BRS: Random Comments on Some Recent PF Threads

[Chris Hillman]

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
<br /> ds^2 = -(1-2m/r) \, dt^2 + \frac{dr^2}{1-2m/r} + r^2 \, d\Omega^2<br />
Then he considers the signature -++ submanifold r=r_0, where r_0 &gt; 2m (a coordinate cylinder in the Schwarzschild chart in the exterior region, if you like):
<br /> ds^2 = -(1-2m/r) \, dt^2 + r_0^2 \, d\Omega^2<br />
Then he appears to claim that a coordinate transformation takes this to the cartesian form
<br /> ds^2=dT^2 - dx^2-dy^2<br />
But this would only be possible if the coordinate hypercylinder r=r_0 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 r=r_0 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 r=r_0 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.

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 \vec{X} \in T_p M based at some point p along a timelike or spacelike curve with unit tangent vector \vec{U} amounts to solving an initial value problem
<br /> \nabla_{\vec{U}} \vec{X} = 0, \; \; \vec{X}(0) = \vec{X}_p<br />
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
<br /> \begin{array}{rcl}<br /> \vec{e}_1 &amp; = &amp; \partial_\theta \\<br /> \vec{e}_2 &amp; = &amp; \frac{1}{\sin(\theta)} \; \partial_\phi<br /> \end{array}<br />
in the trig chart
<br /> ds^2 = d\theta^2 + \sin(\theta)^2 \, d\phi^2, \; \;<br /> 0 &lt; \theta &lt; \pi, \; -\pi &lt; \phi &lt; \pi <br />
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
<br /> d\vec{e}_k = \vec{e}_j \, {\omega^j}_k<br />
(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, {\omega^1}_2 = -\cos(\theta) \; d\phi--- 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):
<br /> \left[ \begin{array}{cc} <br /> 0 &amp; -\cos(\theta) \, d\phi \\<br /> \cos(\theta) \, d\phi &amp; 0<br /> \end{array} \right]<br />
Then, in our simple example, \phi 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 \theta=\pi/4 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 {\omega^1}_2 = -\cos(\theta) \, d\phi 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) \phi=\pi/2[/tex] from origin to \theta=\pi/2, \; \phi=\pi/2<br /> [*] quarter circle arc on equator (great circle arc)<br /> [*] segment (great circle arc) back to origin<br />
  <br />

 

[Dale]

"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.

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.

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.

 

[Chris Hillman]

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
<br /> {\omega^1}_2 = {\omega^2}_1, \; \;<br /> \hbox{but} \; {\omega^2}_3 = -{\omega^3}_2<br />
Compare the straightforward skew-symmetry of {\omega^j}_k = -{\omega^k}_j 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.

 

[Chris Hillman]

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 \theta=\pi/3 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
<br /> [ \vec{e}_j, \; \vec{e}_k ] = \vec{e}_\ell \; {C^\ell}_{jk}<br />
(Thus, these commutation coefficients are antisymmetric: {C^\ell}_{jk} = -{C^\ell}_{kj}.) The dual coframe one-forms are \sigma^j. Then Cartan's first structure equation is
<br /> d\sigma^j = -{\omega^j}_k \wedge \sigma^k<br />
where--- too many opportunities to make a serious error if I don't check everything carefully, so let me get back to you.

 

[George Jones]

"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.

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.

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.

 

[Chris Hillman]

Citations for Parallel Transport

Thanks, George, this is just what we need! I'll try to see if I can find Fecko's book in the library. When I get a chance, I'll try to write a BRS post on parallel transport. The daunting thing is that I should first write posts on Lie groups and Lie algebras, vector fields, connection the Cartan way, Maurer-Cartan form... I know this isn't as hard as I am making it sound, which is making it frustrating. Clearly working some well chosen explicit computations the Cartan way would be an essential goal of the projected BRS thread!

 

[Dale]

The Fecko book is also available on Google books:
http://books.google.com/books?id=vQ...C0Q6AEwAA&safe=active#v=snippet&q=385&f=false

But of course, that is always somewhat hit-or-miss.

 

[Chris Hillman]

I never use Google, for the obvious reasons.

Re what George said about "history of fringe claims", I have to revise my judgement: IMO Anamitra does appear to be behaving like a genuine pseudomath crank, not a troll. The list of bizarre mistakes he has made in the cited thread along include:

• appears to claim spheres can't exist (he's confused by the coordinate singularity at North pole in the standard trig chart)
• appears to claim parallel transport is path in-dependent (oddly enough, he appears to be trying to use frame fields rather than coordinate bases, which would be good, but he's apparently confused by one example where the frame fields he is using happen to parallel transported around the curve he happens to be considering, and he also assumes that "broken line" curves are "improper")

Random selection of weirdities just from the first few pages of just the one thread:

You are getting these results because you are considering a spherical space-time surface [actually, an ordinary round Riemannian sphere] which should not exist in practice.
...
Near the Earth's surface the "geodesics" are great circles possibly due to the impenetrability of the Earth and not due the"strong" curvature of space-time.
...
If we have two geodesics connecting a pair of points A and B , the particle at the point of intersection would be in a state of indecision (as to which spacetime curve it should follow).
...
So if we parallel transport a vector along different paths starting from the same point the components do not change, when referred to the local inertial frames.
...
So if we parallel transport a vector along different paths starting from the same point the components do not change,when referred to the local inertial frames.
...
This "demonstrably false" notion [Anamitra's claim that parallel transport is path independent] arises out of the fact that the singularities north and south poles have been chosen simultaneously.
...
But if one is interested in the parallel transporting a vector at a stretch along a curve it should not be one with sharp bends. In such an instance it cannot be called parallel transport in the totality of the operation.

He's also been very slow to recognize that he is using the word "thread" improperly, and very slow to use the quotation feature of VB and the LaTex features of PF. And he insists on using brackets when others would use parentheses. He refers repeatedly to "physical distance" (in the large), he repeatedly confuses coordinate singularities with geometrical singularities...

All in all, he seems to take the typical crank attitude that "it must be a good idea" [sic] to insist on doing the same incorrect thing over and over again... not to mention his insistence that the world adopt his private terminology, rather than adapting his writings to use standard technical terms correctly.

 

[Chris Hillman]

"Geodesic doubts", plus: local vs ultralocal

Re User:TrickyDicky's thread "Geodesic doubts"

www.physicsforums.com/showthread.php?t=424278

Let's imagine a test particle in outer space not being subjected to any significant force, gravitational(far enough from any massive object) or any other. Its path would be describing a geodesic that follows the universe curvature, right? Would that be an euclidean straight path, or would it follow a curved path, like an ellipse or a hyperbola?

The question as stated is confused almost beyond repair. Ambiguities include:

• gravitational force, in gtr?
• if TrickyDicky meant a test particle in a cosmological model such as FRW model, then dust models have the property that the world lines of the dust particles are timelike geodesics, and it makes sense to consider other timelike geodesics (world line of a particle in inertial motion which is not stationary wrt nearby dust particles),
• as bcrowell noted, in a curved manifold, geodesics are "straight" by definition,
• if TrickyDicky meant to ask about the coordinate equation of a world line, that will only be valid wrt a particular chart and will probably not have a simple geometrical/physical interpretation.

TrickyDicky then restated his question as

I guess what I really meant was what kind of curvature does our spacetime have.

One wonders if he has any idea how frustrated that would make some feel if they tried to answer his first (completely different) question!

And unfortunately, his new question is still ambiguous:

• By "our spacetime", does he mean "best fit FRW model to current observations of our own universe?"
• By "curvature", does he mean the Riemann tensor of the spacetime?
• Or does he mean the Riemann tensor of a spatial hyperslice? If so, there are infinitely many foliations of any spacetime into a family of spatial hyperslices. Does he mean the family orthogonal to the world lines of the dust particles? If so, such a family exists and is unique only if the congruence of world lines of dust particle is irrotational (vorticity tensor vanishes).

Needless to say, I recognize that one needs to try to have patience with people who, due to inadequate understanding of topic T, are unable to clearly express an unambiguous question about some issue regarding topic T.

when I've read descriptions of the Einsten model of the universe, IIRC they talk about a hypersphere embedded in Euclidean ambient space.

There is an Einstein cosmological model, the static lambdadust isometric to R \times S^3, with the obvious -+++ signature, which is not consistent with observations and would in any case be unstable against small perturbations. Most likely he is misremembering someone describing a figure in MTW, Gravitation which represents H^3 as a hyperboloid in E^{1,3}; some FRW models feature H^3 hyperslices orthogonal to the world lines of the dust or fluid particles.

In a nonexpanding spacetime I guess the curvature would be that of the spatial part of the line element, right?

In a typical cosmological model in gtr, i.e. a four-dimensional Lorentzian manifold with a stress-energy tensor describing matter, which satisfies the EFE, there is a gravitational field which is due to the presense of matter, which is often modeled as a perfect fluid or dust (pressure-free perfect fluid). In such models, it makes sense physically and geometrically to compute/describe the expansion scalar, shear tensor, and vorticity vector of the congruence of world lines of dust/fluid particles. As George Jones noted, for an expanding phase, the expansion scalar will be positive. If the vorticity vector vanishes, it makes sense to compute/describe the Riemannian geometry of the (unique) family of three dimensional spatial hyperslices which are everywhere orthogonal to the congruence of world lines of dust/fluid particles. This three dimensional curvature tensor is not the same as the "spatial components" of the Riemann tensor of the spacetime itself, however.

Ich's suggestion of constructing a Riemann normal chart is interesting.. and for the FRW dust with E^3 hyperslices orthogonal to the world lines of the dust particles it should be easy enough.

I was asking for the case of a non-expanding spacetime manifold just to fix concepts before I go into the more geometrically complex FRW metric.

The FRW models are the simplest cosmological models in gtr which resemble our universe even approximately.

"Non-expanding spacetime manifold": is it possible that TrickyDicky is trying to ask about a stationary cosmological moodel? (Timelike Killing vector field, not neccessarily vorticity-free?) If so, the better known candidates include the Einstein lambdadust already mentioned, which is static, and the Goedel lambda-dust (homogeoneous but world lines of dust particles has nonzero vorticity).

Expansion of space is purely coordinate dependent. Or better, as George Jones puts it, "expansion" is a property of a congruence, like the one defined by the canonical observers in a FRW metric. It's not a property of spacetime.

I agree with what George Jones said, but the bolded statement is IMO potentially misleading. In fact I am not sure what technical statement Ich had in mind here. Ich? Or am I talking to myself?

But this seems to be at odds with General Covariance, according to which only those properties that are invariant under changes of coordinates are physically real, so if expansion vanishes just by a change of coordinates as youare claiming, then expansion is a coordinate artifact rather than a physical fact.

I see why he's confused here. I'd reassure him that the acceleration vector, expansion scalar, shear tensor, vorticity vector of a timelike congruence is a coordinate-free geometric description of the relative motion of the particles whose world lines comprise the timelike congruence. In the case of a family of particles in a state of inertial motion, the congruence of their world lines will be a timelike geodesic congruence, so the acceleration vector will vanish identically. Particles in a perfect fluid, or charged particles in an EM field, will generally have nonzero forces acting so will generally have nonzero acceleration (acceleration just means path curvature of a world line, which has units 1/length and is not to be confused with curvature of a surface, a hyperslice, or of spacetime, all of which have units 1/area).

However, if you choose some other congruence in the very same spacetime, you might find completely different acceleration vector, expansion scalar, shear tensor, vorticity vector!

So the freedom George Jones had in mind, I am pretty sure, is the freedom to choose a congruence. Sometimes a choice is "natural", e.g. in the FRW models, it makes sense to single out the congruence of world lines of material particles (dust or fluid, depending upon the model). Also, in some models, some congruences may be distinguished by being particulary symmetrical (in the FRW models, more or less by contruction, this is true for the congruence of world lines of material points).

From what I've been learning from this thread, we have grounds for selecting a 'canonical' coordinate system - i.e. a point of view where the universe appears homogenous and isotropic.

In an FRW model, that turns out to be the same thing as picking out the congruence of world lines of the particles whose mass-energy is collectively producing the gravitational field (curvature of spacetime). If we compute the Einstein tensor wrt the associated frame field, all but one component vanishes; the only nonzero component at p is positive and represents the matter density of the dust, as measured by the dust particle whose world line passes through event p in our spacetime (M,g). It is often convenient to think of tensor fields as smooth sections in an appropriate tensor bundle over M, incidentally. Then we are saying that for our special frame field, the same information is conveyed by the graph of a smooth function on M.

In "The speed of light?", when bcrowell wrote

"Local" is actually very difficult to define rigorously [Sotiriou 2007], but basically the idea is that if your apparatus is of size L, any discrepancy in its measurement of c will approach zero in the limit as L approaches zero.

I'd caution that there is a huge terminological ambiguity in the physics literature (but not the math literature) regarding "local", which has traditionally been tolerated on the grounds that, allegedly, "the intended meaning will be clear from context". However, from my reading of arXiv eprints by persons not experienced in gtr, it is quite clear that this assumption is invalid: real physicists are getting confused, all the time. Matt Visser has started using "ultralocal" to refer to the level of tangent spaces and jet spaces, and "local" to refer to the level of local neighborhoods (consistent with huge math literature on fiber bundles and suchlike, and with the huge literature produced by math-knowledgeable physicists like Witten).

Jet spaces are one of those concepts everyone should know, but few do, even though it would have been well known to any well-educated mathematician of the late 19th century*. The basic idea is very simple: in addition to some function, include its first and second order derivatives (or partial derivatives) as "additional variables". This is just what you need to get started in Lie's theory of the symmetry of a differential equation (ODE or PDE). To be the topic of another long-deferred BRS thread.

*Just noticed the bolded phrase could sound snooty :blush: Actually, I was trying to express wry despair at the fact that the explosive growth of mathematics in the 20th century inevitably meant that many valuable notions like jet spaces and Frobenius cocycles became undeservedly obscure. So that even well educated persons in the 21st century might not know really valuable concepts which would have been well known to a well educated person of the 19th century.

 

[Chris Hillman]

BRS: Roy Kerr at PF, plus Penrose diagrams generally

Re "Can one diagonalize the Kerr metric?"

www.physicsforums.com/showthread.php?t=247794

Is it possible to diagonalize the Kerr metric in the Boyer-Lindquist coordinates? If so then I think calculations with the metric will become easier. I forget under what condition a matrix can be diagonalized. Can anybody remind me?

I was about to answer this when I saw that User:tagsdad gave a correct answer, in fact better than what I would have said which would have been less specific and probably less concise. And appeared to sign "Roy Kerr", although I am guessing this user means he is posting the result of an email inquiry to Roy Kerr.

It might be worth adding that the Doran chart generalizes the Painleve chart for the Schwarzschild vacuum to the Kerr vacuum.

Re "Penrose diagrams in general"

www.physicsforums.com/showthread.php?t=422583&page=2

If Penrose-Carter diagram means:

• compactificated space-time diagram and
• null-like world lines are 45 degree lines

than, every spherically symmetric space-time has Carer-Penrose diagram? If Yes, can you cite a paper about proving the existence?

Else: on the 2D Carter-Penrose diagram only radial motion can be studied, what about 3D=2+1 Penrose-Carter diagrams? Can you draw it? The 2D diagram is simply rotated?

Those are all good questions, which I should have addressed in the BRS thread. I'll say more there, but briefly: "Penrose-Carter" diagram is a somewhat informal term, but in general the idea is to find a chart such that

• the line element is conformal to some easy to understand line element such as Einstein static model (used in the standard diagram for Minkowski spacetime),
• all coordinates have finite ranges, so that the entire spacetime is "mapped" in a finite region, in the sense of Mappa Mundi,
• asymptotically flat sheets should have a similar boundary to the usual diagram for Minkowski spacetime, and similarly sheets asymptotic to some familiar nonflat manifold should (perhaps) exhibit this relation in the chart,
• in particular, asymptotically flat sheets should have loci labeled scri^+, scri^-, i^+, i^0, i^-.

Thus, a Penrose-Carter chart for a four-dimensional Lorentzian manifold is a local coordinate chart, often with coordinates which can be identified as "angular coordinates" on an exterior sheet. Often one suppresses one or two angular coordinates, so that each "point" in the resulting "Penrose diagram" denotes a round sphere or a circle of some radius (larger near the boundary of any asymptotically flat sheet, smaller in the interior, generally).

In the case of boost-rotation symmetric vacuum solutions (see the BRS on Weyl vacuums), it is in fact convenient to draw three dimensional Penrose diagrams in which only one angular coordinate has been suppressed, so that each "point" in the diagram is really a circle.

 

[George Jones]

Re "Can one diagonalize the Kerr metric?"

www.physicsforums.com/showthread.php?t=247794

I was about to answer this when I saw that User:tagsdad gave a correct answer, in fact better than what I would have said which would have been less specific and probably less concise. And appeared to sign "Roy Kerr", although I am guessing this user means he is posting the result of an email inquiry to Roy Kerr.

Actually, there are other indications that tagsdad really is Roy Kerr!

 

[Chris Hillman]

In that case I nominate tagsdad for Science Advisor!

 

[Chris Hillman]

Bell spaceship "paradox" again, oh nooooo!

Here we go again: in "Quick question - has length contraction actually been experimentally confirmed?"

www.physicsforums.com/showthread.php?t=425566&page=2

There is the thought demonstation, first presented by Dewan and Beran, later retold by John Stewart Bell, usually referred to as 'Bell's spaceship paradox'. Two spaceships, connected by an unstretchable tether of length L, tether fully extended, are initially comoving. They synchronize their clocks. At an agreed point in time they commence acceleration, parallel to the tether, both accelerating at exactly the same G-count. For the tether to not break it would have to decrease the separation between the spaceships. However, since the spaceships meticulously maintain the same G-count the tether will snap.

There is only one physical factor that the breaking of the tether can be attributed to: length contraction.

The tether does break, but of course the "physical cause" is that the Bell congruence has nonzero expansion tensor (compare the Rindler congruence). IOW, the tether breaks because nearby Bell observers are moving away from each other. Notice that the expansion tensor approach neatly avoids the issue of "distance in the large": physically, the fact that nearby particles in the tether are moving away from each other means that at some place where the tether has a mechanical flaw, it will snap.

 

[Chris Hillman]

The Term "Locally Flat"

In the mis-titled thread "No globally flat geometry on S²" the OP is asking a question about the topology of the sphere. Unfortunately, User:lavinia just said in

www.physicsforums.com/showthread.php?p=2868002#post2868002

Globally flat means that the curvature tensor is identically zero.

No, a Lorentzian or Riemannian manifold (M,g) is said to be locally flat if the Riemann tensor vanishes. That implies that in a sufficiently small local neighborhood of any point, we can introduce a Cartesian chart and in our neighborhood, the geometry will mimic that of a flat space (or spacetime). But globally the topology could be nontrivial; consider the case of the cylinder R \times S^1. Its Riemann tensor has only one algebraically independent component, which vanishes. Yet its topology is nontrivial.

 

[Chris Hillman]

BRS: Two questions concerning models constructed via "matching"

In "GR Vacuum solutions",

www.physicsforums.com/showthread.php?t=427333

TrickyDicky asks about vacuum solutions

when applying the solution to the Mercury precession problem or the bending of light by the sun problem, we are actually introducing the mass of the sun to solve them, and this to me seems a bit contradictory with the premise that there is no matter in the manifold under consideration.

Boundary conditions, just like solving any "field equation" for a source-free solution. No source term in the interior of the domain, but a source must be assumed on boundary to get a nontrivial solution.

Tom.Stoer replied to TrickyDicky by saying:

The Schwarzschild solution has two patches matching at the Schwarzschild radius. The outer solution is the familiar vacuum solution, whereas there is an inner solution which is NOT a vacuum metric and which therefore differs from the well-known inner vacuum solution used for a black hole. Instead a spherical symmetric, non-rotating, incompressible fluid is used which leads to a regular solution w/o singularity at r=0.

That is not quite correct.

Rather, the Schwarzschild stellar model is constructed by matching a perfect fluid interior (with constant density, so "incompressible" fluid, but nonconstant pressure falling to zero at some r0 &gt; 2m, in fact r_0 &gt; 9/8 \cdot 2m--- see the discussion of Buchdahl's theorem in the textbook by Schutz) to a region of the exterior sheet of Schwarzschild vacuum, where the matching is across the world sheet of the round sphere at r=r0 (the zero pressure surface). The result is indeed a static spacetime with no curvature singularities anywhere, and with a true "center" in the fluid ball, at r=0 (where pressure is maximal).

However, it seems to me that Tom.Stoer was trying to address the issue raised by TrickyDicky, by pointing out that this suggests that the mass parameter in the exterior solution arises from imposing a physically appropriate boundary condition. Astrophysical black holes are formed by the collapse of ordinary matter, which we can idealize as an OS collapsing dust ball, constructing using a similar matching construction, in which again we have a regular center inside the dust ball, right up the moment when the dust ball collapses to a strong spacelike singularity inside the interior region of the newly formed black hole. See the BRS thread on "Conformal Compactifications" for how the causal structure of the OS model differs from the "eternal black hole".

In "GR Dust Cloud"

www.physicsforums.com/showthread.php?t=427311

Austin0 asks (murkily) about essentially the same construction; the answer is that in the exterior region of a model constructed by matching either a (dynamical) collapsing dust ball interior region or static spherically symmetric perfect fluid ball interior region to a static asymptotically flat Schwarzschild vacuum exterior region, the mass parameter in the exterior is the mass of the dynamical collapsing dust ball or static fluid ball respectively. The exterior region doesn't care which.

[EDIT: Heh, Dalespam already made this point while I was composing this post... ]

In general there is a problem with comparing the geometry of two locally nonisometric spacetimes both using a "radial coordinate" labeled r. Typically it is not so easy to compare "what happens at radius r=10m" between two such solutions, or even to compare the physical meaning of the parameter m. But that's probably too sophisticated for Austin0 right now.

 

[Dale]

But that's probably too sophisticated for Austin0 right now.

Definitely, but he is learning so he may get to that point sometime reasonably soon.

 

[Chris Hillman]

BRS: Survey of Conservation Laws in GTR?

In

www.physicsforums.com/showthread.php?t=426479

Ben Crowell asks for a review on the status of conservation laws in gtr, specifically:

What I'm interested in here is general conservation laws that would be valid in any spacetime, not the kind of conservation laws that hold for test particles in a spacetime with some special symmetry expressed by a Killing vector.

Oh boy! This turns out to be a big question which has been intensively studied for nearly one century, since no-one (either mainstreamers or dissident fringe figures) is happy with results known for a long time which show that naive notions of conservation laws (e.g. based upon Gauss's law) won't work in curved manifolds. As a result we have

• some really good ideas which are however at present technically and conceptually difficult to use
• some really bad ideas from fringe figures (often well known for being involved in every controversy they can find)
• some old but still useful ideas which sometimes work for limited purposes in limited circumstances
• nothing noncranky which is fully general, other than
• Noether "charges" from the analysis of variational symmetries in a Lagrangian formulation of a PDE (or Hamiltonian formulation of the geodesic equations, but Ben expressly ruled out anything depending on Killing vector fields)

I should also point out that several researchers appear to have been driven round the bend by trying too hard to do the impossible in this area, so I recommend starting out with the limited ambition of understanding better some of the ideas which have been explored so far, rather than turning gtr on its head.

Before I say anything else, in case some SA/M with little background in gtr, but who appreciates the utility of conservation laws, Gauss's law, etc. in elementary mathematical physics, is intrigued by the news that Gauss's law doesn't work too well in gtr, the canonical nontechnical reference is the UseNet Physics FAQ at Chez Baez:

math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html

Is Energy Conserved in General Relativity?

In special cases, yes. In general -- it depends on what you mean by "energy", and what you mean by "conserved".

The remainder of this post is addressed to those who have studied at least one modern gtr textbook.

First, AFAIK, there is no up-to-date comprehensive survey of the status of conservation laws in gtr. The subject is apparently far to massive for that, no pun intended.

Fool that I am, I will attempt to provide, if not a survey, at least a semi-annotated list of suggested further reading.

I think it is important to go way back and begin with an old review paper, since anything published later will probably assume the reader has read this review (or else it was probably written by an ignoramus):

J. N. Goldberg,
Conservation Laws in General Relativity
chapter in Gravitation: An Introduction to Current Research, ed. by Louis Witten, Wiley 1962.

That is an invaluable book much too hard to obtain from "downsized" university libraries, waanhnh!--- it also contains several other must-read review papers from the early days of the Golden Age of Relativity, including the review of exact solutions by Ehlers and Kundt, a review of ADM, and more. If you can't find it at your uni, ask Los Alamos, they may have the last surviving copy.

Next, several textbooks such as MTW and Carroll have excellent discussions of conservation laws. See in particular Carroll or Hawking & Ellis for a differential law (generally valid in gtr) which explains how Ricci curvature can generate Weyl curvature. As Carroll explains, the well known law {T^{ab}}_{;b} = 0, can easily be mistaken for a "conservation law", but that is not the role it plays in gtr, despite its obvious similarity to {T^{ab}}_{,b} = 0 in flat spacetime.

The failure of conservation laws like {T^{ab}}_{,b} = 0 to make sense in gtr was known to Einstein and other early investigators, and it bugged the heck out of them, as you would imagine. And it still bugs the heck out of most modern researchers.

Here are some of the ideas which have been explored (neccessarily, there is some overlap in the following list):

• Pseudotensors: Einstein and several others came up with various versions of gravitational energy-momentum pseudotensors, which we can think of as a "virtual additional term" in the energy-momentum tensor, a term which purports to track the location and tranfser of the energy and momentum of the gravitational field. They are called pseudotensors because they behave like tensors under some subgroup of the full diffeomorphism group. These pseudotensors fell into disfavor many times as various failings came to light. For example: about 15 years ago some researchers started playing with GRTensorII and found that pseudotensors give hugely inconsistent results for most the familiar simple exact solutions. Despite this, right now they are again being promoted by some researchers.
  
  With suitable caution, you can try
  

  arxiv.org/abs/hep-th/0308070

  The Energy-Momentum Problem in General Relativity
  S. S. Xulu
  

  Energy-momentum is an important conserved quantity whose definition has been a focus of many investigations in general relativity. Unfortunately, there is still no generally accepted definition of energy and momentum in general relativity...quasi-local masses have their inadequacies...in this work we use energy-momentum complexes to obtain the energy distributions in various space-times.
  
  We elaborate on the problem of energy localization in general relativity and use energy-momentum prescriptions of Einstein, Landau and Lifgarbagez, Papapetrou, Weinberg, and M{\o}ller ... The Cooperstock hypothesis for energy localization in GR is also supported.

  (Note that claims for pseudotensors are IMO often overstated, and the suggestion of Cooperstock is quite controversial.)
• Superenergy tensors: these include the well known Bel-Robinson "superenergy tensor", which is a fourth-rank tensor with strong symmetry properties. I don't know of any extensive review, but try
  

  arxiv.org/abs/gr-qc/9912050

  Applications of Super-Energy Tensors
  J. M. M. Senovilla
  

  Abstract: In this contribution I intend to give a summary of the new relevant results obtained by using the general superenergy tensors. After a quick review of the definition and properties of these tensors, several of their mathematical and physical applications are presented. In particular, their interest and usefulness is mentioned or explicitly analyzed in 1) the study of causal propagation of general fields; 2) the existence of an infinite number of conserved quantities in Ricci-flat spacetimes; 3) the different gravitational theories, such as Einstein's General Relativity or, say, $n=11$ supergravity; 4) the appearance of some scalars possibly related to entropy or quality factors; 5) the possibility of superenergy exchange between different physical fields and the appearance of mixed conserved currents.

• Just as important as conserved quantitites are fluxes which track nonconservation, so to speak!
  • Komar mass and momentum are defined in terms of Killing vectors for stationary asympotically flat spacetimes, which often arise models of an "isolated gravitating system" (Ben ruled Killing vectors out, but I think we should rule them back in). One of the most important achievements in classical gtr to date has been the introduction of Bondi mass and momentum, which generalize these to nonstationary situations, and allow us to track globally the energy and momentum carried off to conformal infinity by gravitational radiation (and other massless radiation) from an isolated system.
  • In addition, Killing vector fields and their generalizations give rise to various notions of conserved currents, and you can look for terms like Bach currents. Random example:
    

    arxiv.org/abs/astro-ph/0007046

    Riemannian collineations in General Relativity and in Einstein-Cartan cosmology
    L.C.Garcia de Andrade
    

    Riemannian vectorial collineations along with current Killing conservation are shown to lead to tensorial collineations for the energy-stress tensor in general relativity and in Einstein-Cartan Weyssenhoff fluid cosmology.

  • In addition, for any system of possibly nonlinear PDEs arising as the Euler-Lagrange equations of some Lagrangian (for example, the Ernst equation, whose stationary axisymmetric case gives rise to the Ernst family of all stationary axisymmetric vacuum solutions in gtr), variational symmetries give rise to Noether currents and Noether charges (e.g. you might obtain "for free" a conserved quantity analogous to the energy contained in a solitonic wave).
• Also important are some other quantities, defined in special classes of spacetime models, whose unexpected behavior signals further phenomena to be aware of: in paticular the Misner-Sharp mass appears naturally in various spherically symmetric models, and plays a role in the Israel-Poisson notion of "mass inflation" in the interior of black hole (with some infalling massless radiation). Compare and contrast some remarks I hinted at in the BRS on Weyl vacuums, where I sketchily indicated some related ways in which "mass" behaves unlike Newtonian intuition in gtr.
• Quasilocal notions of energy and momentum: currently these seem to offer the closest approach to what Ben wants; for a review see
  

  relativity.livingreviews.org/Articles/lrr-2009-4/

  Quasi-Local Energy-Momentum and Angular Momentum in General Relativity
  László B. Szabados
  See also
  

  arxiv.org/abs/gr-qc/0004074

  Quasi-Local Conservation Equations in General Relativity
  J.H. Yoon
  

  A set of exact quasi-local conservation equations is derived from the Einstein's equations using the first-order Kaluza-Klein formalism of general relativity in the (2,2)-splitting of 4-dimensional spacetime. These equations are interpreted as quasi-local energy, momentum, and angular momentum conservation equations. In the asymptotic region of asymptotically flat spacetimes, it is shown that the quasi-local energy and energy-flux integral reduce to the Bondi energy and energy-flux, respectively. In spherically symmetric spacetimes, the quasi-local energy becomes the Misner-Sharp energy. Moreover, on the event horizon of a general dynamical black hole, the quasi-local energy conservation equation coincides with the conservation equation studied by Thorne {\it et al}. We discuss the remaining quasi-local conservation equations briefly.

• Another venerable approach seeks "conservation laws" appropriate for working within the ADM formalism and other initial-value approaches to gtr; see for example
  

  arxiv.org/abs/gr-qc/0003019

  Noether Charges, Brown-York Quasilocal Energy and Related Topics
  L. Fatibene, M. Ferraris, M. Francaviglia, and M. Raiteri
  

  The Lagrangian proposed by York et al. and the covariant first order Lagrangian for General Relativity are introduced to deal with the (vacuum) gravitational field on a reference background. The two Lagrangians are compared and we show that the first one can be obtained from the latter under suitable hypotheses. The induced variational principles are also compared and discussed. A conditioned correspondence among Noether conserved quantities, quasilocal energy and the standard Hamiltonian obtained by 3+1 decomposition is also established. As a result, it turns out that the covariant first order Lagrangian is better suited whenever a reference background field has to be taken into account, as it is commonly accepted when dealing with conserved quantities in non-asymptotically flat spacetimes. As a further advantage of the use of a covariant first order Lagrangian, we show that all the quantities computed are manifestly covariant, as it is appropriate in General Relativity.

• Even the mass/momentum of test particles can be problematic; for a review see
  

  relativity.livingreviews.org/Articles/lrr-2004-6/

  The Motion of Point Particles in Curved Spacetime
  Eric Poisson
  Things get worse when one tries to go to the next approximation and consider objects which are small but not that small. In particular, one can search for "conserved quantities" in post-Newtonian formalism; see
  

  arxiv.org/abs/gr-qc/9503041

  Conservation laws for systems of extended bodies in the first post-Newtonian approximation.
  Thibault Damour, David Vokrouhlicky
  

  The general form of the global conservation laws for $N$-body systems in the first post-Newtonian approximation of general relativity is considered. Our approach applies to the motion of an isolated system of $N$ arbitrarily composed and shaped, weakly self-gravitating, rotating, deformable bodies and uses a framework recently introduced by Damour, Soffel and Xu (DSX). We succeed in showing that seven of the first integrals of the system (total mass-energy, total dipole mass moment and total linear momentum) can be broken up into a sum of contributions which can be entirely expressed in terms of the basic quantities entering the DSX framework: namely, the relativistic individual multipole moments of the bodies, the relativistic tidal moments experienced by each body, and the positions and orientations with respect to the global coordinate system of the local reference frames attached to each body. On the other hand, the total angular momentum of the system does not seem to be expressible in such a form due to the unavoidable presence of irreducible nonlinear gravitational effects.

• Another idea: consider asymptotically flat spacetimes and models of "isolated systems", see
  

  arxiv.org/abs/0906.2155

  Null Geodesic Congruences, Asymptotically Flat Space-Times and Their Physical Interpretation
  T. M. Adamo, C.N. Kozameh, E.T. Newman
  (This paper offers an excellent discussion of conservation of Bondi mass and angular momentum of isolated systems, taking account of that lost to radiation carrying away mass and momentum to conformal infinity.)
  

  arxiv.org/abs/0802.3314

  On Extracting Physical Content from Asymptotically Flat Space-Time Metrics
  C. Kozameh, E. T. Newman, G. Silva-Ortigoza
  Next, consider quasigroups which behave nicely "at conformal infinity"; see
  

  arxiv.org/abs/gr-qc/0403044

  Quasigroups, Asymptotic Symmetries and Conservation Laws in General Relativity
  Alexander I. Nesterov
  

  A new quasigroup approach to conservation laws in general relativity is applied to study asymptotically flat at future null infinity spacetime. The infinite-parametric Newman-Unti group of asymptotic symmetries is reduced to the Poincar\'e quasigroup and the Noether charge associated with any element of the Poincar\'e quasialgebra is defined. The integral conserved quantities of energy-momentum and angular momentum are linear on generators of Poincar\'e quasigroup, free of the supertranslation ambiguity, posess the flux and identically equal to zero in Minkowski spacetime.

• Another idea: formulate a general theory of conservation laws for classical field theories; examples include:
  

  arxiv.org/abs/gr-qc/9911095

  A General Definition of "Conserved Quantities" in General Relativity and Other Theories of Gravity
  Robert M. Wald and Andreas Zoupas
  

  In general relativity, the notion of mass and other conserved quantities at spatial infinity can be defined in a natural way via the Hamiltonian framework: Each conserved quantity is associated with an asymptotic symmetry and the value of the conserved quantity is defined to be the value of the Hamiltonian which generates the canonical transformation on phase space corresponding to this symmetry. However, such an approach cannot be employed to define `conserved quantities' in a situation where symplectic current can be radiated away (such as occurs at null infinity in general relativity) because there does not, in general, exist a Hamiltonian which generates the given asymptotic symmetry. (This fact is closely related to the fact that the desired `conserved quantities' are not, in general, conserved!) In this paper we give a prescription for defining `conserved quantities' by proposing a modification of the equation that must be satisfied by a Hamiltonian. Our prescription is a very general one, and is applicable to a very general class of asymptotic conditions in arbitrary diffeomorphism covariant theories of gravity derivable from a Lagrangian, although we have not investigated existence and uniqueness issues in the most general contexts. In the case of general relativity with the standard asymptotic conditions at null infinity, our prescription agrees with the one proposed by Dray and Streubel from entirely different considerations.

  
  

  arxiv.org/abs/hep-th/9608008

  Asymptotic conservation laws in field theory
  Authors: I. M. Anderson, C. G. Torre (Utah State University)
  (Submitted on 1 Aug 1996 (v1), last revised 14 Oct 1996 (this version, v2))
  

  A new, general, field theoretic approach to the derivation of asymptotic conservation laws is presented. In this approach asymptotic conservation laws are constructed directly from the field equations according to a universal prescription which does not rely upon the existence of Noether identities or any Lagrangian or Hamiltonian formalisms. The resulting general expressions of the conservation laws enjoy important invariance properties and synthesize all known asymptotic conservation laws, such as the ADM energy in general relativity.

• Another idea: look for "conserved quantities" in the well-known GEM formulation of linearized gtr; try
  

  arxiv.org/abs/gr-qc/0311030

  Gravitoelectromagnetism: A Brief Review
  Bahram Mashhoon
  

  The main theoretical aspects of gravitoelectromagnetism ("GEM") are presented. Two basic approaches to this subject are described and the role of the gravitational Larmor theorem is emphasized. Some of the consequences of GEM are briefly mentioned.

• Another class of ideas: reformulate gtr in such a way as to remove the mathematical problems entirely. Or rather, formulate an arbitrarily good mimic of gtr which allows one to work in a convenient way with conserved quantities, including a conserved notion of the energy/momentum in the gravitational field itself. One example is the work of Itin on teleparallel gravity. See
  also
  

  arxiv.org/abs/0905.4026

  Conservation of Energy-Momentum in Teleparallel Gravity
  Mariano Hermida de La Rica
• Last but not least, one can formulate a completely new classical gravitation theory which is constructed to ensure that conservation laws can be formulated and proven; the list of candidates which has been proposed (but in almost every case, too little studied) is far too long for me to even think about attempting to itemize them here!

I should stress that scientific controversies (and alas, a generous dollop of pseudoscientific suggestions) are impossible to avoid in such a huge topic, and I am not neccessarily endorsing anything the authors of the above papers say; I am simply trying to briefly indicate the depth and breadth of the work which has been done so far.

 

[Chris Hillman]

"News" from National Geographic, sort of

It seems that a reporter recently attended an astronomy conference and wrote an article for National Geographic recounting a number of things she heard.

In "Rogue black holes?!"

www.physicsforums.com/showthread.php?p=2873886#post2873886

Astronuc asked

Astronomers have long known about rogue black holes?

Yup, for a long time. But not to worry, the galaxy is so large that Earth isn't likely to encounter one in the forseeable future.

How long?

Redmount and Rees 1989 discussed possible recoil effects due to asymmetric emission of gravitational waves during the merger of two black holes, and I believe there are even earlier discussions. So the possibility has been discussed for at least two decades, and during the last decade there have been an increasing number of numerical simulations plus observations supporting the idea that rogue black holes are common.

Rees discussed possible recoil effects in his 1998 review of "Astrophysical Evidence for Black Holes", published as a chapter in the Chandrasekhar symposium at the U of C, and its been well known for a long time in the field.

Kelly Holley-Bockelmann, of Tennessee's Vanderbilt University, and her colleagues were the first to show that the objects could arise from violent mergers.

Using a computer model, Holley-Bockelmann found that two combining black holes rotating at different speeds or of different sizes give the newly merged black hole a big kick.

I don't see anything new or surprising here. All of these features have been discussed previously.

[Gravitational wave recoil effects during a merger] sends the object hurtling in an arbitrary direction at velocities as high as 2,485 miles (4,000 kilometers) per second.

Not arbitrary; the direction of kick is likely to align roughly with the orbital phase at the moment of merge, and there may also be a component orthogonal to the plane of coalescence due to spin-spin interaction (a strong field gravitomagnetism phenomenon).

"This is much higher than anyone predicted," Holley-Bockelmann said. "Even the average kick velocity of 200 kilometers [124 miles] per second is extremely high."

This claim is AFAIK a few years old.

The National Geo article also discusses new estimates of the number of intermediate mass black holes in our galaxy, but again, it has been known for some years that these are after all not uncommon objects.

Is this a new revelation about DM and BHs?

Not the stuff I discussed above. I suspect this is the predictable result of a science reporter attending a meeting, hearing something in a talk which was spun as "new", and not checking with experienced researchers before writing a story.

More often such misinformation comes straight from PR flacks attempting to portray mundane faculty research as a "revolutionary advance"; as everyone here probably knows, many news outlets reprints such press releases verbatim without any pretense of fact checking. In this case, the reporter put her name on the piece so she may be culpable--- especially since it is impossible to think of a more convenient venue for a science reporter to ask some senior people in the field to comment/explain than at a major conference.

The apparent (indirect) observation of what is inferred to be "dark matter" in an accretion disk, now--- that really is new, AFAIK:

They also found hints that dark matter may play an important role in the hot disks that form around companion-consuming black holes.

 

[Chris Hillman]

Shannon entropy, popularity of normal distributions, and Ehrenfest paradox ad nauseum

In "Entropy of a product of positive numbers"

www.physicsforums.com/showthread.php?t=428040

clamshell asks a question about Shannon entropy which doesn't really make sense

I accept that the entropy (Shannon's) of a sum of positive
numbers is the sum of the -P_n * LOG(P_n) for each number
in the sum of numbers where P_n is the average contribution
of a term. IE, if T = A + B, then
S = -(A/T)*LOG(A/T) - (B/T)*LOG(B/T).
But what about the entropy of A*B?

Please don't respond with "why do it?", just give me some idea of how you might attempt to do it.

The Boltzmann entropy can be defined for any partition of some finite set into r blocks X = \union_{j=1}^r A_r. It is simply the logarithm (any base you like) of the multinomial coefficient
<br /> H (\pi) =<br /> \log \, <br /> \left( \begin{array}{ccccc}<br /> &amp; &amp; n &amp; &amp; \\<br /> n_1 &amp; n_2 &amp; \ldots &amp; n_{r-1} &amp; n_r <br /> \end{array} \right)<br />
where
<br /> |X| = n, \; |A_j| = n_j<br />
If we put p_j = n_j/n we obtain a probability measure on the partition, and applying Stirling's approximation under the assumption that all the blocks are large (i.e. all the n_j are large), we obtain the approximation
<br /> H(\pi) \approx -\sum_{j=1}^r p_j \log p_j<br />
We can define this expression to the Shannon entropy H_p(\pi) of a finite partition of any set X endowed with a probability measure p (where at first one should probably think of varying the partition while keeping p fixed). Notice that the blocks need no longer be finite sets, and that both Boltzmann and Shannon entropies characterize the "variety" of the finite partition. The first is a purely combinatorial notion while the second is a probabilistic notion.

More generally, if we have measure-preserving transformation g on X, with a bit more work we can define the Shannon entropy of the transformation g, which is the definition used in ergodic theory. Shannon himself used this definition for Markov chains, and he showed that his entropy satisfies a number of very important formal properties which justify its interpretation as a measure of "the variety of alternatives", the interpretation Shannon stressed in his founding paper, Shannon 1948. The most important of these formal properties states that if you refine some partition, you can compute the new entropy by taking a weighted average of the entropies of the blocks and adding this to the original entropy. In terms of physics, this corresponds to the fact that refining a partition increases the Boltzmann entropy, which is traditionally interpreted in terms of "fine-graining your macrostates". Put in other words, if you can identify a refined classification of microstates into macrostates, your uncertainty concerning which macrostate an unknown microstate belongs to is larger.

Typical questions answered in the theory of entropy include: "if I learn that the unknown state of the system lies in a particular block of one partition, how much information do I gain about which block of another partition it lies in"? See the BRS thread "Exploring the Rubik Cube with GAP"

www.physicsforums.com/showthread.php?t=422410

for a group theoretic interpretation and generalization of Boltzmann entropy. Here, the "alternatives" in question are alternative "motions" under some group action, say a group of transformations of X, and we can ask questions such as these: "if we learn the motion of one subset A under an unknown transformation on X, how much information do we gain about the motion of another subset B?"

Clamshell in effect considers a partition into only two blocks and assigns one of them an arbitrary probability p where 0 < p <1, so that the other block has probability 1-p. Then of course the Shannon entropy of this partition into two blocks is simply
<br /> H(\pi) = -p \log p - (1-p) \log (1-p)<br />
Then he asks about the entropy of the "product" of the two blocks! That simply doesn't make sense. Probably clamshell is trying to ask about the refinement property but is getting himself confused.

One can certainly try to "categorify" these notions of entropy, and John Baez and myself have discussed that in various internet posts.

BTW, re the remarks of marcus in

www.physicsforums.com/showthread.php?p=2876424#post2876424

Shannon entropy in ergodic dynamics is simply the Hausdorff dimension of a certain fractal set, and the group action generalization of Boltzmann entropy is also a generalization of the notion of "degrees of freedom".

Re

www.physicsforums.com/showthread.php?t=427247

striphe (whose user name is unpleasantly suggestive and who has apparently been posting quite a bit of fringe stuff) asked about mean and standard deviation.

FWIW, the emphasis in elementary statistics on mean and standard deviation is mainly due to Gauss in the early 19th century and Fisher in the early 20th century, and as both researchers knew, the reason why these are so convenient is that variance is essentially euclidean geometry in disguise (the Pythagorean theorem). This ensures that variance is easy to compute and work with, and obeys many very simple laws which can be understood in simple geometric terms. See for example M. G. Kendall, A Course in the Geometry of n Dimensions, Dover reprint, for a short book which stresses this fact. In addition, as Chebyshev showed later in the 19th century, variance remains useful even for general (non-normally distributed) "random variables", although of course many theorems fail in general. These and other considerations led Fisher to develop an extensive theory of estimators suitable for deciding which probability measure belonging to some family of measures (typically defined by some formula giving the "density" in terms of a finite list of parameters) best fits given data.

In contrast, while "nonparametric statistics" such as estimators of Shannon entropy have in some sense better justified interpretations, due to the work of Shannon and numerous subsequent researchers such as Kullback, and may enjoy powerful formal properties of their own, and may even support an abstract notion of (highly non-Riemannian) geometry permitting a "geometric intrepretation" of the statistic, there are typically practical problems with using such estimators--- and even legal problems, due to the absurd legal requirements in the U.S. that federally funded medical researchers use only "unbiased estimators", a requirement which more or less ensures bad decisions when working with nonlinear statistics.

Re

www.physicsforums.com/showthread.php?t=428147

I am sooo sick of watching people make silly mistakes while trying to discuss the so-called "Ehrenfest paradox" (which was completely resolved by Langevin c. 1927 in the manner expected by Einstein, using a frame field). Just one which has already appeared: you cannot spin up a rigid body in str. If it is already spinning, it is possible that its component particles might be in Born rigid rotation, but if you try to alter that state of motion, it cannot remain rigid. This makes it nontrivial to compare alleged "identical disks", one spinning and one not! Also, almost everyone confuses submanifolds with quotient manifolds, which is disastrous here. The so-called "spatial geometry of the rigidly rotating disk" refers to a certain Riemannian manifold which arises as a quotient by the congruence of world lines of Langevin observers, not as a submanifold of Minkowski spacetime! As Einstein expected, the geometry of this manifold turns out to be very nearly hyperbolic near the center, with a curvature depending upon the rotation rate of the disk.

 

[Chris Hillman]

BRS: the Ehrenfest thread

That's right, with this post, which I hope won't hijack this running thread, this will be the third concurrently running thread in the BRS related to the so-called "Ehrenfest paradox" and "Bell paradox".

In the very long thread "Ehrenfest / rod thought experiment." (which I haven't been reading--- for reasons of bloodpressure, as someone put it!--- but which seems to actually be mostly more related to the "Bell paradox")

www.physicsforums.com/showthread.php?t=428147&page=5

Ben Crowell asked

Actually, is there a standard definition of a static observer? The books I have seem to define a clear notion of a static spacetime and of static coordinates used to describe a static spacetime (i.e., coordinates in which the metric is diagonal and time-independent). But defining a property of coordinates isn't quite the same as defining a property of an observer, since observers are local and coordinates are global.

Static spacetime (M,g): there exists a timelike vorticity-free (thus, hypersurface orthogonal) Killing vector field \xi. Static observer: the ones whose world lines are integral curves of this Killing vector field.

Note: usually the Killing vector has to be renormalized to make a timelike unit vector field, which is then the tangent vector field to the (proper time parameterized) world lines of the static observers.

In the special case of homogeneous static spacetimes, we actually have many timelike Killing vector fields formed by various linear combinations of "translational" Killing vector fields with a timelike Killing vector field. But Schwarzschild vacuum (in particular) is only static, not homogeneous.

 

[Chris Hillman]

BRS: A very common misunderstanding of how science works

In "Are the foundations stable"

physicsforums.com/showthread.php?t=429435

EnergyLoop asks

We are constantly finding new information that does not fit the current model, but by adding a new constant or variable into the equations it repairs the problem, but gives us new things to look for such as dark matter and dark energy. I can’t help thinking about Aristotle’s crystalline spheres and the Earth centered universe, this was a simple concept initially, until the motion of the planets was realized, then it became a complex mathematical model to try to explain this motion, Copernicus then simplified the problem, and removed the complexity.
I hope we are not heading in that direction again, is the foundation sound, equations are build on previously established equations does anyone re-examine these?

This seems to have become a very common refrain, particularly since the recent discovery of an accelerating expansion. (Cosmologists quickly adjusted; the general public did not!) I'd actually like to see a PF FAQ somewhere which addresses this, mentioning some of the same points I've made so often in the past, including:

• All scientific knowledge is provisional. Constantly finding new information and constantly trying to fit new data or theoretical speculations into the body existing well-established theories is precisely what scientists do on a day-to-day basis. The fact that scientific theories can be tested by comparing quantitative predictions with quantitative data, and the fact that scientific theories are constantly up for revision is what makes science the most powerful tool known to man for the discovery and organization of knowledge about the natural world.
• Scientific knowledge consists of a vast body of experimental/observational data plus the terminology, notation, and theories we use to interpret the data. We make theories and we make predictions from theories using mathematical reasoning, and you need to know the appropriate mathematical background to understand the theories.
• When new data cannot be fit into existing theories, scientists look for explanations. First and foremost, a careful examination of the possibility that the data collection or analysis contained some subtle systematic bias or other flaw. If that fails, then one searches for the change to the theories which is "the least possible".
• The public seems to generally misunderstand the nature of scientific advances: they should be astonished not by how much has changed, but by how little, even in such an extreme revelation as the "accelerating Hubble expansion". That is, one benefit of knowing the data and the theories is that you can appreciate how "introducing nonzero Lambda" is actually the smallest possible change to the theories. Also, the data hasn't changed, only our interpretation. Making a minimal change means that only a very small portion of our interpretation/understanding of the universe changed as the result of that particular "cosmological revolution".
• To repeat: the public seems to generally misunderstand the nature of scientific revolutions. Newtonian gravitation was never "discarded", it is still used, and more often than gtr, which is a bit harder to work with. If and when a successful theory of quantum gravity appears, gtr will still be used (most likely) because the new theory will be (most likely) a bit harder to work with. Similar remarks hold for non-relativistic physics, and various specific theories which are known to be "wrong" but are still useful for limited purposes.

In "Dark Matter Distribution Around Galaxies"

www.physicsforums.com/showthread.php?t=429521

RLutz asks

Is there any reason why galactic black holes might have something to do with dark matter creation? The squashed beachball of dark matter sort of looks like what I would expect say field lines coming out of a pulsar to look like or something.

Ditto Chalnoth, plus a reminder that a black hole of mass M gravitates just like any star of mass M; unless you are very close to it you won't encounter the strong portions of the exterior field which result from the fact that the hole is so much more compact than the star.

In "Black Holes are Tears in Space"

www.physicsforums.com/showthread.php?t=429376

I`m not a scientist, i`m actually a 3D Artist. I just have a lot of faith in science unfortunately didn`t have the attention span to pay attention enough in high school and even more tragic is that my university didn`t offer any science courses!

Actually, that is tragic. Due to the global financial crisis, at least one state university in the U.S. has just eliminated its science majors.

Could it be possible that a black hole is a tear in space? It seems like it could be a way of explaining why some say you could travel through a black hole or worm hole and wind up somewhere else. If space itself was really in a shape we couldn't comprehend then maybe a tear in one place could wind up opening in an entirely different place. Does that make sense?

It's vaguely reminiscent of various possibilities discussed in serious physics like wormholes or curvature singularities, but much too vague to make much sense in a scientific discussion. So the best short answer is probably: a black hole is a region of spacetime characterized by the presence of an event horizon, which you can think of as an imaginary surface which you can fall through, but once you do, you can't ever emerge from behind the horizon, at least not into the same external region of spacetime in which you started. A very good book for poets which IMO can enable persons with only a high school science education to actually understand this, sort of, from (good!) pictures, is Geroch, General Relativity from A to B, University of Chicago Press.

In "Re: Big Bang and PreExisting Void?"

www.physicsforums.com/showthread.php?t=425597&page=4

JDoolin, whose "knowledge" of cosmological models is apparently based upon Wikipedia, not textbooks plus graduate level schoolwork, is arguing with Chalnoth and others about the Milne chart for the Minkowski vacuum, claiming that this chart is "inequivalent" to the cartesian chart. Of course that hinges on the meaning of "inequivalent"; gtr is however based upon Lorentzian geometry, and in Lorentzian geometry, any chart covering a region U in a spacetime (M,g) is just as good as any other.

In "velocity of gravity wave"

physicsforums.com/showthread.php?t=429166

spikenigma seems to think the European Gravity Field and Steady-State Ocean Circulation Explorer project has something to do with comparing the speed of gravitational and EM radiation! Needless to say, the investigators say something very different: from

www.esa.int/esaLP/ESAJJL1VMOC_LPgoce_0.html

GOCE will be gathering data or around 20 months to map the Earth's gravity field with unprecedented accuracy and spatial resolution. The final gravity map and model of the geoid will provide users worldwide with a well-defined data product that will lead to:

• A better understanding of the physics of the Earth's interior to gain new insights into the geodynamics associated with the lithosphere, mantle composition and rheology, uplift and subduction processes.
• A better understanding of the ocean currents and heat transport.
• A global height-reference system, to serve as a reference surface to study topographic processes and sea-level change.
• Better estimates of the thickness of polar ice-sheets and their movement.

• 
  One can use the next section to compute whether or not GOCE will be able to track ballistic missile submarines but in any case, it will be clear to anyone here, I think, that GOCE has nothing to do with "speed of gravity".

 

[Chris Hillman]

In "the arrow of time"

www.physicsforums.com/showthread.php?t=337236&page=11

A-wal expresses in a memorable way one key element of Simple Physics crackpottery:

You think I don’t understand the concepts just because I don’t know how to speak your language.
...
Would you watch your favorite dvd in binary code? The code is necessary but no one cares what it looks like.
...
I really don’t want to think like physicists do. I want to get this straight in my head and still think the way I do. I don’t know how you can do it like that.

Translation:

I refuse to learn to reason scientifically about scientific issues. But I insist upon exercising my constitutional right to make incontestable statements concerning scientific issues any time, any place. My way of muddled "thought" [sic] has just as much "validity" [sic] as all your hi-falutin equations. I don't need no stinkin' math!

He also offers a memorable anti black hole rant (based on willful ignorance, one might say):

I don’t mind if I’m wrong. My ego isn’t tied up in this and I have nothing to prove. I find it difficult to accept what I don’t understand and I’m not convinced by what I’ve been told. How the hell can an object that can never reach the horizon from any external perspective ever cross the horizon from its own perspective? Is not just the light from those objects that’s frozen. How could it be if they could always escape? They’re moving slower and slower through time relative to you because time in that region is moving slower and slower relative to you. If the time dilation/length contraction go up to infinity then no given time can ever long enough and no distance can ever be short enough locally if it’s infinitely length contracted from a distance! Are those inertial coordinates you use to describe an object crossing the event horizon even relative? Does it take into account the fact that you’re constantly heading into an ever increasingly sharpening curve?

 

[Chris Hillman]

Multiple misunderstandings of analysis of Hagihara observers

In

www.physicsforums.com/showthread.php?t=431407

several posters seem to be trying answer this question:

How does the tidal tensor (electroriemann tensor) distort a small sphere of test particles in circular orbit around an isolated massive object?

The easiest way to analyze this is to study an appropriate frame field associated with the Hagihara observers, who move in circular orbits in planes parallel to the equatorial plane, where we are of course in the exterior region of the Schwarzschild vacuum. (We can ask the same question and proceed the same way for suitable observers in the Kerr vacuum, but that situation is more complicated in several ways.)

The good news is that some participants appear to be trying to learn about Hagihara observers. The bad news is that they are reading a Wikipedia article (I myself wrote an early version of that particular article, but we should presume that the current version has been trashed by years of edits by persons who didn't know what they were doing, or didn't care) instead of good textbooks. Thus, everyone participating in that thread (as of 23 September 2010) is badly confused on many many points, e.g.

• "frame": in the WP article, frame means frame field, not "local coordinate chart" or whatever these posters are trying to do to apply the EP on a curved spacetime over a local neighborhood rather than in the tangent space to a single event,
• none of the participants appear to understand the distinction between components wrt a frame field and components wrt a coordinate basis,
• the WP article Lut Mentz cites gives frame field components, not coordinate basis components,
• acceleration vector, expansion scalar, shear tensor, and vorticity tensor all refer to a timelike congruence (not neccessarily geodesic, although many books make that assumption for simplicity, and the Hagihara observers are geodesic observers in the equatorial plane),
• the timelike unit vector in a frame field defines a timelike congruence, but the frame field also involves three unit spacelike vector fields, with all four being mutually orthogonal at each event,
• the posters appear to be confusing a congruence having nonzero vorticity tensor with a spinning frame field,
• the Hagihara observers are only geodesic observers in the equatorial plane; their world lines correspond to circular orbits parallel to the equatorial plane so obviously most of these require some acceleration to exist,
• to quote the components of a tensor, it helps to adopt a frame and to quote the components wrt the frame, but note that one naturally derives first a spinning frame in which \vec{e}_2 points radially outward, and only later finds the nonspinning frame,
• small spheres of Haghihara observers in the equatorial plane shear as they orbit, but don't change volume (vanishing expansion scalar!),
• if one adopts the nonspinning frame, then Lense-Thirring precession shows up (in fact you can derive an exact version of the Lense-Thirring precession formula using this approach!); the nonspinning frame of the Hagihara observers slowly spins wrt the frame of a distant star, so to speak.

For what it is worth, you can derive a frame field adapted to Hagihara observers by starting with the frame of static observers
<br /> \begin{array}{rcl}<br /> \vec{e}_1 &amp; = &amp; \frac{1}{\sqrt{1-2m/r}} \; \partial_t \\<br /> \vec{e}_2 &amp; = &amp; \sqrt{1-2m/r} \; \partial_r \\<br /> \vec{e}_3 &amp; = &amp; \frac{1}{r} \; \partial_\theta \\<br /> \vec{4}_4 &amp; = &amp; \frac{1}{r \sin(\theta)} \; \phi<br /> \end{array}<br />
in the \vec{e}_4 direction, at each event, by an undetermined boost, with the boost parameter depending only on r. Then require that the acceleration of the new frame vanish in the equatorial plane. This gives an ODE for the boost parameter as a function of r which can easily be solved. The result is
<br /> \begin{array}{rcl}<br /> \vec{f}_1 &amp; = &amp; \frac{1}{\sqrt{1-3m/r}} \; <br /> \left( \partial_t + \sqrt{\frac{m}{r}} \; <br /> \frac{1}{r \, \sin(\theta)} \; \partial_\phi \right) \\<br /> \vec{f}_2 &amp; = &amp; \sqrt{1-2m/r} \; \partial_r \\<br /> \vec{f}_3 &amp; = &amp; \frac{1}{r} \; \partial_\theta \\<br /> \vec{f}_4 &amp; = &amp; \frac{\sqrt{1-2m/r}}{\sqrt{1-3m/r}} \;<br /> \left( \sqrt{\frac{m}{r}} \; \partial_t +<br /> \frac{1}{r \, \sin(\theta)} \; \partial_\phi \right)<br /> \end{array}<br />
(Recall that the static frame and the chart are only defined on r > 2m; notice that the Hagihara frame is only defined on r>3m, at best!) Then by construction
<br /> \nabla_{\vec{f}_1} \vec{f}_1 = 0<br />
in the equatorial plane (off this plane, the acceleration is nonzero!), the expansion scalar vanishes, the only independent nonzero component of the shear tensor is
<br /> \sigma_{24} = \frac{-3}{4} \sqrt{\frac{m}{r^3}} \;<br /> \frac{1-2m/r}{1-3m/r}<br />
while the only independent nonzero component of the vorticity tensor is
<br /> \omega_{24} = \frac{-1}{4} \sqrt{\frac{m}{r^3}} \;<br /> \frac{1-6m/r}{1-3m/r}<br />
The tidal tensor is
<br /> \begin{array}{rcl}<br /> E_{22} &amp; = &amp; \frac{-2m}{r^3} \; \frac{1-3m/2/r}{1-3m/r} <br /> \approx \frac{-2m}{r^3} \; \left( 1-\frac{3m}{2r} \right) \\<br /> E_{33} &amp; = &amp; \frac{m/r^3}{1-3m/r} <br /> \approx \frac{m}{r^3} \; \left( 1 + \frac{3m}{r} \right) \\<br /> E_{44} &amp; = &amp; \frac{m}{r^3}<br /> \end{array}<br />
All these expressions are refer to the frame \vec{f}_1, \ldots \vec{f}_4, and are only valid in the equatorial plane. And the acceleration vector, expansion scalar, shear tensor, and vorticity tensor all refer to the timelike unit vector field \vec{f}_1, whose integral curves are the world lines of the Hagihara observers.

So small spheres of inertial observers orbiting very nearly in the equatorial plane are sheared parallel to that plane, but maintain constant volume.

To study precession, you should introduce an undetermined secular rotation about \vec{f}_3, with the rate of rotation depending only on r, and demand that the Fermi derivatives of \vec{f}_2, \ldots \vec{f}_4 with respect to \vec{f}_1 should vanish in the equatorial plane. This gives an ODE for the rotation rate which you can solve, finding that the cumulative angle of rotation (at each event along the world line of a equatorial Hagihara observer) is
<br /> \psi = t \, \sqrt{\frac{m}{r^3}} \; \sqrt{1- 3 m/r}<br />
(The Lense-Thirring precession formula given in many textbooks is only an approximation of the exact result.)

With respect to the new, nonspinning frame, all the tensors just mentioned have different components from what we found for the first frame, in which the "principle axes" are seen to very slowly spin wrt the spatial vectors of the nonspinning frame. But the tensors are defined in terms of \vec{f}_1 which is shared by the two frames, and they are three dimensional tensors, so their traces and quadratic invariants (and higher order invariants) will neccessarily agree regardless of which frame you compute the components in!

You can compare all this with the Lemaitre frame appropriate for studying the physical experience of observers falling in freely and radially. Their tidal tensor wrt the Lemaitre frame is m/r^3 \; \operatorname{diag}(-2,1,1). So:

A small sphere of test particles released from rest inside Fr. Lemaitre's spaceship elongates radially and compresses orthogonally, while keeping constant volume, so forming a prolate spheroid; if Fr. Lemaitre looks out the window, he can see that the long axis of the prolate spheroid is aligned parallel to the direction toward the massive object.

A small sphere of test particles released from rest inside Dr. Hagihara's spaceship behaves similarly, but compresses slightly more than expected orthogonally to the equatorial plane, and slightly less radially, so forming a triaxial ellipsoid. If he looks out the window, he can see that the long axis of this triaxial ellipsoid is parallel with the direction of the massive object. In addition, assuming Dr. Hagihara's spaceship is gyrostabilized, as he keeps returning to "1 January" in his orbit, over time he notices a very gradual precession wrt the distant stars of where he is in his orbit on 1 January by his clock, and he also observes his spaceship to be very very slowly spinning as it orbits.

The tidal and precession effects have very different characteristic time scales, however--- the participants in the thread in question have not yet recognized this.

If Dr. Hagihara looks out the window, he can see that his spaceship, which is gyrostabilized, is nevertheless very slowly spinning wrt the massive object, and also wrt "the distant fixed stars". Looking at neighboring spaceships (also inertial and initially motionless wrt him), he sees that they appear to be very slowly shearing orthogonal to the equatorial plane (because the ships further out are moving more slowly in their orbits, as per Kepler) and that their trajectories are slowly swirling about his ship in the sense of nonzero vorticity.

All of these things can be studied in various limits and are consistent with Newtonian expectation in a suitable slow motion weak gravity limit.

If I am somehow making this sound hard, that is not my intent. This is not hard. It does however involve learning several new concepts (new for most people, or at least not yet understood by most people) and one needs to keep all these things straight: coordinate charts, frame fields, congruences, tensors defined on spacetime independent of any congruence, effectively three-dimensional tensors defined wrt a specific congruence, kinematic decomposition of a congruence, components of a tensor, invariants of a tensor, when orthogonal hyperslices do and do not exist, etc.

In

www.physicsforums.com/showthread.php?t=431572

the OP appears to be enthused by the recent widely publicized claims of Nikodem J. Poplawski which are based on elementary misunderstandings as I explained elsewhere in the BRS, so the short answer there is that Poplawski doesn't yet understand the theory of Lorentzian manifolds sufficiently well to avoid mistakes, and his claims are based upon such mistakes. It is possible that the OP might also be vaguely referring to notions of "baby universes" being born inside black holes, which is an intriguing speculative suggestion, but not one which is currently very well supported even by theoretical arguments, and of course not at all by observation!

 

[Chris Hillman]

Gravity shielding: how to

In "e.p. implies no gravitational shielding?; Feynman?"

www.physicsforums.com/showthread.php?t=432327

Ben Crowell asks how/why gtr forbids gravitational shielding. I haven't yet had a chance to read that thread, but it reminds me of an amusing observation I made a decade or so ago in a UseNet post: there is a large class of static minimally coupled massless scalar field (mcmsf) solutions, in which the spacetime is a curved Lorentzian manifold, but which exhibit "zero gravity", e.g. static observers can hover over a region where the energy of the mcmsf is concentrated without any need to fire their rocket engines!

See "BRS: Static Axisymmetric "Gravitationless" Massless Scalar Field Solutions"

www.physicsforums.com/showthread.php?t=433793

(Mcmsf solutions are exact solutions in which the only contribution to the stress tensor comes from the field energy of a massless scalar field, which is minimally coupled to curvature.)

Exercise: read Post #1 in the above cited BRS. Find the static spherically symmetric solution in this class--- note that it is not given by choosing
w = \frac{1}{\sqrt{z^2+r^2}}
Compute its Komar mass. Think about matching across nested spherical shells to a Minkowski region inside the inner shell, and to a Schwarzschild or Minkowski region (depending upon what you found for the Komar mass) outside the outer shell. (Note that the scalar field "wants" to model a long range interaction, so any such matching will require some explanation: what is the physical reason why the scalar field is nonzero only between the two nested spherical shells?)

 

[Chris Hillman]

Penrose Diagram Confusions, plus: Expanding Space

In "rotating black holes in Penrose diagrams"

www.physicsforums.com/showthread.php?t=433568

the OP is badly confused and I am not up to the task of trying to suggest what to say to him, but its good that several SA/Ms are trying to help. One minor clarification:

The Kerr solution, like the Schwarzschild solution, assumes a perfectly symmetrical collapse.

AFAIK no rotating analog of the Oppenheimer-Snyder collapsing dust ball, which produces a non-rotating hole, is yet known. However, if you look at geodesic congruences of world lines of freely falling test particles, the Doran congruence in the Kerr vacuum is generalizes the Lemaitre congruence from the nonrotating case (Schwarzschild vacuum), and the Doran observers in the equatorial plane do exhibit planar motion as they spiral in.

The Penrose diagram of the interior an actual rotating black hole caused by gravitational collapse is still a matter of some speculation and debate.

AFAIK, the situation remains the same as ten years ago: despite plausible analogies suggesting what to expect, nothing rigorous is known about the interiors of generic rotating holes (e.g. formed by generic collapse of ordinary matter, with some matter or radiation falling into the hole from the exterior region). The very nice paper on mass-inflation cited by JesseM discusses generic charged holes by way of pursuing the analogy.

After having read [the cited arXiv eprint] I do believe that even without [even with?] the mass inflation, there is no white hole.
the structure of the "white hole" if we look at the metric should be exactly the same as the structure of the black hole, and matter could excape because it can be accelerated faster than the speed of light (like matter inside the event horizon) . So this would not be a white hole but a black hole, because the metric would be the same.
Does anyone disagree with this?

I think the appropriate response is: "no-one has any idea what you are trying to say". I bolded some obviously problematic claims (although I can't tell whether he is trying to make a claim or to deny one!).

FWIW, the eternal Schwarzschild and Kerr spacetimes contain both "white hole" and "black hole" event horizons, or more properly, horizons from which particles must emerge from a past interior region into an exterior region, and horizons in which particles which fall from an exterior region into a future interior region cannot re-emerge into the original exterior region. In addition, the eternal Kerr vacuum (and the RN non-null electrovacuum) also contain Cauchy horizons, which must not be confused with event horizons. The eternal Kerr vacuum also contains both and interior asymptotically flat regions (negative Komar mass) as well as exterior asympotically flat regions (positive Komar mass).

BTW, "relativityfan" is making my troll antennae twitch uncomfortably I sense a whiff of Chip on Shoulder, which raises the question of whether this user is using an ironically chosen handle at PF. It's always worth remembering that one reason to "write defensively" when posting in the public areas is that you never know when some putative newbie is planning to cherry pick responses to some "naive question" in order to, say, quote them in some anti-science website. A large number of religiously committed creationists do this quite a bit with regard to anything related to the standard hot Big Bang theory.

In "Geodesics doubts"

www.physicsforums.com/showthread.php?t=424278

the OP is again rather confused (as the poster later admitted). FWIW,

As I'm familiar with the phrase, "space is expanding" is not a coordinate-dependent phenomenon.

Depending on context, it might be reasonable to interpret "space is expanding" to refer, in an exact fluid solution, to the congruence of world lines of the fluid particles, and then the acceleration vector, expansion scalar, shear tensor, and vorticity tensor of this congruence are all geometric, coordinate-free notions. In particular, the expansion scalar gives a coordinate-free notion of whether or not a small ball of fluid particles has expanding or contracting volume. (Oops, just saw that in Post #14, George Jones already said something similar. And yay!, in Post #13, Ben Crowell pointed out that as soon as you start talking about "velocity in the large", things get tricky, and you need to be more precise about what operationally significant notion for defining "distance in the large" you have in mind.)

Expansion of space is purely coordinate dependent.

I think I know why Ich said this, and depending upon how you think of things, that's not wrong either, but wow, this certainly shows why it is so important to introduce some math, or at least to say that without using math, people are likely to wind up talking about different things and thus making apparently mutually contradictory statements.

Another more general set of fundamental observers, independent of FRW symmetries, is defined to be at rest in normal coordinates, maybe one could call them "Einstein observers", as they reproduce the inertial frames of SR if curvature is negligible, on which most people base their intuition.

I see that Ich is thinking of Riemann normal coordinates (on a spatial hyperslice? in a cosmological model?) but I don't understand how he intends to define his observers. If the congruence of their world lines is nonexpanding, however, it will generally be nongeodesic. Also, a rigid congruence is one with vanishing expansion tensor, but not all spacetimes admit such congruences.

From what I've been learning from this thread, we have grounds for selecting a 'canonical' coordinate system - i.e. a point of view where the universe appears homogenous and isotropic.

I think at least two posters are thinking about "preferred timelike congruences" in cosmological models. There are at least two possible interpretations of what this might mean:

• vorticity-free congruence whose spatial hyperslices are homogeneous and isotropic (this puts strong constraints on the spacetime)
• congruence of the world lines of the fluid or dust particles which model galaxies; in a fluid this might not be a geodesic congruence but in a dust solution, it will be a timelike geodesic congruence; however if the vorticity tensor is nonvanishing these world lines will not admit a family of orthogonal hyperslices.

The OP asked Ben

bcrowell, this is what I call gaslight, are you saying that considering expansion as physical fact (as I and many others do) or as a coordinate artifact is just a matter of taste?

The problem is that he still doesn't realize that various notions of "expansion" have been referred to without definition in the thread. If one understand this to refer to the expansion scalar of a timelike congruence having some agreed upon physical significance in a given gtr model, then yes, this is coordinate-free. If one understands "expansion" to mean something else, then it could well be coordinate-dependent. Re what Ben said in his Post #36, if we consider "expansion" to refer to galaxies in a cosmological model, it is reasonable to stipulate that we will use the expansion tensor of the congruence to describe how nearby pairs of galaxies are moving wrt each other. But this only tells us about nearby galaxies, which is probably not what the OP wanted!

Hmm... velocity in the large, visual appearances... as the number of valid but conceptually subtle concepts mentioned in this thread increases, someone who doesn't already understand all of them will probably gain the (false) impression that the signal to noise ratio is increasing. Certainly the OP seems to be becoming increasingly confused, not less so. But maybe that's part of the learning experience.

 

[Chris Hillman]

Tidal tensor of some familiar cosmological models

In "Weyl curvature and tidal forces"

www.physicsforums.com/showthread.php?t=433916

User:TrickyDicky asks about the tidal tensor of the FRW models. He is correct that the Weyl tensor of these models vanishes identically; all their curvature is Ricci curvature, due to the immediate presence of matter (usually taken to be radiation fluid or dust) as per Einstein's field equation. However, the tidal tensor is the electroriemann tensor, not the electroweyl tensor! Thus, a small sphere of initially comoving test particles in an FRW model will contain nonzero mass, so it will contract; by symmetry, the gravitational attraction of nonzero mass outside the sphere cancels out. The tidal tensor shows this isotropic tidal compression.

For example, consider the FRW dust with E^3 hyperslices:
<br /> ds^2 = -dt^2 + t^{4/3} (dx^2 + dy^2 + dz^2), \; \;<br /> 0 &lt; t &lt; \infty, \; -\infty &lt; x, \, y, \, z &lt; \infty<br />
Take the frame of the dust particles
<br /> \begin{array}{rcl}<br /> \vec{e}_1 &amp; = &amp; \partial_t \\<br /> \vec{e}_2 &amp; = &amp; t^{-2/3} \; \partial_x \\<br /> \vec{e}_3 &amp; = &amp; t^{-2/3} \; \partial_y \\<br /> \vec{e}_4 &amp; = &amp; t^{-2/3} \; \partial_z <br /> \end{array}<br />
Then the tidal tensor is
<br /> {E\left[\vec{e}_1\right]}_{ab} = <br /> \frac{2}{9 t^2} \; \operatorname{diag} (1,1,1)<br />
indicating isotropic tidal compression.

For comparison, for the Schwarzschild vacuum in the frame of static observers or Lemaitre observers,
<br /> {E\left[\vec{e}_1\right]}_{ab} = <br /> \frac{m}{r^3} \; \operatorname{diag} (-2,1,1)<br />
(traceless, as must happen for a vacuum solution!), which indicates radial tidal tension and orthogonal tidal compression.

The Jacobi geodesic formula states
<br /> \ddot{\vec{\xi}}^a = -{E^a}_b \, \xi^b<br />
where \vec{\xi} is a connecting vector (spacelike, short, points from fidudical geodesic to a neighboring geodesic as you let parameter run in both proper time parameterized geodesic curves), and where overdot denotes differentiation wrt proper time,
<br /> \dot{(\cdot)} = \nabla_{\vec{e}_1} (\cdot)<br />
Or in terms of matrix algebra, if we think of vectors as column matrices,
<br /> \ddot{\vec{\xi}} = -{\cal E} \, \vec{\xi}<br />
IOW, at each event we have a linear operator which acts on spacelike vectors in the projection of the tangent space to the normal hyperplane element, and this takes each vector to its second derivative wrt proper time, assuming it is connecting our fiducial geodesic to a very nearby one. This only makes sense because the tidal tensor is defined in terms of the Bel decomposition of the Riemann tensor with respect to a particular timelike congruence--- assumed here to be a geodesic congruence.

I say it because I've just read that the Weyl curvature only happens in in empty spacetime, without any gravitational source nearby.

That's not true, assuming "empty spacetime" means a vacuum or maybe electrovacum region. Weyl curvature can happen anywhere, even in a region containing matter. If that weren't true, we could shield against gravitational waves using cardboard boxes.

Two bookmarkable links:

www.math.ucr.edu/home/baez/einstein/
www.math.ucr.edu/home/baez/gr/gr.html

These should help TrickyDicky.

For those of you who use Maxima, here is a Ctensor file you can run in batch mode under wxmaxima, which allows you to easily verify that the Weyl tensor vanishes and that the electroriemann tensor (tidal) tensor has the frame field components I just mentioned:

/* 
FRW dust with E^3 hyperslices; comoving cartesian chart; nsi coframe 

All FRW dusts can be embedded with codimension one.

*/
load(ctensor);
cframe_flag: true;
ratchristof: true;
ctrgsimp: true;
/* define the dimension */
dim: 4;
/* list the coordinates */
ct_coords: [t,x,y,y];
/* define background metric */
lfg: ident(4);
lfg[1,1]: -1;
/* define the coframe */
fri: zeromatrix(4,4);
fri[1,1]: -1;
fri[2,2]:  t^(2/3);
fri[3,3]:  t^(2/3);
fri[4,4]:  t^(2/3);
/* setup the spacetime definition */
cmetric();
/* display matrix whose rows give coframe covectors */
fri;
/* compute a matrix whose rows give frame vectors */
fr;
/* metric tensor g_(ab) */
lg;
/* compute g^(ab) */
ug: invert(lg);
christof(false);
/* Compute fully covariant Riemann components R_(mijk) = riem[i,k,j,m] */
lriemann(true);
/* Compute R^(mijk) */
uriemann(false);
/* Compute Ricci componets R_(jk) */
ricci(true);
/* Compute trace of Ricci tensor */
tracer;
/* Compute R^(jk) */
uricci(false);
/* Compute and display MIXED Einstein tensor G^a_b */
/* For (-1,1,1,1) sig Flip sign of top row to get G^(ab) */
einstein(false);
/* WARNING! leinstein(false) only works for metric basis! */
/* Compute Kretschmann scalar */
rinvariant();
expand(factor(%));
/* Covariant Einstein tensor as matrix */
matrix([-ein[1,1],-ein[1,2],-ein[1,3],-ein[1,4]],
[ein[2,1],ein[2,2],ein[2,3],ein[2,4]],
[ein[3,1],ein[3,2],ein[3,3],ein[3,4]],
[ein[4,1],ein[4,2],ein[4,3],ein[4,4]]);
/* electroriemann tensor */
matrix([lriem[2,2,1,1], lriem[2,3,1,1],lriem[2,4,1,1]],
[lriem[3,2,1,1],lriem[3,3,1,1],lriem[3,4,1,1]],
[lriem[4,2,1,1],lriem[4,3,1,1],lriem[4,4,1,1]]);
/* magnetoriemann tensor */
matrix([lriem[2,4,3,1],lriem[2,2,4,1],lriem[2,3,2,1]],
[lriem[3,4,3,1],lriem[3,2,4,1],lriem[3,3,2,1]],
[lriem[4,4,3,1],lriem[4,2,4,1],lriem[4,3,2,1]]);
/* Weyl tensor vanishes */
weyl(true);
/* Construct NP tetrad for our frame, compute Weyl spinors and Petrov type */
nptetrad(true);
weyl(false);
psi(true);
factor(psi[0]);
factor(psi[1]);
factor(psi[2]);
factor(psi[3]);
factor(psi[4]);
petrov();

For comparison, here is the Schwarzschild vacuum in the frame of the Painleve observers:

/* 
FRW dust with E^3 hyperslices; comoving cartesian chart; nsi coframe 

All FRW dusts can be embedded with codimension one.

*/
load(ctensor);
cframe_flag: true;
ratchristof: true;
ctrgsimp: true;
/* define the dimension */
dim: 4;
/* list the coordinates */
ct_coords: [t,x,y,y];
/* define background metric */
lfg: ident(4);
lfg[1,1]: -1;
/* define the coframe */
fri: zeromatrix(4,4);
fri[1,1]: -1;
fri[2,2]:  t^(2/3);
fri[3,3]:  t^(2/3);
fri[4,4]:  t^(2/3);
/* setup the spacetime definition */
cmetric();
/* display matrix whose rows give coframe covectors */
fri;
/* compute a matrix whose rows give frame vectors */
fr;
/* metric tensor g_(ab) */
lg;
/* compute g^(ab) */
ug: invert(lg);
christof(false);
/* Compute fully covariant Riemann components R_(mijk) = riem[i,k,j,m] */
lriemann(true);
/* Compute R^(mijk) */
uriemann(false);
/* Compute Ricci componets R_(jk) */
ricci(true);
/* Compute trace of Ricci tensor */
tracer;
/* Compute R^(jk) */
uricci(false);
/* Compute and display MIXED Einstein tensor G^a_b */
/* For (-1,1,1,1) sig Flip sign of top row to get G^(ab) */
einstein(false);
/* WARNING! leinstein(false) only works for metric basis! */
/* Compute Kretschmann scalar */
rinvariant();
expand(factor(%));
/* Covariant Einstein tensor as matrix */
matrix([-ein[1,1],-ein[1,2],-ein[1,3],-ein[1,4]],
[ein[2,1],ein[2,2],ein[2,3],ein[2,4]],
[ein[3,1],ein[3,2],ein[3,3],ein[3,4]],
[ein[4,1],ein[4,2],ein[4,3],ein[4,4]]);
/* electroriemann tensor */
matrix([lriem[2,2,1,1], lriem[2,3,1,1],lriem[2,4,1,1]],
[lriem[3,2,1,1],lriem[3,3,1,1],lriem[3,4,1,1]],
[lriem[4,2,1,1],lriem[4,3,1,1],lriem[4,4,1,1]]);
/* magnetoriemann tensor */
matrix([lriem[2,4,3,1],lriem[2,2,4,1],lriem[2,3,2,1]],
[lriem[3,4,3,1],lriem[3,2,4,1],lriem[3,3,2,1]],
[lriem[4,4,3,1],lriem[4,2,4,1],lriem[4,3,2,1]]);
/* Weyl tensor vanishes */
weyl(true);
/* Construct NP tetrad for our frame, compute Weyl spinors and Petrov type */
nptetrad(true);
weyl(false);
psi(true);
factor(psi[0]);
factor(psi[1]);
factor(psi[2]);
factor(psi[3]);
factor(psi[4]);
petrov();

 

[Chris Hillman]

Sectional curvature

In "Weyl curvature and tidal forces"

www.physicsforums.com/showthread.php?t=433916

it looks like the Baez pages cleared up the main confusion, but TrickyDicky introduced a new one!

the spacetimes of constant sectional curvature like Schwartzschild

I don't know how Tricky got that confusion, but its not true.

Take any two-dimensional submanifold S of any Loretzian or Riemannian manifold M. It inherits a metric from the parent and thus has a Gaussian curvature at each point. Choose a point P and consider the tangent plane to S at P. Associated with this is a sectional curvature obtained from the action of the Riemann tensor on bivectors. Computing this wrt a frame field you obtain the Gaussian curvature of S at P.

The metric tensor on M at T_P M induces a metric tensor on the space of bivectors at P. If you use a coordinate basis to compute sectional curvature, you have to divide by a normalization factor which arises from this metric on bivectors. If you use a frame field, you don't need to do that.

Schwarzschild vacuum certainly does not have constant sectional curvature! That is a very rare property for a Riemannian/Lorentzian manifold to have!

My initial guess was that Tricky misread something somewhere concerning a venerable approach to finding initial data with which to start a numerical simulation using an initial value approach to gtr, such as ADM formalism. In this approach, one looks for a conformally flat three manifold to start the evolution; for example, "space at a time" in a time symmetric evolution in which dust is thrown out, hovers momentarily, and falls back---- the "space at a time" where the dust momentarily hovers may be conformally flat. But most spatial hyperslices certainly are not.

BTW, in a Lorentzian manifold, the Bach tensor measures the departure of a given hyperslice from being conformally flat, similarly to the fact that the Weyl tensor measures the departure of the spacetime itself from being conformally flat.

But then Tricky said this:

Schwarschild solution has constant sectional curvature (that of a parabole)

It seems that Tricky misread a WP article (or some idgit munged a WP article). He must be referring to the Flamm paraboloid, an embedding of a t=t_0 hyperslice in the exterior region of Schwarzschild vacuum written in the Schwarzschild chart (only valid on the exterior). But that certainly does not have "constant sectional curvature".

I think this illustrates that people simply cannot hope to understand gtr from reading WP articles or popular books without sufficient mathematical background and ability to understand the math. But maybe I am too pessimistic--- Tricky did seem to learn something valuable from Baez's pages in the end.

the interior Schwarzschild solution (ie. one which contains matter) is conformally flat.

The bolded phrase is ambiguous: newbies typically don't know how to guess correctly whether an author means "Schwarzschild incompressible perfect fluid ball" or "future or past interior region of the Schwarzschild vacuum". Here, atyy means the first.

Penrose is talking about the vacuum Schwarzschild solution, which as far as I understand has non-zero Weyl curvature.

In a vacuum, the Ricci tensor vanishes, so the Weyl tensor agrees with the Riemann tensor in a vacuum region!

The "exact" solution describing our non-uniform universe must somehow contain corrections to all these approximations which must be joined up to each other somehow.

Correct. Just like in any field theory.

is deflection of light outward the sun's rim produced by Weyl curvature or Ricci's?

Weyl. The deflection is an effect of the curvature in the vacuum region outside the Sun (in an idealized model formed by matching a static spherically symmetric perfect fluid (ssspf) ball such as the Schwarzschild perfect fluid across the zero pressure surface to an exterior Schwarzschild vacuum region. Since the curvature there is entirely Weyl curvature, this lensing is a Weyl effect.

In a conformally flat region, by definition there can be no lensing. Thus, no lensing in FRW models, or in the interior of the Schwarzschild incompressible fluid ball (pretending for the sake of argument that light can propagate freely at the speed of light there, which of course isn't true!). Note: most ssspf solutions are not conformally flat, nor are many cosmological models conformally flat.

I was confusing the spatial curvature(paraboloid) of the Schwarzschild metric that is indeed of constan sectional curvature

Not if he is talking about the hyperslice which corresponds to the Flamm paraboloid, and I don't know what other paraboloid he could be talking about.

 

[Chris Hillman]

Re

www.physicsforums.com/showthread.php?t=434034&page=2

an interesting point is that a certain increasingly populous class of satellites, generally launched with little fanfare, are increasingly using various kinds of stealth technology (e.g. nonreflective surfaces), and increasingly engage on a timescale of minutes in various evasive behaviors (e.g.furl sails at certain times, change their geometry in other ways when passing over certain areas, frequently change orbits).

Interestingly, many evasive techniques appear to be aimed specifically at foiling casual observation by amateur astronomers (through telescopes, but certainly these things don't wish to needlessly draw the attention of naked eye observers either). Rather astonishing, considering the expense involved, but apparently true.

Be this as it may, the net result is an increasing frequency with which amateurs notice odd flashes in the night sky, which appear rather unlike flashes from conventional communication satellites in long term, known, stable orbits.

What if anything to say about this in the public areas of PF is, I guess, a matter for discussion in the SA forum.

 

[Chris Hillman]

Precession, Komar integrals, plus Kleinian geometry

Re "Perihelion advance expressed differently"

www.physicsforums.com/showthread.php?t=434694

• As Lut Mentz hinted, one of the nice things about linear field equations is that you can separate out various effects and treat them seperately. In the case of the precession of Mercury, there is a much larger effect due to the influence of the major planets. We can get away with computing the gtr residual or extraNewtonian precession using a simplified model which ignores the outer planets entirely (and the quadrupole moment of the Sun, and...) only because we are using the linearized approximation, which is good enough to obtain the gtr residual to the desired accuracy for Mercury and other cases in our solar system.
• To second order (in an appropriate perturbation expansion) you can think of the motion as Kepler motion on an ellipse in a tranparent piece of plastic which you slowly rotate with constant angular velocity. But higher order terms make the actual motion more complicated, and nonlinear corrections make it even more complicated (in a peturbative approach).
• The perturbation expansion just mentioned refers to the Einstein-Binet equation for timelike geodesics in the exact Schwarzschild vacuum, but Einstein 1915 used a weak field approximate solution instead, which turns out to introduce an unwanted problem. So much better to follow the textbook approach.

Re "Newtonian generalization of Komar mass"

www.physicsforums.com/showthread.php?t=434722

Jonathan Scott (uh-oh!) asks a murky question about Komar mass. I expect Pervect will chime in there, to say (at least) that Komar mass is not defined the way JS suggested. Carroll's textbook has a very nice discussion, but I don't care for his notation! Here's how I'd describe the definition:

Suppose you have an asympototically flat spacetime (Lorentzian 4-manifold) which admits a timelike vorticity free Killing vector \vec{\xi}, i.e. we have an asymptotically flat static spacetime. Although this isn't needed, to simply the discussion let's also suppose we have introduced a Schwarzschild type chart so that our line element has the form
<br /> ds^2 = -f^2 dt^2 + g^2 dr^2 + r^2 \; d\Omega^2<br />
where f,g are metric functions of r only, where we suppose that our Killing vector field \vec{\xi} = \partial_t. Let \vec{N} be the outward pointing unit normal to the spheres r=r0, and let the timelike unit vector field \vec{U} be the normalization of the Killing vector field \vec{\xi}. Then

• Average over the sphere at r = r_0 (Schwarzschild radial coordinate) the quantity
  <br /> \vec{N} \cdot \nabla_{\vec{U}} \vec{\xi}<br />
  (don't forget to use the appropriate Jacobian factor in the integrand!)
• Let r0 tend to infinity

With more thought you can see that this does not actually depend upon adopting a particular coordinate chart, but only upon the assumptions that our spacetime manifold is asymptotically flat and static. Strictly speaking, we don't even need gtr to define any of these notions! It is true that in practice, we need to adopt a chart in which it is mathematically convenient to average over spheres, or at least approximate spheres, provided they become round as r0 -> infty, and provided that our radial coordinate has the required asymptotic properties.

If you apply this to a simple model of an isolated nonrotating star, consisting of a static spherically symmetric perfect fluid ball matched across the zero pressure surface to a portion of a Schwarzschild vacuum exterior region, then you obtain the mass of the spacetime, i.e. the mass of the star. Since the whole point is to take a limit, the Komar mass really only cares about the "shape" of the vacuum exterior, in fact only about the "shape at spatial infinity"! Scott may be thinking of a perfect fluid which has pressure decreasing only asymptotically to zero, and which is asymptotically flat, but if so, not everything which "looks" AF really is, so this needs to be checked. Or he may simply be confused about the definition of Komar mass.

The fun thing about Komar mass is that you can also apply this setup to stationary axisymmetric spacetimes, such as an asymptotically flat sheet (exterior or interior) of the Kerr vacuum, and then you can define and compute Komar angular momentum (about the symmetry axis) in addition to Komar mass. In the case of the Kerr vacuum you obtain the usual mass and angular momentum parameters for the Kerr vacuum solution as usually written (e.g. in the Boyer-Lindquist chart). This requires averaging over approximately round spheres, which turns out to be good enough, as long as they become round in the limit r0 -> infty, and as long as r is an appropriate radial coordinate wrt asymptotic flatness.

These Komar integrals only apply to spacetime models in which we have appropriate Killing vector fields. A more general definition, ADM mass, agrees with Komar mass when the Komar integrals are defined.

In "Is Minkowski space the only Poincare invariant space?"

www.physicsforums.com/showthread.php?t=434555

Arkadiusz Jadczyk (uh-oh!) wrote

If you mean just "a manifold", then you can take any homogeneous space P/H, where P is the Poincare group and H its closed subgroup.

He is correct (that's the basic idea of Kleinian geometry, in fact, one of my fav topics). The OP was probably thinking of four dimensional manifolds. More generally, G/H where G is any Lie group and H any closed subgroup.

Actually, this only gives the smooth manifold portion of a much wider notion of Kleinian geometry. We can in fact take G to be any group, as in "BRS:Exploring the Rubik Group". For example, in order to study things like finite projective spaces over some finite field, we might take G=PGL(d+1,p^n). Then we get finite analogues of various familiar geometries. This turns out to be closely related to the study of finite simple groups--- in This Week, John Baez often discussed various aspects of Kleinian geometry, including these connections.

Minkowski spacetime arises from the case where G is the ten dimensional Poincare group and H is the six dimensional Lorentz group (notice that 10-6=4, as per my posts in "BRS:Exploring the Rubik Group"), and you can obtain discrete quotient spaces of Minkowski spacetime using Klein's approach. It matters very much whether or not we include "improper motions" in G or H!

It is easier to explain the simpler case of the round sphere S^2 and its discrete quotient round RP^3. Let's take G = SO(3), which will give Riemannian isometries on "round manifolds". In particular, S^2, as a homogeneous Riemannian two-manifold, arises as SO(3)/SO(2), while RP^2, as a homogeneous Riemannian two-manifold, arises as SO(3)/O(2) (quotient by a larger one dimensional closed subgroup, a discrete supergroup of SO(2), which gives a discrete two-fold quotient of S^2, by identifying antipodal points). In both cases, each element of SO(3) acts on S^2 or RP^2 as a proper isometry, in the sense of Riemannian geometry.

Other ways of representing "the sphere" give or remove various structural features, as appropriate depending on context, e.g. we might be interested in removing some of the Riemannian manifold structure. In particular we can remove some of the metric space structure, while retaining just enough structure to define conformal geometry on the sphere. To do this we should take G to be the Moebius group, and then we can find a closed subgroup H such that G/H is the sphere, but this is the sphere endowed with conformal geometry rather than Riemannian metric geometry. Now each element of the Moebious group (recall their classification into elliptic, parabolic, hyperbolic, loxodromic elements!) acts on S^2 as a conformal motion.

Twistor theory starts by exploiting the Lie group isomorphism between the Lorentz group and the Moebius group.

So it really matters here whether one thinks of something like the rotation group as SO(3) or O(3), and similarly for how you think of the "euclidean group" (the semi-direct product of the group of translations with the rotation group, which makes the group of translations a normal abelian subgroup of the euclidean group). Similarly for the Poincare group and the Lorentz group.

A great deal is known about the possible isometry groups for various kinds of exact solutions in gtr; see Stephani et al., Exact Solutions of the Einstein Field Equations, Cambridge University Press.

 

[Chris Hillman]

Radiation Pressure

In "Radiation pressure"

www.physicsforums.com/showthread.php?t=434798

TrickyDicky wrote:

my main confusion comes from the fact that if the Stress-energy tensor for electromagnetic radiation is traceless, that would imply the pressure components of the tensor equal zero, and yet it's obvious radiation exerts pressure when absorbed or reflected and radiation pressure plays an important role in star dynamics.

(the bolded phrase is of course incorrect). Replies included:

What nicksauce is saying is that the stress-energy tensor of radiation looks like:
<br /> \begin{bmatrix}<br /> \rho &amp; 0 &amp;0 &amp; 0 \\<br /> 0 &amp; -p &amp; 0 &amp; 0 \\<br /> 0 &amp; 0 &amp;-p &amp; 0 \\<br /> 0 &amp; 0 &amp; 0 &amp; -p<br /> \end{bmatrix}<br />
With p = rho/3, the trace is zero.

Components wrt a frame field, of course!

I think there might be some confusion here. For radiation pressure to accelerate a bit of matter, the radiation cannot be impinging in completely isotropic fashion. But if an electron is moving wrt the CMBR (for example), it does experience a drag force from radiation pressure which scales like the fourth power of the temperature; see Peebles, 5.6.

Also, nicksauce/phyzguy gave the contribution to the stress tensor of a radiation fluid (as in an early epoch in cosmology) rather than the contribution of an EM wave, which might be closer to what Tricky wanted. For a plane wave (components wrt a suitably "aligned" frame field!)
<br /> \begin{bmatrix}<br /> \varepsilon &amp; \varepsilon &amp;0 &amp; 0 \\<br /> \varepsilon &amp; \varepsilon &amp; 0 &amp; 0 \\<br /> 0 &amp; 0 &amp; 0 &amp; 0 \\<br /> 0 &amp; 0 &amp; 0 &amp; 0<br /> \end{bmatrix}<br />
Which is also traceless, and shows a directional pressure term. The off-diagonal terms shows the momentum. A spherical wave looks like a plane wave in a very small region far from the source of the radiation.

Bit rushed, so can't say more right now...