
(r^2+a^2)^2=a^2 \, \sin(\theta)^2 \, (Q^2-2Mr+r^2+a^2)<br />
Thus Maxima:
So the OP is looking for real roots in an appropriate range for a fourth order polynomial. The first thing is to check how many there are, and he can do that using Sturm chains. Note this requires choosing numerical values for the other parameters. Then he can apply the formula for the roots of a quartic to find the roots, then he can choose the particular root he needs. The answer will probably take about a page to write if he writes small. But he can use perturbation theory to find a useful approximation. Even better, he can use perturbation theory from the start and seek an approximate but memorable answer rather than an exact but useless since over complex answer.
There are many wonderful books which offer brief but useful introductions to the elements of perturbation theory, including:
Wilf, Mathematics for the Physical Sciences, 1962, available as Dover reprint, stylish and in good taste in terms of what the author chooses to discuss (one notational quirk: c/a+b for c/(a+b) appears in-line, a not uncommon notation before 1920 or so, but rather odd in a 1962 book and sure to confuse modern students),
Richard Bellman, Perturbation Techniques, 1966, available as Dover reprint; even more stylish!,
Simmond and Mann, A First Look at Perturbation Theory, 1986, Dover reprint; not as stylish but it has a chapter on polynomials which will get the OP where he wants to go,
Richards, Advanced Mathematical Methods with Maple, CUP; one of the very best math methods books, and if you use Maple, definitely the one to obtain,
E. J. Hinch, Perturbation Methods, CUP, excellent for DEs.
All but the last book discuss analytic (i.e. "given by a symbolic formula", not "real/complex analysis") approximations to roots of polynomials, and all discuss numerous applications to ODEs. There not as much overlap as you might guess due to the richness of the subject, however.
Wilf's book also discusses Sturm chains and many other useful things; I have used Sturm chains in some of my old posts to locate real roots of (for example) the "effective potential" for the Schwarzschild-de Sitter lambdavacuum. Sturm's technique is quite useful and well worth learning.
Dover books: I don't think they'll help anyone learn superstring theory, but perturbation theory is never going to go out of style, so Dover books on perturbation theory are not likely to become irrelevant any time soon.
BRS: multiple confusions about FRW models; plus, Maxwell's mechanical model
Re "Conditions for spacetime to have flat spatial slices"
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this thread certainly seems to have been going in circles since JDoolin got involved, and I confess I lack the energy to try to do more than skim it. In the sequel, I think my comments may partially duplicate things George Jones, Ben Crowell, Lut Mentz, and some others have already said to JDoolin, but most are I think new.
Originally, I think Peter Donis was trying to ask something like this: when does a spacetime admit a family of spatial hyperslices which are all locally isometric to E^3? Or even: when does a (vacuum) (perfect fluid) model in gtr admit an irrotational timelike congruence whose orthogonal spatial hyperslices are locally isometric to E^3? If so, there are good partial answers in the research literature; see "the exact solutions book" coauthored by Stephani for starters.
Due to differences between Lorentzian and Riemannian geometry, it is not possible to diagonalize every symmetric matrix using Lorentz transformations (in the tangent space to any point, in a Lorentzian manifold) S \rightarrow \Lambda^t \, S \, \Lambda, although it is possible to diagonalize every symmetric matrix using orthogonal transformations (in the tangent space to any point, in a Riemannian manifold) S \rightarrow O^t \, S \, O. See for example the book by Barrett O'Neill.
Then, JDoolin started talking about lotsa stuff which seems to arise from various misconceptions about curved manifolds in general and FRW models in particular. It is probably no coincidence that he hints that he is a devoted fan of the non-standard (and possibly even incorrect) approach of Lewis Epstein. (I recall once looking at the book in question but can't recall anything now.)
Just one example indicating serious confusion:
JDoolin said:
I'm a little troubled that the Robertson Walker chart is either mapping coordinate time or proper time, depending on who gives me an answer.
I have no idea what he might mean by that, but it would make no sense if he were using "mapping", "coordinate", or "proper time" in their standard senses.
JDoolin said:
You're sometimes saying that the vertical coordinate in the Robertson-Walker diagram represents the proper time of particles. Other times, you're acting like it is the actual time passed by the central observer...The only place where those two definitions can be shared is along the single line representing the worldline of the "stationary" particle.
Well, proper time measured by an observer between two events on his world line is the interval integrated along his world line between those two events.
Lut Mentz mentioned a (valid) coframe field which defines the FRW dust with E^3 hyperslices orthogonal to the world lines of the dust particles, in which differences in the t coordinate does give proper time intervals as measured by any observer riding on a dust particle. So
Peter Donis said:
What Mentz114 is calling the Painleve chart for the FRW spacetime is a different coordinate system used to describe that spacetime, in which the metric looks quite different than it does in the Robertson-Walker coordinate system. In this coordinate system, the "time" coordinate t does *not* directly represent the proper time of "comoving" observers (at least, I don't think it does based on looking at the metric--Mentz114, please correct me if I'm wrong).
Lut is correct; Peter is not. A Painleve type chart is distinguished by the existence of an irrotational timelike geodesic congruence whose orthogonal hyperslices are "nice", even locally flat (locally isometric to E^3), such that differences in the time coordinate corresponds to proper time intervals as measured by any of a certain family of inertial observers--- the ones whose world lines are the integral curves of the irrotational timelike geodesic congruence just mentioned. IOW, the t=t_0 hyperslices are the orthogonal hyperslices of our irrotational timelike geodesic congruence, and these slices are locally flat (or, in a generalized notion of Painleve chart, otherwise "nice").
JDoolin said:
It has to be proper time that he's talking about, because, he then proceeds to do a Galilean Transformation on the diagram.
Part of the problem seems to be that JDoolin doesn't yet understand that "proper time" makes no sense unless referred to a timelike congruence of world lines. No doubt he is thinking of the proper time measured by observers comoving with the dust particles, but even so, by omitting the qualifiers I suspect he is confusing himself.
JDoolin said:
proper time (OF A WORLDLINE) is an invariant quantity, but proper time is NOT A COORDINATE. Coordinates are contravariant; not invariant.
Aha, this is clearly a specific confusion. There are certainly plenty of coordinate systems such that differences in time coordinate correspond to proper time interval measured by an observer riding on one integral curve of a certain timelike congruence (not neccessarly a geodesic congruence). Then, coordinate time intervals certainly do correspond to proper time intervals as measured by observers having the specified world lines. So JDoolin just confirmed my guess about one of his underlying confusions: a clear example where sloppy writing permitted sloppy thinking, which prevented his making progress.
A coordinate is simply a strictly increasing (real valued scalar) function defined on some open neighborhood U of some manifold. That is, a function z such that dz is nonzero everywhere on U. If we have another such function y, and if the two-form dy \wedge dz is everywhere nonzero on U, then our two coordinates form a "partial net" such that the integral curves of \partial_y, \, \partial_z are never tangent in U. (See the nice discussion in Hilbert and Cohn-Vossen, Geometry and the Imagination.) This is the generic situtation, in a sufficiently small U. Continuing, we can add more coordinates until we have a local coordinate chart on U, or some smaller neighborhood contained in U. Then the strictly increasing property means that the n-tuple of values of the coordinates uniquely labels each event in U. That's all there is to it.
Peter Donis said:
The specific example I used, that of FRW spacetime, *does* have the property that a single coordinate patch can be used to cover the entire spacetime
True.
JDoolin said:
The Effect of the Lorentz Transformations are essentially directly proportianal to distance in space and time. i.e. If you Lorentz Transform an event that is 2 light years away, the effect will be roughly twice as much as if you Lorentz Transform an event that is 1 light-year away...If you do a Lorentz Transform on an event that is a billion light years away, the effect is roughly a billion times as much as if you do an LT on an event that is 1 light year away.
This strongly suggests to me that another fundamental problem is very likely that JDoolin has never mastered the geometry of linear transformations. He seems to vaguely grasp part of the meaning of "linearity" but he clearly does not understand that the Lorentz group consists of many linear transformations which have rather distinct geometric properties, e.g. rotations versus boosts versus loxodromic transformations.
JDoolin said:
Check out the discussion page for the Milne Model, because there are some things there that came from the actual book. When I tried to put actual quotes from Milne in the main article, they were removed.
Oh gosh. Well, not to defend anything which may or may not go on in WP (so far we have only JDoolin's side of this story), but as a rule, encyclopedia articles not about "the history of X" are not concerned with what historical figure F said about X 100 years ago, but about anything in modern textbooks attributed to F (because derived from what F actually said) which may be relevant to modern understanding, taking account of the big picture.
In general, in my experience, mathematically weak students (often autodidacts) often decide that "reading the masters" will make up for studying modern textbooks. But when Chandrasekhar and others urge us to read the masters, they really mean that the most mathematically capable students may not require spoon feeding from modern textbooks, but by studying the old masters and occasionally attempting exercises in modern textbooks as a reality check, may efficiently reinvent anything they need to reach the current frontiers. I would modify that slightly: postdocs with time and effort can benefit from reading the masters, but ambitious Ph.D. students need to reach the frontiers ASAP and they'd be well advised to stick to the textbooks and (when they know enough) the research literature.
Time and again I see well-intentioned autodidacts go down this "Great Books of Science" path, which IMO limits them to pseudo-intellectualism, which is pretty sad, since none of this stuff is so very hard if you approach it in the right manner.
Also, not to denigrate Milne, who was certainly a leading astronomer in his day who made important contributions, but his writings on gtr-related stuff are nowhere near as important or relevant as the books by MTW and Chandrasekhar, so anyone wishing to "read the masters" should at the very least know who the masters have been!
JDoolin said:
Pardon me, but does the current model really "FIT" that well? We have no real explanation for inflation. We have no dark energy. We have no dark matter. We have a theory that is inconsistent with quantum mechanics. But we have an equation that matches up really well.
Well, this is a rather childish view. Anyone who knows anything about real science knows that when we approach the frontiers of science, there are always far more questions than answers. What should impress outside observers are the facts that
cosmologists have a theory which explains so much with so little,
cosmologists clearly recognize a large number of issues where they know they can't yet say very much with very much confidence.
JDoolin said:
The thing is, I can show you a conformal mapping between the Milne model and the Robertson Walker Metric, which preserves the speed of light and has exactly the same events, if you're interested. All the same variables are there. All the same events are there, except the singularity is transformed into a lightcone in Milne's version, and it's turned into a plane in the "comoving matter" version.
Not implausible since FRW dusts and Minkowski spacetime ("Milne model") are both conformally flat, so conformal transformations certainly exist. But what on Earth does he think this proves? Maybe this?
JDoolin said:
In the Friedmann-Walker diagram, the light "from the big bang" crosses every single worldline. But in the "comoving particles" diagram, the light just passes a finite number of worldlines.
Oh wow, my gosh, he really is confused! I don't know what charts/frame fields he means by "Friedmann-Walker diagram" or "comoving particles diagram", but I don't have to, in order to know he has made some kind of serious error.
Unless, just possibly, he doesn't realize that the FRW dust with E^3 hyperslices (as in the coframe Lut Mentz wrote down) and the FRW dust with S^3 hyperslices are not locally isometric and thus not physically equivalent, and the second claim refers to an FRW dust with S^3 hyperslices, while the first refers to an FRW dust with E^3 hyperslices. If so, part of his confusion must involve some misinterpretation of a conformal mapping between two such distinct dust models, both conformally flat but with nonzero Ricci tensors and with different global (and local!) properties.
It's frustrating to see someone so confused, and possibly so confused by reading one bad book (at least, it was obviously bad for JDoolin to read that book), but from the posts by JDoolin I've seen so far, I think we can give him benefit of the doubt by assuming he is honestly curious and simply confused by his reading, not one of these idgits who have set out to blow down modern cosmology well before they have a clue what this subject is all about.
In "Special relativity adandons Maxwell's mechanical interpretation of EM?"
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there is a fine old Science article from c. 1979 which very clearly discusses Maxwell's mechanical model. Every five years I need to cite it and lately, find I can't remember author or title, which makes it difficult to find the citation. Sigh... Another good cite here would be Feynman's brilliant discussion of how EM waves propogate, in terms of the partial derivatives appearing in Maxwell's equations written out in conventional vector calculus style.
Re
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the perfect book for him would be Geroch, Mathematical Physics, written from the categorical point of view throughout. Also highly recommended: Lawvere and Schanuel, Conceptual Mathematics (much more sophisticated than you'd guess from the early chapters).
Re
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www.physicsforums.com/showthread.php?t=448388
looks like the intersection of four conics on CP^4, so one can use Groebner basis methods to look for a solution, or at least for information about solutions. One can use Schubert calculus (cohomology of certain Grassmannians) to compute the expected number of solutions. If he wants real solutions rather than complex ones (as I suspect he does), things will probably be a lot more complicated, unless he gets lucky.
Kendall, A Course in the Geometry of n Dimensions would be perfect for the OP. E.g., Kendall derives the distribution of chi-square from n-dimensional euclidean geometry. Most of the other distributions studied by people like Fisher can also be so derived, and Kendall does so. Great book for those interested in understanding the unity of mathematics!
BRS: Hubble expansion (right) versus "expanding Earth" (wrong)
In "Universal expansion"
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Landrew admits
If a little knowledge is a confusing thing, I certainly have the prerequisites to be confused about Universal Expansion.
That is exactly the problem, but he gets points for recognizing this!
Some physicists seem to be saying that all the stars and galaxies are flying apart like shrapnel from a large explosion, and other physicists seem to be saying that space itself is expanding metrically, thereby accounting for the fact that the more distant the object we observe, the faster it seems to be moving away, even apparently exceeding the speed of light.
The language of physics is mathematical reasoning. The mathematics of cosmological models formulated in gtr is unambiguous, but gtr rests upon the mathematics of curved manifolds, which laypeople don't know anything about. Thus when physicists speak to a lay audience they must "dumb down" the truth into statements in natural language ("plain English"). In different contexts, physicists may consider different and apparently inconsistent partial reformulations in natural language to be appropriate, but laypersons should not assume from "obvious contradictions" that there is anything wrong with the actual mathematics. In particular, both of these statements intuitively capture some aspects of the actual mathematics
"stars and galaxies are flying apart like shrapnel from a large explosion"
"space itself is expanding"
but neccesarily, both also miss crucial aspects, and their apparent mutual contradiction is seen to be illusory when one studies the actual mathematics. For example, the first statement suggests that galaxies are "flying apart like shrapnel" from an explosive event located in a particular place, but the Hubble expansion could not be more unlike such an isolated explosion!
If space itself is expanding over time, then matter itself would have to be expanding at the same rate... if the metric expansion model is correct, millions of years ago, our solar system was a smaller scale model of how it is now.
That is a VCM (Very Common Misunderstanding); see
Code:
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otherwise the Earth wouldn't have remained in the "Goldilocks Zone" which has allowed life to exist in this planet for billions of years.
Brooklyn is not expanding. The Earth is not expanding. The Sun is not expanding. The Solar System is not expanding (much). Landrew's body is not expanding either, and he shouldn't expect otherwise, because his body is held together by chemical forces, not gravitation. It is true that planets and stars are held together by gravitation but the Sun, the Earth, and the Solar System are all more dense than the average volume in the current epoch of the Universe, and this has been true throughout their history. Thus it should not be surprising that they are almost immune to cosmological expansion, as mathematical analysis verifies. OTH, on a very large scale, distant pairs of galaxies interact only weakly with each other and these will be subject to the Hubble expansion (on top of various motions "wrt the CMB" they may possesses by chance).
The Earth and Moon are very slowly moving further apart, but this is due to something else entirely. The Hubble expansion has almost no effect on the Earth-Moon system.
If our Earth was indeed smaller, the gravity of our planet would have also been less. The flying dinosaurs would have had less difficulty flying in lesser gravity. Perhaps this explains why when scientists examined their skeletons, they determined that they were built much too heavy to ever get off the ground today.
Nice try, but no. The pteranodons simply had some tricks for getting into the air which the old analyses Landrew refers to did not take into account. The surface of the Earth has never differed from its current value during the 4.5 billion year history of biotic life on Earth.
Or is a better solution to invent a theory that 96% of our universe is invisible dark matter, to make things seem to work out?
Sigh... ignorant indeed. And it is dark energy plus dark matter, not just dark matter. And these are not theories, but inferences drawn from several very well established theories (gtr, hot Big Bang theory). And radioactivity is invisible to the naked eye, but not long after its existence was inferred from chemical reactions (in photographic plates), scientists figured out how to measure the amount and nature of radioactivity from substances like Radium, thus confirming that it does exist, and later devised a now well-established theory explaining why it exists.
Because science is honest by design, as it were, scientists working at the frontiers uncover apparent inconsistencies with previous knowledge, and one of the most characteristic features of science is that science provides a powerful error-correction/inconsistency-resolution method, which may take time but seems to get us there in the end, if we simply work hard enough. One key aspect of the inconsistency-resolution method is that scientists try to make minimal changes to well-established theories in order to resolve apparent contradictions at the frontiers of scientfic knowledge. Their first attempts often involve tentative inference of the existence of something with unexpected properties, followed by attempts to verify that this stuff actually exists. This is exactly what is happening wrt dark matter and dark energy.
For more information about how science works, Landrew should see the UCB website "Misunderstandings of Science"
no, it is simply a function such that the roots of the derivative V'(r_c) = 0 help to organize turning points r=r_c for the radial motion of trajectories. That is, for particular values of E,L (energy and momentum of the test particle), the graph of V typically has a local minimum, and if your particle has energy E just a bit larger than that minimum, when you draw a horizontal line with height E on the graph of V, it will intersection the curve V(r) at two turning points. This means that the radial motion of the particle will oscillate between these two values. Simulataneously, of course, it is has nonzero angular motion, so the result is that a particle with suitable L, E will orbits in a quasi-elliptical trajectory which turns out not to quite return to the same location at the maximal radius--- this is the famous precession of the periastria.
Most gtr textbooks offer very clear explanations of this; see for example MTW.
BRS: Categories & Permutation Groups; plus Destroying the Earth
Re "Category Theory Used in Physics"
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Tom Gilroy said:
I'd first like to mention that the definition of a category that appears on Baez's page is incorrect. A category does not consist of "a set of objects and a set of morphisms." A category consists of a class of objects and a class of morphisms (any set is a class, but not all classes are sets).
Oh for heavens sakes, Baez knows this. Gilroy needs to take account of the fact that he was reading a gentle introduction for students. Baez is a master at starting with an oversimplified presentation and gradually introducing more sophistication, e.g. "actually, we should use classes instead of sets in the definition".
The textbook by Geroch (same University of Chicago professor as in Geroch group in gtr) Mathematical Physics, University of Chicago Press, is a very clear introduction to both category theory and to its use in organizing a host of techniques in graduate level mathematical physics.
Re "Group action on cosets of subgroups in non-abelian groups"
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(I like the fact that the OP actually chose a descriptive title!), nbruneel asks about right cosets.
nbruneel said:
Let G be a non-abelian finite group, S < G a non-normal proper subgroup of index v >= 2, and G/S the set of v right cosets S_1 = S, S_2, ..., S_v, of S in G.
Actually, the set of right cosets is written
<br />
S \!\setminus\! G = \left\{ Sg \; | \; g \in G \right\}<br />
where a typical right coset is written
<br />
S g_0 = \left\{ s.g_0 \; | \; s \in S \right\}<br />
Here, x \rightarrow xg is a right action by G on itself. More generally, writing the identity element as e, a right action x \rightarrow x.g by G on X satisfies
<br />
x.e = x \;\; \forall x \in X, \; \;<br />
(x.g_1).g_2 = x.(g_1 g_2) \;\; \forall x \in X, \; \forall g_1, g_2 \in G<br />
In contrast the set of left cosets is written
<br />
G/S = \left\{ g S \; | \; g \in G \right\}<br />
where x \rightarrow gx is a left action by G on itself. Compare the actions of G on itself by conjugation:
<br />
x \rightarrow g^{-1}xg, \; \; x\rightarrow gxg^{-1}<br />
Which is a right action, and which a left action?
kakarotyjn said:
I want to ask a question first why \phi_g is an element of Sym(v)?why should there is a S_j equals to S_i*g?
One can prove that right multiplication by any g of any coset S g_0 gives either the same coset or a disjoint coset S g_1, \; S g_1 \cap S g_0 = \emptyset. This means that the right action by G by right multiplication on S\G does indeed permute the cosets, which answers the first question. One can also prove that the action is transitive: given two cosets, there is some g such that right multiplication by g of the first coset gives the second coset. This answers the second question.
The proofs are very easy; see almost any group theory textbook, e.g. Fraleigh, A First Course in Group Theory.
Actually, nbruneel is only interested the right action by right multiplication in G. Then G acts on its subgroups by right multiplication, and for each subgroup S the orbit includes all the cosets S\G. That is, G acting on the right cosets of S by right multiplication gives a transitive permutation group on S\G. For example:
if [G:S]=2, the induced permutation group must be S_2 (the unique transitive permutation group of degree 2).
if [G:S]=3, the induced permutation group must be one of S_3 or A_3 (the two transitive permutation groups of degree 3).
if [G:S]=4, the induced permutation group must be isomorphic to one of S_4, A_4, V = C_2^3 (Klein's four element group), D_4 (the eight element dihedral group), or C_4, these five possibilities being the five transitive permutation groups of degree 4,
if [G:S]=r. the induced permutation group must be one of the transitive permutation groups of degree r.
nbruneel said:
what are the conditions for this map to be necessarily surjective?
IOW: "when does the right action by G by right multiplication on the right cosets of a proper subgroup S give an induced permutation group on S\G which is isomorphic to the full symmetric group on [G:S] letters?"
More generally, one can ask: "when does the right action by G by right multiplication on the right cosets of a subgroup S yield a particular transitive permutation group?" The answer is given by certain zeta functions, and it is remarkable that this involves a close connection between this thread and the thread on category theory! Someone asked when small categories (see Mac Lane) or "kittygories" are interesting. One answer is that the category of finite sets arises naturally in enumerative combinatorics; much of John Baez's work over the past decade has involved the generalization of Eulers methods using generating functions to categorical techniques. It turns out that essentially all problems in enumerative combinatorics (e.g. count the number of nonisomorphic (labeled) (unlabeled) binary trees having n vertices, for all n) can be reformulated using an appropriate functor called a structor or "combinatorial species". The theory of functors then turns out to be very closely related to the theory of permutation groups! In fact, these are so tightly related they are more or less different faces of the same phenomenon!
For some hints, see Cameron, Permutation Groups, Cambridge University Press, LMSST series.
Re "Can general relativity be constructed with differential forms?"
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at the time of this post it seems that all the respondents are answering a different question: "can curvature be expressed using differential forms", to which the answer is the one given by MTW and other standard sources (plus arkajad and other posters): "yes, if you use lie algebra-valued exterior forms" (or more prosaically, if you use a matrix of ordinary real valued exterior forms). But this slurs over the question of how to express the Einstein tensor using the formalism of differential forms! Various authors have looked at this, but it's not so straightforward as should be obvious from the fact that the Ricci tensor is symmetric whereas (for example) the form representing the EM field is antisymmetric, so that the field equations of EM can indeed be written in the formalism of exterior calculus.
the book by Szirtes, Applied Dimensional Analysis and Modeling, has a very clear discussion of how to "scale up" the result of laboratory experiment with a small steel pellet and a sandbox to the well-presevered Barringer meteor crater in Arizona (where the impactor was apparently an iron meteorite, with about the same density as steel). From an experiment for a given impact angle, one can scale up to infer the diameter and speed of the iron meteorite which created the crater. Assuming an impact angle of 67 degrees, Szirtes obtained figures of 30 m diameter and 18 km/sec for the Barringer crater impactor. The velocity is reasonable and the circular shape of the preserved crater does suggest a large impact angle.
The same technique applies to the much larger Chicxulub crater in present day central America, except that I believe one should use a pebble because this crater was apparently created by a rocky impactor. But the Chicxulub crater is heavily eroded and partially under water, so this might not be so easy to scale up.
The scaling analysis suggests a nice science fair project: using a small steel pellet and a small pebble, a sandbox, and a device which can propel the pellet and pebble at a known speed (several hundred feet per second--- I envision a typical American high school student who can employ the family artillery pieces), measure for various impact angles the crater size and shape. Reason: scaling analysis suggests studying experimentally the unknown function F in the relation
<br />
d/D = F \left( \phi, \frac{v}{\sqrt{gd}} \right)<br />
where d is the diameter of the projectile, D is the geometric mean diameter of the crater, \phi is the impact angle, v is the impact speed, g is the surface gravity on Earth. The argument
<br />
\frac{v}{\sqrt{gd}}<br />
is the Froude number (I think the name is pronounced "frood", even though Froude was British), a dimensionless number which comes up in almost every phenomeon (in biomechanics, hydrodynamics, astronomy) in which some characteristic speed, distance, and acceleration (often a surface gravity) are involved. The function may be different for steel pellets and pebbles.
It is interesting to estimate the range of meteorite impact velocities we can expect on Earth. Whenever we deal with gravitation in an isolated system with characteristic mass M, radius a, we expect the dimensionless ratio GM/a to be relevant. This partially explains why characteristic speeds of form \sqrt{GM/a} are common in astronomy, and also why we should not be surprised that the speed to escape from a circular orbit is a constant times the speed to stay in that same orbit! Specifically, for an object orbiting the Sun at the mean distance of the Earth a,
<br />
v_{\rm orbit} = \sqrt{GM_{\rm Sun}/a}, \; \; v_{{\rm orbital} \, {\rm escape}} = \sqrt{2GM_{\rm Sun}/a} <br />
which are about 30 and 40 km/sec respectively. Here, the orbital escape speed is the speed which an object falling directly toward the Sun, "from spatial infinity" and initially at rest wrt the Sun, will acquire when it reaches distance R from the Sun. A sideways impact on the Earth would then have speed at most \sqrt{3GM/R}, while a head-on impact would have speed at most the sum of these, or 70 km/sec. This doesn't take account of the additional speed due to falling toward the Earth due to its gravitational attraction. But a particle falling directly toward Earth, "from spatial infinity" and at rest wrt the Earth, acquires speed
<br />
v_{{\rm surf} \, {\rm escape}} = \sqrt{G M_{\rm Earth}/R_{\rm Earth}}<br />
which is about 11 km/sec. Clearly the maximal expected impact speed is less than the sum of these, or about 80 km/sec. On the other hand, the minimal expected impact speed would be about v_{{\rm surf} \, {\rm escape}}. Or so the author of a Wikipedia article suggests--- I suspect that
<br />
v_{\rm min} =<br />
v_{{\rm orbit} \, {\rm escape}} <br />
- v_{\rm orbit}<br />
+ v_{{\rm surf} \, {\rm escape}}<br />
or about 20 km/sec is a more reasonable guess for a scenario in which an object orbiting the Sun with the same sense of rotation as the Earth and in approximately the same plane as the Earth "catches up to the Earth from behind" and impacts the surface of the Earth. I am not sure anyone has thought this through, however.
Similarly, on the Moon we might very crudely expect roughly the same maximal and minimal impact speeds as for the Earth (the orbital motion of the Moon around the Earth being so slow compared to the speeds previously mentioned). I'll leave Mecury, Venus, Mars as exercises.
It is also interesting to use scaling analysis to determine what is required to destroy the Earth. Dimensional analysis suggests that the gravitational binding energy of an isolated object should be proportional to GM^2/R, and indeed, according to Newton the gravitational binding energy of an isolated uniform density ball of mass M and radius R is
<br />
E_{\rm bind} = \frac{3 G M^2}{5 R}<br />
which for the Earth is about 2.24 x 10^32 J. This is the energy required to throughly disperse the material comprising the Earth wrt the center of gravity of the presently existing object (by first giving the outermost layers their surface escape velocity, then giving the next layers the new, smaller, surface escape velocity, and so on until the Earth is all gone). The kinetic energy of an impactor is OTH mv^2/2, where m is the mass of the impactor and v is the impact speed. Setting these equal, for v ~ 80 km/sec we find that a Moon sized object will do the job.
For a more typical impact speed of v ~ 25 km/sec, a direct impact from an object of about 6 Moon masses will suffice to split the Earth in two, and a direct impact from an object of about 11 Moon masses will suffice to completely disperse bits of Earth wrt the Earth's orbit around Sun. More energy would be required to disperse bits of Earth wrt the Sun's orbit around the center of the galaxy; less to simply split the Earth into two or more pieces.
Using the same scaling analysis, assuming the mass of the galaxy is concentrated near the center (which is reasonable), the orbital speed of the Sun about the center is about 250 km/sec. From this we estimate a maximal relative speed for an object coming from our galaxy, but outside our solar system, to be about 600 km/sec. Then an object of about 1/60 Moon masses can completely disperse the material in the Earth wrt the current center of mass of the Earth.
How about destroying the Sun? Well, the Sun is basically a fluid ball, so to pull it apart a massive object need only lift off the Sun a substantial amount of its matter; it need not impact the Sun at all. This suggests estimating tidal disruption of the Sun, and again dimensional analysis suggests the formula we need; Newton's gravitation theory is needed only to determine a multiplicative constant. And his force law is itself an almost trivial consequence of dimensional analysis, so in fact one needs only the value of G, which must be determined by experiment.
BRS: classification of surfaces, Maurer-Cartan forms
In "Introduction to Topology Resources"
Code:
www.physicsforums.com/showthread.php?t=450554
blinktx411 asks for supplementary textbooks, specifically ones discussing the classification of surfaces (in this context, that is shorthand for two-dimensional topological manifolds).
Possibly silly question: any university teaching a course in algebraic topology must have a math library, yes? So what is stopping him from browsing the algebraic topology books on offer there? Likely they include some books devoted to surface theory.
There are many possibilities, but three leap to mind:
Frechet and Fan, Initiation to Combinatorial Topology, recently reprinted classic; the goal of this short book is to explain the classification of surfaces via combinatorial topology (drawing squares and identifying edges, that kind of thing), so it should be just what the OP wants,
Hatcher, Algebraic Topology, available for free download at his website, but IMO worth the price of the Cambridge U Press paperback edition; Hatcher does cover the classification of surfaces, briefly, but I recommend this book for its lovely motivation of both homotopy and homology, by far the best I have seen yet.
May, A Concise Course in Algebraic Topology, recently reprinted classic, should keep any serious student happy in case of boredom.
Also, I second the recommendation of the books by Lee and Massey. And the OP should look at some books on Riemann surfaces, because the motivation for the classification of surfaces comes from that subject.
In "Maurer-Cartan forms in physics"
Code:
www.physicsforums.com/showthread.php?t=450515
Haushofer asks: "what uses do Maurer-Cartan forms have in physics?" Well, they are useful in such topics as de Rham cohomology (which gives for example a crude but easy partial description of the topology of the three-dimesional Lie groups classified by Bianchi, which I expect will interest the OP) and the formulation of gauge theories (mentioned in some other current PF threads) using the exterior calculus of fiber bundles, e.g. electromagnetism and (to some extent) gtr. See Frenkel, The Geometry of Physics for details and further motivation for Maurer-Cartan forms. Also, for those interested in Cartanian geometry (minimal common generalization of Kleinian and Riemannian geometry), this subject is founded upon the Maurer-Cartan form.
BRS: sloppy thinking/writing about singularities and a fringe website
Re "BBT,SLT Order Vs disorder"
Code:
www.physicsforums.com/showthread.php?t=449410
several posters seem to be twisting the words of Science Advisors into "black holes are not mainstream", which is nonsense and should not be left uncorrected, even though it is exhausting to curtail their word games.
Chalnoth's Post #3 is a good answer to the OP. Then Leonstavros asserted
The fact that all physical laws breakdown in a singularity proves disorder to the nth degree
First of all, "singularity" is a mathematical term which is used in mathematical physics according to standard mathematical usage! And "singular" just means "unusual" or "remarkable", so a singular locus is simply a place (locus) where something happens which is somehow "different" from what happens in most places. That gives a great deal of latitude, and there are many kinds of singularity in mathematics, including
in the theory of functions
removable singularities: in w = (z^3-a^3)/(z-a), w -> 3a^2 as z -> a even though the denominator blows up.
pole of order m: in w = 5/(z-a)^3, z=a is a pole of order 3,
essential singularities: w = \exp(1/z), z is neither a pole nor removable,
branch points: in w^3 = z^5, z winds five times about zero for every three times w winds, and this is a remarkable property characteristic of a branch point associated with a Riemann surface,
in linear algebra, a singular value of a matrix A is a square root of an eigenvalue of the square matrix S = A^t A,
in the theory of vector fields on a smooth manifold M, "locally all vector fields are alike", but the congruence of the integral curves of a vector field on M is organized about singular points where the vector field vanishes; this is crucial for the elementary theory of (smooth) dynamical systems
singular integrals involve summation processes which compensate for some kind of blowup in the function being integrated
&c.
In gtr, singularities include
singularities in an expression for the metric tensor, indicating a boundary where a given coordinate chart is no longer valid,
singularities in fields or matter, i.e. places where a field component or matter density blows up (e.g. "shell-crossing singularities" in LTB dust models)
curvature singularities including singularities in various tensors constructed from the Riemann tensor such as Ricci or Weyl; for the Riemann tensor possibilities include:
weak or strong; that is, there is a hierarchy of "destructive power": strong singularities sphaghetiffy everything; progressively weaker ones destroy progressively fewer and fewer unlucky observers, so to speak,
scalar or nonscalar (not all curvature singularties are indicated by blowups in some scalar invariant constructed from the Riemann tensor)
timelike, spacelike, null, or none of these
certain geometrically meaningful singularities which are not curvature singularities, e.g. fold singularities in CPW solutions, or "struts" in Weyl vacuum solutions,
Although it is tiresome to keep pointing this out, I believe it is important that a new generation of SA/Ms get in the habit of objecting when posters exhibit sloppy writing/thinking. Failing to do so encourages students to develop such bad habits, which will greatly limit their effectiveness in coursework (and possible subsequent scholarly pursuits). Worst of all is the possibility that the next generation of scientists might develop/accept various habits of sloppy thinking--- that would clearly be very bad for the future of science itself!
Next, Leonstavros probably is thinking of singularties in matter density or in curvature, which are distinct concepts in gtr. Tossing in concepts from QFT or speculations about the yet unknown quantum theory of gravity only further muddies the waters, which may be Leonstavros's intent (see the italicized sentence above).
Chalnoth said:
Nobody expects the singularity to be real, just an artifact of our incomplete understanding of the early universe. So using it to make any point is just nonsensical.
Agree not worth mentioning this to Leonstavros, whom I suspect of trolling for comments he can twist to impress a friend or something like that, but while there is a common expectation that quantum gravity will banish curvature singularities to an effective field theory approximation (presumably gtr or a very good mimic of gtr), and further that "curvature" may not even make sense in quantum gravity, this need not imply that all "singularities" will be banished. The history of mathematics suggests that "unusual occurrences/places" are fairly ubiquitous and thus quantum gravity is likely to turn out to present a new sequence of even more fundamental puzzles.
Leonstavros said:
We are initially organized into complicated biological entities(entropy decreasing) and then experience aging, illness and finally death(entropy increasing).
Not so fast, almost certainly Leonstavros doesn't understand that the second law is only valid for a closed system and living organisms are not closed systems; to maintain their cytological biochemical organization they require a constant energy flux, which for life on Earth is derived from the Sun irradiating Earth which then radiates waste heat (including the tiny contribution from living organisms) to deep space. During the evolution of complicated life forms, entropy of the system consisting of Sun, Earth, deep space was always increasing. Roughly speaking.
Leonstavros said:
We use math to explain the Universe so when the math breaks down when we approach a singularity tells me that either our math is not good or the physical laws do break down. You mentioned in a previous post that the universe will eventually become a bunch of black holes as proof of entropy increasing but aren't black holes singularities?
Good example of how sloppy writing/thinking increases entr..er, mental confusion
The defining characteristic of a black hole (according to the currently standard definition) is the presence of an event horizon, which is not a curvature singularity but rather a two-dimensional locus which is globally remarkable but locally unremarkable.
Chalnoth then linked to a PDF at
[size=+2]Warning! Fringe site[/color][/size]
Code:
olduniverse.com
Chalnoth quoted from a pdf found there
Adrian Bjornson? said:
...the metric becomes singular and the density becomes infinite. . . In reality, space will probably be of a uniform character, and the present [relativity] theory will be valid only as a limiting case...
Actually, at energy-densities approach the Planck energy-density, Wheeler long ago suggested, spacetime may be replaced by a highly irregular "foam", which would certainly not have a "uniform character"--- that phrase better describes the tangent space near a point in classical gtr, a concept which is of course only valid at regular (nonsingular, heh) events in a given spacetime.
Tanelorn said:
It appears that this "olduniverse.com" site seems to be non mainstream to say the least?
Yes, decidedly fringe, and also quite out of date, even as a fringe viewpoint.
Tanelorn said:
this is quite a revelation for me, I had assumed that singularities and black holes were now mainstream facts, even having experimental measurements. Perhaps black holes could still exist which are also not singularities? Perhaps some other structure of matter prevents them shrinking further from a white dwarf to a complete singularity, but which is still smaller than the schwarzschild radius?
This is a serious (and intentional?) distortion of current mainstream belief in astrophysics. In fact:
black holes are characterized by the presence of an event horizon (and the absence of any material surface) and there is mounting evidence for these two crucial properties; see the sources in the BRS sticky "Some Useful Links for SA/Ms",
in the (near vacuum) outside a realistic black hole, perturbations of the spacetime curvature due to infalling matter, tidal distortions from a massive object passing nearby, etc., will be radiated away in the form of gravitational radiation and that consequently, the geometry of the region outside the hole will closely resemble the geometry of the famous Kerr vacuum solution,
gtr unambiguously states that matter falling through the event horizon cannot re-emerge,
it is expected that the long sought quantum theory of gravitation will not affect the first two items in this list,
it is expected that quantum gravity won't nullify the third item in any meaningful sense, but in the very long term, the unsolved information paradox involves whether in the unimaginably distant future something (surely not unaltered matter which fell in long ago, however), so in this sense quantum gravity might modify the second item in some sense; to be more precise about "what sense" we'd need to possesses and to understand a viable quantum theory of gravitation,
it is expected that inside the event horizon, "mass inflation" of even small amounts of infalling matter and radiation might imply that well inside the horizon, the geometry may be quite different from the geometry of the Kerr vacuum and may even not be describable by gtr at all.
Chalnoth said:
singularities in General Relativity are held up as a reason to think that General Relativity must be an incomplete theory.
True, but not, IMO, by wise physicists! All theories that I know of admit singularities of various kinds and this can even be beneficial! (For example: Dirac deltas are "singular functions" of a kind.) Rather, gtr is a classical theory and therefore incompatible with quantum mechanics; quantum theories generally admit a classical limit for sufficiently low energies and it is reasonable to assume gtr is this limit for the unknown quantum theory of gravity; since quantum phenomena are well established and since theory unambiguously suggests that they should dominate at very high energy densities (in gtr this is equivalent to "very large Riemann curvature components"), gtr is expected to break down at very high energy densities.
The Thanksgiving holidays is a time when many American students may be working take home exams. Several recent posts in the relativity subforum look to me very much like exam questions badly mangled by struggling students. So SA/Ms should be cautious in answering these queries.
Re "general metric with zero riemann tensor"
Code:
www.physicsforums.com/showthread.php?t=451093
the thread does not of course concern the general metric with vanishing Riemann tensor, but the OPs desire to find an explicit coordinate transformation to Minkowski vacuum. He wrote down a metric in a particular chart, found the Riemann tensor vanishes, and concluded
therefor must be isomorphic with minkowski tensor.
He should say: the Riemann tensor vanishes, therefore this spacetime must be locally isometric to Minkowski vacuum.
To find the coordinate transformation: read off the obvious coframe. The dual frame is the frame of the Milne observers, whom we recognize from the facts that the acceleration and vorticity of the timelike unit vector \partial_\tau vanish and the three-dimensional Riemann tensor of the hyperslice t=t_0 is
<br />
r_{2323} = r_{2424} = r_{3434} = -1/\tau^2<br />
i.e. the slices are locally isometric to H^3. So in the Minkowski chart, the integral curves of \partial_\tau appear as straight lines expanding linearly from a particular event, WLOG the origin, while the hyperslices appear as nested H^3. Now a little hyperbolic trig finishes the task.
[EDIT: I think George Jones had the same advice!]
Re "questions about black holes"
Code:
www.physicsforums.com/showthread.php?t=451061
From what I understand a black hole is the result of a tremendous amount of matter being pulled together to a finite point in space and this point creates some kinds of a dip in space. Please correct me if I am wrong?
Not quite right on both points. According to gtr:
a black hole results when an event horizon forms, which happens with any amount of mass-energy is compressed into a sufficiently small region, so anything sufficiently dense must form a black hole,
"the gravitational field" is represented by the Riemann curvature tensor of spacetime; near any massive nonrotating static object this assumes are particularly simple form (which could have been guessed from Newtonian physics!); the components of the curvature turn out to vary like m/r^3 (note the exponent; these components are related to tidal accelerations which also scale like m/r^3 in Newtonian gravitation); the curvature of spatial hyperslices is sometimes indicated by displaying an embedding diagram of such a hyperslice (with one dimension suppressed), but this is only a crude and in many ways misleading representation which is merely intended to suggest that the curvature is spherically symmetric and increases as r decreases.
As Dale Swanson already noted, the jets are associated with matter orbiting outside real black holes in Nature (a feature not included in the simple gtr model just discussed).
Re "ADM Mass for a diagonal metric"
Code:
www.physicsforums.com/showthread.php?t=451031
ditto bcrowell: this metric need not represent a black hole at all, or even a manifold with -1+n signature, and certainly seems to be 1+4 dimensional. It seems clear that the OP is not ready for ADM integrals but should consider first the simpler case of Komar integrals. The definition of Komar mass-energy and Komar angular momentum requires assuming an AF metric (which rules out e.g. cylindrical symmetry or nonzero Lambda), and requires a timelike Killing vector field (for the mass-energy) or spacelike cyclic Killing vector field (for the angular momentum).
d{{}^\ast\!F} = 4 \pi \, {^{}^\ast\!J} (exterior calculus; *J is the three-form dual to current one-form J) or {F^{ab}}_{;b} = 4 \pi \, J^a
(Warning! Rasalhague writes the two-form F as A, which is bad notation since the universal notation is F=dA where A is the potential one-form!)
Is the "4-velocity of the source distribution" that of the worldlines of particles at rest in the centre-of-momentum coordinate system of the sources?
The source consists of charged particles, and their world lines are idealized as a congruence of timelike world lines. The velocity vector field is \vec{v} and at each event, we have a well-defined hyperplane contact element orthogonal to the world line through that event, and thus a well defined charge density wrt the frame comoving there with the charged particle having that world line. Thus \vec{J} = \sigma \, \vec{v} is well defined.
And is comoving volume the spatial (3-dimensional) volume (rather than a 4d volume of spacetime), as measured in these coordinates?
Yes. Note that this is a question about the definition of densities generally in relativistic physics, not about electromagnetism. Rasalhague should think about how changing to another frame field will affect the components (wrt the frame) of a vector, a one-form, a two-form.
It seems that these equations just don't apply (become meaningless) in the case of a single, discrete charge following one world line, because, for events not on the world line of such a source, no value is defined for the field at events remote from any source.
This is one of those places where the Dirac delta "function" is a really useful fiction (and not even fictitious once you know about distributions in the sense of Laurent Schwartz).
DrGreg said:
But I've never been happy with Dirac deltas because they're not actually functions. Maybe someone who understands this better could comment.
Try Rudin, Functional Analysis. A lot of preliminaries, but once you know enough about linear operators on function spaces, the lovely theory of tempered distributions is one of the nicer things you get almost "for free", as Rudin explains. (Hmm... there must be a shorter path, but right now I can't suggest one.)
Try MTW for more about formalisms for writing the curved space Maxwell equations. As always, components are simpler and easier to interpret if you use a frame field. Also, differential forms work the same way (locally) on any manifold, so you don't really need to learn any new techniques if you already know the exterior calculus formalism for E&M on flat spacetime.
Re "Spaces with constant curvature"
Code:
www.physicsforums.com/showthread.php?t=451076
asks if R^m \times S^n, \; \; S^m \times S^n are spaces of constant curvature. In the sense of the old term of Clifford ("space forms') the answer is "no, they are direct products of spaces of uniform curvature but do not themselves have uniform curvature". Reason: fix any point P. Some 2-surfaces passing through P have Gaussian curvature different from others.
Re "Layman's question about the application of the curvature to space"
I understand that the force of gravity is more accurately described as space curvature. I.e., a massive object like the sun or Earth can be visualized as a bowling ball placed on a rubber sheet, creating a curvature.
Standard remarks apply:
spacetime not space curvature (see first chapter of MTW for why that's so important),
rubber sheet analogy merely suggestive, not accurate
Objects passing nearby on a straight trajectory will then assume a curved trajectory. I am wondering if the same thing applies to stationary objects on the surface, like a person standing on the earth. How?
I'll try to rephrase the question: "Curvature effects include geodesic deviation. In a nonvacuum static model, such as the interior of a static perfect fluid, does gtr still say that geodesic deviation will occur?" Short answer: yes, but this may not have the same clean interpretation which null geodesics enjoy in vacuum, electrovacuum, or dust solutions, in the geometric optics approximation.
Re "ADM Mass for a diagonal metric"
Code:
www.physicsforums.com/showthread.php?t=451031
now praharmitra claims that his metric is a "black hole". First of all, unless he goofed in writing down his metric function, that spacetime does not have vanishing Einstein tensor. He never said whether he is thinking of E^5 or E^{1,4} signature, but it doesn't matter: the Einstein tensor does not vanish!
Furthermore, when we write down the obvious static 3-spherically symmetric metric Ansatz
<br />
ds^2 = -A \, dt^2 + B \, dr^2 + C \, d\Omega^2<br />
where A, B, C are functions of r only, and where
<br />
d\Omega^2 = d\chi^2 + \sin(\chi)^2 \; ( d\theta^2 + \sin(\theta)^2 \, d\phi^2 )<br />
gives the metric of a unit S^3 (in polar 3-spherical chart), then when we demand that the Einstein tensor vanish, we are led to two ODEs for A,B in terms of C. Choosing C = r^2, we immedialty obtain the Tangherlini vacuum
<br />
A = 1-M/r^2, \; B = 1/A, \; C = r^2<br />
The choices for A,B,C offered by praharmitra do not give a vacuum black hole, even if one assumes he forgot to say that the signature is E^5 rather than E^(1,4)--- and in the former case, "black hole" probably wouldn't make sense, since a black hole should have an event horizon. It is not obvious from studying just one chart valid only in the static exterior, but the Tangherlini vacuum does have an event horizon at r=m, which is topologicially S^3, so clearly "black hole" is apt in this case. See
Roberto Emparan and Harvey S. Reall,
"Black Holes in Higher Dimensions" Living Reviews in Relativity
Code:
relativity.livingreviews.org/
The generalization to
<br />
T^{ab} = T_{EM}^{ab} + T_{\Lambda}^{ab}<br />
where
<br />
T_{EM}^{ab} = \epsilon \, \operatorname{diag}(1,-1,1,1,1), \; \;<br />
T_{\Lambda}^{ab} = \Lambda \, \operatorname{diag}(1,-1,-1,-1,-1)<br />
are contributions with the expected form for EM and Lambda terms is
<br />
A = 1 \; -\; \frac{M}{r^2} \; + \; \frac{Q}{3 \, r^4}<br />
\; + \; \frac{\Lambda}{6} \, r^2<br />
Then
<br />
G^{ab} = \Lambda \, \operatorname{diag}(1,-1,-1,-1,-1)<br />
\; + \; \frac{Q}{r^6} \operatorname{diag}(1,-1,1,1,1)<br />
In the expression for A, notice that the M,Q terms have different powers than in E^{1,3} but the Lambda term has exactly the same form for any E^{1,d}. The answer for any dimension is just what you would guess by comparing the Schwarzschild and Tangherlini solutions in their respective Schwarzschild exterior charts.
Note: M,Q, Lambda might not have quite the same interpretation in higher dimensions, so I reserve the right to change any of these by some positive constant multiplicative factor after further thought!
The choices given by praharmitra are much more complicated than these and appear not to have the property he claims. His A is asymptotically
<br />
r^2 + 1 + Q/3 - \frac{M + \hbox{stuff}}{r^2} + O(1/r^4)<br />
rather than
<br />
\Lambda/6 \, r^2 + 1 - M/r^2 + O(1/r^4) <br />
so his claim about "asymptotically de Sitter" appears to be... well, possibly correct if he's using a strange convention about where the "gravitational red shift" is unity (the standard choice is the one I used above), but he still needs to explain what his parameters mean physically.
From what I understand, the solutions of a differential equation form a manifold. Is that correct?
In some cases it might be reasonable to regard a solution space for a system of DEs as a finite or (more likely) infinite dimensional topological manifold, but I think most experts would agree that the most successful theory to date regards the solution space (for the kind of boundary value problems for systems of PDEs which often arise in mathematical physics) as a Sobolev space, a notion which requires a background in functional analysis. Most good graduate level textbooks on real analysis or PDEs contain a discussion of Sobolev spaces, and a readable introduction to this point of view can be found in Robinson, Infinite Dimensional Dynamical Systems, which focuses on boundary-value problems in the parabolic family (e.g., diffusion equations).
Re
Code:
www.physicsforums.com/showthread.php?t=453104
the OP asks about the boundary/initial value problem
<br />
\begin{array}{rcl}<br />
u_{tt} & = & a^2 \, u_{xx} + t x, \; \; 0 < x <l; \; t>0 \\<br />
u(0,t) & =& u(l,t)=0 \\<br />
u(x,0) &= & u_t(x,0)=0<br />
\end{array}<br />
That is a linear equation; the general solution has the form
<br />
u(x,t) = F(x+t) + G(x-t) - \frac{t \, x^3}{6}<br />
and Sturm-Liouville theory then gives the standard solution to the stated IBVP. Even better is the integral transform approach which leads to the result stated by Polyanin (author of eqnet) for the more general case where tx is replaced by any "reasonably nice" \Phi(t,x)).
Re
Code:
www.physicsforums.com/showthread.php?t=451822
my, what an admirable rant I decry spending all day asking Google or Amazon to "just tell me the answer" without even considering the possibility of visiting the university library, but never mind that. FWIW, when I taught linear algebra I actually tried hard to give an intuitive explanation of matrix multiplication based on a counting problem, and the students (somewhat to my surprise) seemed to understand and appreciate the explanation.
The kind of problem I suggested has the following form: suppose we are building an apartment house complete with furnishings. Suppose the house has two luxury apartments and eight budget apartments, each having different types of furniture (chairs, beds, desks, possibly in deluxe or budget models). And each type of furniture requires certain numbers of screws and brads. How many screws and brads do we need to order to make the furniture for the apartment house? To find out, it is natural to first represent the given data in the form of three tables and then to realize that we should multiply them matrix-fashion to find the answer we need!
When matrix multiplication is introduced this way, students may not be so surprised when they are told that matrix multiplication is not in general commutative. At least my students seemed to take this much better than I had expected.
Re "Solutions to Killing's equation in flat spacetime"
Code:
www.physicsforums.com/showthread.php?t=454237
A solution to Killing's equation is a flow corresponding to an infinitesimal "rigid motion". In E^3 a rigid motion (as is proven in elementary analytical geometry plus group theory) consists of a rotation composed with a translation. In E^{1,3} a rigid motion comsists of a Lorentz transformation composed with a translation. More precisely, the Lie groups in question E(p,q) are the semidirect product of a normal Lie subgroup (the translation group) with a Lie subgroup which is isomorphic as a Lie group to O(p,q).
The expression the OP is asking about simply says "the solution of the Killing equation in Minkowski spacetime is the result of composing an infinitesimal Lorentz transformation with an infinitesimal translation".
Remember, the Killing equation deals with vector fields which correspond to infinitesimal motions and which live in the Lie algebra of vector fields on the manifold. Exponentiating these gives motions, elements of the Lie group whose tangent space at the identity corresponds to the Lie algebra. In particular, in terms of matrix Lie groups, exponentiating a "Minkowski-antisymmetric" matrix results in a matrix belonging to SO+(1,3), the connected componenet of the full Lorentz group. So the expression quoted by the OP is additive, while after exponentiation we are dealing with a noncommutative Lie subgroup of the group of rigid motions, i.e. the group of self-isometries.
Re "Black Hole time dilation + biological paradox"
Code:
www.physicsforums.com/showthread.php?t=453962
moocownarf (why, MUD me, a narf!) assumes
If a spaceship housing humans were to travel near a black hole, time would slow down due to the increased gravity.
That is not what gtr says at all and doesn't even make sense (slow down wrt what?).
Rather, due to geodesic deviation owing to curvature (nonzero gravitational field), light signals sent from a nearby world line to a more distant world line will typically diverge so that the distant observer finds by this light signal comparison that the clock of the nearby observer "is running slow" wrt his own clock. But this effect depends on their relative motion (difficult to describe in curved spacetime without getting very precise about how "distance in the large" is measured!) as well as the gravitational field, so to state predictions about frequency shifts you need to specify
metric tensor
two specific world lines
(possibly) specific null geodesics corresponding to the light signals
The last arises because due to gravitational lensing in a nonzero gravitational field typically a signal sent from event A can arrive at event B by two or more routes.
Re
Code:
www.physicsforums.com/showthread.php?t=453956
but the teacher is really bad at making the bridge between the maths and the physics.
Or the class is ill-prepared? And maybe the instructor is a junior faculty member who was not even given the opportunity to choose his own textbook?
General advice to those with time to try to learn this stuff properly: it can be very helpful to first learn representation theory for finite groups which is much, much easier than for finite dimensional Lie groups (infinite dimensions is a whole new world of trouble and unexpected beauties). In the theory of representations of finite groups, be sure to learn the close connection with the theory of invariants of finite groups and Groebner basis methods for computing them. See Ideals, Varieties and Algorithms, one of the great books produced so far by Homo sap in my opinion.
Uhm... symmetry group S3? Does he mean the symmetric group on three letters? (If so, the instructor must have had the same idea I did--- first teach the theory for finite groups.) The rotation group SO(3)?
the naivety of the OP who assumes that context is irrelevant astonishes me--- but probably only because I've been doing math so long.
Anyway, if these groups are finite groups and if the context is permutation groups, the notation he mention probably refers to distinct permutation representations of certain "abstract" symmetric groups. In particular, S_4[2] might mean the degree six permutation representation of S_4, i.e. a certain 24 element subgroup of S_6 which is isomorphic as a group to S_4.
Play around with GAP for hundreds of thousands of further examples of similar notation.
My own brain seems to be rotating and I am losing my own train of thought here.
Mueiz is correct that in a rotating reference frame the spacetime is flat, so how does that jive with my claims that the measured geometry is not Euclidean? Am I making a mistake in my assertions?
Mueiz is correct that in a rotating reference frame the spacetime is flat, so how does that jive with my claims that the measured geometry is not Euclidean? Am I making a mistake in my assertions?
I realize that the question was to CH, but anyway, there is a distinction between the curvature of space and the curvature of spacetime. The relevant notion of curvature of space is given by a purely spatial metric determined by radar measurements carried out by comoving observers. I have a derivation of the spatial metric here: http://www.lightandmatter.com/html_books/genrel/ch03/ch03.html#Section3.4 (subsection 3.4.4)
BRS: Posters who are years away from being ready to try to learn gtr?
This post is addressed to Ben Crowell and other SAs who may want to try to respond to three baffled newbies who don't seem to recognize how much background they currently lack.
Re
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In the case of Lorentzian four-manifolds, as everyone here knows, you can find a coordinate chart such that at a specific event E the metric tensor takes the form
<br />
\left[ \begin{array}{cc|cc}<br />
0 & 1 & 0 & 0 \\<br />
1 & 0 & 0 & 0 \\<br />
\hline<br />
0 & 0 & 1 & 0 \\<br />
0 & 0 & 0 & 1<br />
\end{array} \right]<br />
(where the first two coordinates are null). But there is more: in a neighborhood of E, you can find a chart in which the metric tensor takes the form
<br />
\left[ \begin{array}{cc|cc}<br />
0 & \exp(f) & u & p \\<br />
\exp(f) & 0 & v & q \\<br />
\hline<br />
u & v & \exp(g) & 0 \\<br />
p & q & 0 & \exp(g)<br />
\end{array} \right]<br />
where f,g,u,v,p,q are six functions of the four coordinates (again, the first two coordinates are null). Notice that the case f=g=u=v=p=q reduces to Minkowski vacuum. Also notice that the chart gives two foliations into E^{1,1} and E^2 submanifolds; the E^{1,1} and E^2 submanifold passing through an event E' do not have orthogonal tangent spaces at E'.
In other words, given any event in any Lorentzian four-manifold, there some coordinate transformation such that the metric tensor in the new coordinates takes the given form for specific functions f,g,u,v,p,q. Similarly for other dimensional cases, but notice that odd and even dimensions are a bit different!
Such assertions are closely related to the uniformatization theorem, but in general require different and significantly harder methods of proof. In two dimensions, our semi-canonical charts look like
E^{1,1} case
<br />
\left[ \begin{array}{cc}<br />
0 & \exp(f) \\<br />
\exp(f) & 0 <br />
\end{array} \right]<br />
E^2 case:
<br />
\left[ \begin{array}{cc}<br />
\exp(g) & 0 \\<br />
0 & \exp(g) <br />
\end{array} \right]<br />
The second case says that (locally) every Riemannian two-manifold can be given an isothermal chart, which is a real form of the uniformatization theorem familiar from a course in complex variables. See the book by Steven Krantz for more about relationships between Riemannian geometry and complex variables.
Next up:
E^{1,2} case
<br />
\left[ \begin{array}{cc|c}<br />
0 & \exp(f) & u \\<br />
\exp(f) & 0 & v \\<br />
\hline<br />
u & v & 1<br />
\end{array} \right]<br />
E^3 case
<br />
\left[ \begin{array}{cc|c}<br />
\exp(f) & 0 & u \\<br />
0 & \exp(f) & v \\<br />
\hline<br />
u & v & 1<br />
\end{array} \right]<br />
You get the idea. But a very simple counting argument may help: in d dimensions, a coordinate transformation to a semicanonical local coordinate chart can (locally) remove d degrees of freedom, so
in two-manifolds, 3-2 = 1
in three-manifolds, 6-3 = 3
in four-manifolds, 10-4 = 6
in five-manifolds, 15-5 = 10
...
where 3,6,10,15,... are the number of independent components of the metric tensors.
There are other semicanonical charts; see such books as the monograph by Stephani et al, Exact Solutions of the Einstein Field Equations. See also Bondi radiation coordinates in textbooks such as D'Inverno, Introducing Einstein's Relativity, for an example of a chart in which the components of the metric tensor have a more direct geometric/physical meaning. (Actually, the form preferred by Penrose exhibits the meaning even more clearly than the one used by Bondi; the two forms of radiation charts are however essentially equivalent, the Penrose version is just a bit nicer in some respects.)
The chart I sketched above is only "semicanonical" because the above form does not determine a unique chart having the given form! For details on this particular semicanonical chart, search for an eprint by Gu in the arXiv giving a generalization of the Eddington-Kerr chart in Kerr vacuum to any Lorentzian manifold.
In
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the OP is obviously very confused
I don't know what R^00 means or the scalar curvature by that matter
Which is why one needs to know something about Lorentzian manifolds before trying to understand, much less use, gtr.
A brave SA could point out that written out in full generality, the Einstein field equations are a system of ten coupled second order PDEs for ten variables (functions of four coordinates), the components of the metric tensor, in terms of ten more variables, the components of the matter tensor (given functions of four coordinates). Compare the Maxwell field equations, which can be written as a system of coupled PDEs giving the components of the EM field in terms of the current density four-vector. Unlike the Maxwell equations, the field equations of gtr are nonlinear.
Beginners in gtr usually find it easiest to initially focus on trying to understand the vacuum field equations, in which the matter tensor is assumed to vanish. It is usually helpful to adopt some simple metric Ansatz, which is a condition both on the form of the coordinate chart and on the geometry of spacetime. For example, the Schwarzschild vacuum is commonly derived by writing down the metric tensor of a static spherically symmetric spacetime in terms of a Schwarzschild coordinate chart; this reduces the field equations to two coupled ODEs which are easily solved.
Since g^00 equals negative 1
Only in a chart comoving with certain observers.
In
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this question is closely related to the others just discussed. I am not sure what to advise telling these posters since it seems apparent to me that they are years of formal study away from being able to make a reasonable attempt at trying to understand how to use gtr to make valid mathematical models and correctly derive predictions.
If anything I said seems confusing, bear in mind that a coordinate on a smooth manifold M is simply a monotonic function x, i.e. dx \neq 0 on some neighborhood U, a notion which requires only the smooth structure on M. In a p-dimensional manifold, if we can find p coordinates on U such that the exterior product of their gradient one-forms does not vanish on U, we have a chart on U. Then you impose a Riemannian or Lorentzian metric tensor on M and write down its components in terms of this chart, and similarly for other tensor fields.
Lurking in the background here are other important questions which have been intensively studied by Frobenius, Darboux, Caratheodory, Cartan, and other geometers, such as charts in which a given exterior form assumes a nice appearance. Here, the case of one-forms stands apart, which plays a key role in thermodynamics; given a one-form, an adapted chart can be found in which it appears like one of the following:
<br />
du, \; u \, dv, \; dw + u \, dv, \; \dots<br />
Whereas given any vector field, an adapted chart can be found in which it looks like a coordinate vector field
<br />
\partial_u<br />
You might be worried about the duality between vector fields and one-forms. But there is no contradiction (exercise).
One should also be aware of the Lorentzian analogue of the circle of ideas introduced in Riemann's famous lecture introducing Riemannian geometry, in which Riemann showed that in an open neighborhood of any point P in any Riemannian manifold, there is a coordinate transformation to a chart of a particular form (not uniquely specified) such that, up to second order in Riemannian distance from P, the components of the metric tensor can be given in terms of the components of the Riemann curvature tensor evaluated at P. This clarifies the relationship between the Riemann curvature tensor and the metric tensor.
Indeed, I suspect that all three posters are struggling to express questions such as: how much information is required to specify an arbitrary Lorentzian four-manifold? An arbitrary solution of the vacuum field equation in gtr? In some other metric gravitation theory such as Brans-Dicke? (Turning this around: how many Lorentzian manifolds are filtered out by restricting to vacuum solutions in gtr or a competing theory?) And how much information is required to specify an arbitrary axisymmetric vacuum solution? An arbitrary dust solution? Such questions were already considered by Einstein in terms of a counting argument introduced by Riemann himself in his famous talk, and have been taken up by later researchers such as Sachs and Siklos.
For what it is worth, here are some rough answers (one can be more precise):
to specify an arbitrary Lorentzian four-manifold requires specifying six functions of four variables, plus eight functions of three variables and six functions of two variables; that is, the Riemann wealth is
<br />
6 \cdot (4|2) + 8 \cdot (3|2) + 6 \cdot(2|2)<br />
where these functions are specifying certain second partials on certain submanifolds,
to specify an arbitrary vacuum solution in gr requires specifying four functions of three variables, plus six functions of two variables; that is, the Riemann wealth is
<br />
4 \cdot (3|2) + 6 \cdot (2|2)<br />
where again we are specifying second partials on certain submanifolds.
The fact that to a first approximation, we need only four functions versus six is related to the fact that there are two polarization modes for gravitational plane waves, each described by a complex function (so four real functions in all). The reduction from four to three variables is related to the fact that the vacuum Einstein field equations can be rewritten in an initial value formulation, so that the second derivatives of our four functions are specified on a Cauchy hyperslice--- or in a variant formulation of Ray Sachs, on two null halfspaces. Strictly speaking we must also specify second derivatives of six functions of two variables (think of prescribing values on the intersection of our two null sheets). Solving the IVP from this data then recovers the full metric tensor "above" the two null halfspaces. Compare the Bondi radiation formalism, where we work on a conformal compactification, start at future null infinity (prescribing values on a certain two-submanifold) and work backwards (prescribing values on a forward light cone) by solving an IVP to recover the metric inside a "light cone". Roughly speaking.
In the case of arbitrary Lorentzian four-manifolds, you might be worried that up above I sketched a chart requiring six functions of four variables, no partials need apply, but then I said "six functions of four variables plus some more stuff". But there is no contradiction (excercise).
The notion of Riemann wealth rests upon a slightly more sophisticated notion of counting than the simple argument mentioned above. Riemann's idea was to use power series whose terms count the number of independent partial derivatives of some variable (see the book by Wilf, Generatingfunctionology).
A good example to begin with is the ordinary wave equation in E^{1,2}. Starting with
<br />
-u_{tt} + u_{xx} + u_{yy} = 0, <br />
\; u(0,x,y)=f(x,y),<br />
\; u_t(0,x,y) = g(x,y)<br />
take the Laplace transform t \rightarrow \omega, then the Fourier transform (x,y) \rightarrow (k,\ell). Then solve a simple algebraic equation to find
<br />
({\mathcal FL} u)(\omega,k,\ell) = <br />
\frac{\omega \, ({\mathcal F} f)(k,\ell) <br />
+ ({\mathcal F} g)(k,\ell)}<br />
{\omega^2 + k^2 + \ell^2}<br />
Take the inverse Laplace transform, then the inverse Fourier transform (the two types of transform commute, but this order is more convenient). This operation gives the general solution of the IVP in the form
<br />
u = g \ast \psi + f \ast \psi_t<br />
where we used the convolution product and where
<br />
\psi(t,x,y) = \frac{1}{\sqrt{t^2-x^2-y^2}}<br />
is the fundamental solution of our wave equation. The Riemann wealth of our wave equation is
<br />
(2|0) + (2|1)<br />
(one function of two variables, f, plus a second function of two variables,g, both specifying values on a Cauchy slice, with g representing a first partial) reflects this way of obtaining the general solution, but applies even to PDEs where no general solution is available.
Another good example is Maxwell's equations of EM; here too the Riemann wealth clearly reveals the possibility of an initial value formulation.
I think it is fair to say that while much is known, truly definitive answers to all such questions are not yet available. This is because answers would come close to giving a kind of parameterization of infinite dimensional solution spaces--- even local parameterizations ("local" in the sense of "local neighborhood", not "ultralocal" in the sense of jet spaces) are hard to come by, particularly for nonlinear PDEs.
(I wish there were some way to make posts in this thread older than say two weeks vanish, since otherwise the "Recent" in the title makes no sense... Perhaps some of you will continue to lobby for expiration dates on sensitive or time-limited threads/posts after I have left PF.)
I wouldn't want them to disappear. There have been many times when I have referred back to some conversation on PF more than a year after the conversation. However, it would certainly be easier to read these posts on my Blackberry if each new "Recent PF Thread" that were commented on had its own BRS thread. Then the old ones would automatically get pushed down the list without needing to be deleted.
as I think most of you know, the strongest type of gravitational radiation, mass quadrupole radiation, results when the second time derivative of the quadrupole moment of the source (an isolated gravitating system) is nonzero; for a Kerr hole and for most isolated spinning objects (geometrically close to an oblate spheroid) and indeed for any almost axisymmetric object with spin axis aligned with the axis of symmetry, such as a spinning disk, this is zero or very close to zero. For a spinning bar, on the other hand, it is nonzero, so an isolated spinning bar emits gravitational radiation, according to gtr. Over time, this means that, according to gtr, its spin rate decreases in a specific manner, as gravitational radiation gradually carries off energy from the system. Similarly, the orbit of a binary system changes over time in a specific manner as gravitational radiation gradually carries off energy from the system.
Many of the sources in the sticky thread "Useful Links for SA/Ms" explain quite clearly why the strongest expected sources of gravitational waves should come from very distant and very rare events--- the merger of two supermassive black holes. Also, what to expect from improved LIGO/VIRGO and from LISA, which are sensitive to somewhat different frequency bands. In particular, LISA should be able to detect gravitational radiation from certain "nearby" binaries (containing at least one compact object like a nuetron star or stellar mass black hole) not yet in the death throes of a merger event, but not LIGO/VIRGO. Which should however detect other kinds of events, at least once the improved instruments are operational.
I wish that SA/Ms would make greater use of that sticky thread. I left it in a very incomplete state, but nonetheless I think it should be very handy in responding to a great many gtr-related threads.
No and no. The BRS thread on "Conformal Compactifications and Penrose-Carter Diagrams" might be helpful in explaining why not. To get a jump on the more ambitious among the amateur dissenters, interested SA/Ms can also look for some very clear recent eprints which compare and contrast the traditional definition (see Hawking and Ellis) of "event horizon" with attempts to concoct a practical quasilocal definition which should be more useful for several purposes, and should avoid the "teleological paradox" I explained in that BRS using the example of a collapsing spherical shell of massless radiation which forms a Schwarzschild black hole in what was originally a locally flat region of spacetime.
Re "Proof of GR"
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through post #10, both User:thetexan and respondents are missing the point: the lightbending prediction of gtr (and competing theories) is quantitative and these quantitative predictions can be tested in many situations, including
stars passing near the limb of the Sun (observed with optical telescopes during a solar eclipse),
quasars not neccessarily passing very near to the Sun (observed with radio telescopes, not neccessarily during a solar eclipse),
and the quantitative prediction of gtr has passed every test of the light-bending formula given by Einstein--- not just a single number, but an entire curve, has been very well tested with impressive positive results. And there are many other independent tests of gtr, including some like lensing which are related to Einstein's light bending formula, all of which gtr has passed with increasingly impressive precision. Because its competitors posit additional "tunable parameters", whereas gtr has no tunable parameters, this very well-established experimental/observational accuracy over a very wide range of conditions is even more impressive.
I have repeatedly urged SA/Ms to make a habit of replying in such threads, not by attempting to debunk a particular error, but by explaining how a particular error illustrates one or more common misconceptions about science. In this case, IMO, the most important point by far is that an alarming number of apparently otherwise intelligent laypersons--- whose (mis)-information comes entirely from popular science magazines and YouTube videos of unknown provenance--- entirely fail to appreciate that science is quantitative and that quantitative predictions play a critical role in the evaluation, refinement, and technological/medical application of scientific theories. Indeed, the word quantitative is an essential part of the very definition of science!
To be fair, several respondents to this thread did try to point out that "thetexan" revealed another major misconception by speaking of "proof by experiment" rather than "disproof by experiment".
There are various technical errors in the thread through Post #10, but the deficiencies I have just pointed out are of far greater importance.
I have repeatedly urged SA/Ms to compile a list of common general misconceptions about science, with clear explanations of the crux of the errors involved, perhaps annotated with references to threads which illustrate examples of these general misconceptions. I believe that if SA/Ms had at hand a sticky thread they could draw upon in composing replies to particular befuddled newbies (or cranks), PF would be much more efficient in its educational mission. I fear that the result is that PF is squandering the time and talents of its most valuable resource, the SA/Ms, by making them waste their energy arguing with ignoramuses over specific (non)-issues founded upon specific misconceptions in the mind of an ignoramus, rather than in explaining how popsci books often leave a dangerously inaccurate impression of how science works and why it has proven so successful.
IMO, it is crucially important for every scientific society, and indeed for every professional scientist, to devote some time to ensuring that the public which supports science with tax monies does not continue to develop more and more inaccurate misconceptions about how science works, lest they fail to continue to support it, not out of a well-reasoned decision to redistribute the allocation of limited resources, but out of profound ignorance about why science is so essential to the well-being of the people. However, on the evidence of what actually happens at PF, no one is listening, so... I give up.
In his Post #1, yuiop appears to claim to have found a new explicit static spherically symmetric perfect fluid (ssspf) solution of the EFE which is distinct from the Schwarzschild ssspf:
yuiop said:
I was playing around with the Schwarzschild interior solution when I came up with this interesting solution that I think would be fun exploring.
But in his Post #4 yuiop reveals that he was trying to state the Schwarzschild ssspf solution in terms of the Schwarzschild chart. The expression he gives is correct, as can be verified by any SA/M who has installed Maxima (or better, GRTensorII under Maple), but this is hardly new and he later gives a citation to the book where he found it.
Another reason why the title of the thread is completely inappropriate is that black hole models are of course very different from what we have here, an idealized model of a possibly compact isolated nonspinning and spherically symmetric object which is certainly not a black hole, because it has a surface and the Schwarzschild vacuum matches across this surface to a static spherically symmetric perfect fluid solution. You could not possibly find anything more remote from a black hole model, conceptually!
In his Post #3, Lut Mentz writes down an expression due to Letelier but forgot to say that f is a function of r. But with that stipulation understood, his expression for the metric tensor does give a class of ssspf solutions, written in a spatially isotropic chart--- note well that the radial coordinate in such a chart is quite different from the Schwarzschild radial coordinate used in a Schwarzschild chart! Confusingly, Mentz114 and yuiop are using the same letter for two distinct coordinates.
Any ssspf solution can be written down using either a Schwarzschild or a spatially isotropic chart (among other possibilities).
Mentz114 said:
If the potential has a singularity then this could cause a singularity in the metric, maybe.
Lut has confused matters unneccessarily by failing to specify what kind of singularity he has in mind. See the eprint where he found the Letelier metric.
yuiop said:
I posted a link for non-uniform density version of the interior solution
Presumably he means either a general expression for ssspf solutions, or a particular example having nonuniform density. The Schwarsschild ssspf is characterized by having uniform density ("incompressible fluid ball" [sic]) although it has of course pressure varying with radius (and vanishing at the surface), like any other ssspf solution.
yuiop said:
For an arbitrarily large shell the finite forces can be made arbitrarily small, so in principle such an artificial black hole shell that does not collapse could be constructed without requiring material of infinite strength.
Not true. In his Post #11, Peter Donis says
if the shell outer radius is 9/8 times the Schwarzschild radius (2M) or less, I believe there is *no* static solution
Correct! This result is known as Buchdahl's theorem and is proven, for example, in the textbook of Schutz. This is a general result which applies to any ssspf solution.
yuiop said:
If we have a hollow shell with an outer radius of 9M/4 and an inner radius of 2M so that the mass enclosed within r is zero
Excluded by Buchdahl's theorem, as should be obvious from studying the metric and Einstein tensor.
yuiop said:
Buchdahl's theorem states that for any distribution of matter or equation of state, that the pressure term becomes infinite somewhere within the sphere for R \leq 9 \, r_s/8 and that the sphere should collapse due to infinite gravitational force. However it is obviously not true for a sphere with a vacuum cavity,
Not true! Stipulating a perfect fluid in the interior region allows the possibility of vacuum regions; a vacuum void is just a special kind of fluid EOS from this perspective (zero pressure and density!).
It is possible to consider a hollow void inside a legitimate ssspf solution, but of course this requires a thin shell supporting the weight of the overlying fluid, and the stresses on the inner shell exclude any but a rather modest fluid ball. It would be unrealistic to neglect the mass of this hard shell, which is what happens if one simply carries out the matching and declares the mismatch in the extrinic curvature tensor (negative of expansion tensor defined by spacelike congruence of outward pointing normals to the E^{1,2} hypersurfaces r=r_0) across the inner surface to result from the presence of a hard shell under stress. (See Poisson, A Relativist's Toolkit.)
"Hard science fiction" writers might like to work out the details in Newtonian gravitation, ideally using a thin but not infinitesimally thin elastic solid shell, in case sufficient water and steel were available in some convenient location in some solar system that some advanced civilization wanted to construct an artificial "water world" minor planet. Argue from the breaking limit of steel that gtr is not needed for the largest possible water worlds. Argue that for a shallow uniform sea, you can in principle built a steel shell and cover it with water, but then show that the water would tend to evaporate due to fairly small gravity. With more work I think you could show that the resort would need a constant infusion of water to be viable. Further, this water world couldn't be orbiting a massive planet without introducing additional tidal stresses on the steel shell.
Ignoring all these considerations, it is true (by spherical symmetry, basically) that a spherical void inside a ssspf + inner hard shell type solution would have "zero gravity" in the void, in gtr just as in Newtonian theory, as is easily verified by carrying out a double matching (Mink interior to ssspf at inner surface, and ssspf to Schwarzschild exterior at outer surface, where the matching across the inner surface shows the presence of the hard shell). But this has nothing whatever to do with black holes!
In his Post #6, yuiop once again fails to simply write down the metric tensor he has in mind, although one can guess that he is using a Schwarzschild chart and is giving the g_{tt} component in a very odd an inappropriate way, but he fails to state what g_{rr} is supposed to be! It's like talking to Tom Van Flandern, he keeps changing the ground without ever saying quite what model he is discussing at any given moment. This makes it impossible to debunk his claims, because one can only guess at what his claims might be!
Also, while
<br />
3/2 \, \sqrt(1-2m/R} + 1/2 \, \sqrt(1-2*m/r \, (r/R)^3}<br />
makes sense (with the surface at r=R),
<br />
3/2 \, \sqrt(1-2m/R} + 1/2 \, \sqrt(1-2*m/r \, p(r)/p(R) \, (r/R)^3}<br />
does not because p(R) = 0 by definition. As often happens when people become deranged by some unworkable obsession, yuiop appears to be unable or unwilling to slow down long enough to spot any problems with his claims.
Peter Donis in his Post #10 draws attention to further suspicious features of yuiop's claims.
yuiop said:
Negative pressure means repulsive gravity.
Such claims are meaningless without further elaboration, and IMO should be avoided as misleadingly naive.
yuiop said:
As long as the outer surface is at 2.25M,
Impossible by Buchdahl's theorem, unless the interior is not a perfect fluid (includes possible vacuum void).
yuiop said:
pressure is everywhere negative and independent of r so is uniform throughout the fluid.
To be fair, the gtr literature is filled with bad papers postulating (without the slightest pretense of robust theoretical motivation, much less observational evidence, except in the large scale cosmological scenario, which is completely different from the situation considered here) negative pressure in non-quantum arenas. As with any theory, "garbage in, garbage out". If you postulate wild conditions with no relation to anything which we know or suspect to be physically possible under the conditions relevant to your model, you can come up with pretty much any Lorentzian manifold, which renders gtr useless. To see why, consider turning the EFE on its head and simply defining the stress tensor to be whatever you compute from the Einstein tensor of an arbitrary Lorentzian manifold. Obviously, in almost all cases, the alleged "stress tensor" is completely unphysical--- if that were not so, the EFE would be useless in excluding impossible situations.
yuiop said:
Nudging the outer surface out beyond 2.25 loses the event horizon which is the item of interest
This is the real problem, you'll never convince yuiop now that he has not "disproven the BH". Barf.
yuiop said:
Of course I recognise that the equations I have used are for static geometries with uniform mass density and the sequence I have described is very dynamic. I am hoping that the tensor experts on this forum may be able to come up with some answers for a similar thought experiment with more realistic parameters. Possibly such an analysis would require a finite element simulation on a university computer, but the spherical symmetry should ease that task.
He started off with a fundamentally wrong-headed idea and now he wants us to fix it up for him? Good grief.
Thanks to pervect in particular for taking the time to debunk some of yuiop's claims, although as I noted above, since yuiop still not clarified what metric he is talking about, we can't even verify that he is even talking about a ssspf solution in the weakest possible sense: stress-tensor inferred from Einstein tensor shows isotropic pressure in the frame field defined by the static fluid elements. Even if he were to come up with some expression for g_{rr} further considerations would show that the pressure and or density exhibit wildly unphysical characteristics. He would then need to come up with a solid theoretical argument from solid state physics or whatever justifying stuff like negative pressure on stellar scales, which seems to be a very unlikely outcome.
Because yuiop is refusing to specify what model precisely he has in mind, I think the thread discussed in this post illustrates some of the points I tried to make in the preceding post. Above all: rather than trying to debunk specific claims about an alleged "mathematical model" [sic] which some poster refuses to fully define using terminology/notation standard in contemporary math/physics, focus on trying to explain why his claims illustrate various general misconceptions about how mathematical models work in physics.
BTW, here is a Ctensor file you can run in batch mode under Maxima which computes some stuff for the Schwarzschild ssspf in the Schwarzschild chart in the notation used by yuiop (which he found in some book); this makes it easy to verify that the stress tensor does have the form appropriate for a ssspf and you can also plot the pressure to verify that it is positive and falls to zero at the surface r=R:
Code:
/*
Schwarzschild ssspf; Schwarzschild chart; static coframe
Covers the region 0 < r < R.
Surface (zero pressure sphere) at r=R
There the metric tensor becomes
ds^2 = -(1-2m/R) dt^2 + dr^2/(1-2m/R) + R^2 dOmega^2
which matches to Schwarzschild exterior vacuum with mass parameter m (the region R < r < infty)
ds^2 = -(1-2m/r) dt^2 + dr^2/(1-2m/r) + r^2 dOmega^2
The ssspf region is conformally flat; each t=t0 slice is a 3-spherical cap.
*/
load(ctensor);
cframe_flag: true;
ctrgsimp: true;
/* define the dimension */
dim: 4;
/* list the coordinates */
ct_coords: [t,r,theta,phi];
/* list a constant */
declare([R,m],constant);
/* define background metric */
lfg: ident(4);
lfg[1,1]: -1;
/* rows of this matrix give the coframe covectors */
/* only need enter the nonzero components */
fri: zeromatrix(4,4);
fri[1,1]: -3/2*sqrt(1-2*m/R)+1/2*sqrt(1-2*m*r^2/R^3);
fri[2,2]: 1/sqrt(1-2*m*r^2/R^3);
fri[3,3]: r;
fri[4,4]: r*sin(theta);
/* setup the spacetime definition */
cmetric();
/* compute a matrix whose rows give frame vectors */
fr;
/* metric tensor g_(ab) */
lg;
/* compute g^(ab) */
ug: invert(lg);
christof(false);
/* Geodesic equations */
# cgeodesic(true);
/* Compute fully covariant Riemann components R_(mijk) = riem[i,k,j,m] */
lriemann(true);
/* Compute R^(mijk) */
uriemann(false);
/* Compute Ricci componets R_(jk) */
ricci(true);
/* Compute trace of Ricci tensor */
tracer;
/* Compute R^(jk) */
uricci(false);
/* Compute and display MIXED Einstein tensor G^a_b */
/* For (-1,1,1,1) sig Flip sign of top row to get G^(ab) */
einstein(false);
/* WARNING! leinstein(false) only works for metric basis! */
/* Einstein tensor as matrix */
matrix([ein[1,1],ein[1,2],ein[1,3],ein[1,4]],
[ein[2,1],ein[2,2],ein[2,3],ein[2,4]],
[ein[3,1],ein[3,2],ein[3,3],ein[3,4]],
[ein[4,1],ein[4,2],ein[4,3],ein[4,4]]);
/* electroriemann tensor */
matrix([lriem[2,2,1,1], lriem[2,3,1,1],lriem[2,4,1,1]],
[lriem[3,2,1,1],lriem[3,3,1,1],lriem[3,4,1,1]],
[lriem[4,2,1,1],lriem[4,3,1,1],lriem[4,4,1,1]]);
/* magnetoriemann tensor */
matrix([lriem[2,4,3,1],lriem[2,2,4,1],lriem[2,3,2,1]],
[lriem[3,4,3,1],lriem[3,2,4,1],lriem[3,3,2,1]],
[lriem[4,4,3,1],lriem[4,2,4,1],lriem[4,3,2,1]]);
/* Compute Kretschmann scalar */
factor(rinvariant());
/* Compute Weyl tensor; shows ssspf region conformally flat */
weyl(true);
Figures: for Schwarzschild ssspf with m=1, R=5, frame field of the fluid elements, geometric units (reciprocal area) for curvature components:
Density (red curve) and pressure (blue curve). Note that the constant density, 0.048, is larger than 0.008, the central pressure.
Tidal tensor; the nonzero components agree with each other throughout (in a more general ssspf, they need only agree at the center).
In this example, the Buchdahl limit is 9/4 = 2.25 < 5. If the unit of length is about a kilometer, then after converting from geometric units to standard units, this example is seen to be a very crude model of neutron star (neutron stars are typically not very far above their Buchdahl limits!).
great answer, and if harcel's post count rises above 600 with no problems visible, Marcel is a shoo-in for SA
Re "Gravitational waves due to acceleration"
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the OP is expressing a very common confusion betweeen
changing tidal field of a moving source (decays like 1/r^3, where r is "the" distance from the source using any appropriate notion of distance, and where "direction" to source changes as source moves)
gravitational radiation (moves at speed v=1, decays like 1/r)
For more advanced students, one can compare/constrast EM radiation versus Coulomb electric field with gravitational radiation (weak field approximation) (Petrov N component) versus "Coulomb" tidal field (Petrov D component). In NP formalism, for a well chosen NP tetrad (equivalent to a frame field but written using some complex variables), in a vacuum (so Riemann agrees with Weyl tensor), the Petrov D component is given by the spinor component Psi_2 while the outgoing radiative component is given by component Psi_0.
Re "Gravitational lense"
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three seemingly contradictory (but not really) answers:
there is an extensive theory of weak-lensing which can be expressed using concepts like "focal planes", which treats various rough models of galactic mass distributions as mathematically analogous to various shaped optical lenses
in gtr--- this is particularly relevant in cosmology--- due to the mathematics of curved Lorentzian manifolds, there are infinitely many distinct notions of "distance in the large", so terms like "distance" have to be interpreted with care or confusion and error will result,
strong-lensing is needed to study optical effects near an isolated compact massive object like a neutron star or black hole, and this involves more subtle concepts that weak-lensing.
Re "big bang ordinary explosion, evidence for expansion of space?"
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jostpuur said:
But then I'm been told that this cannot be so. The red shift cannot be result of an ordinary explosion (meaning an explosion which has center somewhere in flat space), but this must be because the universe itself is a manifold that is expanding. What real evidence do you have for this claim?
Don't try to tell me that if we were in an ordinary explosion, and not in the center of it, then we would see...
This poster is a veteran fringe-theory proponent at PF (yuk), and IMO shows insufficient ability to learn gtr or cosmology, but his particular confusion can be addressed with a flat space Newtonian discussion of certain transformations in affine geometry. In terms of matrices, one can represent any element of the affine group AGL(n,R) as block matrix with an element of GL(n,R) in upper left block (nxn), a one in lower right block (1x1), and a column vector in upper right block, where these matrices act on row vectors from the right. Then the plane x_{n+1} = 1 in R^{n+1}, acted on by multiplication from the right of elements of GL(n+1,R), including its subgroup AGL(n,R), give affine transformations of this plane, which is identified with R^n endowed with affine geometry. Now you can compare
dilation from a point \vec{x}-\vec{x_0} \rightarrow c \cdot (\vec{x}-\vec{x_0}), \; c>0
dilation from a line
dilation from a plane
See the figures below for Hubble's law for a dilation from any point (illustrated for planar affine geometry). The point is that in the first case, we actually have uniform expansion, i.e. an observer riding on any "marked point" finds that distances to other marked points from himself increases the same way as do distances from "the origin". Hence uniform expansion in Newtonian terms. This is also applicable to understanding the Milne model in Minkowski spacetime, with some changes due to fact that the spatial hyperslices orthogonal to the (timelike geodesic) congruence of Milne observers have H^3 geometry (curvature decreasing over time), not E^3 geometry. While the Milne model neglects gravity (expansion continues at uniform rate; no slowdown due to gravitational attraction of model galaxies on each other), these simple examples can clear up one of the most basic confusions about what the standard hot Big Bang theory actually says.
Re "Project in general relativity"
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haushofer's suggestion is IMO ludricously underambititous. Assuming that the OP is serious, ie. he
is a registered uni student
intends to devote a lot of summer time to his project
seeks to master classical gtr with a view towards future work in cosmology or astrophysics related to gravitational radiation and other classical gtr phenomena,
I would advise him to set up a reading course in gtr, if possible, and to design his reading course around the following goals:
review and refactor all his math/physics notes from previous schoolwork, with focus on gtr,
buy at least three and borrow at least three more (try ISL from local public library if no bricks and mortar university) from a list of the best modern gtr textbooks, eg.
Lee, Intro to Smooth Manifolds (study everything, for background in manifold theory)
D'Inverno (for example, radiation formalism)
MTW (study everything)
Stephani (for example, far field versus weak field)
Plebanski and Krasinksi, Intro to GR and Cosmology (for example, frame fields)
Poisson, Relativists's Toolkit (for example, congruences and their decomposition)
Griffiths and Podolsky, Exact Spacetimes for Einstein's GTR
and study material new to him, compare different treatments of material he already learned, take notes, revise, reorganize, write out proofs of important facts like Raychaudhuri formula--- put aside for the future anything which seems to require background he doesn't yet know, later he will figure out what he needs and where to find it,
use the bibliography of MTW, in particular, to look up, copy, and read some of the "classic" review papers and possibly some "classic" research papers (but don't neglect studying modern textbooks in order to read, say, Kerr 1963)
visit arXiv daily and try to read interesting looking recent eprints in gr-qc section
increase his depth/breadth knowledge of topics in "pure math" useful for gtr, e.g. by
buy student Maple (if registered student), and learn to use Maple "built-in" facilities, e.g. from Richards, Advanced Mathematical Methods with Maple,
study as much as possible about smooth dynamical systems (including vector fields on manifolds and standard results in dynamical systems defined using systems of ODEs)
study as much as possible about the theory of PDEs (including good old harmonic functions in R^3)
study perturbation theory from some of the many fine introductions,
study Lie's theory of the symmetries of DEs using e.g. Olver, Applications of Lie Groups to Differential Equations
[*] learn to compute with tensor gymnastics by practicing computer verification of facts like Raychaudhuri formula
[*] install GRTensorII and learn to solve the EFE and study solutions, using Griffiths and Podolsky as a guide; also, check claims in recent arXiv eprints
[*] take any opportunities to learn other areas of classical physics, e.g. Sofer, Classical Field Theory
I consider the above outline achievable for a serious student working steadily throughout summer. If the OP proves unwilling to buy three and borrow three more textbooks, he can forget about trying to master basic gtr ever. Owning personal copies of some of the most important references (I am thinkng MTW, Griffiths and Podolsky, and Plebanski and Krasinsky might be a good list of three textbooks to buy) is essential because he will need to refer to them repeatedly as he continues to learn. Hopefully he will be receptive to the suggestion. Anything less than the above will result at best in proportionately less than full mastery of basic gtr. And if he lacks good judgement and/or ability, results will be unsatisfactory--- feedback from faculty in his uni will be essential indications of whether he is really learning stuff as well as he thinks.
If his ultimate interests like in quantum gravity or string theory, the above should still be very useful, and could even ultimately place him at an advantage since many physicists who work in string theory or dabble in quantum gravity appear not to know the gtr foundations as well as they ought, but his ultimate interests will no doubt influence what topics he chooses to focus on.
I'd have to recommend against a serious student spending much time with Wikipedia or other web resources--- serious students study challenging mainstream textbooks and "classic" papers, hard, and if a "student" has to be told this, IMO he/she is probably not good material for a future scholar.
Figures: in affine geometry, in a dilation from any point, say "the origin" (original positions of marked points indicated with larger open circles, new positions of marked points with smaller filled circles):
linear Hubble's law holds for increases in distances of marked points from "the origin"
linear Hubble's law holds for distances of marked point from any marked point
Pervect and Peter Donis correctly recalled reading that the Kinnersley-Walker photon rocket, an null electrovacuum solution providing a simple model of an isolated massive object which accelerates by emitting asymmetrically directed EM radiation, does not emit any gravitational radiation. An easy way to verify this is to run in batch mode under Maxima the following Ctensor file:
Code:
/*
Kinnersley-Walker null dust outflux; Student psph chart; slowfall coframe.
An exact null electrovacuum solution which models a massive object (mass m) which accelerates due to asymmetric emission of EM radiation.
The "photon rocket" has Weyl tensor of Petrov type D
showing no gravitational radiation is emitted.
The Kretschmann scalar is
48*m^2/r^6
just like Schwarzschild vacuum.
*/
load(ctensor);
cframe_flag: true;
ratchristof: true;
ctrgsimp: true;
/* define the dimension */
dim: 4;
/* list the coordinates */
ct_coords: [u,r,theta,phi];
/* define background metric */
lfg: ident(4);
lfg[1,1]: -1;
/* Declare the dependent and independent variables */
depends(h,u);
depends(m,u);
depends(p,u);
declare(a,constant);
/* Define the coframe covectors */
/* only need enter the nonzero components */
fri: zeromatrix(4,4);
fri[1,1]: -1+m/r+p*r*cos(theta);
fri[1,2]: -1;
fri[2,1]: -m/r-p*r*cos(theta);
fri[2,2]: 1;
fri[3,1]: p*r*sin(theta);
fri[3,3]: r;
fri[4,4]: r*sin(theta);
/* setup the spacetime definition */
cmetric();
/* compute a matrix whose rows give frame vectors */
fr;
/* metric tensor g_(ab) */
lg;
/* compute g^(ab) */
ug: invert(lg);
christof(false);
/* Compute fully covariant Riemann components R_(mijk) = riem[i,k,j,m] */
lriemann(true);
/* Compute R^(mijk) */
uriemann(false);
/* Compute Ricci componets R_(jk) */
ricci(true);
/* Compute trace of Ricci tensor */
tracer;
/* Compute R^(jk) */
uricci(false);
/* Compute and display MIXED Einstein tensor G^a_b */
/* For (-1,1,1,1) sig Flip sign of top row to get G^(ab) */
einstein(false);
/* WARNING! leinstein(false) only works for metric basis! */
/* Einstein tensor as matrix */
matrix([ein[1,1],ein[1,2],ein[1,3],ein[1,4]],
[ein[2,1],ein[2,2],ein[2,3],ein[2,4]],
[ein[3,1],ein[3,2],ein[3,3],ein[3,4]],
[ein[4,1],ein[4,2],ein[4,3],ein[4,4]]);
/* electroriemann tensor */
matrix([lriem[2,2,1,1], lriem[2,3,1,1],lriem[2,4,1,1]],
[lriem[3,2,1,1],lriem[3,3,1,1],lriem[3,4,1,1]],
[lriem[4,2,1,1],lriem[4,3,1,1],lriem[4,4,1,1]]);
/* magnetoriemann tensor */
matrix([lriem[2,4,3,1],lriem[2,2,4,1],lriem[2,3,2,1]],
[lriem[3,4,3,1],lriem[3,2,4,1],lriem[3,3,2,1]],
[lriem[4,4,3,1],lriem[4,2,4,1],lriem[4,3,2,1]]);
/* Compute Kretschmann scalar */
rinvariant();
expand(factor(%));
/* Construct NP tetrad for our frame, compute Weyl spinors and Petrov type */
nptetrad(true);
weyl(false);
psi(true);
petrov();
The Weyl tensor shows the only nonzero Weyl spinor component (wrt the given NP tetrad) is \Psi_2 = m/r^3 (the sign discrepancy with GRTensorII result is due to differing sign conventions), which means that the field is pure Coulomb (the field of an isolated object); the acceleration is due soley to the EM radiation (which appears in the Einstein tensor and has the form appropriate for a radiative EM field, i.e. the principle Lorentz invariants of the EM two-form vanish).
In principle, a massive object could accelerate by emitting asymmetrically pure gravitational radiation, and there is an approximate solution indicating that this is possible, but it is not nearly as nice as the KW solution.
passionflower said:
Actually solutions that do not have any gravitational waves are more the exception than the rule in GR.
Somwhat true for exact solutions of the EFE discovered so far, and very likely true as a statement about the solution space of the EFE.
passionflower said:
Solutions where masses accelerate must obviously have gravitational waves.
Wrong, as the example shows.
passionflower said:
And solutions where masses are moving wrt each other will typically also generate gravitational waves.
He is correct about that, as one can see from say Schutz's discussion of weak field theory.
Consider a simple weak-field model in which a massless rod (sic) with a massive ball (mass m) on each each end expands and/or contracts along its length. This does produce gravitational radiation because the quadrupole moment has nonzero second derivative wrt time.
Consider next two massive objects which are moving in inertial motion in almost flat spacetime. Approximately, they move linearly along straightlines in space, and the quadrupole moment of the system consisting of these two objects (we neglect contribution of anything else to gravitational field) has nonzero second derivative, and we find amplitude of order m v^2/R, where R is the distance from the observer to the COM of the system and where v is positive but much less than c=1.
Reason: to compute I_(xx), integrate rho x^2, where we have two delta masses, mass m, at x=a-vt/2, and x =a + vt/2, say, where 0 < v << 1, so that COM of system is at the origin. Then we find the traceless quadrupole moment includes terms like m v^2 t^2 whose second derivative wrt t is nonzero.
some idgits are insisting (without argument) that "black holes can't exist" [sic] because "the gravitational influence of the singularity can't get outside the horizon"--- this last part is true, but the point is that it is irrelevant! See the link already mentioned by Lut Mentz.
skeptic2 said:
This reference makes no mention of how a singularity may have a causal effect on the event horizon.
"skeptic2" (sockpuppet check, anyone?) is missing the point: the singularity need not causally affect the horizon or the exterior in order for the gravitational field of a Schwarzschild or Kerr hole to be nonzero in the exterior! The fact that the field is nonzero in the exterior for these solutions shows that their claim is false about simple mathematical models in gtr of black holes, but more importantly, there is a simple physical reason why one should not be puzzled that black holes have nonzero gravitational fields in the exterior: in our universe, according to current mainstream judgement, all black holes are formed by the complete collapse of some massive object.
Consider a simple OS model of the formation of a nonrotating black hole by the collapse of an initially "infinite radius" dust ball, This is constructed by matching a portion of the FRW dust with E^3 hyperslices, which we can write using the frame of the dust particles, whose world lines form a vorticity-free timelike geodesic congruence orthogonal hyperslices locally isometric to E^3, to the Schwarzschild solution, which we can write in the Painleve chart, using the frame of Lemaitre observers whose world lines again form a vorticity-free timelike geodesic congruence with orthogonal hyperslices locally isometric to E^3. Thus, we can give the OS model a frame associated with a vorticity-free timelike geodesic congruence whose orthogonal hyperslices are globally isometric to E^3, and we can visualize the dust ball as an ordinary ball whose "radius" shrinks with time. (If we use the suggested coordinates, the time coordinate measures changes in proper time of any dust particle in the interior, or any Lemaitre observer in the exterior). Now the exterior field is roughly speaking systematically formed as the ball shrinks; between t and t+dt it doesn't change outside R(t) but evolves a larger vacuum region on R(t+dt) < R(t) according to the covariant differential equation discussed in the textbook of Carroll (which shows how changing Ricci curvature can create nonzero Weyl curvature). Eventually R(t) shrinks past the event horizon, and after this we have a black hole. The influence of changes inside the horizon cannot propagate outside the horizon, but that is not necessary because the entire exterior field has already been "created" according to the EFE, and it is static, so should not and does not change when the mass which was responsible for creating the exterior field can no longer send signals to the exterior region.
IOW, the exterior fields of black holes are, according to gtr, simply the "fossil remnants" of the field formed when some massive object (which has since undergone complete gravitational collapse resulting in an event horizon) was formed by collecting and concentrating material in some "smallish" (on cosmological scales) bounded region.
BRS: "Interesting" Exact Solutions (off the Top of my Head)
Happy New Year!
In
Code:
www.physicsforums.com/showthread.php?t=459863
George Jones asked for "interesting solutions" of the EFE.
First let me say that PAllen is correct: while an exact solution modeling two mutually orbiting bodies would unquestionably be very interesting indeed, AFAIK none is yet known. (It would not be stationary and it would contain gravitational radiation, which would break axisymmetry, so I don't think a solution of this kind could have any Killing vectors.)
There are many simple exact solutions (and some not so simple ones) which can be (mis?)-characterized as models of the gravitational fields of two objects, such as the "double Kerr" modeling two spinning Kerr objects held apart by some combination of a "massless strut" (aye, there's the rub!) and spin-spin interaction, and various Weyl vacuums representing two objects held apart by a massless strut (or massless cables "stretching off to spatial infinity"). In the sequel I will avoid solutions containing "massless struts".
Of the classes of solutions extensively discussed by Plebanski and Krasinski, George forgot to mention the particularly important LTB family dust solutions (spherically symmetric time-varying dust solutions), which includes many interesting special cases. And of the classes discussed by Griffiths and Podolsky, George forgot to mention the particularly important family of colliding plane wave (CPW) spacetimes.
I think it is best to try fit the solutions discussed by Griffiths and Podolsky (and indeed, MacCallum et al.) into a poset showing which are specializations of others, and indicating other noteworthy relations; e.g. some are locally isometric to portions of others, or conformally related. However, this screen is too small to contain my diagram, so I'll try to create a similar effect using words and a tiny bit of mathematical notation.
Perhaps the most basic way to start organizing many well known exact solutions is to begin with spacetimes admitting a two dimensional abelian Lie algebra of Killing vector fields (possibly a subalgebra of an even larger Lie algebra of Killing vector fields). This category includes several large families of exact solutions, including
stationary axisymmetric non-null electrovacuums (KVs generating commuting time translation and rotation)
cylindrically symmetric nnevacs (KVs generating commuting spatial translation and rotation),
boost-rotation symmetric nnevacs (two KVs generating commuting boost and rotation),
Gowdy spacetimes (KVs generating two commuting spatial translations)
These families are all known in terms of solutions to certain families of second order quadratically nonlinear PDEs with two dependent variables and two independent variables--- the coordinates not associated with Killing vector fields, in an appropriate chart. A third dependent variable appearing in the metric tensor is determined by quadrature from these two.
A great deal of effort has gone into solving these nonlinear systems of PDEs (with two independent variables and two dependent variables), as you might expect. Mathematically, methods inspired by the inverse scattering transform in the theory of solitons have enjoyed success; typically these allow a kind of nonlinear superposition of seed solutions, but unfortunately the result often seems to include massless struts. There are also some closely related methods inspired by extensions of Lie's theory of the symmetries of differential equations to include infinite dimensional algebras of more general symmetries. There are also various known Baeklund transformations which can be used to obtain new solutions from old ones belonging to certain families. And there has also been considerable work on "algebraico-geometric" approaches. There is considerable overlap between all these notions.
Each of the families I mentioned contains large subfamilies, especially vacuum solutions with the same metric symmetries, and these families also contain examples which can be generalized outside the family, e.g. by adding nonzero Lambda. These families can generally also be extended to include minimally coupled massless scalar fields in addition to an EM field.
The family of all stationary axisymmetric nnevacs includes the family of all stationary axisymmetric vacuums (Ernst vacuums). The Ernst vacuums includes the family of all static axisymmetric vacuums (Weyl vacuums).
Examples of Ernst vacuums include the Kerr and Taub-NUT vacuum solutions. The Kerr vacuum plays a physically important role, as we know, as the quiescent state of any rotating black hole; the Taub-NUT vacuum is important pedagogically; as a local solution given on a certain region, its alternative maximal extensions are particularly noteworthy.
The Ernst vacuums are governed by a quadratically nonlinear system of two PDEs in two dependent variables (functions of two indepedent variables). The Ernst system can be rewritten as a single PDE expressed in terms of complex variables (the Ernst equation, which comes in equivalent versions exhibiting either the SL(2,R) or SU(1,1) internal symmetries). However, I prefer to write it in ordinary vector notation for real variables in Minkowski spacetime, like this:
<br />
\begin{array}{rcl}<br />
p \, \Box p & = & \| \nabla p \|^2 - \| \nabla q \|^2 \\<br />
p \, \Box q & = & 2 \, \nabla p \cdot \nabla q<br />
\end{array}<br />
where p,q are now functions of t,x,y,z. Even more memorable, perhaps, is the Lagrangian from which this system arises:
<br />
L = \frac{\| \nabla p \|^2 + \| \nabla q \|^2}{p}<br /> Axisymmetric, time-independent solutions (p,q) of this system then completely define each Ernst vacuum. Here, p serves as the metric function roughly corresponding to Newtonian gravitational potential; one next produces a second metric function using q as a "twist potential", and finally one obtains the third metric function by quadrature from the first two. For this procedure to make sense, we require that the curved spacetime wave operator (restricted to operating on time independent axisymmetric functions) agree (when written out explicitly using the canonical chart) with the flat spacetime wave operator, and it does.
The Ernst system has a ten dimensional group of point symmetries in the sense of Lie, and because it arises from a Lagrangian, some of these are Noether symmetries, each of which gives rise to a conservation law, much as happens for the usual wave equation. Interestingly, the same master system arises directly (no "twist potential" needed) in the definition of CPW models! Indeed, in some sense, if you can solve any of these systems you should be able to solve all the others.
As explained in the BRS thread on this family, each Weyl vacuum solution is generated by an axisymmetric harmonic function (in Newtonian gravitation, the potential of an axisymmetric static field). IOW, the master system reduces to \Delta p = 0, so that instead of looking for solutions (p,q) to the Ernst system, we set q=0 and look for solutions of a much simpler system, a single linear PDE. In the BRS on Weyl vacuums, I explained why this linearity does not contradict the fact that the EFE is nonlinear.
The Weyl vacuums include many examples of independent interest, most notably the Schwarzschild vacuum. Also noteworthy are the static cylindrically symmetric vacuums, which were found very early by Levi-Civita; in general their Weyl (=Riemann) tensors have algebraic symmetries which are Petrov type I, but special cases have more symmetries and occur as examples in some of the families mentioned so far.
The Schwarzschild vacuum can be generalized to the static spherically symmetric lambda vacuum (Schwarzschild-de Sitter) solution, whose possible global structures are noteworthy, and also to the spherically symmetric null dust (Vaidya null dust). The Vaidya null dust can be further generalized to Kinnersley-Walker photon rocket, and to Robinson-Trautman null dusts, which include local solutions with notable alternative maximal extensions. Also, Schwarzschild vacuum can be fit into the OS collapsing dust ball model, which can be generalized to LTB collapse models.
The Kerr vacuum arises as a limiting case of a particularly important stationary axisymmetric exact solution, the Neugebauer-Meinel model, in which a "rigidly rotating" thin disk of dust is matched to an exact vacuum exterior. This solution was found after a decade long search, by a spectacular application of "elementary" ideas from PDEs plus special functions. It is difficult to express simply, but a previously known solution, the Bardeen-Horowitz vacuum, which originally arose as a limiting case of the Reissner-Nordstrom "throat", is a simple exact vacuum solution which also arises in a limit from the Neugebauer-Meinel model.
As noted in the BRS on Weyl vacuums, in terms of global structure, a local solution drawn from the Weyl family typically represents the static exterior region of a larger spacetime which also includes a dynamic future interior which is locally isometric to a boost-rotation symmetric vacuum. The C vacuum and its generalization to the Bonnor-Swaminarayan vacuum illustrate how this works.
Examples of the family of all stationary axisymmetric electrovacuums include the Kerr-Newman electrovacuum and the Melvin electrovacuum. A familiar example in the subfamily of static axisymmetric electrovacuums is the Reissner-Nordstrom electrovacuum.
Gowdy models have proven to be very important in attempts to better understand the nature of the solution space of the EFE; one of the more important discoveries of the past 15 years has been the appearance of "spikes", which along with BKL type oscillations may or may not turn out to be "generic" features of some "regions" of the solution space. (Several international groups have been trying for decades to extend the success of modern methods of studying PDEs into the realm of gtr, using notions such as Sobolev spaces.) Gowdy spacetimes are also very closely related to CPW models.
Also pedagogically important are Levi-Civita's type D static vacuums, which include the Schwarzschild vacuum and various others, of which the most important are the plane symmetric Taub vacuum (this is not really analogous to the plane symmetric gravitational field in Newtonian theory, unless you fancy negative mass infinite planar sheets!), which is closely related to the plane symmetric Kasner vacuum, and the C vacuum, which is the simplest example of a boost-symmetric vacuum.
Turning to dust solutions: a large family of dust models are constructed by assuming homogeneous hyperslices locally isometric to some three-dimensional Lie group. These are usually called Bianchi I dusts (Kasner dusts), Bianchi II dust, ... Bianchi IX dust (mixmaster models). These are all completely defined by certain systems of ODEs (independent variable is the "comoving" time coordinate used in appropriate charts, comoving with the dust particles and adapted to the symmetries of the constant time hyperslices).
Some of these families of Bianchi dusts are noteworthy for the fact that they exhibit the famous feature of the mixmaster models, an infinite cascade of quasiperiodic oscillations in the curvature tensor (very roughly: contraction along x,y with expansion in z, suddenly transitions to contraction along x,z with expansion in y, suddenly transitions to... ). The famous BKL conjecture states in part that something similar should occur during the approach to many future strong spacelike curvature singularities which arise in gtr, including the interior of generic black hole solutions. The FRW dusts (and as already noted, the Kasner dusts) occur as special cases which do NOT exhibit BKL oscillations.
Other noteworthy dust models include the Szekeres dust (no Killing vectors at all), the Koutrosh-McIntosh dust (another large family of dust solutions), the Ellis-MacCallum families of dust solutions (one example can be matched to part of the Schwarzschild vacuum), the cylindrically symmetric stationary Van Stockum dust (pedagogically valuable for its distinguished locus "in space" and for comparision with Goedel lambdadust), the Bonnor dust (models a "rigidly rotating" ball of dust), and as already mentioned, the LTB dusts. There are also dust models which--- like the Van Stockum dust, but cosmologically more reasonable--- model large scale rotation; these are useful in arguing that observational evidence so far is inconsistent with such large scale rotation, and also for comparision with the different notion of "rotation" involved in the Goedel lambdadust.
The Goedel lambdadust has a five dimensional Lie algebra of Killing vectors; it is homogeneous (unlike Van Stockum dust) and contains CTCs, among other interesting properties. It fits into a larger family of spacetimes with related properties.
Radiation plays a central role in any classical field theory, so not surprising that wave solutions are particularly important.
A very large class (known in terms of solutions to certain PDEs) is the class of Kundt null dust waves, which include null electrovacuum and vacuum subclasses. The Weyl tensor of a Kundt wave generally has Petrov III and Petrov N components--- that is, wrt a suitable NP tetrad, the Weyl spinor has only two components, \Psi_3 (pure longitudinal shearing) and \Psi_4 (type N; transverse, spin two). It has recently been proven that the Kundt waves provide all counterexamples to the natural (but false) expectation that all non-flat spacetimes should have some curvature invariant (possibly constructed using some higher order covariant derivatives of the Riemann tensor, e.g. R_{abcd;ef} \,R^{ab;c} \, R^{de;f}) nonvanishing (Penrose pointed out many decades ago that wave solutions provide counterexamples to this notion).
Another important subclass of the class of Kundt waves is the family of pp-waves, which also includes null electrovacuum and vacuum subclasses. The Weyl tensor of the pp-waves contains only type N radiation. The pp-waves can be classified by the structure of the Lie algebra of their Killing vector fields; this can range from dimension one to dimension seven. One of the most important, EK4 vacuum pp-waves (two dimensional isometry group), consists of the axisymmetric gravitational waves. A subclass of EK4 waves is the class of EK6 waves (three dimensional isometry group); this is a one-parameter family which consists of all stationary axisymmetric gravitational waves.
The pp-waves include the family of plane waves, which reduces in weak-field theory to the usual linearized gravitational waves studied by Einstein himself. Generic vacuum plane waves form the symmetry class EK9, which has a five dimensional Lie algebra of Killing vector fields. Examples include the exact monochromatic linearly polarized gravitational wave.
A host of further examples of EK9 pp-waves (aka vacuum plane waves) are interesting because they illustrate various kinds of "strengths" of null curvature singularities (loci where some components of the Riemann tensor diverge); some of these are highly destructive but others are "weak" in the sense that, roughly speaking, the curvature measured by a typical inertial observer increases so rapidly that at least some congruences of timelike geodesic world lines don't have time to create singularities in their expansion tensor; even weaker singularities have the property that the curvature diverges so quickly that the metric tensor does not develop singularities either; an encounter with such a singularity would appear to be survivable by a sufficiently small object, but gtr shrugs and declares itself unable to predict what happens after the encounter, because these null curvature singularities are also Cauchy horizons.
The subclass EK11 has a six dimensional Lie algebra of Killing vector fields, and consists of the family of circularly polarized monochromatic gravitational waves.
Among the more symmetric examples of null dust plane waves, the class SG15 (six dimensional isometry group) consists of the conformally flat plane waves, which contain no gravitational radiation, but allows for time varying (but spatially uniform) amplitude of null dust. The class SG16 (seven dimensional isometry group) is a one parameter family consisting of the conformally flat plane waves with amplitude independent of time. The Bonnor beam is constructed by matching an SG16 interior across the world sheet of a cylindrical surface to an EK6 exterior; this models an isolated, intense, confined beam of incoherent EM radiation.
BRS: "Interesting" Exact Solutions (off the Top of my Head)
(continued)
CPW models are noteworthy because (along with the boost-rotation symmetric vacuums, one could argue) they are the only known large family modeling physical interactions. Specifically: the nonlinear interaction of the "tails" of two plane waves.
In a CPW model, two plane waves (each typically containing both null dust, e.g. incoherent EM radiation, and gravitational radiation) approach each other in an initially flat region, moving of course "at the speed of light", and collide with each other, leaving behind a curved region (the "interaction zone") containing partially backscattered radiation. Typically, each wave focuses astigmatically the integral curves of the wave vector of the other wave.
Another noteworthy feature of the family of CPW spacetimes is illustrated by two of the simplest and most important examples of CPW spacetimes: the nonlinear interaction zone of the Ferrari-Ibanez CPW is locally isometric to the outer portion of the Schwarzschild vacuum future interior, while the nonlinear interaction zone of the Chandrasekhar-Xanthopoulos CPW is locally isometric to a portion of the Kerr vacuum future interior. (A certain Baecklund transformation produces the CX CPW from the FI CPW.) For this reason, CPW models can be used to study "interesting" black hole interiors.
A third noteworthy feature is that when two gravitational waves (Petrov type N) collide, the interaction zone is typically Petrov type I; this happens because the interaction zone includes, as already mentioned, backscattered radiation. Similarly, when two EM waves (null electrovacuum regions) collide, the interaction zone is typically non-null electrovacuum. This is illustrated by another simple example in which two exact EM waves collide to produce an interaction zone locally isometric to the Bertotti-Robinson electrovacuum.
A fourth noteworthy feature: the global structure of CPW spacetimes reveals a new kind of geometric singularity, the "fold singularity", which is not a curvature singularity but which is also not merely an artifact of mathematical description. Physically, generic null geodesics in a CPW spacetime avoid the fold singularities, but a measure zero subset of null geodesics run into a fold singularity. The ones which avoid this fate typically run into a future strong spacelike singularity which finishes the evolution of the interaction zone. But some CPW models do not develop such a strong spacelike singularity, but rather a weak null singularity or even a mere Cauchy horizon. In the latter case there are, of course, arbitrarily many possible extensions through the Cauchy horizon, and gtr declares itself unable to say which alternative to choose. Our predictions have to allow for all of them because classically there is simply no way to guess what might happen after encountering a Cauchy horizon.
It is natural (and desirable) to extend CPW spacetimes to colliding pp-waves, or even colliding axisymmetric pp-waves (which would include models of two steady laser beams bending each other due to their mutual gravitational attraction), or even coaxially colliding axisymmetric pp-waves. However, little progress appears to have been made since the pioneering work of Szekeres which founded the study of CPW many decades ago.
Notable electrovacuums not yet mentioned include the Mamjumdar "conformastat" nnevac and the Bertotti-Robinson nnnec--- which is, remarkably, isometric to the Cartesian product S^{1,1} x S^2. Similarly, the Nariai lambdavac is the Cartesian product S^{1,1} x H^2. These are the only two solutions which arise as direct products!
Notable vacuum solutions not yet mentioned include the Petrov vacuum, which has a four dimensional Lie algebra of Killing vector fields and is homogeneous and arguably the closest thing in gtr to a "plane symmetric gravitational field", but physically it is really nothing like Newton's plane symmetric gravitational field. (The Weyl vacuum arising from the Newtonian plane symmetric potential is not plane symmetric and is also rather unlike Newton's plane symmetric field, except in the weak-field limit when all candidates agree approximately. The Taub plane-symmetric vacuum, and the Kasner plane-symmetric vacuum, are also rather unlike a Newtonian plane-symmetric gravitational field.)
Among the perfect fluid solutions, few interesting rotating fluids are yet known, but all the static spherically symmetric perfect fluid solutions are known more or less explicitly in several formulations. Typically one must solve a nonlinear ODE for one dependent variable which then determines another, and these specify the solution, for example by giving both pressure and density as a function of the Schwarzschild radial coordinate r (a spherically symmetric function on the spacetime characerized by the locus r=r0 being a geometric two-sphere with area 4 pi r0^2). Visser and his students have discovered some particularly interesting internal symmetries of the governing ODEs which enables one to generate one or two parameter families of ssspf solutions from a "seed" ssspf solution. In particular, both physically and mathematically it is convenient to parameterize specific ssspf solutions by the central pressure and density values. Some but not all admit equations of state functionally relating pressure to density. Among the most interesting simple examples the Tolman IV ssspf is particularly noteworthy.
Other notable perfect fluid solutions include Kantowski-Sachs fluids, the Wahlquist rotating perfect fluid (Weyl tensor is Petrov type D, but cannot serve as an interior solution suitable for matching to a portion of Kerr exterior), Szekeres-Szafron fluids, Senovilla fluid, McVittie fluid ("interpolates" between Schwarzschild vacuum and FRW dust).
I should also mention radiation fluids (equation of state \rho = 3p) such as the Klein radiation fluid, and of course the FRW radiation fluids, which can be used to model the early universe "pre-recombination". Many authors discuss fluids with equation of state \rho=p, which can often be interpreted as portions of mcmsf solutions and IMO should otherwise be rejected as unphysical.
All the solutions mentioned so far are either vacuum solutions or have matter tensors corresponding to well understood fields (EM fields) or states of matter (perfect fluids). I could have mentioned mixed models containing charged dust, or two interpenetrating dust congruences, etc. (not nonsensical if one things of the dust particles crude models of stars which are not actually in physical contact with each other, but like all dust solutions plagued by the appearance of shell-crossing singularities in the stress tensor--- which howeverneed not be accompanied by curvature singularities in the same locus).
In addition to these, minimally coupled massless scalar field solutions are particularly easy to find, and they can be readily combined with EM fields or dust, etc., to form more elaborate models. Noteworthy mcmsf solutions include the Janis-Winacour mcmsf, the Roberts mcmsf (used to construct the Maeda wormholes), and the Ellis mcmsf (used to construct the Morris-Thorne wormhole).
Venturing outside gtr, some spacetimes are remarkable for occurring as solutions which can be compared in interesting ways with corresponding gtr solutions. Also, some spacetimes are vacuum solutions both to the EFE and to the field equations of other theories, such as the pp-waves.
George asked for references, and various textbooks/monographs do discuss many of these in detail. In particular, Stephani's textbook discusses pp-waves, Bertotti-Robinson nnevac, and Robinson-Trautman vacuums, among others. Islam, Introduction to Mathematical Cosmology, discusses the Senovilla fluid, Ellis-Madsen mcsmf, and some others. Several textbooks discuss Kasner dusts and the vacuum subclass (the Kasner vacuums). Others discuss the Goedel lambdadust.
Griffiths, Colliding Plane Waves in General Relativity, discusses the Khan-Penrose CPW, the Ferrari-Ibanez CPW (collision of two particular linearly polarized gravitational planew waves with aligned polarization, resulting in interaction zone locally isometric to Schwarzschild "shallow interior"), Chandrasekhar-Xanthopoulous CPW (collision of two particular linearly polarized gravitational plane waves, with nonaligned polarization, resulting in an interaction zone locally isometric to Kerr "shallow interior"), and the example of Griffiths (collision of two particular EM plane waves, resulting in an interaction zone locally isometric to Bertotti-Robinson non-null electrovacuum), and many other notable examples of CPW spacetimes.
In addition, several major review papers cited in the sticky BRS thread "Some Useful Links for SA/Ms" discuss in detail the Taub-Nut vacuum, Weyl vacuums, pp waves, and other important examples.
Of the classes of solutions extensively discussed by Plebanski and Krasinski, George forgot to mention the particularly important LTB family dust solutions (spherically symmetric time-varying dust solutions), which includes many interesting special cases. And of the classes discussed by Griffiths and Podolsky, George forgot to mention the particularly important family of colliding plane wave (CPW) spacetimes.
I certainly don't have nearly the comprehensive knowledge that you do, but I did have these solutions in mind when I made my original post. I purposely posted a truncated the list (but long enough enough to include the usual suspects covered in introductory GR course) because I am curious about what other posters will suggest.
Forgot to say: Schutz's textbook includes a discussion of the Heintzmann ssspf, another of the better known ssspf solutions. The review by Lake and the papers by Visser at al. on ssspf solutions mention some other interesting examples, including a rather recent solution of Martin.
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the notions of first and second order contact may help:
the tangent line to curve C at point P makes first order contact with C, and if P is an inflection point or if the path curvature vanishes there for any reason, the tangent makes second order contact
the tangent plane to euclidean surface S at point P makes first order contact with S, with quadratic deviations related to Gaussian curvature of S at P
This generalizes to higher dimensions, and the intrinsic curvature turns out not to depend upon the embedding, although this is not obvious from this approach.
In any case, the first order contact of tangent line with a curve C doesn't prevent the extrinsic curvature of C from varying, nor does the fact that the tangent space at any point P of any manifold M makes first order contact with M prevent the intrinsic or extrinsic curvature of M from varying.
A good reference is Berger's book on Riemannian geometry. See also the BRS on euclidean surfaces of revolution.
The surface of revolution described by the OP is very similar to an example in MTW. It has positive Gaussian curvature on 0 < r < 3^{-1/4}, negative Gaussian curvature on 3^{-1/4} < r < 1, and vanishes outside the unit disk; see the figures below. On the latitude circle r=3^{-1/4} \approx 0.7599 the tangent planes make second order contact with the surface of revolution, so the Gaussian curvature vanishes along this circle.
(To make uploadable images, I used the plot2d command of Maxima to plot the curves and then used ksnapshot (with capture mode "window under cursor") to export the Maxima figures as .png files. Ideally, SA/Ms would often provide figures this way.)
Figures:
the height z=\exp(-1/(1-r^2)) of the surface of revolution z=f(r) considered by the OP (zero outside the unit circle),
the Gaussian curvature on 0 < r < 1 (zero outside the unit circle); notice the curvature is negative on 0.76 < r < 1.
BRS: Why do first covariant derivatives of the metric tensor vanish?
Again re "Metric tensor of a non-homogeneous universe"
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andrewkirk (addressing bcrowell) said:
If I understand you correctly the covariant derivative of a tensor tells us whether or not the tensor is changing relative to the metric tensor. In other words it tells us the changes in the tensor "net of any changes attributable to the change in the metric tensor". So no matter how many different types of spaces we "sew" together, with all sorts of associated radical changes to the metric between different parts of the manifold, the covariant derivative of the metric will always be zero because it is "changes in the metric tensor net of any changes attributable to changes in the metric tensor", which will be zero by definition.
Think of a McLaurin series expanding the metric wrt any event on any Lorentzian four-manifold. To first order it should always look just like the metric tensor of E^{1,3}; curvature (which distinguishes between Lorentzian manifolds which are not locally isometric to each other) represents second order deviations from the metric tensor of flat spacetime.
The very same issue arises the same way for Riemannian manifolds; this has nothing to do with physics per se, but with the assumptions Riemann made in formulating the notion of a Riemannian (or Lorentzian) manifold! See the excellent discussion of Riemannian two-manifolds in Berger, A Panorama of Riemannian Geometry, a readable romp through a wonderful subject. (Note that many of the topics discussed by Berger in this book are special to Riemannian viz. Lorentzian geometry, but there is also considerable overlap.)
The identity component of a (nontrivial) topological group G is the connected component of the identity, which is always much larger than the trivial subgroup (consisting of the identity element e in G)! It is always a closed normal subgroup; see Cohn, Lie Groups, Theorem 2.4.1. (Since the mapping class group is a group and the identity map is its identity, this greatly generalizes the result desired by the OP.) In the case when G is a Lie group, this implies that it is a Lie subgroup of G. Example: SO(3) is the identity component of O(3). The identity component has the same dimension as G when G is a finite dimensional Lie group.
Note: in general topology, the path component of a point p in X need not be quite the same thing as the topological component of p. (See a good textbook on topology for the standard definitions of these notions.) But IIRC, the distinction doesn't much matter in this context.
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Let M be an n-manifold and N an n-dim. subspace of M, of possibly lower dimension than M , then M is knotted, or N is a knot in M if there are non-isotopic embeddings of N in M.
This definition would regard both an "unknotted" circle and the trefoil knot as knotted one-dimensional submanifolds of S^3, for example, which seems somewhat perverse. Surely one should say that "N is knottable in M" if at least two nonisotopic embeddings exist. Or some such terminology. Because the property of a submanifold being knotted in its parent is extrinsic, not intrinsic.
conversely, if f,g, are two non-isotopic embeddings of N in M, does it follow that H_n(M) is not trivial?
Consider N = S^1.
Re "Affine plane, block design"
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The website of Peter Cameron (author of numerous books on combinatorics, group theory, &c) includes at least two complete sets of lecture notes dealing with the geometries associated with classical groups, including their finite dimensional relatives
The unfinished (indeed, hardly even begun) BRS on the Rubik group is also devoted to finite geometries from the point of view of Klein's Erlangen Program.
According to Klein, each geometry on a "naked set" X is associated with a "generalized isometry" group G, consisting of those bijective transformations X \rightarrow X which preserve the geometrical structure. Subgroups of G correspond to "more rigid" alternative geometrical structures which can be placed on X. And conjugacy classes of subgroups enumerate the "geometrical concepts" of a geometry and display the interrelationships between these elements.
In particular, it helps to start with G=PGL(d+1,q), where q = p^n for some prime p, which is the "isometry group" for the d-dimensional projective geometry over field GF(q). This has a subgroup H=AGL(d,q) which is the "isometry group" for d-dimensional affine geometry over GF(q). (More precisely, G has two conjugacy classes of subgroups isomorphic to H, which turns out to be geometrically significant). H has further interesting subgroups defining even more rigid geometries, e.g. analogs of euclidean metrical geometry.
We can consider various actions by these groups and study the resulting lattice of stabilizers and its Galois dual, the lattice of fixsets. In the case of geometries which arise by "decorating" (adding structure to) projective geometry, it helps to start with the action on the points of projective space. Studying the stabilizer lattice we can identify certain fixsets which correspond to the k-flats. We can consider these as subsets and study the action on the k-flats, which provide additional orbits in the action on points, lines, 2-flats,... (d-1)-flats (or hyperflats).
When we restrict from G to H, some of these orbits break up into smaller orbits, and the stabilizer lattice becomes more complicated under the restricted action. That is, as a rule, more rigid geometries have more "geometrical concepts" than less rigid geometries. For example, in projective geometry all points are equivalent, and all hyperflats are equivalent, but in affine geometry, there are two kinds of points, ordinary points and ideal points, and also two kinds of hyperflats, ordinary hyperflats and a unique ideal hyperflat (the "hyperflat at infinity").
Studying the stabilizer lattice of the action of the basic geometrical elements (various kinds of k-flats for k=0...d-1) by H=AGL(d,q), we can identify a conjugacy class of stabilizers which corresponds to the parallel lines. The stabilizers in this class correspond to the sets of parallel lines, and also shows how each such set relates to various other geometrical concepts of affine geometry.
In general, because group theory provides so many powerful tools, as Galois himself recognized, it is very helpful to transform counting questions (and other questions) into the realm of the lattice of subgroups of some group, where such questions are usually easier to answer. This is one advantage of Klein's point of view, but there are many others--- I have long been intrigued by the way that simply writing down an action by a group leads naturally to the correct "geometrical concepts" and their interrelationships, and indeed to an information theory generalizing the highly successful information theory of Shannon and in some sense unifying it with classical Galois theory.
Examples of useful facts from the theory of groups include:
Given H a finite index subgroup of G, [G] is the size of the right coset space H\G, so if H is the stabilizer of some "geometric configuration" implicit in some action by G on some set X, there are [G] geometric motions of the given configuration.
The number of conjugates of H in G is [G:N_G(H)], where N_G(H) is the normalizer in G of H, so if H is the stabilizer of some "geometric configuration" implicit in some action by G on some set X, there are [G:N_G(H)] configurations which are geometrically equivalent to the given configuration.
H is normal in N_G(H), and the right coset space (also a factor group) H\N_G(H) is the internal symmetry group of the configuration. So in the chain H < N_G(H) < G, the right coset space N_G(H)\G corresponds to "external motions" while the right coset space (a group in its own right) H\N_G(H) corresponds to "internal motions".
andreass asked about AGL(2,2), but finite geometries over GF(2) are "geometrically atypical" in many ways (basically, because "lines have too few points"), so I suggest studying the stabilizer lattice of PGL(3,3) and its subgroup AGL(2,3) instead. Especially in conjunction with PGL(3,R) and AGL(2,R), the corresponding real projective and affine geometries.
In the figures below, you can see that the conjugacy class of the stabilizer of 3 parallel (ordinary) lines has 4 elements, so there are 4 sets of 3 parallel lines. Furthermore, each line is included in a unique such triple, and each triple contains three lines. Well, duh!, in this case, but we also see relations which might not be so obvious when we study more complicated geometries using Klein's approach. In the figure, notice that the class consisting of 36 C2 is the pointwise stabilizer of a line (which arises in the action on points) whereas the conjugacy class of 12 S3^2 is the setwise stabilizer of a line. And the class of the stabilizers of the 4 triples of parallel lines is intermediate between the pointwise and setwise stabilizers of a line--- which makes sense! Also, |AGL(2,3)| = 432, so there are 432/18 = 24 motions of each triple of parallel lines, consisting of translations by various amounts ("distance" is not a concept of affine geometry, so I am avoiding that word) in various directions, plus certain "reflections". Again, comparing with AGL(2,R) is helpful!
For actions by finite groups G on finite sets X, the logarithm of the indices [G] (i.e. the logs of the sizes of the right coset spaces H\G) behave just like Shannon entropies. For actions by finite dimensional Lie groups G on finite dimensional manifolds X, the dimensions of the right coset spaces H\G behave just like Shannon entropies. To each stabilizer H < G which arises in some action by G on some set X, there corresponds the right coset space H\G, or complexion, which measures the variety of motions of the "geometric element" or "geometric configuration" which is stabilized by H. So these entropies measure our uncertainty about which of the possible motions will be chosen in some "random process". The coset space formed from the intersection of two stabilizers measures our uncertainty about the joint motion of two configurations, and restricting from the action by G to the action by one of the stabilizers H on this joint coset space gives a conditional complexion, where the corresponding conditional entropy measures our uncertainty concerning a motion of the second configuration after we are told the chosen motion of the first configuration. Generally, in finite geometries, such motions are not entirely independent (due, if you like, to the failure of the Hilbert hotel phenomenom familiar from bijections on infinite sets), so this is actually significant information.
Furthermore, if \varphi: X \rightarrow Y is some G-hom (morphism in the category of G-sets, for a given group G; compare the category of R-modules, for a given ring R, for example), the stabilizer of preimages of a point \varphi(x) \in Y under the given action on Y is a subgroup of the stabilizer of x in X under the given action on X. Even better, it is a normal subgroup, so if we combine two transitive G-sets, one the G-homomorphic image of the other, into a single G-set, which means that we regard \varphi as an G-endomorphism mapping one orbit onto another orbit, then conditional complexion measuring our uncertainty about the motions preimages of any point y in the second orbit given the motions of y generalizes the notion of Galois group from classical Galois theory. Similarly for other G-endomorphisms. This is also related to the notion of cellular automata and certain shift spaces studied in symbolic dynamics.
Returning to the figure, we can see other notable stabilizer subgroups. For example, the class of 12 C3 consists of groups of shears which fix one line pointwise and fix the two parallel lines setwise. The class of 36 D6 corresponds to the 36 flags (ordinary point on an ordinary line), while the class of 54 D2 corresponds to the 54 intersections of a pair of ordinary lines.
The class of 9 GL(2,3) give the stabilizers of the 9 ordinary points, and choice of one of them corresponds to choice of origin, which then implies a restriction from affine transformations to linear transformations. In a linear geometry on the affine plane, one ordinary point is distinguished as "the origin", a concept which simply makes no sense in affine geometry.
Taking a wider view, there happen to be two conjugacy classes of subgroups isomorphic to AGL(2,3) in PGL(3,3), each consisting of 13 subgroups. These correspond to the stabilizers in the action on projective points and projective lines, so as you would expect from projective duality, the stabilizer-fixset lattice of these two actions look the same. Choice of a particular stabilizer in the action on lines corresponds to choosing one of the 13 lines of the projective plane over GF(3) as the ideal line, leaving 12 ordinary lines. This also chooses one of the equivalent "affine structures" which can be placed on the projective plane. The orbit of 13 points under PGL(3,3) breaks up under our chosen subgroup isomorphic to AGL(2,3) into 4 ideal points (the points lying on the ideal line) and 9 ordinary points.
Furthermore, there is a conjugacy class in PGL(3,3) of 117 subgroups isomorphic to GL(3). In projective geometry, these are the stabilizers of a configuration consisting of a line and a point off that line. Choosing one of them amounts to
choosing an ideal line (placing an affine structure on the projective plane)
designating a particular ordinary point as "the origin" (placing a linear structure on the just defined affine plane).
Again, this sketchily illustrates why we say that linear geometry is more rigid than affine geometry, which is in turn more rigid than projective geometry.
There are in all 46 conjugacy classes of subgroups of AGL(2,3), so in the action on points and lines, only a small fraction of the totality of subgroups appear as stabilizers. However, we can consider many derived actions, e.g. actions of subsets of various sizes, and in this way, by going back and forth between the geometry and the abstract structure of its isometry group, we can eventually identify each subgroup with some possibly subtle geometrical property or configuration, e.g. a labeled or colored configuration of some kind.
GAP4 can be very useful in exploring small finite geometries. If you want to explore,
PrimitiveGroup(9,7) gives the action by AGL(2,3) on the 9 ordinary points of the affine plane over GF(3)
TransitiveGroup(12,157) gives the action by AGL(2,3) on the 12 ordinary lines of the affine plane over GF(3)
and you can write a routine to combine these two orbits into an intransitive permutation group. (Make sure you use the correct generators in constructing the combined action--- IsomorphismGroups is the tool you need--- and of course you will need to reindex one of the orbits.)
The OP asked about block designs, a notion introduced by R. A. Fisher, who needed to design experiments which would efficiently explore all possible relationships between certain variables in agricultural experiments. However, the notion turns up in many places in combinatorics, and turns out to have many unexpected applications in science (and the original applications remains quite important, e.g. in medial research). There are indeed many relationships between finite geometries and block designs, and this has been a major topic of research in combinatorics for almost a century! Cameron is an expert on this subject so his website should be very helpful.
There are also many relationships between the classical groups PGL(d+1,F), AGL(d,F), etc., and the theories of linear representations, invariants, Lie algebras, reflection groups, regular polytopes, Schubert calculus, multiply-transitive groups, and finite simple groups. Many of the best of the expository series of John Baez, This Week in Mathematical Physics, were devoted to exploring one or another aspect of these relationships. One which is particularly relevant here is the q-calculus, in which one constructs q-analogs (for q = p^n as above) for binomial coefficients. The q-analog of Pascal's triangle then counts the number of k-flats in d-dimensional projective space over GF(q), for given q. This is then closely related to things like Bruhat partial order on Young diagrams, invariant tori, parabolic subgroups, Schubert calculus...
More generally, enumerative geometry can be approached via the theory of structors (certain functors, also called "combinatorial species"), and it turns out that the theory of structors is very closely related to the theory of finite permutation groups (which is in turn very closely related to the theory of actions by finite groups on finite sets). And it greatly generalizes the wonderful counting formula of Polya. There is a natural generalization to actions on infinite sets, the oligomorphic actions (see the website of Peter Cameron for more about these). And this is in turn closely related to model theory and various topics in mathematical logic. And to close the circle, notions from topology such as compactness turn up naturally here and play an important role. The unity of mathematics is a wondrous if sometimes bewildering thing!
It is intriguing that I seem to see more questions relating in some way to Kleinian geometry in the past six months at PF, and unfortunate that there seems to be little interest among the SA/Ms in exploring this wonderful topic, which happens to be a more serious interest of mine than general relativity--- I only yak about gtr so much because I happen to know that theory and people ask about it constantly at PF, and their naive questions so often require considerable sophistication to answer well, so inevitably I seem to keep getting sucked back into trying to help students learn more about gtr.
Figures: for the action of AGL(2,3) on the points and lines of the affine plane over GF(3) (orbits: 9 ordinary points and 12 ordinary lines; not shown: orbit of 4 ideal points and trivial orbit of 1 ideal line)
Stabilizer subgroups, up to conjugacy (inclusion runs upwards)
Fixsets, up to affine motions (inclusion runs downwards)
Notice the Galois duality of these two lattices, which generalizes the well known duality between subgroups and fixed fields from classical Galois theory.
BRS: I'd rather be yakking about Kleinian geometry!
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Such a project would be much too hard for an undergraduate. But looking towards graduate work, he should make sure to learn about symplectic integrators for Newtonian multibody simulations, and if his mathematical sophistication were up to integro-differential equations (which apparently is not the case), he can study the theory of the Vlasov equation in a Newtonian context.
The idea here is to model stellar clusters/galaxies as "dust, Newtonian style", using the probablistic methods of statistical mechanics. Try
C. C. Lee, "Dynamics of Self-Gravitating Systems: Structure of Galaxies", in
Studies in Applied Mathematics,
edited by A. H. Taub,
MAA Studies in Mathematics, 1971.
(A. H. Taub is the Taub in Taub-NUT vacuum.)
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I think Goldbeetle and Dark_knight90 need to be warned (I predict Goldbeetle will ignore the warning but others might not) that context is crucial in reading scientific papers, which is the most important reason why trying to learn gtr by following the history guarantees failure. One must master gtr as currently understood from modern textbooks, and read some good modern papers, before possibly reading a book like Pais's biography which does attempt to trace the historical development of gtr. But to really understand the history, one must study the leading textbooks a generation prior to Einstein 1916, and the most important scientific papers in the years around 1916, and one must understand the issues which concerned physicists of the time both in modern terms and in terms of how contemporaries understood or misunderstood them. Indeed, to really understand the history, one must also know something about the politics of the day (the early development of gtr occured, after all, in the context of a global war, then post-war chaos, then a global depression...). And one must come to know the personalities involved as closely as is possible after so many years. (Pais was personally acquainted with Einstein and knew another leading figure well, Niels Bohr.)
But I can't stress this enough: one must master the modern theory of gtr before (possibly) attempting any historical analysis. The productions of would-be historians of science who have written dreadful nonsense owing to failing to obey this rule speaks for itself. Indeed, rather incredibly, there are even historians who have overlooked the political context of the early development of gtr.
I really don't know where so many people seem to have obtained the false notion that one can avoid mathematics by studying the historical development of gtr. It's really sad that PF doesn't do more to try to prevent students or interested laypersons from going down this path, because nothing good can come of that.
I often vehemently disagree with Cleonis, but in his Post #10 in the Goldbeetle thread, he does make an important and valid point: the modern understanding of the Principle of Equivalence is that this is a simple consequence of the assumption that "spacetime" should be modeled by a Lorentzian manifold, and it simply amounts to saying that the tangent space to each event E is a real vector space equipped with E^{1,3} binary form, which is identified with the metric tensor at E.
I generally disagree with the historical interpretations of Cleonis, but in his Post #6 he again surprised me somewhat by drawing attention to a valid point about the interaction between Nordstrom and Einstein; from the modern point of view, N was quite correct to challenge AE to refine his overly vague understanding of the POE. In the early days, it was far from clear that gtr was above all a theory of gravitation, or even that the Riemann curature tensor defines "the gravitational field". AE wanted gtr to be more than a theory of gravitation; in a sense, the modern understanding has eventually come to a superficially similar point of view: roughly speaking, there are two phenomena common to all (classical) theories of fundamental physics, energy and gravitation; the metatheory of energy conversion/transport which applies to all theories of specific interactions is thermodynamics; the metatheory of gravity is gtr. In order to use thermodynamics, one adopts a model of "matter" (e.g. an equation of state for a gas); in order to use gtr, one writes down a Lagrangian, obtains a model of matter, specific non-graviational fields, "exotic matter", etc., finds the appropriate energy-momentum tensor, and attempts to find spacetime models in which the Einstein tensor matches the energy-momentum tensor and all the fields satisfy their field equations (use covariant derivatives!) on the resulting curved spacetime. This is the modern way of understanding the "universal" character Einstein sought, and it makes it plausible that there should be a deep relation between thermodynamics and gravitation.
Re
Code:
www.physicsforums.com/showthread.php?t=462327
My gosh... good illustration, sad to say, of what I meant when I said that the kooks are using Maple/Maxima too, so that there is an arms race. Clearly this poster is hopelessly confused by mathematical notation, has no knowledge of classical physics, much less relativistic physics, and probably has a language barrier too. And has no clue about CODE tags.
IMO, the only reasonable reply is a polite reformulation of this:
brutal version of possible reply said:
Your post is incoherent, but clearly shows that you lack the math/physics background to use the computer tools you are trying to use. You'll never be able to say anything useful/intelligible, so give up your interest in physics forthwith, or expect to be labeled a "cranky ignoramus" if you persist.
Re "Coordinate radius [r]"
Code:
www.physicsforums.com/showthread.php?t=462307
Another earnest newbie, who also suffers from a deficit of mathematical sophistication (and a language barrier), but maybe fixable if he/she can be persauded to study hard modern textbooks for several years.
It is my understanding that the coordinate radius [r] is defined in terms of the ‘reduced circumference’, i.e.
coordinate radius r = LaTeX Code: circumference/2 \\pi
As such, a number of texts seem to describe calculating [r]
I think PF should tell such posters
brutal version of possible reply said:
If you cannot be bothered to cite your sources you are too lazy to either learn or converse about mathematical physics. If you insist on keeping secret what textbooks you are reading, you will raise the suspicion that you are trying to pull a fast one on your professors. In any case, what you claimed your textbook says makes no sense.
Re "Tensor Rank of Stress Tensors"
Code:
www.physicsforums.com/showthread.php?t=462258
this poster is confusing simple two-forms (a subclass of antisymmetric second rank tensors) with symmetric second rank tensors. Happy to say that in this case I see hope that this student can overcome his difficulties.
The stress tensor is commonly given in terms of a rank two tensor - the tensor appears to be composed with the components of the force density vector over a given differential area, and *the normal vector of that differential area*.
I think I know what he is trying to say, and he is confused, yet shows some insight here. One possible response would be to suggest that if he learns to use frames and to compute frame components of tensors, his immediate confusion will go away and he'll understand stress tensors much better. In a coordinate basis, the two-dimensional area element does occur as a denominator in many general expressions for coordinate basis components for curvature quantitites, but this is irrelevant to physical understanding at the appropriate level of stress tensors.
the stress tensor should be a rank three tensor, with one rank for the force vector, and the other two for the differential area two-vector. When I set it up this way, I can transform the stress arbitrarily, and get results that make physical sense
This is wrong. Instead of the force acting on a particle, consider the acceleration as a function defined on the world line of that particle. This is simply the path curvature, which is a vectorial quantity defined along the world line, which is given by the covariant derivative of the tangent vector along itself. Technically, it is much easier to consider your world line just one integral curve of a timelike congruence and to work with the congruence.
Re "How to make two frames purely Galilean"
Code:
www.physicsforums.com/showthread.php?t=462154
the OP is evidently a fringer hoping to "prove" [sic] that Galilean relativity can replace str. Mathematically and geometrically, Galilean relativity can be understood as a sensical mathematics/geometry associated with a degenerate bilinear form E^{0,3} and a symmetric group E(0,3), and this group even arises naturally as the point-symmetry group associated with certain simple ODEs, as does the distinct symmetry group E(1,3), the Poincare group, which is associated an indefinite but nondegenerate bilinear form E^{1,3}. But Galilean and Minkowski geometries are completely different as Kleinian geometries, so their "curved manifold" elaborations (as Cartanian geometries) are also distinct. Thus, there is no hope of "showing" [sic] that they are mathematically or geometrically equivalent, and thus no hope of "showing" [sic] that they are "physically equivalent" [sic].
User:chinglu1998 is explicity cranky. User:grav-universe is IMO simply being coy and I have no doubt the underlying crankiness will soon become evident.
Re
Code:
www.physicsforums.com/showthread.php?t=461941
ShiroSato needs to give some context, e.g. by quoting from a cited book or paper. All one can say without context is that "degrees of freedom" is a somewhat archaic term for the dimension of some kind of parameter space. E.g. the Lorentz group has "six degrees of freedom" which can be understood as consisting of three independent rotations and three independent boosts. The Poinare group has "ten degrees of freedom", adding time translation and three spatial translations.
Re
Code:
www.physicsforums.com/showthread.php?t=461717
I think TrickyDicky is trying to ask: what does the contribution of the CMBR to the energy-momentum tensor look like? Answer: a superposition (using weak-field theory is appropriate here) of null dust terms associated with "waves" coming from all directions, so adding up to a very tiny radiation fluid term, I think. Checking this carefully would be a good exercise.
I note that above Kleinian geometry made another appearance, and reiterate that this is really more my thing anyway...
but in any case, it will be clear to anyone here, I think, that GOCE has nothing to do with "speed of gravity".
I sense a whiff of Chip on Shoulder, which raises the question of whether this user is using an ironically chosen handle at PF. It's always worth remembering that one reason to "write defensively" when posting in the public areas is that you never know when some putative newbie is planning to cherry pick responses to some "naive question" in order to, say, quote them in some anti-science website. A large number of religiously committed creationists do this quite a bit with regard to anything related to the standard hot Big Bang theory.
I advise one and all to avoid using "friendship" features at social networking sites, but any curious SA/M knows what to do if they want the answer to Edward's question: shoot me an encrypted PM. (See the BRS thread "PKI Cryptosystems for SA/Ms: A Tutorial", which should have been titled "Personal Cryptography: a Tutorial for SA/Ms", but too late to change it now.)
] is the size of the right coset space H\G, so if H is the stabilizer of some "geometric configuration" implicit in some action by G on some set X, there are [G
