BRS: Random Comments on Some Recent PF Threads

In summary, this conversation discuss the pros and cons of using a notation convention that is purely a matter of convention. The pros of using this convention are that it is easier to read Cayley and Schreier diagrams and that it is generally neater with regards to interface with category theory and other topics. Some cons are that it is harder to use with "functional notation" and that some operations become harder to notate.
  • #1
Chris Hillman
Science Advisor
2,355
10
Re
Code:
www.physicsforums.com/showthread.php?t=403487
Actually, there is a move back to writing multiplication left to right. And while it is not wrong that this purely a matter of convention and symmetric within a narrow context, in a wider context, this notational choice acquires very real consequences. IMO, a major stumbling block to developing certain fields in mathematics has always been that, due to historical accidents like a seemingly innocuous notational choice made in a narrow context, it is impossible to find a notation which "behaves nicely" with all other standard notations.

Some pros:
  • GAP uses this convention
  • much easier to read Cayley and Schreier diagrams (e.g. for combinatorial group theory) if use this
  • generally, neater interface with category theory and other topics
  • most operations no harder to notate
Some cons:
  • much harder to use with "functional notation"
  • some operations become harder to notate
Choose your poison!

Re
Code:
www.physicsforums.com/showthread.php?t=408525
ditto Lavinia, but even more emphatically! IMO, the principle reason why people start off wrong and get more and more confused when they try to learn, say, Lorentzian geometry, is that they haven't first studied curves in E^2, E^{1,1}, E^3, E^{1,2} and surfaces in E^3, E^{1,2}. And as just one example where classical surface theory continues to provide inspiration for topics of current interest, consider the classic problem of constructing uniform negative curvature surfaces in E^3, which has inspired many late 20th century developments related to the theory of solitons and is now being tied up with string theory and other physics stuff. See Rogers and Schief, Backlund and Darboux Transformations, Cambridge University Press.

petergreat said:
In addition, classical differential geometry lacks the techniques that are widely applied in theoretical physics, such as differential forms.

That is of course completely incorrect, and in fact the classical setting is a good place to explore differential forms. See the classic textbook by Flanders, Differential Forms with Applications to the Physical Sciences, which contains chapters on curve theory and surface theory.

Re
Code:
www.physicsforums.com/showthread.php?t=405899
This is a classic question closely related to a subject initiated by Erdos and Turan: the distribution of various properties (such as cycle decomposition, order) of randomly chosen elements of S_n, n large. A few hints can be found in section 14.4. of Bollobas, Random Graphs, 2nd edition, Cambridge University Press, 2001. There is a huge literature on these topics which is unfortunately hard to find on-line, but you can try
Code:
groupprops.subwiki.org/wiki/Probability_distribution_of_number_of_cycles_of_permutations
Years ago I argued (in Wikipedia policy discussion pages) that although Wikipedia's model has proven very successful at growing an alleged "encyclopedia", it is fundamentally flawed wrt reliability, which rather munges the whole point of making the alleged "encyclopedia". So how to design a wiki which can grow rapidly while maintaining high quality? I suggested that the best approach may be to building a "universal on-line encyclopedia" may be to continuously aggregate and render in uniform "wikiskin" articles drawn from thousands of specialist encyclopedic wikis ("wikispedias"?) which focus on specialized technical subjects, such as "group theory", authored by graduate students and faculty in those subjects, and edited by leading experts.

In the past few years, some projects have appeared which follow my advice about the neccesity of restricting article creation/modification to recognized subject matter experts (the other half, about the essential role of the "editor" in the true sense of that word, which wikipedia.org has largely succeeding in debasing by conflating it with "author"). One problem here is that the existing solution to authenticating identity/credentials on-line, the gpg "web of trust", is underutilized in academic circles, which is tragic because this is hampering grass roots efforts such as the projects I am describing. Some projects have even attempted to emulate traditional peer review (Citizendium), although I tend to feel that simply restricting authorship to known individuals with good credentials, and restricting editorship to recognized leaders, may be enough, at least initially, to ensure good growth coupled with good quality. Other projects, such as the subwiki.org "wikispedias", have focused on the fact that it is possible to manage ("edit" in the true sense of the word) a specialist wikipedia (e.g. by ensuring reasonably coherent notation/terminology, making wise choices for internal linkings, resolving any differences of opinion on how to present scientific controversies); compare the chaos at Wikipedia.

I hesitate to suggest linking to the subwiki.org wikispedias in the public areas, because like some other promising projects, they are currently open to anyone, and could quickly be ruined. Also, I don't want to overload their servers. But it's a very promising project and I hope that if we cautiously spread the word to serious people, eventually the universities/government will fund the project to allow for servers keeping pace with growth. (It would be tragedy of they vanished behind paywalls.)

I love the fact that graduate students can produce hilarious essays like
Code:
groupprops.subwiki.org/wiki/The_promise_of_freedom
But imagine the mess when the cranks discover that anyone can sign up and create/modify articles in this wiki, oh my! One of the nice things about keeping out the idgits is that authors can write with style, while still keeping within the bounds of terminological/notational conventions.

Given the importance in so many areas of mathematics of actions by large finite groups, and the properties of "generic" elements of large symmetric groups in particular, why is it so hard to find good information on-line? One might almost think there must some kind of suppression. And here's a surprise, maybe--- there is some kind of suppression! Can you think why? In a future BRS I may explain, maybe even tell you a few things citizens need to know, but are not "allowed" to know. Mathematical censorship can hurt you!

Re
Code:
www.physicsforums.com/showthread.php?t=406011
Lavinia should have said "all two-dimensional Riemannian manifolds" to avoid possible confusion. It saves time in the end to try to write out a bit more to avoid confusion.

Hmm...some more possible future candidates for SA: lavinia, shoehorn, Martin Rattigan? (shoehorn and I seem to disagree on the value of open source, but hopefully that is not a serious conflict!)

Re
Code:
www.physicsforums.com/showthread.php?t=405407
It's important to raise from specific to general when confronting an OP based upon multiple misconceptions. In this case, the most important misconception is that a black hole of mass m actively sucks stuff straight in like a powerful vacuum cleaner, rather than attract like any other object of mass m. In gtr, rougly speaking, the gravitational field itself gravitates, which turns out to mean that any object of mass m attracts other mass-energy just a bit more strongly than in Newtonian theory. This effect is neglible in must situations, so that a black hole of mass m interacts with other objects pretty much like any other object of mass m (think elliptical and hyperbolic trajectories as in Newtonian gravitation), unless the other object approaches it very closely. But because compact objects of mass m (neutron stars and black holes) are so much smaller than ordinary stars of mass m, another object can get much closer to a compact object and thus experience a stronger gravitational field, where the "extra attraction" becomes significant. That said, the answer the OP probably wanted is that, roughly speaking, in the range 2m < r < 6m, independent of mass m, the effects of gtr become highly significant. Further out, it becomes harder and harder to detect the difference.

Examples:
  • light bending near the limb of our Sun is a small but detectable effect, but near the limb of a neutron star (or near the "cross section" of a black hole, the "dark disk" astronomers are trying to detect for Sag A*) optical effects can be much more dramatic.
  • when two ordinary stars pass close by each other, their shape becomes deformed and one or both may even be pulled apart (tidal disruption); there are many factors involved here, not all directly involving gtr, but very roughly speaking, when an ordinary star happens to pass close by a compact object, parts of the ordinary star may encounter a strong gravitational field and thus are more likely to be pulled off; astronomers are studying examples of real stars which are apparently being disrupted by specific supermassive black holes
  • when two compact objects happen to have a close encounter, an interesting gtr phenomenon which can have a dramatic effect is "spin-spin" interaction, which can result in the two objects having a highly non-Newtonian interaction, as if they had been "kicked" during the close encounter.

seto6 said:
black hole can destroy them self when they interact whit each other, there are other possibility too like merge,,one gets kicked out of orbit

Black holes cannot be pulled apart by tidal disruption, although in a sense their horizons can be distorted during a close encounter. When two black holes collide, they merge. When they have a close encounter which does not result in a collision, their trajectories may behave in non-Newtonian ways as mentioned above. Astronomers are studying black holes which appear to have been kicked out of their parent galaxies by spin-spin effects (this can happen as the result of a close encounter, but also when two holes merge in such a way that the initial burst of strong gravitational radiation is highly asymmetric--- think action and reaction).

Re
Code:
www.physicsforums.com/showthread.php?t=404319
Oh my, passionflower arguing with Chalnoth, scratch flower off the list of potential future SA candidates.
buckethead said:
Let me restate that to say a body will radiate grav waves when and only if it either changes direction or linearly accelerates. So this would include an orbiting body as well as a linearly accelerating body.
This is not even wrong due to ambiguity of what buckethead means by "changes direction" and "accelerates".

In linearized gtr we can treat approximately the generatation of gravitational radiation by configurations of mass-energy and mass-energy currents (momentum and angular momentum). Then it turns out that the decisive role is played by the multipole moments of configuration as a function of time. The monopole moment gives the "Coulomb field" while the dipole moment can be removed by choosing an appropriate comoving coordinate system (this is possible because of the spin two character of gravitational radiation). The second time derivative of the mass quadrupole moment gives rise to the strongest gravitational radiation. Examples (using Newtonian language to suggest the intended picture):
  • two pointlike objects falling directly toward each other
  • two pointlike objects in circular orbits around their COM
  • an object rotating in a nonaxisymmetric manner (e.g. a rod rotating about an axis making nonzero angle with its central axis)
Nonexamples:
  • a spherically pulsating star
  • an axisymmetric disk rotating about its axis of symmetry
The next two strongest contributions are from the mass octupole moment and the current quadrupole moment. The latter is interesting because it can arise from Rossby waves in neutron stars, and may be detectable by LISA. Gtr makes rather precise predictions about various effects which should dominate as a young neutron star cools. Roughly speaking, in far field theory, when using a chart obeying the "Lorenz gauge condition", the angular momentum wrt "the center of mass" determines the Komar spin and the major frame-dragging/gravitomagnetism effects, but the current dipole is transformed away when using a chart obeying the stronger "transverse traceless condition".

An excellent review is
Schutz and Ricci
Gravitational Waves, Sources and Detectors
Code:
arXiv.org/abs/1005.4735

The study of tensor multipole moments, especially covariant definitions of these (the weak-field theory just described uses sensible-in-context but noncovariant notions of multipole moments), is highly developed and fascinating. Unfortunately, I know of no adequate review paper on-line.

Re
Code:
www.physicsforums.com/showthread.php?t=408367
oh good, Mentz114 already posted the necessary correction! :smile:

(Before you ask, I think that some years back, Mentz114 declined SA-ship, and I think shoehorn may also have done so. Even worse, I fear I might have been unhelpfully involved in at least one of those :sad:)

Re
Code:
www.physicsforums.com/showthread.php?t=408273
oldman said:
First consider a uniform spherical cloud of non-interacting test masses falling radially toward a central mass. As it falls the sphere will become distorted by tidal accelerations that change
inter-particle separations, into an ellipsoid of revolution whose axis is radial, as described and
illustrated by Roger Penrose in The Road to Reality,Section 17.5, p.396,397.
RP is describing the tidal deformation of a small initially spherical cloud of test particles by the Coulomb field due to a stationary massive object.

oldman said:
If instead of a cloud of test particles the sphere were a isotropic solid, it would be strained by tidal forces (to a degree depending on its proximity to the central mass) into an ellipsoid of revolution
Not that simple. A good exercise is to run the computation (see my old thread "What is the Theory of Elasticity?"

oldman said:
The internal stresses that develop are compressions perpendicular to the ellipsoid axis and tensions along this axis. It looks to me that the radial compressive tidal forces are very like (but opposite in direction) the centripetal forces that would make the solid rotate about its radial axis (say spin about this axis), and that the tensile forces are very like (but opposite in direction) the centripetal forces that would make the solid rotate about any axis perpendicular to the solid’s radial axis. I’ll take the liberty of labeling these tidal forces anti-spin and anti-rotation forces because that's what they look like to me.

Terrible terminology because this has nothing to do with gtr per se, and conflicts with terminology used for gtr effects.

This appears to be a murky reference to gravitational torque--- a small aspherical object can experience a torque as it moves through an ambient gravitational field, and thus change its orientation)--- which is best studied in a Newtonian context before tackling gtr. If the small object is spinning about some axis, this gets much more complicated even in Newtonian theory.

oldman said:
If the solid were to rotate with an appropriate angular velocity about an axis perpendicular to its radial ellipsoid-of-revolution axis, the centripetal accelerations generated by such rotation might exactly cancel the tensile tidal accelerations.

I think oldman may be trying to suggest that under some circumstances, the deformation of a spinning perfect fluid body or elastic solid due to "centrifugural forces" might briefly just cancel the tidal deformation due to an ambient gravitational field. If so, he should run some computations, in Newtonian gravitation.

oldman said:
As in the case of The Moon presenting the same side to us as it orbits the Earth?

Tidal coupling, not what oldman appears to think.

A possible reference is Murray and Dermott, Solar System Dynamics.
 
Last edited:
Physics news on Phys.org
  • #2
Some 9 June 2010 threads in relativity subforum

Re
Code:
www.physicsforums.com/showthread.php?t=408422
Passionflower said:
Can anyone point me to a good source using the Schwarzschild metric that gives Einsteins predicted result of the measurement in Principe?
Translation: where can one find a good discussion of the lightbending formula?

There are four classical solar system tests which involve formulas (gtr predictions) into which we can plug numbers suitable for various solar system situations:
  • gravitational red shift
  • light-bending
  • extra-Newtonian precession of quasi-Keplerian orbits
  • light delay
A particularly readable account may be found in D'Inverno's textbook.

The light bending formula says that when a massive object O passes near background stars, these will appear to be displaced radially outward from O on the celestial sphere by specific angles.

It is probably worth mentioning that the light-bending formula has been verified many times since the observations which Eddington made at Principe during the 1919 solar eclipse. See MTW Box 40.1 for nice figure from measurments made in Western Australian during the 1922 solar eclipse. Since 1968 it is more common to use radio telescopes, since then we don't need to wait for a solar eclipse. In fact, it is routine to verify that quasars not particularly near the Sun are displaced by just the right amount, as "seen" by radio telescopes.

Passionflower said:
some kind of physical distance?
...
then when does he do...to relate the physical distance to the Schwarzschild r?
...
where does he go from the physical distance to the Schwarzschild r coordinate?
...
don't we need to know the value of LaTeX Code: r_0 to get a physical result?
...
those r values are not physical radii.
...
the physical distance

Such language always reflects a major confusion concerning
  • the arbitrariness of coordinates
  • failure to recognize the multiplicity of operationally significant notions of "distance in the large"
in Lorentzian manifolds. However, Passionflower is struggling to ask a good question: if the definition of "radial coordinate" is arbitrary, how can we say that statements like "the tidal force decays like 1/r^3" make sense in gtr?

Let's back up a bit.

A coordinate x is simply a monotonic smooth function defined on some simply connected open neighborhood U in M, i.e. dx \neq 0 on U. If we have four coordinates such that [itex]dt \wedge dx \wedge dy \wedge dz \neq 0[/itex] on U, they form a "net" on U, and we have a coordinate chart on U. Needless to say, we can always find infinitely many charts on any given such U!

So what do we mean by "radial coordinate" in Lorentzian geometry? Well, suppose M happens to admit a three dimensional Lie subalgebra A of spacelike Killing vector fields isomorphic to so(3), such that the integral curves of each Killing vector field in A are circles. Then suitable hyperslices in M admit a "spherically symmetric" family of two-spheres which are orbits under the SO(3) action given by A. In general, neither the hyperslices nor the family of nested spheres need contain a "central point", however. Nonetheless, the surface area of these nested spheres can be used to define a radial coordinate via A = 4 pi r^2. Even simpler, the Gaussian curvature of the spheres is 1/r^2, so r is simply the reciprocal square root of the Gaussian curvature. (Don't confuse this with components of the Riemann curvature tensor of spacetime, or the components of the Riemann curvature of a spatial hyperslice, or with components of the extrinsic curvature tensor of some slice!)

But in the Schwarzschild vacuum exterior region we have the principle outgoing null geodesic congruence, in which spherical wavefronts expand at the speed of light. The optical expansion scalar of this congruence is 1/r, just like in flat spacetime with the usual radial coordinate. Thus, r has immediate physical and geometrical significance.

However, it is not the same as the "proper radial distance" obtained by integrating
[tex]
ds = \frac{dr}{\sqrt{1-2m/r}}
[/tex]
along radial rays. I guess that Passionflower wants to do something like this:
  • choose a static perfect fluid solution, with zero pressure surface at some sphere with area A = 4pi a^2.
  • construct a static stellar model by matching across r=a to a region of Schwarzschild exterior with appropriate mass parameter (which will be the Komar mass)
  • hyperslices t=t0 will now allow [itex]0 < r < \infty[/itex], where r is defined on both interior and exterior by the nested spheres having Gaussian curvature 1/r^2, so define a "proper distance radial coordinate" by integrating from r=0 outwards.
Then the point is to convince him that this new radial coordinate will not differ very much from r.

One simple and popular static spherically symmetric perfect fluid solution is the so-called Tolman IV fluid, which has the line element
[tex]
ds^2 = -C^2 \, (1+r^2/A^2) \; dt^2
\; + \; \frac{1}{1+r^2/A^2} \; \frac{1+ 2 \, r^2/A^2}{1 -r^2/B^2} \; dr^2
\; + \; r^2 \; d\Omega^2,
\; \; 0 < r < r_s
[/tex]
where A, B are positive constants which, as it turns out, can be expressed in more physical terms as
[tex]
A = \frac{1}{\sqrt{4\pi/3 \; (\varepsilon_0 + 3 p_0)}}, \; \;
B = \frac{1}{\sqrt{4\pi/3 \; (\varepsilon_0 - 3 p_0)}}
[/tex]
where [itex]\varepsilon_0, \; p_0[/itex] are the central mass-energy density and central pressure, respectively. Notice that r is the usual Schwarzschild radial coordinate; the zero pressure surface is r=r_s where
[tex]
r_s^2 = \frac{B^2-A^2}{3} =
\frac{3}{2 \pi} \; \frac{p_0}{\epsilon_0^2 - 9 p_0^2}
[/tex]
Notice that A,B have the dimensions of length while C is dimensionless.

Now we want to match this ssspf solution across r = r_s to
[tex]
ds^2 = -(1-2m/r) \; dt^2 + \frac{dr^2}{1-2m/r} + r^2 \; d\Omega^2,
\; \; r_s < r < \infty
[/tex]
for a suitable choice of the mass m. The conditions we need to satisfy are that the induced metric tensor and the expansion tensor associated with the E^{1,2} slices r=r_0 are continuous across r=r_s (see Poisson, A Relativist's Toolkit for these matching conditions). It turns out that to obtain a matching we require C= A/B and then
[tex]
m^2 = \frac{(B^2-A^2)^3}{27 \, B^4}
= \frac{1}{\pi/6 \, (\varepsilon_0 - 3 p_0)}
\; \left( \frac{p_0}{\varepsilon_0 + 3 p_0} \right)^3
[/tex]
So
[tex]
B^2 = \frac{r_s^3}{m}, \; \;
A^2 = r_s^2 \; (r_s / m - 3)
[/tex]
gives an "exterior" parameterization of our stellar model.

Each reasonable ssspf has both "exterior" and "central" parameterizations, but not all obey a specific equation of state giving pressure as a function of density. But it turns out that the Tolman IV fluid does obey a certain (strange!) equation of state, but not which is physically well motivated, so this model shouldn't be taken too seriously. In particular, I would be surprised if it can be matched very closely to our best guesses about the density and pressure as functions of r in our Sun, but since these functions are not very well known, I don't think that should prevent us from trying to plug in values for the mass and surface radius and solve for the central pressure and density, and hoping that these turn out to be not too far from reasonable guesses for the central pressure and density of our Sun! (I've done this for the more interesting case of neutron stars, and found that the Tolman IV fluid does surprisingly well, particularly considering that more complicated models often do worse.)

The values we need are
[tex]
m= 1.48 \times 10^5 {\rm cm},
\; \; r_s = 6.96 \times 10^{10} {\rm cm}
[/tex]
So, plugging into the formulae found above, we should try... oh gosh, numerical instability. The expressions found above show that the Tolman IV ssspf is adapted to compact objects for which r_s is not too much larger than 2m, since otherwise A is almost equal to B, which makes m, r_s dependent on very small changes in either A or B.

Well, A = 15.15 km, B = 23.29 km gives m = 2.09 km, r_s = 10.21 km, vice m=2.01 km, r_s = 11.91 km for a sophisticated numerical model of a neutron star of about 1.3 solar masses.

Re
Code:
www.physicsforums.com/showthread.php?t=407909
the so-called "centrifugal reversal" has been well known for decades. This is a real effect, as one can verify by cranking through the math, but IMO "centrifugal reversal" is a very bad way of thinking about it. There are number of other effects like this which have resulted in much ink spilled over the question (nonproblem, perhaps?) of how to "intuitively interpret" them. I tend to feel that proposed "intuitive interpretations" are never consistent or a reliable guide in all situations, and thus do more harm than good, but obviously, some authors have very strong views in favor of their preferred "intuitive interpretation".

Re
Code:
www.physicsforums.com/showthread.php?t=405847
pbk137 said:
In GR it is often desireable to find coordinates in which the metric is flat, at least locally.
pbk is probably trying to ask about something like a Gaussian normal chart, in which the connection is made to vanish (by construction) at one event, but of course the curvature will generally not vanish.

pbk137 said:
My question is, given a general, non-diagonal metric tensor, how to diagonalize it,

Levels of structure to the fore! At the level of a single tangent space, one can try to find a new linear basis (not neccessarily an ON frame) which diagonalizes a symmetric tensor. But careful! The well known theorem about diagonalizing symmetric matrices in linear algebra refers to inner product spaces in which the inner product is positive definite! There is an analogous but more complicated theorem for indefinite metrics which is discussed in some textbooks; it turns out that not all symmetric matrices allow Lorentzian ON bases in which they are diagonalized. Thus, not every energy-momentum tensor can be "diagonalized at a given event".

pbk137 said:
and how this process relates to a new coordinate system in which the metric is locally flat.

Warning: in gtr, "locally flat" means a region in which the curvature tensor vanishes identically.

The OP may be trying to ask about various "nice" coordinate charts which can be defined for Lorentzian manifolds. For example, one can show that one can always find a "warped product" chart in which the metric tensor looks like this:
[tex]
\left[ \begin{array}{cc|cc}
0 & \exp(f) & m & n \\
\exp(f) & 0 & p & q \\
\hline
m & p & \exp(g) & 0 \\
n & q & 0 & \exp(g)
\end{array} \right]
[/tex]
That is, the line element is
[tex]
ds^2 = 2 \, \exp(f) du \, dv \; + \; \exp(g) (dx^2+dy^2)
\; + \; 2 \, du \; ( m \, dx + n \, dy) \; + \; 2 \, dv \; ( p \, dx + q \, dy)
[/tex]
where [itex]\partial_u, \; \partial_v[/itex] are null and [itex]\partial_x, \; \partial_y[/itex] are spacelike. There are six metric functions of four variables here; since a generic chart will involve ten metric functions of four variables, we have removed four of the ten degrees of freedom by changing to special coordinates. However, we still have a great deal of remaining freedom, so these coordinates are certainly not unique. Notice that the idea of these coordinates is to take advantage of the facts that
  • any E^2 manifold can be given isothermal coordinates in which the line element has form [itex]ds^2 = \exp(g(x,y)) \; (dx^2+dy^2)[/itex]
  • any E^{1,1} manifold can be given isothermal coordinates in which the line element has form [itex]ds^2 = 2 \, \exp(f(u,v)) \; du \, dv[/itex]

Another kind of nice chart which looks a bit messy but which has many advantages was introduced by Bondi, and generalizes outgoing Eddington coordinates from Schwarzschild vacuum to arbitrary AF spacetimes. One variant I like particularly, which is similar to a chart used by Gu, comes with the frame field
[tex]
\begin{array}{rcl}
\vec{e}_1 & = & \partial_u + h \, \partial_r - k \, \partial_x - j \, \partial_y \\
\vec{e}_2 & = & -\partial_u + (1-h) \, \partial_r + k \, \partial_x + j \, \partial_y \\
\vec{e}_3 & = & \frac{1+x^2+y^2}{2r} \exp(-b/2) \;
\left( \exp(-f/2) \, \partial_x + g \, \exp(f/2) \, \partial_y \right) \\
\vec{e}_4 & = & \frac{1+x^2+y^2}{2r} \exp(-b/2+f/2) \partial_y
\end{array}
[/tex]
Here x, y should be thought of as stereographic coordinates for a family of nested compact surfaces which are "wrinkled spheres", and the six metric functions (of all four variables) have the following geometric/physical interpretations:
  • b controls the expansion scalar of the outgoing principle null geodesic congruence given by [itex]\partial_r[/itex]; specifically, the expansion scalar turns out to be [itex]1/r + b_r/2[/itex],
  • f controls the plus mode of type N radiation; specifically, the Riemann tensor (components taken wrt the frame) has terms [itex]f_{uu}/r[/itex] representing type N plus polarized radiation,
  • g controls the cross mode; specifically, the Riemann tensor (components taken wrt the frame) has terms [itex]g_{uu}/r[/itex] representing type N cross polarized radiation,
  • f, g together control the shear scalar of the outgoing principle null geodesic congruence
  • h (with b,f,g) controls the acceleration of "static" observers and the "squared norm" of dr
  • k controls the "inner product" of dr with dx
  • j controls the "inner product" of dr with dy
  • together k,j influence the expansion tensor of the "static" observers
This chart is useful for obtaining the "news functions" (two, one for each mode of type N radiation) in the Bondi radiation formalism. All the Bondi type charts are a bit tricky in that one starts at scri^+ and works backwards and inwards, and typically the charts break down as one approaches r=0 because the integral curves of the principle null geodesic congruence begin to intersect. See the discussion of Bondi radiation formalism in D'Inverno's text.
 
Last edited:
  • #3
Of course I have not intentionally "contributed" to Schiller's website!

Re
Code:
www.physicsforums.com/showthread.php?t=409201
JustinLevy said:
I noticed Christoph Schiller's textbook, called "Motion Mountain", lists some people from this forum as contributors:
Code:
www.motionmountain.eu/help.html
For example Jonathan Scott, Chris Hillman, and probably others. Also are included quite a few respectable physicists names I recognize.

Specifically, Schiller says (in his website):
Schiller said:
Important material was provided by...Chris Hillman...plus a number of people who chose to remain unnamed.

I have certainly not knowingly "contributed material" to Schiller's project, and between us, from conversations about a decade old which I only vaguely recall, I was not the only one who unsuccessfuly requested that Schiller rephrase the sentence in question to avoid suggesting that all these people are active collaborators with him.

Between us,I reported the post asking that my name be deleted (and suggesting that Scott's name also be deleted). It seems clear that Justin Levy's intention was good, but I think (and I as I recall, the others I discussed this with thought) that the best way of handling such possibly unintentional misrepresentations is to keep quiet and hope that nobody makes it into a big issue :rolleyes: (Barbra Streisand effect, eh?)

So to repeat, I am trying to keep a low profile on the web and desire that my name not be mentioned in the public areas at PF. TIA.

Re
Code:
www.physicsforums.com/showthread.php?t=409265
to study the physical experience of an observer radially falling into a Schwarzschild hole, one should use the ingoing Lemaitre frame field attached to the world line C of one ingoing Lemaitre observer. Ultralocal effects such as tidal tensor are easily computed. Local (as in local neighborhood!) effects such as optical appearance is trickier. Ideally one would compute analytically the unique null geodesic congruence consisting of all null geodesics which terminate on C. In practice its easier to find them numerically. As you would expect, there are strong lensing effects as the infalling observer nears r=3m, which differ from those observed by a static observer hovering outside the horizon (which I have described in detail in past posts). As the infalling observer continues to fall, other effects occur, which I hope to describe in a future BRS post.
 
Last edited:
  • #4
Re my Post #3, between me and the SA/Ms, I did PM JustinLevy explaining my view that the best response to Schiller's implications is to avoid making a big deal of it, and he agreed to delete his post, so that's solved (unless some other PF poster makes a point of bringing it up again).

In the SA forum thread, "Teaching while Debunking"
Code:
www.physicsforums.com/showthread.php?t=407907
I suggested responding to incorrect physics fringe claims by writing a correction written for non-cranky lurkers, addressing first and foremost the most fundamental misconception in sight. I listed some common misconceptions which often underlie incorrect beliefs on the part of physics fringe proponents, and argued that one of the most fundamental is failure to recognize that all physics discussion is theory-dependent. Unfortunately, most cranks and many laypersons/newbies/students share the incorrect impression that "theoretical assumptions" are somehow strange or suspect, when of course you need to make assumptions to even start talking.

For example, in
Code:
www.physicsforums.com/showthread.php?t=410780

Jonathan Scott said:
I'm sure that gravitational changes propagate at c, and the rate of loss of energy of the Hulse binary pulsar, matching the predictions of gravitational wave theory, is one of the strong pieces of evidence for that. However, I think it's a bit misleading to simply state that the speed of gravity has been experimentally verified, in that as far as I know it has not been directly measured in a way which doesn't rely on other theoretical assumptions. In particular Kopeikin and Fomalont claimed to have measured the speed of gravity from effects on the apparent position of Jupiter, but others reckon that their logic was effectively circular because of the theoretical assumptions they used in the calculation.

The implication that Scott believes that making "theoretical assumptions" makes a claim suspect is more subtle in this post than in some of his other posts, but I think it's clearly there. A selection of points one can try to make to posts like this might include:
  • Every discussion in physics is theory-dependent.
  • The biggest problem with many naive discussions of [itex]c_{\rm grav}, \, c_{\rm EM}[/itex] is that such discussion has little meaning if one hasn't first formulated a self-consistent theory with two fundamental speeds, which is quite hard,
  • There's a simple picture one can draw in E^{1,1} showing why it is not easy, for purely mathematical reasons, to have more than one fundamental speed (investigate "grav-radiation cones" wrt the "EM-radiation" Lorentzian metric and the "light cones" wrt the "grav-radiation" Lorentzian metric versa),
  • There is no experimental/observational reason to hypothesize two fundamental speeds, especially in view of the explanatory power of theories like gtr which assume just one fundamental speed,
  • Kopeikin's analysis was flawed, but this hardly implies that the gtr textbooks must be all wet,
  • Scott is corrrect that it has not yet been directly confirmed that gravitational and EM radiation travel at the same speed, in fact, it has not yet been directly confirmed that gravitational waves even exist,
  • The continued observations of orbital decay rates in various binary pulsars exactly matches the gtr predictions; since in gtr (which has so much other support) this decay is due to energy carried off by gravitational radiation from the binary system, this is strong indirect evidence that gravitational radiation exists and carries off energy at the rate predicted by gtr,
  • the excitement about the expected advent gw-astronomy is based in decreasing order on
    • the possibility of "surprises" as we first glimpse a totally unexplored world of information about extreme distant events carried by a kind of radiation never before directly observed,
    • "hearing" events which we think occur but have been obscured to all EM telescopes for fundamental physics reasons
    • directly confirming properties of gravitational radiation (massless radiation, spin-two transverse) as predicted by gtr but not some of its competitors (not that anyone really expects anything else),
    • directly confirming (e.g. by comparing EM and GW observations of a new supernova) that gravitational and EM radiation travel at the same fundamental speed (not that anyone really expects anything else).
 
  • #5
Re
Code:
www.physicsforums.com/showthread.php?t=411825
The Weyl family of vacuum solutions consists of all the static axisymmetric vacuum solutions, and they are all known explicitly, and correspond (not one-one!) to ordinary harmonic functions on R^3. The canonical chart is
[tex]
ds^2 = -\exp(2u) \, dt^2 + \exp(2v-2u) \, (dz^2+dr^2) + \exp(-2u) \, r^2 \, d\phi^2
[/tex]
where u,v are functions on z,r only and
[tex]
u_{zz} + u_{rr} + \frac{u_r}{r} = 0
[/tex]
i.e. u is an axisymmetric harmonic function independent of time (harmonic both wrt the background metric and wrt the metric given by the line element for the Weyl vacuum solution itself).

The Weyl vacuum solution corresponding to the Newtonian potential of a point
[itex]u = m/\sqrt{z^2+r^2}[/itex] is actually the Chazy-Curzon vacuum which is not spherically symmetric!

Passionflower is asking about the solution corresponding to the Newtonian potential a static uniform rod of length 2L, where the Newtonian potential of a static rod is--- well, see the BRS thread "The Weyl vacuums" for details and discussion
Code:
www.physicsforums.com/showthread.php?t=378662
Note in particular that the Weyl canonical chart for an asymptotically flat Weyl vacuum only covers an exterior region; these AF Weyl vacuums model the gravitational field of isolated static axisymmetric configurations of matter and should be thought of as matched across some E^{1,2} world sheet to a matter-filled solution. Some can be interpreted as models of scenarios such as a pair of charged black holes accelerating under an ambient electrical and/or gravitational field, and these admit extensions to nonstatic past and future regions as well as a second exterior sheet.

Passionflower said:
Does the Schwarzschild solution refer to a removed rod or to a removed point mass?

The special case L=m turns out to be the Schwarzschild vacuum solution. I can give the required transformation if anyone cares.

Re
Code:
www.physicsforums.com/showthread.php?t=409241
I lack energy to try to figure out if anyone already gave the correct answer, but FWIW, the easiest way to determine both the orbital velocity (written in the usual Schwarzschild exterior chart) and the physical experience of an observer in a stable circular orbit (these observers are called Hagihara observers since they were first studied by Hagihara in his three volume celestial mechanics textbook) goes like this:

Start with the static frame field
[tex]
\begin{array}{rcl}
\vec{e}_1 & = & \frac{1}{\sqrt{1-2m/r}} \, \partial_t \\
\vec{e}_2 & = & \sqrt{1-2m/r} \, \partial_r \\
\vec{e}_3 & =& \frac{1}{r} \, \partial_\theta \\
\vec{e}_4 & = & \frac{1}{r \, \sin(\theta)} \, \partial_\phi
\end{array}
[/tex]
written in the standard Schwarzschild exterior chart, with line element
[tex]
ds^2 = -(1-2m/r) \, dt^2 + \frac{dr^2}{1-2m/r} + r^2 \, d\Omega^2
[/tex]
Now boost this frame field, at each event, in the [itex]\vec{e}_4[/itex] direction, by an undetermined boost parameter ("rapidity") depending only on r. Compute the acceleration vector of the timelike unit vector field [itex]\vec{f}_1[/itex] in the boosted frame and set this to zero in the equatorial plane. This gives a differential equation which one solves to find the desired Hagihara frame field. There are two solutions, one for counterclockwise and the other for clockwise orbits; I'll use this one:
[tex]
\begin{array}{rcl}
\vec{f}_1 & = & \frac{1}{\sqrt{1-3m/r}} \; \left( \partial_t
- \frac{\sqrt{m/r}}{r \, \sin(\theta)} \, \partial_\phi \right) \\
\vec{f}_2 & = & \vec{e}_2 \\
\vec{f}_3 & =& \vec{e}_3 \\
\vec{f}_4 & = & \frac{1}{\sqrt{1-3m/r}} \; \left(
- \, \frac{\sqrt{m/r}}{\sqrt{1-2m/r}} \, \partial_t \, + \,
\frac{\sqrt{1-2m/r}}{r \, \sin(\theta)} \, \partial_\phi \right)
\end{array}
[/tex]
This is inertial in the equatorial plane where
[tex]
\vec{f}_1 & = & \frac{1}{\sqrt{1-3m/r}} \; \left( \partial_t
- \sqrt{m/r^3} \, \partial_\phi \right)
[/tex]
so that (denoting the proper time of the Hagihara observer under study by s):
[tex]
\frac{d\phi}{dt} = -\sqrt{m/r^3}, \; \;
\frac{dt}{ds} = \frac{1}{\sqrt{1-3m/r}}, \; \;
\frac{d\phi}{ds} = -\frac{\sqrt{m/r^3}}{\sqrt{1-3m/r}}
[/tex]
Notice that [itex]d\phi/dt[/itex] is precisely the Kepler value; this is the orbital angular velocity as measured by a "very distant" static observer. But we should hesitate to say that [itex]d\phi/ds[/itex] is the orbital angular velocity measured by the Hagihara observer himself, because his world line belongs to a timelike congruence with nonzero vorticity, i.e. admitting no orthgonal spatial hyperslice, so that it is tricky to determine when he "comes back to his original spatial position", at least when trying to analyze the situation in terms of the physical experience of the Hagihara observer rather than a distant static observer. To avoid having to concoct a precise operational definition of "orbital period" as determined by an observer in a circular orbit (the problem also arises for the Langevin observers in Minkowski vacuum), we can say that the orbital period as measured by a distant static observer agrees with the Kepler value, but the elapsed time between two events on the world line of the Hagihara observer, as measured by his ideal clock, is related to the elapsed time measured by the ideal clock of the distant observer by dt/ds, with the necessary comparision carried out via appropriate null geodesics between the world lines of these two observers.

On the equatorial plane, the expansion tensor of our Hagihara congruence (components taken wrt the frame) is
[tex]
{H\left[\vec{f}_1\right]}_{ab} = \frac{3}{4} \, \sqrt{\frac{m}{r^3}} \;
\frac{1-2m/r}{1-3m/r} \;
\left[ \begin{array}{ccc}
0 & 0 & 1 \\
0 & 0 & 0 \\
1 & 0 & 0
\end{array} \right]
[/tex]
which shows the expected "pure shear" (off diagonal term only). That is, closer observers are moving faster about their circular orbit, so temporarily neighboring Hagihara observers "slide past each other" in the equatorial plane, just as we would expect from Keplerian intuition. The vorticity vector (on the equatorial plane) is
[tex]
\vec{\Omega}\left[\vec{f}_1\right] = \frac{1}{4} \, \sqrt{\frac{m}{r^3}} \;
\frac{1-6m/r}{1-3m/r} \; \vec{e}_3
[/tex]
which shows that the world lines are twisting about each other in a manner aligned with the frame vector pointing out of the equatorial plane and orthogonal to the radius.

The electroriemann tensor of our Hagihara observer (components wrt the Hagihara frame) is
[tex]
{E\left[\vec{f}_1\right]}_{ab} = \frac{m}{r^3} \;
\left[ \begin{array}{ccc}
-2 \, \frac{1-3m/2/r}{1-3m/r} & 0 & 0 \\
0 & \frac{1}{1-3m/r} & 0 \\
0 & 0 & 1
\end{array} \right]
= \frac{m}{r^3} \;
\left[ \begin{array}{ccc}
-2 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right]
\; + \; \frac{m^2}{r^4} \;
\left[ \begin{array}{ccc}
-3 & 0 & 0 \\
0 & 3 & 0 \\
0 & 0 & 0
\end{array} \right]
\; + \; O\left( \frac{1}{r^5} \right)
[/tex]
and the magnetoriemann tensor is
[tex]
{B\left[\vec{f}_1\right]}_{ab} = 3 \, \sqrt{\frac{m^3}{r^7}} \;
\left[ \begin{array}{ccc}
0 & \frac{\sqrt{1-2m/r}}{1-3m/r} & 0 \\
\frac{\sqrt{1-2m/r}}{1-3m/r} & 0 & 0 \\
0 & & 0 & 0
\end{array} \right]
= 3 \; \sqrt{ \frac{m^3}{r^7}} \;
\left[ \begin{array}{ccc}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 0
\end{array} \right]
\; + \; 6 \; \sqrt{ \frac{m^5}{r^9}} \;
\left[ \begin{array}{ccc}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 0
\end{array} \right]
\; + \; O\left( \frac{1}{r^5} \right)
[/tex]
So the tidal accelerations are augmented transverse to the direction of motion (in a circular orbit) of our Hagihara observer. (The fact that the tidal accelerations of Lemaitre observers agree with those measured by static observers is something of an anomaly, as MTW stress.) The nonzero components of the magnetoriemann tensor show that gyroscopes carried by our Hagihara observer are tugged by tiny forces (real forces, measurable in principle) depending on the orientation of their spin axis. This is a truly new feature of gtr viz. Newtonian gravitation. Recall that for static (noninertial) observers and for Lemaitre observers (inertial and radially infalling "from rest at r=infty"), the magnetoriemann tensor is zero. The fact that it is nonzero for our Hagihara observer is analogous to how in Maxwell's theory of EM, some observers can measure a magnetic field while others, in a different state of motion, do not. In EM, this can happen when one of the principle Lorentz invariants of the EM field tensor vanishes (see Landau & Lifschitz):
[tex]
F_{ab} {{}^\ast \!F}^{ab} = 0
[/tex]
In gtr, the analogous phenomenon can occur when one of the principle Lorentz invariants of the Riemann tensor vanishes:
[tex]
R_{abcd} \; {{}^\ast\!R}^{abcd} = 0
[/tex]
In contrast, for the Kerr vacuum, this invariant is nonzero, indicating the presence of "intrinsic magnetogravitism" (see Ciufolini & Wheeler, Gravitation and Inertia).

Note that the frame field we derived is only inertial on the equatorial plane; in planes parallel to the equatorial plane we have noninertial circular orbits. Note too that our frame field blows up on r=3m. With further work, it can be shown that the innermost stable circular orbit is located at r=6m, so the Hagihara frame field is actually defined only on [itex]6m < r < \infty[/itex]. Recall that the vorticity vanishes exactly on the inner edge of this domain.

Note too that this frame field is inertial but spinning; that is, the Fermi derivatives wrt [itex]\vec{f}_1[/itex] are not all zero. Specifically, the frame vectors are spinning wrt gyroscopes carried by our Hagihara observer (in the equatorial plane) about the [itex]\vec{f}_2[/itex] axis. By introducing a suitable rotation at each event by an undetermined rotation rate depending only on r, and demanding that the Fermi derivatives of the new frame vanish, we obtain a "despun" Hagihara frame which is nonspinning inertial. Then the components of the electroriemann and magnetoriemann tensor reflect the spinning, and we also obtain an exact expression for the Lense-Thirring precession of our Hagihara observer.

For those of you who have installed Maxima, here is a Ctensor file you can run in batch mode:
Code:
/* 
Schwarzschild vacuum; Schwarzschild chart; Hagihara coframe (spinning!)

*/
load(ctensor);
cframe_flag: true;
ratchristof: true;
ctrgsimp: true;
/* define the dimension */
dim: 4;
/* list the coordinates */
ct_coords: [t,r,theta,phi];
/* define background metric */
lfg: ident(4);
lfg[1,1]: -1;
/* Declare the dependent and independent variables */
constant(m);
/* Define the coframe covectors */
/* only need enter the nonzero components */
fri: zeromatrix(4,4);
fri[1,1]: -(1-2*m/r)/sqrt(1-3*m/r);
fri[1,4]: -sqrt(m/r)/sqrt(1-3*m/r)*r*sin(theta);
fri[2,2]:  1/sqrt(1-2*m/r);
fri[3,3]:  r;
fri[4,1]:  sqrt(m/r)*sqrt(1-2*m/r)/sqrt(1-3*m/r);
fri[4,4]:  sqrt(1-2*m/r)/sqrt(1-3*m/r)*r*sin(theta);
/* setup the spacetime definition */
cmetric();
/* compute a matrix whose rows give frame vectors */
fr: radcan(factor(expand(fr)));
/* metric tensor g_(ab) */
lg: radcan(factor(expand(lg)));
/* compute g^(ab) */
ug: radcan(factor(expand(trigsimp(invert(lg)))));
christof(false);
/* Compute fully covariant Riemann components R_(mijk) = riem[i,k,j,m] */
lriemann(false);
/* Compute electroriemann tensor */
EX: matrix(
[
factor(radcan(expand(lriem[2,2,1,1]))),
factor(radcan(expand(lriem[2,3,1,1]))),
factor(radcan(expand(lriem[2,4,1,1])))
],
[
factor(radcan(expand(lriem[3,2,1,1]))),
factor(radcan(expand(lriem[3,3,1,1]))),
factor(radcan(expand(lriem[3,4,1,1])))
],
[
factor(radcan(expand(lriem[4,2,1,1]))),
factor(radcan(expand(lriem[4,3,1,1]))),
factor(radcan(expand(lriem[4,4,1,1])))
]);
/* Compute magnetoriemann tensor */
BX: matrix(
[
factor(radcan(expand(lriem[2,4,3,1]))),
factor(radcan(expand(lriem[2,2,4,1]))),
factor(radcan(expand(lriem[2,3,2,1])))
],
[
factor(radcan(expand(lriem[3,4,3,1]))),
factor(radcan(expand(lriem[3,2,4,1]))),
factor(radcan(expand(lriem[3,3,2,1])))
],
[
factor(radcan(expand(lriem[4,4,3,1]))),
factor(radcan(expand(lriem[4,2,4,1]))),
factor(radcan(expand(lriem[4,3,2,1])))
]);
/* Compute R^(mijk) */
uriemann(false);
/* Compute Ricci componets R_(jk) */
ricci(false);
/* 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);
matrix(
[
factor(radcan(expand(ein[1,1]))),
factor(radcan(expand(ein[1,2]))),
factor(radcan(expand(ein[1,3]))),
factor(radcan(expand(ein[1,4])))
],
[
factor(radcan(expand(ein[2,1]))),
factor(radcan(expand(ein[2,2]))),
factor(radcan(expand(ein[2,3]))),
factor(radcan(expand(ein[2,4])))
],
[
factor(radcan(expand(ein[3,1]))),
factor(radcan(expand(ein[3,2]))),
factor(radcan(expand(ein[3,3]))),
factor(radcan(expand(ein[3,4])))
],
[
factor(radcan(expand(ein[4,1]))),
factor(radcan(expand(ein[4,2]))),
factor(radcan(expand(ein[4,3]))),
factor(radcan(expand(ein[4,4])))
]
);

Re
We assume a kind of gravitation force which propagates at light speed.
That doesn't lead to a self-consistent theory, which is actually what the poster is showing (see MTW).

One can modify the question by asking "assuming gtr, are binary systems stable?" The answer is that they are not stationary and in fact the two objects slowly spiral in towards each other due to the emission of gravitational radiation from the binary system. This effect is very small except in binaries in which at least one member is a compact object like a neutron star or black hole, and is neglible even over billions of years in our Solar system.

Re
Code:
www.physicsforums.com/showthread.php?t=410780
Jonathan Scott said:
The important point is that the strength of the experimental result is constrained by the strength of the theoretical assumptions on which it is based.
I think this claim is profoundly incorrect; putting the cart before the horse, as it were. IMO it is much more appropriate to say
  • all experiments must be interpreted in terms of at least one theory (two is better than one, and the more the merrier),
  • all theories incorporate motivating imagery and theoretical assumptions, which may appear more or less plausible in the context of contemporary physics generally,
  • not all theories are equally plausible in light of currently available evidence,
  • just to emphasize: the last two points are very much time dependent, and science is constanly "churning" and evolving new theoretical ideas/proposals, developing new theories, and synthesizing revised "big pictures".

Code:
www.physicsforums.com/showthread.php?p=2772885#post2772885
Jonathan Scott said:
But if you want to directly measure the speed of gravity, as a test of GR (as Kopeikin and Fomalont claim to have done), you can't rely on GR theory for your calculations, as GR already assumes it is c.
There he goes again. In various recent PF threads (and various past ones too), posters have cited the possibility that someday LIGO/VIRGO may observe a (distinctive!) burst of gravitational radiation from a supernova, which should arrive at the same time as EM radiation providing visual evidence of the supernova. That would be a direct confirmation that gravitational and EM radiation share the same fundamental speed, c=1. This assumption is incorporated into gtr and most other relativistic classical field theories of gravitation, but the interpretation of the hypothetical scenario just described as a direct confirmation that there is a shared fundamental speed does not depend upon this assumption.

Re
Code:
www.physicsforums.com/showthread.php?t=402170
kseto6 said:
there is dark matter because through gravitational lensing [in the Bullet cluster]..we see things made bigger but we are unable to see what is causing it
Actually this mostly misses the point:
  • observations of lensing interpreted via gtr give estimates not only of the total mass but also the distribution of mass in the foreground galaxy, and in the Bullet cluster, it turns out that we can deduce the existence of a halo of massy stuff surrounding the visible galaxy,
  • in either Keplerian dynamics or gtr, the (mostly) "flat rotation curve" observed for stars as a function of radius as they orbit the COM of the galaxy, we deduce the existence of a halo of matter surrounding the visible galaxy.

Re
Code:
www.physicsforums.com/showthread.php?t=405332
this thread is a good example of a common misconception which overlooks the possibility that mathematically speaking we can have two (or more!) different ways of thinking about the same phenomenon, and then it is a matter of taste which one to use; in many cases, it makes sense to make this choice on a case by case basis, according to which point of view is simpler or more fruitful in a given context. See also the "Teaching while Debunking" thread in the SA subforum.

Re
Code:
www.physicsforums.com/showthread.php?p=2772698#post2772698
the most important point to make is that this thought experiment is impossible in either Newtonian gravitation or gtr because it violates conservation of mass, which both these theories insist upon (with some subtleties in the case of gtr). However, one can imagine a closely related thought experiment in which the mass distribution of the Sun suddenly changes, due to hypothetical nongravitational physics which causes an aspherical explosion or something like that. This changes the roughly spherically symmetric distribution of mass in the Sun into a highly asymmetrical distribution of mass in The Object(s) Formerly Known as the Sun, and creates gravitational radiation which propagatees outward at the speed of light, so Earth's orbit is affected at the same time that the visual news that the optical appearance of the Sun has drastically changed arrives by EM radiation.

As for the Earth-Moon system, these two objects are gravitationally bound so both in Newtonian gravitation and gtr can be thought of in first approximation as an isolated system if one uses coordinates with origin at its COM and doesn't look outside the boundaries of the system. So the orbit of the Moon around the Earth won't change very much if described wrt the COM of the Earth-Moon system.

Re
Code:
www.physicsforums.com/showthread.php?t=411959
the most important misconception here, IMO, is failure to recognize that at the relevant scales/curvatures, Newtonian gravitation and gtr give pretty much the same result: if one uses these theories, one finds that there must be matter we can't see optically in some kind of halo surrounding the visible galaxy. This prediction has been checked independently via gravitational lensing (in the case of the gtr prediction) and so far it seems to hold up quite well. Which leaves open the question of just what the inferred Dark Matter might be.
 
Last edited:
  • #6
Re my comments on the "orbital velocity" thread (stable circular orbits in exterior of Schwarzschild vacuum) in the previous post, Mentz114 posted a Maxima script which uses Ctensor() with the coordinate basis flag set (i.e. uses coordinate basis rather than frame field, but the same coordinate chart that I used) to compute thegives the same result by the direct method of solving the geodesic equations; see
Code:
www.physicsforums.com/showpost.php?p=2775189&postcount=117
Mentz114 set [itex]\dot{\phi}=0n \; \dot{r}=0[/itex] and solved for [itex]d\theta/dt = \pm \sqrt{m/r^3}[/itex] whereas I set [itex]\theta=\pi/2[/itex] and ensured [itex]\dot{r}=0[/itex] and obtained [itex]d\phi/dt = \pm \sqrt{m/r^3}[/itex], where as usual I put c=G=1. So same result.

It's gratifying to see that PF regulars are beginning to take advantage of Ctensor(), which is Maxima's primitive but still useful package for working with certain tensorial objects in specific manifolds. (The package Itensor() does index gymastics on unspecified manifolds.) See the BRS thread "Using Maxima for gtr computations"
Code:
www.physicsforums.com/showthread.php?t=378991
 
Last edited:
  • #7
Re
Code:
www.physicsforums.com/showthread.php?t=415710
passionflower said:
one can only calculate a physical radius
I've only seen one other PF user employ this term, which is nonsensical. (Except for Stephen Crothers, who briefly posted in PF several times some years ago.) This raises the disturbing question: is passionflower a sock for that PF user? Or (even worse) a newbie infected with some of his wrong ideas?

Much of the discussion in the thread seems to involve an attempt to construct a physically/geometrically meaningful notion of "distance in the large". But even in a specific curved Riemannian or Lorentzian manifold, there is never a unique geometrically significant notion of "distance in the large". This is a fundamental point which is stressed in textbooks on Riemannian geometry, where many interesting theorems are known which depend upon positive definiteness, so they don't hold in Lorentzian geometry. But while the tools required to study "geodesics/distance/etc in the large" are different in Riemannian and Lorentzian geometry, many qualitative features are similar, in particular the inherent nonunicity.

Reframing this for gtr: even in a specific spacetime model such as the Schwarzschild vacuum (a Lorentzian manifold), there is never a unique physically significant notion of "distance in the large". Depending upon the details of how you define/measure a given notion of "distance in the large", you will obtain distinct notions which give different results when applied to the same scenario.

passionflower said:
[in an ingoing Painleve chart] the spatial section is not flat and thus the r coordinate does not relate to the circumference as it does in flat space.

Actually, the spatial hyperslices are locally isometric to E^3, so geometrically flat, and the r coordinate (as a function on M) is that same as the r coordinate in the Schwarzschild chart--- which Crothers and his fans prefer to call something like "Droste-Weyl-Hilbert" chart, although it is contained in an appendix to Schwarzschild's 1916 paper introducing his vacuum solution.

To be specific, the Schwarzschild chart (valid on the right exterior region) is
[tex]
\begin{array}{rcl}
ds^2 & = & -(1-2m/r) \; dt^2 + \frac{dr^2}{1-2m/r} + r^2 \; d\Omega^2 \\
&& 2m < r < \infty
\end{array}
[/tex]
To transform this to the ingoing Painleve chart (valid on the same region, and extensible to a larger region including the future interior region), we must transform the time coordinate, not the radial coordinate:
[tex]
d\tau = dt + \frac{\sqrt{2m/r} \; dt}{1-2m/r}
[/tex]
which gives
[tex]
\begin{array}{rcl}
ds^2 & = & -(1-2m/r) \; d\tau^2 - 2 \; \sqrt{2m/r} \; d\tau \, dr + dr^2 + r^2 \, d\Omega^2 \\
&& 0 < r < \infty
\end{array}
[/tex]
Here, on any spatial hyperslice [itex]\tau = \tau_0[/itex], we have [itex]d\tau = 0[/itex] so that the line element reduces to
[tex]
ds^2 = dr^2 + r^2 \, d\Omega^2
[/tex]
which is the line element of E^3 written in polar spherical chart.

Please see
Code:
www.physicsforums.com/showpost.php?p=2723459&postcount=5
for some pictures of the regions covered by these charts, plus some hyperslices from the Painleve chart. See also the website of Andrew Hamilton for his tutorial on coordinate charts in the Schwarzschild vacuum; note that to construct the Painleve chart, you can imagine pulling down the t=constant slices (keeping constant proper time intervals along vertical segments between t=t1 and t=t2, but letting the segments slide past one another) so that you straighten out the hyperslices orthogonal to the infalling Lemaitre observers. This is in fact how I rediscovered the Painleve chart before I knew any integral calculus, by using trignometry to derive the relationship [itex]d\tau = dt + \frac{\sqrt{2m/r}}{1-2m/r}[/itex] which is all one needs to find the line element written in Painleve coordinates. Then I noticed that the hyperslices [itex]\tau = \tau_0[/itex] are locally flat, which was my very first independent observation in gtr (although Painleve had noticed this somewhat earlier, in 1921!)

The meaning of the coordinate r is plain from the shared [itex]r^2 \, d\Omega^2[/itex] term in the line elements as written in the two charts. In addition -1/r is the expansion scalar of the principle ingoing null geodesic congruence.

A coordinate y on M is simply a "monotonic" function on some region in M, ie. [itex]dy \neq 0[/itex]. If, in a four manifold M, the wedge product [itex] du \wedge dv \wedge dp \wedge dq \neq 0 [/itex] on some region U in M, then the four coordinates [itex]u, \, v, \, p, \, q[/itex] form a local coordinate chart on U, and some nonzero scalar multiple of this four-form will be the volume form (which is of course a coordinate free notion).

When I say that the same radial coordinate appears in both the Schwarzschild and Painleve charts, I mean that there is a function r on M which is used as a coordinate in both charts. And the level surfaces of this coordinate are the surfaces r=r_0 which have meaning independent of coordinate chart (note that given a transformation to another chart you can repress these surfaces in the new chart).

In this sense, because a monotonic function can be written in any local coordinate chart (local in the sense of "local neighborhood"!), it is a coordinate free notion, so in this sense, "nice" coordinates can have coordinate free significance. Such is the case for r; it has at least three coordinate-free characterizations which uniquely define it using its geometrical properties.

Mentz114 said:
An interesting difference between GP and Schwarzschild coords is that a Minkowski observer in Schwarzschild coords has zero expansion ( ie the ball of coffee grounds does not change volume) but in GP coords the expansion is negative, so it gets smaller with decreasing r. Neither oberver experiences any shear velocities.

Several mistakes here, the first rather serious:
  • the expansion scalar of a timelike congruence (the one in the Raychauduri equation as described e.g. in Baez & Bunn's exposition "The Meaning of the Einstein Field Equation") is a coordinate-free notion, so it does not depend upon the chart used! (expansion scalar: 1/3 the trace of the expansion tensor of the congruence, which is a three-dimensional tensor defined on spatial hyperplane elements orthogonal to the congruence),
  • the property of being a geodesic congruence (or not) is also coordinate free,
  • his description refers to two distinct timelike congruences in the Schwarzschild vacuum, one nongeodesic (the static observers) and one geodesic (Mentz114 probably had in mind the LeMaitre observers who fall in "from rest at spatial infinity")

nutgeb said:
In Schwarzschild coordinates, an object plunging radially toward the massive origin at exactly escape velocity also measures the local space to be spatially flat.

Wrong way around; the outgoing Painleve chart is based on inertial observers who start from the past strong spacelike singularity, transverse the past interior region and emerge in finite proper time though the past ("white hole") horizon into the right exterior region, and asympototically approach "rest at spatial infinity". The ingoing Painleve chart is based on inertial observers who asymptotically start from "rest at spatial infinity", fall radially inwards and eventually pass through the future ("black hole") horizon into the future interior region and after finite proper time encounter the future strong spacelike singularity.

nutgeb said:
Taylor & Wheeler refer to this as the "rain frame" metric in their textbook "Exploring Black Holes."

Painleve-Gullstrand coordinates generalize this same effect to the coordinate chart as a whole. It has been referred to as the "river" model because it can be interpreted as if space itself is flowing toward the massive origin at exactly the escape velocity at each coordinate location.

This is badly garbled; the "rain frame" (bad name) is the ingoing Painleve chart.

Don't confuse frame fields with coordinate charts.

But consider a rod A>-----<B free falling in such a way. I think that you are saying that both the A and the B end will 'see' the space as flat, right? If so, how do we explain the tidal effects, as without it the rod ought to behave as a Born-rigid rod.

At second glance, it may appear hard to even define what one means by "a rod which maintains constant length as it falls". But at third glance, it may occur to use the expansion tensor of a congruence formed by the world lines of bits of matter in the rod, or better yet, a smallish cube of material. If it vanishes, each bit of matter is maintaining constant distance wrt its neighbors. But due to the multiplicity of operationally significant notions of "distance in the large", this cannot be true "in the large" for every physically reasonable notion of defining/measuring "distance in the large". The simplest such method is radar distance and this is in fact a good case to examine in detail--- in past PF posts, I and Pervect did just that!

Since the expansion tensor of the Painleve congruence is nonzero, this congruence clearly cannot be used to describe material in falling "rigid rods" or "rigid plates". And we know from the discussions of Pauli, Ehrenfest, Einstein and Born that "rigid motion" is inadmissable in any event, even in flat spacetime. Although with great caution some elements of our intuition can be resurrected in special circumstances.

In the scenario in which we imagine a rigid rod falling into a Schwarzschild hole, an appropriate model would be to identify the center of mass of the rod (much less massive than the hole) with a Painleve geodesic; then the world lines of other bits of material in the rod will diverge from neighboring Painleve geodesics, as required to keep the expansion tensor zero in the rod. That means there are tensile stresses in the rod which prevent the rod from behaving as a collection of freely falling bits of matter--- to behave like a collection of freely falling test particles, the rod would have to elongate (assuming its falling in oriented along a radius) as dictated by the appropriate component of the tidal tensor.

"Spatial curvature" is something of a red herring here; the physical effects experienced by a family of ideal observers or bits of matter inside some object are controlled by spacetime curvature (the gravitational field) plus any nongravitational fields and (in a phenomonological model of matter) the material properties (e.g. elastic tensor in a linear elasticity model) of any bodies present.

Re
Code:
www.physicsforums.com/showthread.php?t=415937
I'm afraid to even try to read the thing, but bravo, Dalespam! I'm glad someone's on the case :-/

Buried somewhere in there I am sure, even without reading the thing, are at least two very common misconceptions concerning the coordinate-free definition of frequency shifts and the coordinate-free definition of acceleration on the part of the posters arguing with Dalespam, which may not yet have been recognized. The correct definitions are:

  • frequency shifts always involve a pair of world lines and a family of null geodesics connecting these two world lines; then by comparing proper time intervals (as measured along the two world lines) between pairs of null geodesics, one can define/compute a frequency shift for an ideal signal (imagine a monochromatic laser signal emitted from one world line which is artfully directed to reach the second world line),
  • the acceleration vector of a world line is just the path curvature vector of the curve, the covariant derivative of the tangent vector taken along itself as per MTW's discussion). Path curvature has geometric units of 1/length and is completely distinct from spacetime curvature components, which have geometric units of 1/area, the same units as energy density, mass density, and pressure. Which is necessary for the EFE to even make sense.
 
Last edited:
  • #8
Chris Hillman said:
nutgeb said:
In Schwarzschild coordinates, an object plunging radially toward the massive origin at exactly escape velocity also measures the local space to be spatially flat.
Wrong way around;[...] The ingoing Painleve chart is based on inertial observers who asymptotically start from "rest at spatial infinity", fall radially inwards and eventually pass through the future ("black hole") horizon into the future interior region and after finite proper time encounter the future strong spacelike singularity.
I think nutgeb is correctly referring to the ingoing chart. The objects are moving inwards at escape velocity - maybe he should have written "escape speed" or "negative escape velocity".
Chris Hillman said:
In the scenario in which we imagine a rigid rod falling into a Schwarzschild hole, an appropriate model would be to identify the center of mass of the rod (much less massive than the hole) with a Painleve geodesic; then the world lines of other bits of material in the rod will diverge from neighboring Painleve geodesics, as required to keep the expansion tensor zero in the rod. That means there are tensile stresses in the rod which prevent the rod from behaving as a collection of freely falling bits of matter--- to behave like a collection of freely falling test particles, the rod would have to elongate (assuming its falling in oriented along a radius) as dictated by the appropriate component of the tidal tensor.
Yeah, but nutgeb asks to separate the effects of "local spatial curvature" from "the normal tidal stresses and stresses introduced by borne rigidity effects." From experience, I'd say it's better to let the thread die than trying to answer. Maybe others think they are up to the task?
 
  • #9
Hi Ich,

I must confess to being exhausted and depressed, so I lack energy to read/write as carefully as I usually try to do.

Re nutgeb's comment: I agree that the outgoing Lemaitre observers, who start in the past interior, transit the past horizon ("white hole horizon"), emerge into the right exterior region, and radially move outwards from the hole, asympotically coasting to a halt as they approach spatial infinity, correspond to observers who are moving just fast enough to avoid falling back in, so a fair definition of "escape velocity" in this context. The ingoing Lemaitre observers are falling in radially with velocity vector with same magnitude but oppositely directed (in the exterior, as measured by static observers). That's all I was trying to say. If I am still missing some deeper point, that must be due to exhaustion.

maybe he should have written "escape speed" or "negative escape velocity"

I really must be exhausted, just noticed that you already expressed (better) what I just tried to say!

Re
Code:
www.physicsforums.com/showthread.php?t=415710

Mentz114 gives correct definition of electroriemann tensor wrt U but there is terminological issue which deserves mention: most authors seem to use "tidal tensor" for electroweyl tensor (defined same way with Weyl tensor in place of Riemann tensor), which is not the same in nonvacuum. I happen to think there is good reason to think of electroriemann as tidal tensor rather than electroweyl in electrovacuum spacetimes, however. But Mentz114 uses poorly chosen notation since his T_(ab) could be confused with stress-energy tensor! The components he gives are correct frame field components
[tex]
m/r^3 \; \operatorname{diag}(-2,1,1)
[/tex]
but he has labeled them as if coordinate basis components, which is also confusing. What he says about the trace is correct. Then in a later post he says he got he signs wrong in the components! This could be due to his changing some notational convention without noticing, but I am certainly too exhausted to try to guess how that might have happened. I can say he is using Maxima's Ctensor plus some Maxima scripts he has written, and there are some serious and tricky issues with converting the non-standard conventions of Ctensor to standard notations--- which pretty much means "the notation of MTW". (And some books also use nonstandard notations.)

Full disclosure: I have been trying to tutor Lut (Mentz114) for some time via PM and while I think he's learned a lot, and I may bear some blame (poor teaching?) for any remaining confusions... sigh... and maybe for his "cocky" tone which might be somewhat misplaced since he has not made entirely correct statements in that thread. (I am planning to PM him when I can find energy to write diplomatically.)

Mentz114 said:
The problem with an extended body is that only some of it will be on geodesics,

Correct and a good insight (he certainly didn't get that from me since I haven't yet PM'd him about the thread).

and the acceleration of different parts wrt to the rest have to found by calculating the geodesic deviations.

Uh oh... geodesic deviation refers to geodesics, i.e. world lines of particles which are not experiencing any acceleration!

Mentz114 said:
It could be that every Minkowski observer in the Schwarzschild spacetime sees that tidal tensor, I'm not sure. The dust cloud will stretch along the r direction and shrink in the orthogonal directions while preserving its volume.

Is it possible that he is using "Minkowski observer" to mean "inertial observer"? If so, both static and radially infalling observers do find the same tidal tensor components (wrt an appropriate nonspinning frame field), but (for example) Hagihara observers (stable circular orbits in the exterior) measure different ones (wrt a nonspinning frame field), so it is not true even for inertial observers that they all measure the same components.

Re
Code:
www.physicsforums.com/showthread.php?t=415937&page=7
Altabeh said:
I don't believe that we have such thing as "proper acceleration" in GR and all I hit on the internet about the kind of definition you use of this quantity in GR was the Wiki's article where the book or reference from which the material has been taken is unknown; leading me to doubt the validity of this writing.

(Wail) What is PF to do with those who never crack a book but rely on Wikipedia for their "authority"?

[EDIT: oh, I see, he says he did crack some books, but apparently he didn't read them very carefully...]

Of course there is a notion of proper acceleration vector for a timelike curves: the path curvature vector! In practice one is often interested in a congruence of timelike curves and then we can apply the machinery of the kinematic decomposition explained in Hawking & Ellis, the book by Poisson, and many modern gtr textbooks.

Altabeh said:
I have wormed partially through Schutz, Letctures on GR by Papapetrou, D'inverno, Weinberg, the first part of Wald, MTW and recently David McMahon's Demystified Relativity but neither of them defines such thing and actually there isn't even a hint at a way one can deal with a generalized "proper" 4-acceleration in GR.

Those books (except maybe for the last which I have not seen) use the standard definition of the acceleration vector of a conguence of timelike curves with unit tangent vector field U, namely [itex]\nabla_{\vec{U}} \vec{U}[/itex]. If I were not exhausted I'd give page references. For a single differentiable timelike curve, this reduces to the path curvature vector.

Regarding the definition of "local flatness", this is a standard term (no doubt used by Poisson, Schutz, and other authors in places--- if I were not exhausted I'd check and give page refs): a spacetime M is locally flat in a region U if the Riemann tensor vanishes on U. Similarly, M is locally Ricci flat on U if the Ricci tensor vanishes on U.

Note that these authors correctly use "locally" to indicate "local neighborhood" as in topological manifolds. "Region" or "domain" or "local neighborhood" is usually understood as a simply connected open set.

Altabeh said:
In the Eddington-Finkelstein form of the Schwarzschild metric, the singularity of metric disappears which suggests that you're no longer seeing any singularity by doing a coordinate transformation. Now a degenerate metric has been found out to be non-degenerate but what about its physics!?

Needless to say, Altabeh is confusing an inessential coordinate singularity in the Schwarzschild chart--- which is not a geometric or physical feature of the spacetime!* --- with a geometric singularity (e.g. a curvature singularity). This confusion was always common and becomes more so when posters consistently fail to say what kind of "singularity" they are talking about and no-one bothers to object! But I know from experience that constantly objecting to such lazy thinking/writing (which in my experience always causes further confusion as people start to talk past one another) is utterly exhausting and in fact promotes SA/M burnout.

*Although, confusingly, in this spacetime it happens to occur at r=2m, where a certain vorticity-free Killing vector field becomes null (it is timelike on the exterior, where it expresses time translation symmetry or "static" property, and spacelike on the interior, where the spacetime is no longer static but dynamic).

Altabeh said:
So when we introduce Kruskal coordinates, one can behold a completely strange physical feature (still an open problem)

I sense another PF regular is about to start promoting the fundamental error of Crothers :rolleyes:

I reported some of Altabeh's worst misstatements and suggested that some Mentor should put that thread out of its misery before I read your comments, so it seems we are thinking the same way regarding the fate of that thread. Many of the regulars (kev, JS, Altabeh) seem to be talking past one another (and rejecting correction of various of their misstatements by you, Dalespam, and George Jones), so locking it seems appropriate IMO.
 
Last edited:
  • #10
Chris Hillman said:
a spacetime M is locally flat in a region U if the Riemann tensor vanishes on U.

This what I had in mind in (should have written "region" or "subset" instead of space)

https://www.physicsforums.com/showthread.php?p=2805278#post2805278
https://www.physicsforums.com/showthread.php?p=2805339#post2805339,
Chris Hillman said:
Regarding the definition of "local flatness", this is a standard term (no doubt used by Poisson, Schutz, and other authors in places--- if I were not exhausted I'd check and give page refs): a spacetime M is locally flat in a region U if the Riemann tensor vanishes on U. ...

Note that these authors correctly use "locally" to indicate "local neighborhood" as in topological manifolds. "Region" or "domain" or "local neighborhood" is usually understood as a simply connected open set.

Unfortunately, this is not how Poisson and Schutz use the term local "local flatness".
Poisson said:
1.6 Local Flatness
For any given point [itex]P[/itex] in spacetime, it is always possible to find a coordinate system [itex]x^\alpha'[/itex] such that

[tex]g_{\alpha' \beta'} \left(P\right) = \eta__{\alpha' \beta'}\\[/tex]
[tex]\Gamma^\alpha' {}_{\beta' \gamma'} \left(P\right) = 0[/tex]

... The physical intrepretation of the local flatness theorem is that freely falling observers see no effect of gravity in their immediate vicinity ...
 
  • #11
George is right; I should have checked my memory before posting

George Jones said:
Unfortunately, this is not how Poisson and Schutz use the term local "local flatness".

Oh nooo... you are right. Even worse, I now seem to be recall that we've been through this at least once before, and I had forgotten all about that. Must be some subconscious refusal to recognize that my heroes could use such obviously awful terminology, but a quick check (a day too late) showed
  • MTW uses the phrase "locally flatness" in the title of section 7.5 to refer to something like some kind of normal coordinate chart,
  • ditto Stephani, section 3.4,
  • ditto Poisson section 1.6,
  • Schutz section 6.2 seems to use the phrase similarly, or to refer to the fact that (by definition) every point in a Lorentzian manifold has a tangent space, or (later) in still another sense,
  • I couldn't find the term in the index of Wald, Carroll, and a number of other books including the Exact Solutions monograph by Stephani et al., and the new field guide by Griffiths and Podolsky, which are generally reliable guides to terminology used in the research literature on classical gravitation.
I maintain that a careful reading of textbooks on differential geometry and of the research journals on classical gravitation would support my view that there is a consensus among experts that local means local neighborhood, and that a term such as "ultralocal" should be used as shorthand for jet spaces or whatever (in particular, in defining normal charts), but checking the validity (or not) of this claim would take more time than it is worth (I guess).

Apologies to the moderators generally for my report which contained the same misinformation George just corrected... sigh... rather undercuts my moralizing about misinformation from others...
 
  • #12
Wishing on a low note not to end

Re
Code:
www.physicsforums.com/showthread.php?t=143404
dbacle said:
1)What is the relation between Moebius transformations and the Moebius
strip, if any.?. Maybe the same guy worked on both.?. I know the
Moebius Maps are the automorphisms of the Riemann Sphere,aka,
S<sup> 2</sup> (as the 1-pt compactification of the complexes).

No doubt SA/Ms know that Moebius introduced all these concepts (Moebius maps and Moebius transformations are the same thing, as it turns out).

dbacle said:
But I don't see a relation.

This would probably go over the head of the OP, but for any SA/M who is interested in an easy puzzle:

Recall that the Moebius group (the group of Moebius transformations, aka conformal automorphisms) acts on the two-sphere isomorphically to the way the Lorentz group acts on the optical appearance of the celestial sphere. Now consider the Hopf map [itex]S^3 \rightarrow S^2[/itex]. What the preimages of a point, the N & S poles, the equator, other latitude circles? The Moebius strip arises naturally as an embedded surface in S^3 in such a way--- see if you can figure out how.

Re
Code:
www.physicsforums.com/showthread.php?t=416842
showing my age here, but if the OP happens to be the Last Living Reader of Books, there are a number of articles in this book which would probably be perfect for him:

Ergodic Theory, Symbolic Dynamics, and Hyperbolic Spaces
Tim Bedford, Michael Keane, and Caroline Series
University of Oxford Press, 1992

The OP didn't say what kind of "dynamics" he has in mind, but from context he probably means the geodesic flow on H^2 and discrete quotients of same, which is what the book is about. A lovely observation due to the great French polymath mathematician Jacques Hadamard (a contemporary of Henri Poincare) is that geodesics on such surfaces can be studied using symbolic dynamics. (Symbolic dynamical systems are the most abstract type of dynamical system, and are usually studied as measure-theoretic or topological dynamical systems, or both, but as this application shows, they arise in Riemannian geometry as well as many many other phenomena in mathematics.)

For those who love analytic number theory, one of the fascinating things about geodesics on compact Riemannian two-manifolds is that if you enumerate the closed geodesics in order of increasing length, and look at how quickly the number of closed geodesics of length at most L grows with L, there is a natural analogue of the prime number theorem which governs how many prime number at most L exist!

Re
Code:
www.physicsforums.com/showthread.php?t=410614&page=3
Just read the last post and out of context, but tiny quibble: I wouldn't go so far as to say other universes or fairies are literally equally likely based upon current knowledge. Probably the point was that our current knowledge is somewhat modest, and based entirely on theory rather than direct experience (and of course, even if some explorer flew into a BH, theory says he'd have a hard time radioing back his observations inside the horizon!).

FIW: I think it often helps in these threads to distinguish between what gtr (a theoretically solid theory) predicts and what gtr plus "theoretically less than perfectly well founded guestimates of what a theory of quantum gravity might say" predicts. Interestingly, the best current guesses using gtr plus an absolute minimum of "general expectations" about what the yet unknown quantum theory of gravitation might turn out to say suggests that the interior of an astrophysical black hole--- any astrophysical hole should be expected to be "generic" in these that word is used in the research literature--- might be dominated by the phenomenon of "mass inflation", a purely classical effect which seems to suggest that unexpected "counterstreaming" associated with starlight and other stuff falling in might lead to rapid approach to Planckian energies well before an infalling observer reaches r=0. Unfortunately, it has been an open problem for decades to extend heuristic notions of mass inflation (known rigorously for special cases) to generic rotating holes (for simplicity, say electrically uncharged). See the nice review by Andrew Hamilton and (I think) a student, on the arXiv.

(Just saw Dmitry67 refer to "blue sheet", aka Penrose's suggestion of a Cauchy horizon in the interior of a generic rotating electrically neutral hole--- this concept is related to mass inflation but not really exactly the same thing.)

BTW, very glad to see that several SA/Ms are doing very nicely answering various black hole related questions in the relativity subforum! Pat on the back and all that.

Re
Code:
www.physicsforums.com/showthread.php?t=415395
Well, for example, I have this variety in [itex]\mathbb{C}^9[/itex]:
[tex]
\begin{array}{rcl}
x_1^5+x_3^5+x_5^5+x_7^5+x_9^5 & =& 0 \\
x_1^4+x_2 x_3^4+x_4 x_5^4+x_6 x_7^4+x_8 x_9^4 & = & 0 \\
x_1^3+x_2^2 x_3^3+x_4^2 x_5^3+x_6^2 x_7^3+x_8^2 x_9^3 & = & 0 \\
x_1^2+x_2^3 x_3^2+x_4^3 x_5^2+x_6^3 x_7^2+x_8^3 x_9^2 & = & 0 \\
x_1+x_2^4 x_3+x_4^4 x_5+x_6^4 x_7+x_8^4 x_9 & = & 0 \\
x_2^5+x_4^5+x_6^5+x_8^5+1 & =& 0
\end{array}
[/tex]
(this actually arose from an equation in [itex]\mathbb{CP}^9[/itex], where I took a patch in which one of the coordinates equals 1 (it's easy to see where the coordinate was in the original equation).

Now in a generic point, the Jacobian of these polynomials has rank 6, and thus, one would expect that the dimension of the variety is 3 (provided that the polynomials form a radical ideal - is that the case here?). But now in a point like, e.g. [itex] x_1=0, \, x_3=0, \, x_5=0, \, x_6=0, \, x_8=0 [/itex], the rank of the Jacobian is just 4 and thus, one would expect that these points are singular and that there are several irreducible components going through this points. How can one say how many they are and what their dimension is?


a decade ago, I would have unwisely tried to explain how to do this via Mathematica/Maple/MacCaulay2 and probably some others, and given citations to a bunch of books which explain the theory behind the computations. Instead I'll wisely plead exhaustion and... either say nothing or come back when I'm less exhausted and say just a little.
 
  • #13
BRS: Trying to help by providing a tutorial on making/using Penrose diagrams

Re the thread "Schwarzschild metric not stationary inside the horizon?" in the relativity subforum
Code:
www.physicsforums.com/showthread.php?t=423545
I just want to reiterate that the intended purpose of the BRS thread on conformal compactications and Penrose-Carter diagrams
Code:
www.physicsforums.com/showthread.php?t=403956
is to help SA/Ms painlessly master some memorable pictorial images which are essential for understanding the definition of static and stationary (and other things which arise frequently in the relativity subforum).

I'd like to also stress that two superb and highly relevant references are:
  • "the exact solutions catalog": Stephani et al., Exact Solutions to Einstein's Field Equations, 2nd edition, Cambridge University Press, 2001. A comphrehensive monograph.
  • "the field guide to exact solutions": Griffiths and Podolsky, Exact Space-Times in Einstein's General Relativity, Cambridge University Press, 2009. Covers the most important solutions in much greater detail, laying particular stress on explaining their global features.
My discussion (so far incomplete) in the BRS thread on Penrose diagrams largely follows the example of the second book.

Some specific comments:

To clarify: as of August 2010, Ben Crowell is not yet a SA/M, but it seems clear that he will be nominated as soon as his post count is high enough, and I'd expect he'd be badged soon thereafter. (He is a physics Ph.D. on the faculty at a college in the US, and author of an expository website, so clearly well-qualified.)

Ben Crowell is quite correct that usually one should speak of static/stationary regions, because it may not happen that an entire spacetime is either static or stationary. Indeed, while certain regions of common black hole models are either static or stationary (namely, one or more "exterior sheets" occurring "outside" the event horizon), black hole solutions always contain dynamical (nonstationary) regions inside the event horizon.

Introductory textbooks to gtr differ widely in their coverage and mathematical sophistication, and many authors choose not to go into much detail about the definition of static and stationary. The correct definition, employed by all researchers in this area, is the one given in section 18.1 of the "catalog" and in section 2.1.4 of the "field guide":

A spacetime, i.e. a 4-dimensional Lorentzian manifold (M,g), is stationary (on a region U) if (M,g) possesses a timelike Killing vector field on U, call it [itex]\xi[/itex]. It is static if in addition this timelike Killing vector field [itex]\xi[/itex] is irrotational, i.e. if the vorticity tensor (in the kinematic decomposition) vanishes.

The condition of vanishing vorticity is equivalent to the condition that [itex]\xi[/itex] be hypersurface orthogonal, i.e. that U admit a foliation into spatial hyperslices such that each slice is everywhere orthogonal to [itex]\xi[/itex]. This means that if (M,g) is static on U and we adopt a timelike coordinate t on U such that [itex]\partial_t[/itex] is a scalar multiple of [itex]\xi[/itex], then the metric tensor, expressed in these adapted coordinates will be independent of t on U. If (M,g) is stationary on U but not static, no such spatical hyperslices exist.

Please see the figure below for a visual image which I hope will clarify the distinction between "static" and "stationary" regions.

I highly recommend the field guide to all SA/Ms who regularly contribute to the relativity subforum at PF!

bcrowell said:
The Schwarzschild metric, described in Schwarzschild coordinates, has a Killing vector [itex]\partial_t[/itex]. This vector is timelike outside the horizon, but spacelike inside it.

Correct. There are four independent Killing vector fields, three spacelike ones associated with rotations of the nested two-spherical surfaces associated with "spherical symmetry", and a fourth which is timelike in the exterior sheets and spacelike in the interior regions.

There is something potentially confusing about writing [itex]\partial_t[/itex] on both the exterior regions and the interior regions, which arises from the fact that "interior Costa chart" on an interior region is formally almost identical to the usual Schwarzschild chart on an exterior sheet, but these are defined on disjoint domains. So better to write the Killing vector field as [itex]\xi[/itex], meaning a vector field defined on the maximal analytical extension, or if you want to write it explicitly in terms of an (almost) global chart, to use the Kruskal-Szekeres chart or the Penrose chart (conformal compactification of K-S chart).

Is anything wrong with the following argument? The form of the metric is the same on both sides of the horizon, so its components are still independent of t below the horizon.

Right idea, but the argument is wrong for the reason I just hinted at.

Therefore I would think that a Schwarzschild spacetime should not be considered stationary (which also means it can't be static).

The so-called "eternal black hole" (the maximal analytic extension of an exterior sheet, which is the "local solution"--- local as in "local neighborhoods"--- found by Schwarzschild in 1916) has two static regions, the two asymptotically flat exterior sheets and the past and future interior regions, which are non-static. Many other exact solutions modeling some kind of "isolated object" (e.g. other Weyl vacuum solutions) have similar features.

This would be despite common usages such as "a static black hole."

Right.

In the research literature, including monographs, you'll find that people speak of static and nonstatic regions. One should not expect that a maximally extended vacuum solution is likely to lack any non-static regions.

One of the things which irks me is that there is no standard agreement on how to use the word "solution" in gtr. I advocate speaking of "local solution" when referring to what elementary methods typically come up with (for example, deriving the Schwarzschild vacuum solution by assuming a static spherically symmetric form for the metric tensor in a suitable coordinate system easily leads to the local solution on the exterior sheet, but of course cannot include any part of the dynamical regions, so to discover that these are present you have to do more work.

Below I've described what I've seen in several different textbooks, which leads me to think there may be a difference in definition between different authors.

"The fact that the Schwarzschild metric is not just a good solution, but is the unique spherically symmetric vacuum solution, is known as Birkhoff's theorem. It is interesting to note that the result is a static metric." The second sentence seems to me to be false inside the horizon.

In fairness to Sean Carroll, his statement occurs before he discusses the maximal extension, so it's reasonable to interpret it as referring the local solution defined on one exterior sheet. In section 5.7 he could have perhaps stated explicitly that the future interior region is non-static, but this is implicit when Carroll writes "you cannot stop yourself from moving in the direction of decreasing r, since this is simply [a] timelike direction".

Figure: a timelike Killing vector field in a spacetime which is static (left), stationary but non-static (right); see also the discussion of the Frobenius theorem on hyperplane elements in Lee, Smooth Manifolds, Springer.
 

Attachments

  • static_vs_stationary.png
    static_vs_stationary.png
    5 KB · Views: 537
Last edited:
  • #14
"Anamitra" Thread on Einstein-Maxwell Theory

Re
Code:
www.physicsforums.com/showthread.php?t=423983
Anamitra appears to be another in a long line of PF users whose confidence vastly exceeds his/her current level of competence :rolleyes: and it seems pretty clear to me that this poster doesn't know enough about the elements of gtr yet to understand gravitational waves, EM on curved spacetime, or black holes. In particular, this user's Post #1 appears to me to make little sense in the context of gtr.

However, it is true that in principle gravitation and EM can interact by purely classical means, to some extent, in gtr. In particular, mass-energy leads to curvature which changes the propagation of EM waves.

The most important points to bear in mind are probably these:
  • There is a well-established theory for doing EM in the context of gtr, which of course involves simultaneously solving
    • the EFE, including the contribution of the EM field (and any charged matter) to the stress-energy tensor which stands on the RHS of the EFE,
    • the curved spacetime Maxwell equations on the resulting spacetime geometry (M,g),
    Here, solving the EFE yields a metric tensor g on M and thus a Lorentzian manifold (M,g) or spacetime model, and also the gravitational field (Riemann curvature tensor) on our spacetime, while solving the Maxwell equations yields the EM field on M.
  • Electrovacuum solutions are spacetime models in gtr in which the only fields present in the domain of the solution are the gravitational field and a source-free Maxwell field, i.e. all masses and charges are shoved onto boundary conditions. The simplest example is the Reissner-Nordstrom electrovacuum (spherically symmetric static gravitational plus electric field), which is characterized by two parameters, mass and charge of the spherically symmetric object (this object is not included in the model obtained from our electrovacuum solution since it has shoved onto the boundary conditions, but it is implicit). Here, q must be smaller than m, and in realistic models it is much smaller than m.

    To begin to understand how strong curvature affects EM in gtr, it is natural to try to compare the decay wrt r of the components of the tidal tensor with and without charge (for example). E.g. some tidal tensor components change from m/r^3 - q^2/r^4 to m/r^3, that kind of thing. (See maxima file below if you want to play with this.)

    But now you have to think carefully about the radial coordinate: you are comparing small effects depending on r, but the physical meaning of r as "radial distance" has also undergone small changes when you killed off q, although the geometric meaning of r in terms of surface area of certain nested spheres has not. So you need to be careful in comparing "decay wrt r".

But I have no idea how anyone could try to explain this or anything else to Anamitra, because I have the strong impression that this user has demonstrated a lack of interest in listening to those who know more than he/she does! :rolleyes:

Anamitra said:
Changes in the electric and the magnetic fields in response to changes in space-time curvature should have played an important role in the early cosmological processes

IMO, this poster appears to suffer from CoS (chip-on-shoulder) wrt gtr, and appears likely to try to manufacture alleged "discrepancies" or "problems" or "lies, damned lies, and gtr papers" :rolleyes: which exist only in his/her imagination, but FWIW, there is a large body of work in the arXiv on, in particular, possibly significant effects of strong magnetic fields in the early universe. This is a difficult area and much of this work involves approximations, and approximations are also much trickier than exact solutions, so best left to genuine experts, because not only does one need to have mastered many subtleties and overcome many conceptual blocks from simple exact gtr, one needs to understand what can be neglected in a given situation, which is a skill set an order of magnitude beyond mastering simple exact solutions.

Any curious SA/M can look for papers coauthored by George F. R. Ellis (coauthor of the monograph by Hawking and Ellis, Large Scale Structure of Spacetime) and some of his students.

HTH.

P.S. Here is a Maxima file for the RN electrovacuum, which you run in "batch mode" under the wxmaxima front end to Maxima:
Code:
/* 
Reissner-Nordstrom nnevac; exterior psph; static coframe 

This models exterior of spherically symmetric electrically charged object
Given coframe corresponds to static observers who use their rocket engines
to hover over this object.  The coordinate r is Schwarzschild coordinate,
defined such that area of spheres r=r0 is %pi*r0^2

From Killing vectors we obtain three first integrals for t1,theta1,phi1.
Plugging these into epsilon = ds^2 we obtain one for r1.  This gives
general geodesic up to integration over proper time.
WLOG, consider null geodesic in the equatorial plane theta=Pi/2:
	t1 = E/(1-2*m/r+q^2/r^2)
	r1 = +/-sqrt(E^2-(L/r)^2*(1-2*m/r+q^2/r^2)
	theta1 = 0
	phi1 = L/r^2
where variable is affine parameter s (unique up to constant scalar multiple).
Note that r_min is positive real root of
	E^2-(L/r)^2*(1-2*m/r+q^2/r^2
Thus the unparameterized world line in "graphing form" is
	dr/dt = sqrt(1-(L/E/r)^2*(1-2*m/r+q^2/r^2))*(1-2*m/r+q^2/r^2)
	dphi/dt = L/E/r^2*(1-2*m/r+q^2/r^2)
Note this depends only on ratio L/E, so the "world line of a wave packet"
is indpt of frequency!  Also
	dphi/dr = L/E/r^2/sqrt(1-(L/E/r)^2*(1-2*m/r+q^2/r^2))
Integrating from r=-infty to r=r_min and doubling gives exact light bending
as an elliptic integral.

*/
load(ctensor);
cframe_flag: true;
ratchristof: true;
ctrgsimp: true;
/* define the dimension */
dim: 4;
/* list the coordinates */
ct_coords: [t,r,theta,phi];
/* 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]: -sqrt(1-2*m/r+q^2/r^2);
fri[2,2]:  1/sqrt(1-2*m/r+q^2/r^2);
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] */
/* Note: for each j,m, cdisplay(riem) displays riem[i,k,-,-] as matrix */
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);
cdisplay(ein);
/* WARNING! leinstein(false) only works for metric basis! */
/* electroriemann tensor */
print("electroriemann tensor");
expand(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 */
print("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(%));
/* Compute NP tetrad, Weyl spinors, and Petrov type */
weyl(false);
nptetrad(true);
psi(true);
petrov();
And here is the Melvin electrovacuum (infinite cylindrically symmetric magnetic field):
Code:
/* 
Melvin nnevac; cylindrical chart; static coframe 

Models gravitational field of a cylindrically symmetric static magnetic field
Given coframe corresponds to static observers who use their rocket engines
to hover over the region near r=0 where magnetic field is concentrated

Weyl tensor is Petrov type D!
	Psi_2 = -2 q^2 (1-q^2 r^2)/(1+q^2 r^2)^4
*/
load(ctensor);
cframe_flag: true;
ratchristof: true;
ctrgsimp: true;
/* define the dimension */
dim: 4;
/* list the coordinates */
ct_coords: [t,z,r,phi];
/* define background metric */
lfg: ident(4);
lfg[1,1]: -1;
/* define the coframe */
fri: zeromatrix(4,4);
fri[1,1]: -1-q^2*r^2;
fri[2,2]:  1+q^2*r^2;
fri[3,3]:  1+q^2*r^2;
fri[4,4]:  r/(1+q^2*r^2);
/* 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);
cdisplay(ein);
/* WARNING! leinstein(false) only works for metric basis! */
/* Compute Kretschmann scalar */
factor(expand(factor(rinvariant())));
/* WARNING! leinstein(false) only works for metric basis! */
/* electroriemann tensor */
print("electroriemann tensor");
factor(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]]));
print("magnetoriemann tensor");
factor(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]]));
/* Construct NP tetrad for our frame, compute Weyl spinors and Petrov type */
nptetrad(true);
weyl(false);
psi(true);
factor(psi[0]);
factor(psi[2]);
petrov();
 
Last edited:
  • #15
"Madhatter106" thread on "EM and Gravity"

Re
Code:
https://www.physicsforums.com/showthread.php?t=424079
oh gosh, another one!

"Reference frame" is ambiguous (should mean frame field, i.e. quadruple of mutually orthogonal unit vector fields, one timelike, three spacelike, defined on some open neighborhood, but probably Madhatter106 is thinking, not quite correctly, about local coordinate charts and/or tangent spaces at a single event).

If a gravity well was strong strong enough would a passing E field fold upon itself and generate a b field?

The OP appears to admit he/she spoke without thinking in this post, and I think he/she may be confusing
  • the notion that a passing EM wave might alter the motion of an observer who might then measure magnetic as well as electric fields--- in principle, that could happen, although EM wave so strong as to have really strong gravitational effects are not yet known
  • the old (and naive) notion that gravity and EM might be "unified" using some classical field theory "beyond gtr".

this also makes me wonder if the EM spectrum is part of a larger field that includes gravity. I could imagine then that the folding of the E field then creates or adds to the 'gravity' but that doesn't explain how the gravity well existed before the passing E field. and what would prevent such a an interplay from setting up a chain reaction in which the gravity well feeds itself to the point of being singularity.

Just to pick up on one of the more minor confusions implicit in this, because I don't see regulars in the relativity forum drawing attention to this confusion any more, and I think that's a bad thing: most newbies confuse coordinate singularities (which are very roughly analogous to "removable singularities" in elementary complex variables, i.e. the theory of a holomorphic function of one variable) and curvature singularities. And most newbies incorrectly assume that "black hole" is defined by the presence of a curvature singularity, when in fact this concept is defined by the presence of an event horizon.

Just to make things more confusing, in the Schwarzschild vacuum, the event horizon happens to be a coordinate singularity and thus a boundary on the domain of the usual Schwarzschild chart, which is only valid outside the horizon. But changing to another chart such as the ingoing Eddington chart removes the coordinate singularity and allows us to study infall into the interior, etc...
 
Last edited:
  • #16
"mysearch" thread on C. S. Unnikrishnan "Cosmic Relativity"

Re
Code:
www.physicsforums.com/showthread.php?t=423985
Sigh... do I really need to say more?... sigh...

Well, it is true that in simple FRW models you can easily introduce a frame field adapted to a family of inertial observers who are moving wrt the dust particles whose mass provides the gravitational field of the FRW model. Such observers will measure a "dipole anisotropy" of the CMBR, and also a momentum term in the Einstein tensor. But this certainly does not challenge anything in the foundations of special relativity!

"Mysearch" seems to be, IMO, another poster whose enthusiasm for "overturning gtr" greatly exceeds his/her grasp of the reasons why that task is so formidably difficult.

Here is a Maxima file for a frame field adapted to observers comoving with the dust particles in the FRW dust with E^3 hyperslices:
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);
cdisplay(ein);
/* WARNING! leinstein(false) only works for metric basis! */
/* Compute Kretschmann scalar */
rinvariant();
expand(factor(%));
/* 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[2]);
petrov();
When we boost the timelike vector and one spacelike vector, with boost parameter f a function of t only, we obtain this:
Code:
/* 
FRW dust with E^3 hyperslices; comoving cartesian chart; 
nsi coframe comoving with particles moving wrt the dust particles
Must compute acceleration and set equal to zero to find form of f.

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];
/* Declare the dependent and independent variables */
depends(f,[t]);
/* define background metric */
lfg: ident(4);
lfg[1,1]: -1;
/* define the coframe */
fri: zeromatrix(4,4);
fri[1,1]: -sqrt(f^2+1);
fri[1,2]: -f*t^(2/3);
fri[2,1]:  f;
fri[2,2]:  sqrt(f^2+1)*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);
cdisplay(ein);
Notice that
  • to boost the old frame field to obtain the new frame field, I applied (in the tangent space to each event, to each of the four frame vector fields) the transformation represented in our coordinates by the matrix
    [tex]
    \left[ \begin{array}{cccc}
    \sqrt{1+f^2} & f & 0 & 0 \\
    f & \sqrt{1+f^2} & 0 & 0 \\
    0 & 0 & 1 & 0 \\
    0 & 0 & 0 & 1
    \end{array} \right]
    [/tex]
    where f is an undetermined function of t only,
  • metric tensor unchanged, so same spacetime, just a new frame field,
  • new nonzero components appear in the Einstein tensor (don't forget to flip sign of top row in the displayed matrix): momentum and an "isotropic pressure" term--- just what we should expect!--- see for example Schutz's textbook.
To find the undetermined function f, we must compute the acceleration vector for the timelike unit vector field in the new frame, set this to zero, and solve for f.

I repeat this at every opportunity: Maxima is free and open source software, and Maple is actually based upon a common predecessor. Maple is far more developed than Maxima (particularly in DE solvers and Groebner stuff, plus many specialized packages), but Maxima is already quite powerful and IMO all SA/Ms should use it, unless they already use Maple or Mathematica. And Maxima comes with a package, Ctensor, which--- like GRTensorII for Maple--- can compute with frame fields! The examples show how. See also the BRS thread "Using Maxima for gtr computations"
Code:
www.physicsforums.com/showthread.php?t=378991

However, Ctensor currently lacks the offer from GRTensorII to define your own tensorial quantities to compute. In particular, I don't know how to coax it to compute acceleration vector, expansion tensor, vorticity tensor (of a timelike congruence) or the optical scalars of a null geodesic congruence. These are serious limitations because the expansion tensor directly expresses e.g. "Hubble law" type phenomena. In the example above, when we change to our new frame field we obtain a new timelike congruence, and its expansion tensor components wrt the new frame are different from the expansion tensor components of the old congruence wrt the old frame, in an interesting way.
 
Last edited:
  • #17
"Anamitra" thread on "Curved Space-time and Relatitive Velocity"

Re
Code:
www.physicsforums.com/showthread.php?t=423334&page=4
kudos and thanks to DrGreg, DaleSpam and JesseM for taking on the task of trying to explain this to "Anamitra" &c.!

I've notice that you three (plus bcrowell) have been trying to correct him/her on some elementary misconceptions, e.g. parallel transport. I wish you'd been having better luck with "Anamitra", but its good that you have gotten into explaining some important stuff--- maybe a latter read will see and appreciate that discussion!

Regarding the geometric interpretation of the Riemann curvature tensor in terms of parallel transport of a vector around a small closed path (not neccessarily a purely spacelike arc): Penrose emphasizes this interpretation in his excellent but unfortunately obscure expository paper, which appeared as one chapter in Mathematics Today, ed. by Lynn Arthur Steen.

Re JesseM and DaleSpam Posts #52, 53:

In Minkowski vacuum, it does make sense to say that two different events P,Q are "timelike-separated" or "spacelike-separated". In the first case there is a unique timelike line segment between P,Q and in the second case, a unique spacelike line segment. So we can say that in the first case the interval is the elapsed time measured by an observer whose world line coincides to said timelike segment between P,Q. But this picture runs into problems when we take a discrete quotient of Minkowski spacetime, so the local vs. global distinction is important here even for locally flat spacetimes.

In curved spacetimes, there will often be more than one geodesic arc between two events P,Q. And I agree with JesseM that researchers avoid talking about "the" spacetime interval between two different events in a curved spacetime--- precisely because integrating ds along two different curves will give different intervals, so the "interval" is no longer an invariant characterization of the metrical relationship between P,Q.

It looks to me as though many of Anamitra's posts are based upon too many elementary misconceptions to be worth trying to correct in detail, but FWIW:

In Anamitra Post #56:
Anamitra said:
We consider a metric of the type shown below:

ds^2=g(00) dt^2-g(1,1) dx^2 - g(2,2) dy^2 - g(3,3) dz^2

ds^2= dT^2- dL^2 [dT--->Physical time, ds----> physical distance]
Anamitra must not be "getting it", because the notion that there is some "physical time", not otherwise specified, or even worse, some "physical distance", not otherwise specified, is one of the more elementary misconceptions underlying the naive notion that there "should" be some unique method of defining "distance in the large" and thus "relative velocity in the large" in curved spacetimes, which is simply not the case, for purely mathematical reasons.

Also, ds gives the Minkowsi interval on the tangent space to some event. If you integrate along an everywhere timelike arc C the resulting number can be intrerpreted as the "elapsed time" between the two endpoint events on C. If you integrate along an everywhere spacelike arc, the result can be interpreted as a kind of "length" of a spacelike arc (not neccessarily a spacelike geodesic arc), but trying to interpret this geometric notion (referring to the spacetime manifold) to a reasonable physical measurement process is usually an exercise in futility.

In "Anamitra" Post #6:
I have been ,for quite some time, trying to explore the possibility of breaking the speed barrier within the "confines of relativity"? Locally we cannot do it. The laws are very strong in this context.The only option would be to explore the matter in a "non-local" consideration.
Anamitra is very far from being able to read arXiv eprints on gtr, but FWIW, a relevant paper by Robert Low tends to throw cold water on this hope, as does most of the large literature related to "time machines", "warp drive models", and "traversable wormholes". (There are Lorentzian manifolds modeling such things, but these cannot be called "solutions of the EFE" in any reasonable sense, and most of the work in this field suggests that such things are not admissable in gtr plus "effective field theory" motivated hypotheses for the matter tensor, where the "effective field theories" arise from the tricky and sometimes questionable semiclassical approximation.

In "Anamitra" Post #15:
Can a space-time surface be exactly spherical?
The question is ambiguous, but FWIW
  • no spacetime in the sense of a four dimensional Lorentzian manifold (M,g) can have spherical topology in gtr,
  • many spacetimes (M,g) enjoy spherical symmetry, which means that they possesses a family of nested spherical surfaces, each of which is exactly spherical in the sense that restricting the metric tensor g to any of these surfaces gives the metric tensor of a round two-sphere as a Riemannian two-manifold.
In Anamitra Post #17, he/she misspelled the name "Hartle"--- although that's the most minor error he/she made it is consistent with my impression that this poster is more interested in yelling at gtr knowledgeable PF posters than in learning from them :frown:
 
Last edited:
  • #18


Chris Hillman said:
In Anamitra Post #56:
Anamitra said:
We consider a metric of the type shown below:

ds^2=g(00) dt^2-g(1,1) dx^2 - g(2,2) dy^2 - g(3,3) dz^2

ds^2= dT^2- dL^2 [dT--->Physical time,ds----> physical distance]
Anamitra must not be "getting it", because the notion that there is some "physical time", not otherwise specified, or even worse, some "physical distance", not otherwise specified, is one of the more elementary misconceptions underlying the naive notion that there "should" be some unique method of defining "distance in the large" and thus "relative velocity in the large" in curved spacetimes, which is simply not the case, for purely mathematical reasons.
You may not have noticed Anamitra used this same notion of "physical" time & distance in another thread ("On the Speed of Light Again!"
Code:
https://www.physicsforums.com/showthread.php?t=422548
) and I attempted to make some sense of this in post #18 of that thread (this was before the parallel transport argument arose and Anamitra's inabilities became apparent to me):
DrGreg said:
I haven't come across Anamitra's technique before, but I think this is what is happening.

Given a metric of the form

[tex]ds^2 = g_{00}\,dt^2 + g_{ij}\,dx^i\,dx^j[/tex]​

(where i, j take values 1,2,3 only) define two new metrics:

[tex]dT^2 = -g_{00}\,dt^2[/tex]
[tex]dL^2 = g_{ij}\,dx^i\,dx^j[/tex]​

T is being called "physical time" and L is being called "physical length". Both are evaluated by integrating along the same spacetime worldline that you would integrate ds along. And both are dependent on your choice of coordinate system.

It should be clear that if you were to evaluate T along a worldline of constant x1,x2,x3 it would equal proper time. If you were to evaluate L along a curve of constant t it would equal proper length. But for an arbitrary worldline you evaluate both along the same worldline.

If you calculate |dL/dT| along a null worldline you get 1.
Did what I said make sense? Obviously it is coordinate-dependent and probably only meaningful in a static spacetime. In view of the subsequent discussion on parallel transport, I am now suspicious there is some flaw in this approach. Does dL/dT have any significant meaning along a worldline?

Inspired by that I went on to consider a non-static but stationary spacetime in another thread ("deducing some GR from SR?"
Code:
https://www.physicsforums.com/showpost.php?p=2845772&postcount=53
)
DrGreg said:
...That seems to be closely related to what we're discussing here. In our case we have a more general metric of the form

[tex]ds^2 = g_{00}\,(dt - w_i\,dx^i)^2 + k_{ij}\,dx^i\,dx^j[/tex]​

Rindler shows any stationary metric can be written in this form. So we can define metrics

[tex]dT^2 = -g_{00}\,(dt - w_i\,dx^i)^2[/tex]
[tex]dL^2 = k_{ij}\,dx^i\,dx^j[/tex]​

to define "physical time" and "physical length". We are effectively decomposing a 4-vector ds as sum of dT parallel to the Killing vector field and dL orthogonal to dT.
Do these approaches make sense, or are they a waste of time?
 
  • #19


Was it Pascal who wrote: "I regret that I lack the time to be brief"? :wink:

DrGreg said:
You may not have noticed Anamitra used this same notion of "physical" time & distance in another thread ("On the Speed of Light Again!"
Code:
www.physicsforums.com/showthread.php?t=422548
)

Oh gosh, I guess I did miss that and it's even worse than I thought!

I attempted to make some sense of this in post #18 of that thread (this was before the parallel transport argument arose and Anamitra's inabilities became apparent to me):

Did what I said make sense? Obviously it is coordinate-dependent and probably only meaningful in a static spacetime.

I am tired and I am not sure I understand what you had in mind. So please forgive me if I misunderstood or am about to say something obtuse!

[EDIT: I did, later I try to distinguish between Anamitra's procedure and Dr. Greg's procedure...]

FWIW, a few general comments which came to mind:
  • It seems Anamitra is assuming a coordinate chart with the property that there are no cross terms dtdx and so forth, so that the hyperslices t=t0 are orthogonal to the integral curves of the coordinate vector [itex]\partial_t[/itex] (such charts are often used in cosmological models, especially when the integral curves are the world lines of the matter responsible for the gravitational field of the cosmological model).
  • It seems that neither you nor Anamitra have yet proposed an operationally significant procedure by which ideal observers could try to make measurements needed to compute a physical speed according to your schemes; compare say "radar distance" which suffers from numerous defects but at least is motivated by a fairly clear intuitive idea for a possible measurement procedure.
  • Your procedure might make some sense geometrically (but only referred to a specific chart), but it appears to be defined for some curve C rather than for a pair of curves C,C' representing the world lines of the observer and the moving target. This in itself probably makes it a suspect notion since "speed in the large" should be a relative notion referring to a pair of world lines.

A good way of getting some idea of whether your procedure leads to an operationally significant notion of "speed in the large" might be to consider a simple example:

The Painleve influx chart for the right exterior and future interior regions of the Schwarzschild vacuum has line element
[tex]
\begin{array}{rcl}
ds^2 & = & -(1-2m/r) \, d\tau^2 + 2 \, \sqrt{2m/r} \, d\tau \, dr + dr^2
+ r^2 \, \left( d\theta^2 + \sin(\theta) \, d\phi^2 \right) \\
&& -\infty < -\tau < \infty, \; 0 < r < \infty, \;
0 < \theta < \pi, \; -\pi < \phi < \pi
[/tex]
This is not an orthogonal local coordinate chart because of the cross term, but the world lines of the Lemaitre observers (who fall in freely and radially "from rest at r=infty") are the integral curves of the very simple timelike unit tangent vector field
[tex]
\vec{u} = \partial_\tau - \sqrt{2m/r} \, \partial_r
[/tex]
We can write the proper time parameterized geodesics as
[tex]
\begin{array}{rcl}
\tau & = & s \\
r & = & \left( \sqrt{r_0^3} - \sqrt{9m/2} \, s \right)^{2/3} \\
\theta & = & \theta_0 \\
\phi & = & \phi_0
\end{array}
[/tex]
The transformation to the usual Schwarzschild chart (valid only on the exterior r > 2m) is
[tex]
d\tau = dt + \frac{\sqrt{2m/r} \, dr}{1-2m/r}
[/tex]
or
[tex]
\tau = t + \sqrt{8mr} - 4 m \, \operatorname{arctanh}( \sqrt{r/2/m} )
[/tex]
Plugging in the proper time parameterized geodesic representing world line of a Lemaitre observer,
[tex]
\begin{array}{rcl}
t & = & s - \sqrt{8m} \, (\sqrt{r_0^3}-\sqrt{9m/2} \, s)^{1/3}
+ 4 m \, \operatorname{arctanh}
\left( \frac{(\sqrt{r_0^3}-\sqrt{9m/2} \, s)^{1/3}}{\sqrt{2m}} \right) \\
r & = & (\sqrt{r0^3}-\sqrt{9m/2} \,s )^{1/3} \\
\theta & = & \theta_0 \\
\phi & = & \phi_0
\end{array}
[/tex]
Can you work out your procedure for this proper time parameterized geodesic? What notion of "physical speed" does it give relative to a static observer in the exterior region? What measurement process yields the same result?

By the way, the line element for the ingoing Painleve chart
can be rewritten
[tex]
ds^2 = -d\tau^2 + \left( dr + \sqrt{2m/r} d\tau \right)^2
+ r^2 \, \left( d\theta^2 + \sin(\theta) \, d\phi^2 \right)
[/tex]
It can also be written in pseudo-cartesian form with the frame field of the Lemaitre observers becoming
[tex]
\begin{array}{rcl}
\vec{e}_1 & = & \partial_\tau - \sqrt{2m/r} \;
\left( x/r \, \partial_x + y/r \, \partial_y + z/r \, \partial_z
\right) \\
\vec{e}_2 & = & \partial_x \\
\vec{e}_3 & = & \partial_y \\
\vec{e}_4 & = & \partial_z \\
&& -\infty < \tau, \, x, \, y, \, z < \infty
\end{array}
[/tex]
where [itex]r = \sqrt{x^2+y^2+z^2}[/itex], which gives the line element
[tex]
ds^2 = -(1-2m/r) \, d\tau^2
+ 2 \; \frac{\sqrt{2m}}{(x^2+y^2+z^2)^{3/4}}
\; \left( x \, dx + y \, dy + z \, dz \right) \; d\tau
+ dx^2 + dy^2 + dz^2
[/tex]

In view of the subsequent discussion on parallel transport, I am now suspicious there is some flaw in this approach. Does dL/dT have any significant meaning along a worldline?

Inspired by that I went on to consider a non-static but stationary spacetime in another thread ("deducing some GR from SR?"
Code:
https://www.physicsforums.com/showpost.php?p=2845772&postcount=53
Do these approaches make sense, or are they a waste of time?

Can't say yet because I am too tired or confused to be sure I understand what the approach is yet. See the figure below where I tried to sketch what I think you think Anamitra might be trying to propose (a kind of "integrated coordinate speed"?)

As for the expression given by Rindler, that actually dates back to the work of Weyl and Lewis on stationary vacuum solutions in the 1920s. Oversimplying to make a point, consider the coframe field
[tex]
\begin{array}{rcl}
\sigma^1 & = & -(dt - a \, dx - b \,dy - c \, dz) \\
\sigma^2 & = & dx \\
\sigma^3 & = & dy \\
\sigma^4 & = & dz
\end{array}
[/tex]
and its dual frame field
[tex]
\begin{array}{rcl}
\vec{e}_1 & = & \partial_t \\
\vec{e}_2 & = & \partial_x - a \, \partial_t \\
\vec{e}_3 & = & \partial_y - b \, \partial_t \\
\vec{e}_4 & = & \partial_z - c \, \partial_t
\end{array}
[/tex]
So the integral curves of our timelike Killing vector field will look like vertical lines in this chart. (Example: in the Born chart on Minkowski vacuum, which is adapted to Langevin observers--- a Langevin observer moves in a circular orbit with constant angular velocity about the axis of cylindrical symmetry--- their world lines appear as "vertical lines" in the Born chart, but as helices of constant pitch in the standard cylindrical chart.)

But if you were expecting to construct spatial hyperslices orthogonal to those curves, which will look like the figure below... oh no! Because in the case of stationary but not static spacetimes, our timelike Killing vector field will be irrotational, so in fact no family of orthogonal hyperslices will exist!

I'll have to stop here because we are getting into the same damnable stuff which proved so frustrating when I tried to explain why people tend to confuse a dozen distinct concepts at every step of (careless, bad, misleading, useless, wrong) discussions of that old chestnut, "the [sic] spatial geometry [sic] of the [sic] rotating disk [sic]". In particular, some issues which arise include:
  • multiplicity of operationally significant notions of "distance in the large" and thus "speed in the large" even in flat spacetime (when considering pairs of world lines at least one of which is not inertial),
  • difficulties in defining "rigid bodies" even in Minkowski vacuum, except for case of constant angular momentum (which precludes "spinning up" a nonspinning disk),
  • difficulties in defining and working with a suitable notion of elastic bodies in gtr (unavoidably this will mean nonlinear elasticity, which is not easy even in Newtonian physics),
  • radar distance, pedometer distance, on a rotating rigid [sic] "test disk" in Minkowski vacuum,
  • local versus global problems in trying to define "space at a time" for a "rigid" rotating test disk in Minkowski vacuum,
  • difficulties in trying to define "one orbital period" for Langevin observers (noninertial observers who move in circular orbits in Minkowski spacetime), much less inertial observers moving in circular orbits in say Kerr vacuum,
  • neccessity of appealing to kinematic decomposition (acceleration vector, expansion tensor, vorticity tensor) even in Minkowski vacuum,
  • quotient manifolds versus submanifolds,
  • naive attempts to compare a configuration (nonspinning massive disk) in one spacetime (a Weyl vacuum, static axisymmetric) with a configuration (spinning "but otherwise identical" [sic] massive disk) in another spacetime (an Ernst vacuum, stationary axisymmetric),
  • memory effects from detail of spin-up phase in a scenario in which an intially nonrotating disk is spun up while we look away (because we don't want to try to model elastic bodies in gtr),
  • &c.

There are some old PF threads from 2007 or so, back when I still posted in the public areas. I don't think I can improve on what I said back then, or even earlier, in some Wikipedia articles I wrote (the current versions are no doubt completely different and probably I would say they are quite wrong).

FWIW, before I wrote the above, I tried to read the first page or so of the Anamitra thread you mentioned, and had these rather harsh comments on Anamitra's posts:

From Anamitra's Post #1:

Anamitra said:
We consider a point in curved space-time and a "local inertial frame" associated with it.
Now by some suitable transformation we move to some other reference frame at that that point.This frame in general could be a non-inertial one.

Frame field is a completely different concept from coordinate chart. Confusion is inevitable when newbies use the same word for two completely different concepts. You can just barely get away with this if you consider only inertial frame fields, only cartesian charts, only in Minkowski vacuum, but that won't get us into gtr!

"Move" is a terrible choice of words. Anamitra doesn't realize it, but he/she is thinking of boosting/rotating to a new frame at the same event E (wnere "frame at E" means four orthonormal vectors living in the tangent space at E). No translations allowed!

Anamitra said:
Speed of light in the local inertial frame
[ ds^2 ] =[g(1,1)dx1^2 +g(1,1)dx2^2 + g(1,1)dx3^2]/g(00)dt^2=1 [c=1 in the natural units]

Speed of light in the non inertial frame:
[ ds^2 ] =[g'(1,1)dx1'^2 +g'(1,1)dx2'^2 + g'(1,1)dx3'^2]/g'(00)dt'^2=1 [c=1 in the natural units]

I misread that first time through (Anamitra really needs to start using LaTex!) and now I'm too tired to think about it...

Anamitra said:
It is important to note that the concept of the Rindler Coordinates has been used incorrectly wherever and whenever dx/dt [as we find in the Rindler Coordinates] has been interpreted as the physical speed of light[or the physical speed of light and the coordinate speed of light have been used in an interchangeable way] This is serious mistake which can only serve the purpose of propagating errors with carefree abandon.

Ironic, because the error here is that Anamitra hasn't recognized that to define a "physical speed of light" valid over a local neighborhood in a curved spacetime, you must define a measurement process, and it turns out that in curved spacetimes, different measurement procedures can give numerically distinct results for the same scenario.

The simplest procedure is generally radar distance, which makes sense for computing an estimated distance from one observer to another, and repeated computations can lead to a notion of "speed" (wrt the proper time kept by the observer with the radar gun) but doesn't make sense for measuring the "speed of a photon" over an null geodesic arc in some local neighborhood.

Anamitra said:
Are all local frames inertial?The answer is no.
Of course we can find frames that are "locally inertial". Also by suitable transformations we may find frames that are "locally non-inertial".
[In fact the Rindler coordinates relate to uniformly accelerating frames in flat space-time
Code:
[PLAIN]http://en.wikipedia.org/wiki/Rindler_coordinates
[/PLAIN] ]

Same confusion between local coordinate charts and frame fields.

Physical and Coordinate speeds
Let us consider a pair of points a and b lying on the x1-axis. The physical distance[this is the distance as we know in the physical world] between a and b along the x1 axis is given by:

physical distance= integral [from a to b] g(1,1)dx1

If a particle travels from a to b along the x1 axis the physical time is given by:

physical time= integral [from t1 to t2 ] g(0,0)dt

This is incorrect. Anamitra hasn't recognized that he/she needs to specify the world line C of some observer O (not neccessarily inertial) to integrate along C between two events A and B on C in order to obtain "the elapsed time between A,B as measured by O". And that's the easy part. Defining a "spatial distance" is much trickier. Probably Anamitra has something very murkily in mind like this:
  • foliate (M,g) into arbitrarily hyperslices such that each slice is orthogonal to C where C intersects each slice
  • choose some spacelike curve C' on a slice S passing through event E = C intersect S and integrate along
  • maybe you get lucky and C' is a geodesic of S, but this need not mean C' is a geodesic of (M,g)!
Even worse, while such a procedure may have geometric meaning (with more foresight and sophistication, one could choose a "totally geodesic hyperslice" S whose geodesics are also geodesics of the spacetime itself), it rarely has operational physical meaning, i.e. rarely corresponds to any natural method of measurement.

BTW, Anamitra badly needs to read about how to use LaTex in VB!

Anamitra said:
We consider a complicated metric:

ds^2=g(0,0)dt^2-g(1,1)dx1^2-g(2,2)dx2^2-g(3,3)ds^2

The above metric is not a flat space-time metric.Can we convert such a metric to a flat space-time metric by some suitable transformation?The answer is "yes".

Anamitra probably meant "dx3^2", but that's a minor point. It is not true that one can transform the metric tensor of a curved manifold to the metric tensor of a locally flat manifold, simply by changing to a new local coordinate chart!

Anamitra said:
The fact that gravity is equivalent to an acceleration is an interesting fact.

The "ultralocal vs. local vs global" distinction is absolutely crucial here. The equivalence principle is really talking about ultralocal structure (tangent space or maybe a jet space at a single event). No-one is claiming that physics cannot tell the difference between curved and locally flat spacetimes on a local neighborhood!

Anamitra has badly misunderstood the text by Weinberg. It's also striking that his/her citations are to 1963 papers!

I finally lost patience with Anamitra's Post #12 so I didn't even try to parse that mess.

Figures:
  • Attempt to sketch what I think Dr. Greg thinks Anamitra might have in mind: the triangles are clearly coordinate dependent and I don't see why we should expect them to be related in any simple way to a measurement procedure which can be carried out by a distant observer.
  • Attempt to sketch naive picture of the appearance of some integral curves of the timelike Killing vector field and their alleged "orthogonal hyperslices" [sic] in a Weyl-Lewis-Papapetrou type chart for a stationary spacetime, as per Dr. Greg's procedure from the thread "deducing some GR from SR?"... I think.
 

Attachments

  • 2010.08.25.Dr_Greg.png
    2010.08.25.Dr_Greg.png
    3 KB · Views: 528
  • 2010.08.25b.Dr_Greg.png
    2010.08.25b.Dr_Greg.png
    435 bytes · Views: 524
Last edited by a moderator:
  • #20


Thanks for your response.

Chris Hillman said:
It seems that neither you nor Anamitra have yet proposed an operationally significant procedure by which ideal observers could try to make measurements needed to compute a physical speed according to your schemes; compare say "radar distance" which suffers from numerous defects but at least is motivated by a fairly clear intuitive idea for a possible measurement procedure.
True. I was trying to make some sense of Anamitra's proposal. If you have a coordinate system in which one observer is at rest, then I suppose this procedure gives you a way of defining a "velocity" of another particle. But you have a choice of many such coordinate systems, and in general no good reason to choose one above the others, so this isn't going to give you a frame-independent value.

Chris Hillman said:
But if you were expecting to construct spatial hyperslices orthogonal to those curves, which will look like the figure below... oh no! Because in the case of stationary but not static spacetimes, our timelike Killing vector field will be irrotational, so in fact no family of orthogonal hyperslices will exist!
Indeed, yes, I had already grasped that!
 
  • #21
On the significance of operationally significant notions

I think it is very important to force yourself to try to provide a possibly idealized but specific operational scheme by which whatever notion of "velocity in the large" you are trying to define can in principle be measured. This often reveals multiple problems with the original intuition and shows that the notion is not yet well defined.

Since I am a mathematician by training (with no training in physics at all!) it always feel strange when I try to urge physicists to aim for more than "mere geometry" :wink: In a first or second year graduate course on gtr it is no doubt appropriate to urge students to learn to use tools from differential geometry and to avoid over-reliance upon using a specific coordinate chart. But in the followup course, which I guess is life as a physicist, I think it is important to bring back in physical considerations. In this case, to insist upon introducing only operationally significant notions of "distance in the large" and thus "velocity in the large".

See
Code:
www.physicsforums.com/showthread.php?t=407145
for a debunking of a new eprint arXiv which claims that Hagihara observers (those in circular orbits) "do not exist" [sic]. The connection is that IMO the authors fell into error in part by being way too impressed with their alleged exact solution of the geodesic equations using elliptic functions (an old idea, by the way). Even better, they ranted against people who use faulty approximation schemes to draw incorrect conclusions, but that seems to be exactly how they fell into error themselves! Arrrrghghgh!
 
Last edited:
  • #22
Conformal Compactifications aka Penrose-Carter charts

Re the thread "Penrose diagrams in general" started by User:Mersecke
Code:
www.physicsforums.com/showthread.php?t=422583
FWIW, the BRS thread "Penrose-Carter Conformal Compactifcations of Spacetimes"
Code:
www.physicsforums.com/showthread.php?t=403956
was intended precisely to help with such questions.

Mersecke said:
It is possible to draw "precise" well-defined Penrose diagrams for every spacetime?

This is a rather informal term, but fair short answer is "in principle, yes".

Given a spacetime (M,g), we seek local coordinates charts such that the line element is conformal to a flat spacetime line element (e.g. written in cylindrical or spherical chart). Such a chart will have the property that null geodesics will be appear as coordinate lines. Such a chart is not possible for most spacetimes, however.

But in the case of an asymptotically flat sheet, we can try to find a chart which makes radial null geodesics appear as coordinates lines in our chart. Penrose-Carter charts are such charts, in which we impose the additional requirement that the asymptotically flat sheet be mapped onto a compact region. In particular, we demand that "future null infinity, spacelike infinity, and past null infinity" all become loci which form part of the boundary of a compact region in our chart. More technically, we demand that these become sufficiently smooth regions in the "conformally compared metric" such that we can integrate over these "regions at infinity". This is important for, e.g., discussions of Bondi mass loss due to gravitational radiation.

Usually one finds two dimensional representations of the picture given by such charts, drawn "blockwise", for spacetimes which have asymptotically flat exterior sheets, including typical black hole solutions and simple cosmological models. More rare are discussions of such charts for say pp-wave spacetimes.

In such two dimensional representations, in the compact region representing an exterior asymptotically flat sheet, points represent two-spheres, and lines of slope +/-1 represent radial null geodesics. The projections of other inward going null goedesics (with angular dependence suppressed) will in general curve upwards from the line of slope -1 (radially inwardly going null geodesic). See the sketch below. Note that the curves issuing from the event E actually represent surfaces made of two-spheres (large areas near "conformal infinity", small areas near the event horizon and even smaller near the spacelike curvature singularity), so for a non-radial null geodesic C issuing from E, various events on C lie in various such spheres. One of the tricks in "reading" Penrose-Carter compactified charts involves remembering that "curves" stand for three dimensional submanifolds, or more generally, as here, for some curve on said three-dimensional submanifold.

Certainly one can transform to an explicit Penrose chart for the Kerr vacuum, and that is sketched in various textbooks. In general, one shouldn't expect to find transformations given entirely in terms of elementary functions, however--- even in the Schwarzschild case, the W function is needed.

Mersecke said:
Time-like linear lines on the Carter-Penrose diagram of the extended Schwarzschild space-time has some special meaning? They are not geodetic curves, are they?

In general, no.

JesseM said:
these geodesics all have finite length in the diagram

In both the original curved spacetime metric and the "conformally compared" flat spacetime, integrating along a null curve gives zero interval.

JesseM said:
I would guess that straight timelike worldlines in a Penrose diagram don't need to be geodesics.

Right, that would only happen in very special cases. I pointed out in the BRS thread cited above that in the Penrose diagram for Minkowski vacuum, some vertical line segments represent the world lines of Rindler observers, for example (nonzero constant path curvature, so non-geodesic curves!).

Figure:
  • sketch of absolute future of event E in an exterior sheet of (maximal analytic extension of) Schwarzschild vacuum; note that null geodesics issuing from E which approach but remain just outside the event horizon are likely to wind around the event horizon several times before escaping to future null infinity, so quite a bit can be going on the angular coordinates which have been supressed in this two-dimensional sketch!
 

Attachments

  • Schwarz_vac_Penrose_Light_cone_sketch.png
    Schwarz_vac_Penrose_Light_cone_sketch.png
    2.4 KB · Views: 470
Last edited:
  • #23
Alleged "paradox" for null geodesics in curved spacetimes

Re
Code:
www.physicsforums.com/showthread.php?t=424870
this poster appears to have rediscovered the notion of "conjugate points" (see Hawking and Ellis). This is not a paradox but just one more instance of the local (as in "local neighborhood" on a topological manifold) versus global distinction.

Try the link under "Propagation" (of massless radiation) in the BRS sticky thread "Some Useful Links for SA/Ms":
Code:
www.physicsforums.com/showthread.php?t=378653
The textbook by John Stewart, Advanced General Relativity, Cambridge University Press, 1990, is a graduate level introduction to the propagation of massless radiation in general relativity, using NP formalism and ending with a discussion inspired by "catastrophe theory" (terrible name for a lovely and perfectly rigorous theory, as applied here).

Figure:
  • World lines of two "photons" emitted at A which meet again at B
 

Attachments

  • Two_null_geodesics.png
    Two_null_geodesics.png
    1.9 KB · Views: 438
Last edited:
  • #24
BRS: Visualizing sheaves

In
Code:
www.physicsforums.com/showthread.php?t=425082
a newbie asked about visualizing sheaves.

The example any newbie should begin with is no doubt a sheaf of germs of holomorphic functions as in the theory of complex variables. There is an old A. M. Monthly expository article on sheaves which he can look for.

The notion of non-Hausdorff sheaves came up in my research on generalized Penrose tilings, where it turns out to be convenient for some purposes to consider tiling spaces as non-Hausdorff sheaves rather than as branched manifolds. The difference lies in the nature of the local neighborhoods; see the figure. (Well, there's more: the sheaves turn out to have all kinds of powerful formal properties.)

  • Comparing open neighborhoods on a branched manifold with a non-Hausdorff sheaf
 

Attachments

  • Open_Nghbds_BranchedManifold_Sheaf.png
    Open_Nghbds_BranchedManifold_Sheaf.png
    1.8 KB · Views: 546
Last edited:
  • #25


Chris Hillman said:
There is an old A. M. Monthly expository article on sheaves which he can look for.

J. Arthur Seebach, Linda A. Seebach, and Lynn A. Steen
"What is a Sheaf?"
American Mathematical Monthly, Sept. 1970: 683-703

Note that the sheaf of germs of holomorphic functions is somewhat unusual in that, as a topological space, it is Hausdorff. Many sheaves which arise naturally in other contexts are non-Hausdorff, and we must and should not be "afraid" of this, since this property is in fact just what we need.

The figure I provided above is intended to suggest how the non-Hausdorff reality differs from their Fig. 8; the authors remark that this is "difficult to visualize on Hausdorff paper", but I say, "not really!"). In my figure, at left, any open neighborhood of a "branching point" has a "Y shaped cross section", so our space is not a topological manifold there. At right, there are points P,Q which cannot be separated because any open neighorhoods of P,Q must overlap, so this space is non-Hausdorff.

Non-Hausdorff sheaves arise naturally in algebraic geometry (via the theory of local rings) and also in symbolic dynamics (the most abstract part of the theory of dynamical systems). Why does symbolic dynamics embrace the theory of tilings? Well, you can think of a space of two-dimensional tilings as a natural generalization of a space of "two dimensional sequences"; typically the first is acted upon by R^2 while the second is acted upon by Z^2, and similarly for higher dimensions.

One dimensional sequence spaces (acted upon by the group of integers, Z, under addition) offer a very simple example of nonHausdorff sheaves if the diagrams often used to define the so-called "shift spaces of finite type" are reinterpreted slightly. Shift spaces are spaces of binary sequences (typically) which are defined by placing constraints upon which symbols can follow other symbols. (Thus, they are a "topological" generalization of Markov chains; any shift space can be turned into a Markov chain by introducing a suitable Z-invariant probability measure defined on the sequence space.) A very simple example is the Golden Mean Shift, which consists of all infinite binary sequences in which a 1 must be followed by 0, but each 0 can be followed by either 0 or 1; see the figure below.

See Lind and Marcus, Symbolic Dynamics and Coding, Cambridge University Press, 1995, for more about symbolic dynamics.

Now unwrap the defining diagram and intrepret the new diagram as follows: replace zeros and ones by unit length tiles placed end to end on the real line, and interpret each vertex in the unwrapped diagram as the "transition" (going left to right) from one tile to the adjacent tile. This naturally gives rise to a non-Hausdorff sheaf. The difference between thinking of a one-dimensional tiling as a branched manifold vs. thinking of it as a non-Hausdorff sheaf is very simple: consider the locus where two tiles meet; then
  • branched manifold: the tiles meet at a common point
  • non-Hausdorff manifold: the tiles are disjoint but the right boundary of the tile at left and the left boundary of the adjacent tile at right consitute a pair of non-separable points.
Similarly for higher dimensional tiling spaces. I claim the sheaf viewpoint is more intuitive, when you think of constructing a tiling by placing copies of "prototiles" of some color and shape, because the sheaf viewpoint says that when tiles are placed adjacent to each other, some part of their boundaries are "in the same place" but retain their individual identity as belonging to one tile or the other.

This non-Hausdorff sheaf point of view is convenient because one of the most important phenomena in tiling theory is that it is possible that some "local patch" of tiles on some compact domain cannot be extended to a global tiling; for examples in the space of Penrose tilings, see Grunbaum and Shephard, Tilings and Patterns. From the point of view of sheaves, we are saying that some local sections cannot be extended to a global section. This point of view has close connections with important constructions in mathematical logic!

For more information, see Mac Lane and Moerdijk, Sheaves in Geometry and Logic : a First Introduction to Topos Theory, Springer, 1992.

This phenomenon, where local patches cannot be extended to global tilings, does not occur in finite type shift spaces (by definition), but it does occur in other shift spaces. For example, consider a "ribbon" of tiles in a two-dimensional Penrose tiling and recode it as a symbolic sequence. In this way you obtain various sequence spaces hiding inside the space of Penrose tilings. These turn out to be the opposite end of the spectrum from shifts of finite type in that such a space is the closure of the Z-orbit of any sequence in the space (to see this is not true for the Golden Mean shift, consider the sequence consisting of all zeros). The full space of Penrose tilings has a similar character, and (in the non-Hausdorff sheaf point of view) it admits many local sections which cannot be extended to global sections.

Additionally, when you think of sequence or tiling spaces as non-Hausdorff sheaves, they are objects in a category with powerful formal properties: e.g. not only do pullback squares and pushout squares exist (thus, product and coproducts) but also exponential objects (i.e. the space of morphisms from one object to another forms another object in the category) and a classifying object, in the sense of topos theory. In fact, these categories form elementary topoi.

Figures:
  • the Golden Mean Shift Space; at left, the graph which defines it by giving constraints on which symbols can follow each other; on the right, an "unwrapping" of this graph; a particular binary sequence in the Golden Mean Shift Space corresponds to a "threading" from left to right of this unwrapping--- or, from the sheaf viewpoint, a global section.
 

Attachments

  • Golden_Mean_Shift.png
    Golden_Mean_Shift.png
    1,010 bytes · Views: 531
Last edited:
  • #26
BRS: isotropy subgroup of the torus

In
Code:
www.physicsforums.com/showthread.php?t=425222
some posters seem to have misunderstood what "a friend" read in the Lee, Riemannian Manifolds: an Introduction to Curvature, Springer, 1997. Additionally, they have not yet grasped the local versus global distinction. Nor do they yet appear to understand the distinction between topological and Riemannian manifolds.
 
  • #27
BRS: spherically symmetric metrics, plus: connection and curvature

In the thread "spherically symmetric metric form"
Code:
www.physicsforums.com/showthread.php?t=423660
User:mersecske asked
mersecske said:
And Kruskal-Szekeres or Eddington-Finkelstein like coodinates can be used in general to any spherically symmetric metric? What is the form of a general spherically symmetric metric, which is not singular anywhere?
Oh no, yet another possible confusion between curvature singularities (and other geometric singularities) with coordinate singularity! If he insists upon lack of curvature singularities in a Lorentzian manifold which arises as a solution to the EFE, such things are rare.

Ignoring that issue: good question. And if memory serves, there is a nice discussion of spherical symmetry in Plebanski and Krasinski, Introduction to General Relativity and Cosmology, Cambridge University Press, 2006, which would be a good place to start reading.

Here's a thought off the top of my head: spherical symmetry of a spatial hyperslice (which we could demand be free of curvature singularites) should mean a foliation of a Riemannian 3-manifold by nested 2-spheres, without excluding the possibility of multiple "throats". Suppose you try do embedd your manifold as a spherically symmetric surface in E^4. Then, if you can embedd it as indicated in the surface at left in the figure below (one angular coordinate suppressed in the figure), you can use the "height" variable in the embedding as a monotonically increasing label of the nested spheres, which is all you need for a "radial coordinate". (Of course, this coordinate will have no relation to any reasonable notion of "radial distance"!) But if you try this with a surface which "curls over" when you try to embedd it with spherical symmetry, oh no!

(But if you are lucky, and if, in the embedding on the right in the figure below, the horizontal tangent plane "makes high order contact" with the embedded surface, then I think I see a trick which could still give you a nice chart if you think of your manifold as C^k rather than smooth.)

Assuming no difficulties arise in defining a smooth monotonic function labeling the nested spheres (which we can use as our "radial coordinate"), then the next question is: in a spherically symmetric spacetime with such hyperslices, can you construct a Painleve type chart?

In the thread "Potentials, connections and curvature"
Code:
www.physicsforums.com/showthread.php?t=425742
orbb asked
orbb said:
I have a question related about the relation between potentials, connections and curvature in gauge theories.
The answer he seeks is that in the gauge theory approach, in both EM and gtr, the "potential" is a connection and the curvature is constructed from the connection to form the "field strength".

But of course there is a tricky point regarding how this works in gtr; he can look for Usenet posts by Steve Carlip which explain this point very clearly.

Figures:
  • Sketch of two spherically symmetric Riemannian 3-manifolds partially embedded in E^4 as spherically symmetric hypersurfaces (one angular coordinate suppressed in the figure!); can we use "height" as a "radial coordinate"? Note that both manifolds feature multiple "throats".
 

Attachments

  • sph_sym_surface_embedded.png
    sph_sym_surface_embedded.png
    1.7 KB · Views: 386
Last edited:
  • #28
BRS: Stephani fluids, or, User:AWA needs a time out

Re AWA's Post #95 in "The big bang"
Code:
https://www.physicsforums.com/showpost.php?p=2862367&postcount=95
AWA (yelling at Chalnoth) said:
There is a growing bibliography(Sylos-Labini, Pietronero,Mittal,Barrett) with very good observational support that points to a fractal structure of the universe on large scales. And a fractal dispositon of matter may indeed be isotropic and not homogenous. Ever heard of Mandelbrot "conditional cosmological principle?
Not only that, there is a whole family of spacetimes (Stephani) that includes the FRW universes that also allows inhomogenous isotropic solutions. You call yourself "science advisor"?. Why do yo make such categorical assertions when they are not backed up by sound science? That shows either ignorance if you don't know or dishonesty if you choose to ignore those facts that disprove your arguments.

Oh brother. Needless to say, I have reported AWA for violating the proscription against ad homs. Unfortunately for AWA, he/she is not even correct in his claims:
  • Claims about "fractal universe" are very old (Benoit Mandelbrot gave an interesting survey starting with the ideas of Fournier which actually helped inspire BM in his Fractal Geometry of Nature, first edition), but to date, I think it is fair to say that claims about observational support tend to be overstated.
  • Quite a few "fractal universe" models out there are flat out cranky, and their proponents tend towards a blind faith in their favorite models.
  • AWA hasn't defined what he/she means by "fractal disposition of matter", and in fact there are many things one could mean by that. One easy notion is that the density function of a dust (noninteracting massive matter in the context of gtr) should be supported on a fractal set, wrt an appropriate measure consistent with our Lorentzian manifold structure.
  • The Stephani perfect fluids are a well known family of cosmological models which are in general both inhomogeneous and anisotropic, but they have nothing whatever to do with "fractal cosmology"; these models are defined using several arbitrary smooth functions of one variable, and the density and pressure of the perfect fluid varies smoothly from one event to another. In a "cartesian" type chart, the line element can be written
    [tex]
    \begin{array}{rcl}
    ds^2 & = &
    \frac{1}{V^2} \; \left( \frac{-9 \, \dot{V}^2}{f_6^2} \; dt^2
    + dx^2 + dy^2 + dz^2 \right) \\
    && V = f_4 + x \, f_1 + y \, f_2 + z \, f_3 + \, (x^2+y^2+z^2)/2 \, f_5 \\
    && 0 < t < \infty, \; -\infty < x, \, y, \, z < \infty
    \end{array}
    [/tex]
    where [itex]f_1, \, f_2, \, f_3, \, f_4 ,\, f_5, \, f_6[/itex] are six arbitrary smooth functions of t, and dot denotes differentiation wrt t. They do include the FRW models as a special case. Notice that in the given chart, the integral curves of the coordinate vector [itex]\partial_t[/itex] do give the world lines of fluid particles (because the Einstein tensor has the correct diagonal form in the given frame), but t is not the elapsed proper time measured by an observer riding with a fluid particle. See eq (37.45) in Stephani et al., Exact Solutions to the Einstein Field Equations, Cambridge University Press, 2nd edition, 2003.
  • There are many exact solutions (with the nature of cosmological models, featuring an initial Big Bang type strong spacelike curvature singularity) which are homogeneous but anisotropic. The family of Stephani fluids include examples of this, but a simpler example is given by the plane symmetric Kasner dust
    [tex]
    \begin{array}{rcl}
    ds^2 & = & -dt^2 + (t-K)^{4/3} \; (dx^2+dy^2)
    + \frac{(t+K)^2}{(t-K)^{2/3}} \; dz^2 \\
    && K < t < \infty, \; -\infty < x, \, y, \, z < \infty
    \end{array}
    [/tex]
    where [itex]K > 0[/itex] is a constant. (This dust solution has a Weyl tensor of Petrov type D. It is a special case of a more general two paramter family of Kasner dusts, which plays a role in motivating the BKL conjecture. There is a nice discussion of the Kasner dusts in Hawking and Ellis, Large Scale Structure of Spacetime, Cambridge University Press, 1972.) AWA claimed there exist models which are inhomogeneous but isotropic, but that can't occur in the context of smooth Lorentzian manifolds. (See for example Peacock, Cosmological Physics, section 3.1.)
Is it just me, or does it seem as though several PF users, who happen to be regular proponents of misguided fringe viewpoints, have just gotten back to school and are for some reason in a very foul mood?

In any case, for those of you who use Maxima, here is a Ctensor file you can run in batch mode under wxmaxima which will compute the density and pressure of a Stephani fluid:
Code:
/* 
Stephani perfect fluid; comoving cartesian chart; nsi coframe 

The Stephani fluid is a nonlinear perturbation of the FRW models.
It is both inhomogenous and anisotropic.

Note the solution is defined by six -arbitrary- functions of t
The function V is given by
subst( f4+f1*x+f2*y+f3*z+(x^2+y^2+z^2)/2*f5, V, %);
*/
load(ctensor);
cframe_flag: true;
ratchristof: true;
ctrgsimp: true;
/* define the dimension */
dim: 4;
/* list the coordinates */
ct_coords: [t,x,y,z];
/* define background metric */
lfg: ident(4);
lfg[1,1]: -1;
/* declare the dependent and independent variables */
depends([f1,f2,f3,f4,f5,f6],t);
/* define the coframe */
fri: zeromatrix(4,4);
fri[1,1]: -3*diff(f4+f1*x+f2*y+f3*z+(x^2+y^2+z^2)/2*f5,t)/(f4+f1*x+f2*y+f3*z+(x^2+y^2+z^2)/2*f5)/f6;
fri[2,2]:  1/(f4+f1*x+f2*y+f3*z+(x^2+y^2+z^2)/2*f5);
fri[3,3]:  1/(f4+f1*x+f2*y+f3*z+(x^2+y^2+z^2)/2*f5);
fri[4,4]:  1/(f4+f1*x+f2*y+f3*z+(x^2+y^2+z^2)/2*f5);
/* 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(false);
/* Compute R^(mijk) */
uriemann(false);
/* Compute Ricci componets R_(jk) */
ricci(false);
/* 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]]);
/* Weyl tensor shows conformally flat */
/* Construct NP tetrad for our frame, compute Weyl spinors and Petrov type */
nptetrad(true);
weyl(true);
Not a fractal in sight!

And here is a Ctensor file for the Kasner dusts:
Code:
/* 
Kasner dust with E^2 symmetry; comoving cartesian chart; nsi coframe 

This is a homogenous but anisotropic dust solution.
Four dimensional Lie algebra of Killing vector fields
	@_x, @_y, @_z
	-y @_x + x @_y
*/
load(ctensor);
cframe_flag: true;
ratchristof: true;
ctrgsimp: true;
/* define the dimension */
dim: 4;
/* list the coordinates */
ct_coords: [t,x,y,z];
/* 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-K)^(2/3);
fri[3,3]:  (t-K)^(2/3);
fri[4,4]:  (t+K)/(t-K)^(1/3);
/* NOTE WELL declare K to be a constant */
/* NOTE WELL cannot use k since that is already defined */
declare(K, constant);
/* 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) */
factor(lg);
/* compute g^(ab) */
ug: factor(invert(lg));
christof(false);
/* Compute fully covariant Riemann components R_(mijk) = riem[i,k,j,m] */
lriemann(false);
/* Compute R^(mijk) */
uriemann(false);
/* Compute Ricci componets R_(jk) */
ricci(false);
/* 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 */
factor(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 */
factor(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 */
factor(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]]));
/* 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();
/* geodesic equations */
cgeodesic(true);
 
Last edited:
  • #29
Multiple confusions in "curved space/LP spaces"

In "curved space/LP spaces"
Code:
www.physicsforums.com/showthread.php?t=425988
granpa asked
granpa said:
My question is what happens in the curved space of a black hole? Is it like the curved surface of the Earth or like the intrinsic curvature (no, that probably isn't right. maybe 'intrinsic distortion from L2 space) of LP space? Does a small enough piece of space near a black hole behave like regular flat L2 space (yes i know, technically it would be minkowski space but i am only interested in the space component and nonrelativistic speeds)
Unfortunately, someone answered with nonsense
nickthrop101 said:
no one knows the answer to that question, mathmatically then yes it should work it depends if you are past the event horizon ( point of no return) yet becouse at the center of a black hole, the singularity, then there is said to be infinate mass and therefore infinate curviture, so physically there you could not work like that :)
and granpa apparently accepted the misinformation.

I guess this illustrates the dangers of persons who know too little to appreciate how little they know, or even how little some others know.

L^p spaces occur naturally in functional analysis, the study of linear operators on infinite dimensional vector spaces. You should think of L^p spaces and Hilbert spaces as spaces of operators. Hilbert spaces behave much like finite dimensional euclidean inner product spaces; in particular, they have a notion of euclidean inner product and hence euclidean angle. L^p spaces are normed vector spaces but not Hilbert spaces; in particular, while they have a concept of "norm" they lack any concept of angle. See Yosida's textbook for an interesting discussion of "almost orthogonality" in normed vector spaces, however.

In addition, Lorentzian inner products are significantly different from euclidean inner products. The latter are in the language of algebra, positive definite bilinear forms, while the former are indefinite bilinear forms. Geometrically that has profound consequences: for example, a curve can turn around and intersect itself in euclidean space but a timelike curve cannot turn around and intersect itself in Minkowksi space!

Spaces of operators do arise in gtr when one studies functions on manifolds, QFTs on manifolds, and so forth. But granpa is asking about the geometry of spatial hyperslices (orthogonal to static observers?--- he doesn't say) just outside the horizon. Thus, a good short answer is "there is no connection between the geometry of L^p spaces and the kind of geometry you are asking about".

In
Code:
www.physicsforums.com/showthread.php?t=423695
nickthrop101 said:
there is no scientific proof of white hole, and it has been proven that black holes do release a bit if energy in the form of heat, about a nano degree above absolute zero, so energy that is absorbed in a black hole doese return to our universe, also by the laws of concervation then it is impossiable for a universe to loose energy but to only convert it into other forms
Wow, nicktrhop101 seems to be trying to say that black holes have been proven to emit Hawking radiation. If that were true, Hawking would have won a Nobel Prize.

Incidently, Hawking uses "white hole" to denote the time reversal of a black hole formed by gravitational collapse (e.g. in an OS type model), whereas older literature mostly uses this term to denote the past interior of the so-called "eternal hole", or "Kruskal-Szekeres spacetime", or more properly, the maximal analytical extension of the local solution found on the exterior region by Schwarzschild 1916.
 
  • #30
Hi Chris,

In this post:
https://www.physicsforums.com/showpost.php?p=2862816&postcount=107

Anamitra tries a very klugey way to develop a coordinate system defined along some arbitrary path in which the Christoffel symbols along that path are all 0 so the covariant derivative is equal to the ordinary derivative and therefore parallel transported vectors have constant coordinates along that path.

What she wants to do is legitimate (although not what she thinks it means), but she is not doing it right. Unfortunately, I don't know the correct way to do it or what it is even called.

Without knowing better, the way I would approach it would be to construct an orthonormal basis at the beginning of the path and then parallel transport those basis vectors along the path. That way, at any point along the path you would have a set of basis vectors which define a coordinate system where parallel transported vectors have constant components, as well as an easy way to transform back to the original coordinates.

Is there a better way, and does it have a name that I can look up and study?
 
  • #31
Anamitra: still wrong after all these years

... it feels like years, anyway, and I'm not even talking to this poster directly, so you all have more fortitude than I!

"Anamitra": I'm reading that as one word, not "Ana Mitra". Unless you have reason to believe otherwise, I presume User:Anamitra is male. (In at least one post, he seems to sign himself " Anamitra Palit".) I have to say, I also think there is a good chance User:Anamitra is trolling PF, although I don't know what his motivation might be. But even if not, posters who are clueless in math/physics are often clueless regarding malware too, so especially likely to pass on a nasty infection. So I wouldn't open any pdfs from User:Anamitra under any circumstances--- pdf is one of the most common vectors for malware of all kinds, and to some extent the dangers are platform independent. 'Nuff said.

Some quick comments on Anamitra's post:

He is using +--- signature and pointlessly putting m=1/2, but I'll use -+++ signature and write
[tex]
ds^2 = -(1-2m/r) \, dt^2 + \frac{dr^2}{1-2m/r} + r^2 \, d\Omega^2
[/tex]
Then he considers the signature -++ submanifold [itex]r=r_0[/itex], where [itex]r_0 > 2m[/itex] (a coordinate cylinder in the Schwarzschild chart in the exterior region, if you like):
[tex]
ds^2 = -(1-2m/r) \, dt^2 + r_0^2 \, d\Omega^2
[/tex]
Then he appears to claim that a coordinate transformation takes this to the cartesian form
[tex]
ds^2=dT^2 - dx^2-dy^2
[/tex]
But this would only be possible if the coordinate hypercylinder [itex]r=r_0[/itex] has vanishing curvature tensor, since only a locally flat manifold can be given such a cartesian chart. But this submanifold actually has nonzero curvature tensor (easily checked with Ctensor under Maxima or GRTensorII under Maple). So, he's already made a fatal error. (He seems to think that because [itex]r=r_0[/itex] appears in the Schwarzschild chart as a "hypercylinder", it is locally flat, but that is wrong.)

In his other remarks, he seems to think he is constructing a path and parallel transporting a frame along a path, but seems confused about what the -++ submanifold [itex]r=r_0[/itex] has to do with that, and in any case, he clearly has no idea how to work with covariant derivatives.

So Anamitra is doing this all wrong, but to be fair, when I looked over some well known textbooks hoping for a quick cite, I found that none of them say very much about how to do parallel transport in practice! So I've added this to my to do list for the BRS.

Dalespam said:
Without knowing better, the way I would approach it would be to construct an orthonormal basis at the beginning of the path and then parallel transport those basis vectors along the path. That way, at any point along the path you would have a set of basis vectors which define a coordinate system where parallel transported vectors have constant components, as well as an easy way to transform back to the original coordinates.

That's the idea, yes! In particular, it suffices to say how to parallel transport any frame vector along a given path, in order to know how to parallel transport any vector along that path. The connection one-form is defined exactly to say how to parallel transport the frame vectors along any path.

Until I can try to explain this properly, some remarks off the top of my head:

Ultimately, parallel transport of a vector [itex]\vec{X} \in T_p M[/itex] based at some point p along a timelike or spacelike curve with unit tangent vector [itex]\vec{U}[/itex] amounts to solving an initial value problem
[tex]
\nabla_{\vec{U}} \vec{X} = 0, \; \; \vec{X}(0) = \vec{X}_p
[/tex]
That should make us think of integrating a one-form taking values in a Lie algebra, which upon exponentiation gives an element in a Lie group. In a Riemannian n-fold, this Lie group is SO(n), and in a Lorentzian 4-fold, it is SO_+(1,3). (These are simply connected Lie groups.) That is, we should think of an element of so(n) as an "infinitesimal rotation" and an element of so(1,n-1) as an "infinitesimal boost/rotation", whose exponential is an element of the proper orthochronous Lorentz group.

Pursuing this line of thought leads to a picture of (in the Riemannian case) an SO(n)-bundle over M, in which the connection tells how to move "vertically" along the fibers as we move along a curve in the base space M. See figure below, and see Chapter 7 of Isham, Modern Differential Geometry for Physicists, World Scientific, 1999.

As you would probably expect, following Elie Cartan, I claim that the easiest way of computing parallel transport in practice is to use frame fields. In textbooks which discuss frame fields in just enough detail to make them seem like inscrutable beasts, but not enough detail to reveal them as the simple-minded creatures they really are*, you'll see that the usual expression for the Levi-Civita connection in terms of a coordinate basis (an alternating sum of three first derivatives of the metric) must be supplemented by another three terms. This makes things look more complicated, but it is only because when we define a frame field, we have the freedom to choose how to rotate the frame as we move smoothly from place to place. And we can use that freedom to simplify problems!

*Cite suppressed to protect the guilty? No, I just can't think of one right now!

For example, in the familiar case of parallel transport of a frame around a spherical triangle covering an octant of the unit sphere (see the figure below), the standard frame
[tex]
\begin{array}{rcl}
\vec{e}_1 & = & \partial_\theta \\
\vec{e}_2 & = & \frac{1}{\sin(\theta)} \; \partial_\phi
\end{array}
[/tex]
in the trig chart
[tex]
ds^2 = d\theta^2 + \sin(\theta)^2 \, d\phi^2, \; \;
0 < \theta < \pi, \; -\pi < \phi < \pi
[/tex]
is already parallel transported around the loop we want, with a mismatch at the origin where the frame (and the chart) is not defined. So in this case Cartan's structure equation
[tex]
d\vec{e}_k = \vec{e}_j \, {\omega^j}_k
[/tex]
(think right multiplication of a row vector by a matrix), which says how to parallel transport the frame vectors using the connection one-forms--- in our example we have only one to worry about, [itex]{\omega^1}_2 = -\cos(\theta) \; d\phi[/itex]--- says "just follow the frame!"

(I realize that sounds just a bit like what Anamitra may have been trying to say, but I really don't see how to massage his post into a munged version of what I just said.)

We should really think of the "connection one-forms" as a single one-form taking values in the Lie algebra so(n), or in our example so(2):
[tex]
\left[ \begin{array}{cc}
0 & -\cos(\theta) \, d\phi \\
\cos(\theta) \, d\phi & 0
\end{array} \right]
[/tex]
Then, in our simple example, [itex]\phi[/itex] is constant on the first and third arcs, and the cosine vanishes on the equator. Even though we didn't rotate wrt the frame vectors as we traveled around this clockwise arc, since the frame itself has a mismatch at the origin (by construction), the result is a net clockwise rotation by one quarter turn. Which happens to agree (not by chance!) with the result of integrating the constant curvature one over the spherical triangle!

In a slightly more elaborate example, apparently the one Anamitra had in mind, replace the middle arc by a nongeodesic arc on the latitude [itex]\theta=\pi/4[/itex] and shorten the first and last arcs appropriately (strictly speaking, no longer a spherical triangle because one side is not a geodesic arc). Then the cosine factor is a nonzero constant on that arc, and we can integrate to obtain a nonzero counterclockwise rotation by a certain fraction of a quater turn wrt the frame vectors. The result when we transport a vector around in the new clockwise closed loop is a net clockwise rotation by another fraction of a quarter turn.

So in the example of the unit sphere using trig chart, the connection one-form [itex]{\omega^1}_2 = -\cos(\theta) \, d\phi[/itex] is telling us that the given frame is parallel transported when we move along radial lines (great circle arcs) and also when we move along the equator (also a great circle arc), but when we move along a general latitude (not a geodesic arc), we have a rotation rate given by exponentiating an element of the Lie algebra so(2) to obtain an element of the Lie group SO(2).

In a positively curved surface, parallel transport around a clockwise loop results in a clockwise rotation. In a negatively curved surface, parallel transport around a clockwise loop results in a counter-clockwise rotation. In a surface with curvature positive in some places and negative in others, you have to integrate the curvature over the region bounded by the loop, using the appropriate volume form, to see which sign wins.

If I am making this sound hard, that is only because I haven't yet thought very hard about how to explain it!

I recommend Flanders, Differential Forms with Applications to the Physical Sciences, Dover reprint, org. published 1963, even though Flanders doesn't mention the formula I referred to above, because this book offers a brief and intuitive introduction.

I can't resist adding that in Cartanian geometry, the common generalization of Riemannian geometry and Kleinian geometry, we allow more interesting fibers than just SO(n) (Riemannian geometry) or SO(1,n-1) (Lorentzian geometry). For example, we can allow the group G in our G-bundle to include euclidean homotheties as well as rotations. Then parallel transport in a loop can result in a vector coming back rescaled as well as rotated! That was pretty much Weyl's original gauge theory, introduced as a (failed) attempt to unify classical electromagnetism and gtr. We also obtain a notion of the curvature of a connection which gives an appropriate notion of "locally flat" manifold for such a G-geometry.

Figures (left to right):
  • Schematic picture of parallel transport in the bundle picture (Riemannian case): fibers are copies of SO(n), base space is Riemannian manifold (M,g), parallel transport in a loop generally results in a nonzero rotation of a vector carried around the loop.
  • Simple example of parallel transport of a frame around a loop on unit sphere
    • segment (great circle arc) [itex]\phi=\pi/2[/tex] from origin to [itex]\theta=\pi/2, \; \phi=\pi/2[/itex]
    • quarter circle arc on equator (great circle arc)
    • segment (great circle arc) back to origin
 

Attachments

  • parallel_transport_bundle.png
    parallel_transport_bundle.png
    1.9 KB · Views: 408
  • S2_parallel_transport_octant.png
    S2_parallel_transport_octant.png
    1.2 KB · Views: 366
Last edited:
  • #32


Chris Hillman said:
"Anamitra": I'm reading that as one word, not "Ana Mitra". Unless you have reason to believe otherwise, I presume User:Anamitra is male. (In at least one post, he seems to sign himself " Anamitra Palit".)
That is certainly possible. I was also seeing it as one word, but I am fluent in Spanish, so if you end a word in "a" I automatically think female.


Chris Hillman said:
So Anamitra is doing this all wrong, but to be fair, when I looked over some well known textbooks hoping for a quick cite, I found that none of them say very much about how to do parallel transport in practice! So I've added this to my to do list for the BRS.
Yes, I found that in the case of Wald's book. The discussion about parallel transport was not very practical nor were the homework problems. I want Anamitra to do some practical examples so that she can generate her own counterexamples to her claims.


Chris Hillman said:
That's the idea, yes! In particular, it suffices to say how to parallel transport any frame vector along a given path, in order to know how to parallel transport any vector along that path.
So is there a correct name for this process? E.g. "Hillman frame transport". It sure would help to have a reference that I can point Anamitra to. I can't tell her that what she is doing is wrong, but if I can't muster some authority on how to do it right I doubt that she will accept any correction.
 
  • #33
Unfortunately I do not have at hand a copy of Lee, Riemannian manifolds, but that might be a good place to look for a chapter and section citation. I would expect that he would offer good intuition and exercises. Maybe some kind SA/M with a copy of that book can post the relevant section?

Lorentzian manifolds are bit different from Riemannian manifolds in how the connection works, but fortunately this is one case where the differences are fairly minor. The main point--- slurred over in MTW, incidently--- is that to find the connection one-form, you need to use the Lorentzian adjoint rather than the euclidean adjoint in order to determine the one-form from Cartan's equation "by guessing". In MTW that shows up in stuff like
[tex]
{\omega^1}_2 = {\omega^2}_1, \; \;
\hbox{but} \; {\omega^2}_3 = -{\omega^3}_2
[/tex]
Compare the straightforward skew-symmetry of [itex]{\omega^j}_k = -{\omega^k}_j[/itex] in the Riemannian case: in both cases, the connection form is skew-adjoint (so that the exponential is an isometry), the difference is only whether you use euclidean or Lorentz adjoint, which is determined by the respective bilinear forms used on to endow the tangent spaces with an inner product.

Let me see if I can come up with some other section citations to some widely available textbooks. Then at least you or a mentor can cut off the discussion by advising Anamitra to go away and study, which we all agree he needs to do. Although, he's been so insistent about doing things all wrong that I am not optimistic he will heed advice to start over and learn it right from a good book. Another thing he obviously needs to pay more attention to is learning enough LaTex to use the VB tools available at PF instead of uploading pdfs, which I strongly feel should be proscribed at PF, for security reasons if none other--- but 'nuff said.

Terminology: if it helps, I have been discussing the Levi-Civita connection defined by a Riemannian or Lorentzian structure, as a special case of a Cartan connection (the version for which Cartan's approach to the curvature of the connection is most straightforward), also as a special case of a Kozul connection (the one discussed in most modern differential geometry textbooks). I also mentioned the principle G-bundle over M where G is the isotropy group of M, in which the Cartan connection is seen as a g-valued one-form where g is the Lie algebra of the Lie group G, and the curvature is a g-valued two-form.

Other notational issues: sign conventions can get tricky depending on signature, left or right invariant forms, conventions for defining the Riemann tensor, and so forth.
 
  • #34
Some references for connection and parallel transport

Some textbook explanations (to be fair, I must say that they tend to be rather murky!) of why Anamitra is mathematically incorrect regarding the issue of path dependence of parallel transport:
  • Carroll, Spacetime Geometry, Fig. 3.2 "On a curved manifold, the result of parallel transport can depend on the path taken". (The figure illustrates the spherical triangle covering one octant of the sphere.)
  • Wald, General Relativity, Fig. 3.2 "In the case shown here of a closed curve consisting of three mutually orthogonal segments of great circles, the vector comes back rotated by 90 degrees".
  • not really a textbook, but FWIW, Penrose, Road to Reality, Fig. 14.4, "path dependence of parallel transport".

The best textbook I can find for Anamitra might be Millman and Parker, Elements of Differential Geometry, Prentice-Hall, 1977, whose cover features a picture of parallel transport on the sphere! See Section 4.6 and Fig. 4.16 in particular which shows a vector transported around the latitude [itex]\theta=\pi/3[/itex] returning rotated. (One reason I suspect Anamitra of trolling is that I sense he has already seen this or similar discussion, in which case he presumably knows well that he is spouting nonsense, so be careful.) Part of the motivation for this recommendation is that I think first studying parallel transport on surfaces in E^3 should motivate confused but open-minded students to accept the way connections are defined in abstract Riemannian geometry.

Some textbook exercises:
  • Ohanian and Ruffini, Gravitation and Spacetime, Exc. 4 on p. 313: asks the reader to compute Christoffel symbols for sphere in trig chart and verify the same two properties I mentioned in previous post (parallel transport along meridians and equator)--- not very helpful since those are the trivial cases!

I normally recommend for weak students that they learn some curve and surface theory in E^3 from Struik, Lectures on Classical Differential Geometry, before tackling Riemannian and Lorentzian geometry. But this book doesn't discuss parallel transport at all! Ditto Lipschitz, Differential Geometry, Schaum Outline series, McGraw Hill, 1969.

Then I reached for that lovely baroque survey, Berger, A Panorama of Riemannian Geometry (and here we really do mean Riemannian geometry, since the best stuff in this book doesn't carry over to Lorentzian geometry), but his treatment of parallel transport is murkier than I hoped. Ditto (shock!) for the five volume masterpiece of Spivak, A Comprehensive Introduction to Differential Geometry.

For mathematically mature readers seeking additional insight and computational skills, Chapter 10 in Misner, Thorne, and Wheeler, Gravitation. See Section 11.4 for parallel transport in a closed curve.

For a deeper appreciation, concise surveys include
  • Chapters 9, 15, 18, 19 of Frankel, Geometry of Physics, Cambridge University Press, 1997.
  • Chapters 7, 9, 10 of Nakahara, Geometry, Topology, and Physics, IOP, 1990.

Making the frame field way seem harder than the coordinate basis way, when in fact other way around: the frame vector fields are simply vector fields, so their commutators are defined and since the frame vectors span each tangent space we can expand the commutators to define the commutation coefficients
[tex]
[ \vec{e}_j, \; \vec{e}_k ] = \vec{e}_\ell \; {C^\ell}_{jk}
[/tex]
(Thus, these commutation coefficients are antisymmetric: [itex]{C^\ell}_{jk} = -{C^\ell}_{kj}[/itex].) The dual coframe one-forms are [itex]\sigma^j[/itex]. Then Cartan's first structure equation is
[tex]
d\sigma^j = -{\omega^j}_k \wedge \sigma^k
[/tex]
where--- too many opportunities to make a serious error if I don't check everything carefully, so let me get back to you.
 
Last edited:
  • #35
Chris Hillman said:
"Anamitra": I'm reading that as one word,

This is correct; Anamitra is a South-Asian name.

http://in.linkedin.com/pub/anamitra-palit/5/28b/1b6

He has a history of fringe stuff.
Chris Hillman said:
Unfortunately I do not have at hand a copy of Lee, Riemannian manifolds, but that might be a good place to look for a chapter and section citation. I would expect that he would offer good intuition and exercises.

Pages 60-62.
Chris Hillman said:
Some explicit textbook explanations of why Anamitra is mathematically incorrect regarding the issue of path dependence of parallel transport:
  • Carroll, Spacetime Geometry, Fig. 3.2 "On a curved manifold, the result of parallel transport can depend on the path taken". (The figure illustrates the spherical triangle covering one octant of the sphere.)
  • Wald, General Relativity, Fig. 3.2 "In the case shown here of a closed curve consisting of three mutually orthogonal segments of great circles, the vector comes back rotated by 90 degrees".

For mathematically mature readers seeking additional insight and computational skills, Chapter 10 in Misner, Thorne, and Wheeler, Gravitation. See Section 11.4 for parallel transport in a closed curve.

For a deeper appreciation, concise surveys include
  • Chapters 9, 15, 18, 19 of Frankel, Geometry of Physics, Cambridge University Press, 1997.
  • Chapters 7, 9, 10 of Nakahara, Geometry, Topology, and Physics, IOP, 1990.

Another book worth looking at is Differential Geometry and Lie Groups for Physicists by Marian Fecko,

https://www.amazon.com/dp/0521845076/?tag=pfamazon01-20,

which has an unusual format. From its Preface,
A specific feature of this book is its strong emphasis on developing the general theory through a large number of simple exercises (more than a thousand of them), in which the reader analyzes "in a hands-on fashion" various details of a "theory" as well as plenty of concrete examples (the proof of the pudding is in the eating).

The book is reviewed at the Canadian Association of Physicists website,

http://www.cap.ca/BRMS/Reviews/Rev857_554.pdf.

From the review
There are no problems at the end of each chapter, but that's because by the time you reached the end of the chapter, you feel like you've done your homework already, proving or solving every little numbered exercise, of which there can be between one and half a dozen per page. Fortunately, each chapter ends with a summary and a list of relevant equations, with references back to the text. ...

A somewhat idiosyncratic flavour of this text is reflected in the numbering: there are no numbered equations, it's the exercises that are numbered, and referred to later.

Personal observations based on my limited experience with the book:

1) often very clear, but sometimes a bit unclear;
2) some examples of mathematical imprecision/looseness, but these examples are not more densely distributed than in Nakahara;
3) the simple examples are often effective.

S^2 is used for a concrete example of parallel transport in exercises 15.3.8, 15.3.9, and 15.3.10.
 
Last edited by a moderator:

Similar threads

  • Special and General Relativity
Replies
12
Views
356
  • Special and General Relativity
Replies
9
Views
987
  • Special and General Relativity
Replies
8
Views
992
  • Special and General Relativity
Replies
3
Views
1K
  • Special and General Relativity
Replies
9
Views
1K
  • Special and General Relativity
Replies
18
Views
1K
  • Special and General Relativity
2
Replies
43
Views
2K
  • Special and General Relativity
Replies
5
Views
2K
  • Special and General Relativity
Replies
25
Views
2K
Replies
1
Views
337
Back
Top