Covariant global constants of motion in GR?

[tom.stoer]

We know that in GR it is not possible for arbitrary spacetimes to define a conserved energy by using a 3-integral. There are some obstacles like

• the covariant conservation law DT = 0 (D = covariant derivative; T = energy-momentum-tensor) does not allow for the usual dV integration (like dj = 0)
• only with a timelike killing field k one can define a conserved energy-momentum 4-vector t with t = kT and dt =0
• ...

Now let's forget about this specific case and ask the following more general questions:

Suppose we have a 4-dim. pseudo-Riemannian manifold M with spacelike foliations F, F', F'', ... For each F one can define a family of 3-volumes V_F(t) covering M where t indicates a timelike direction (coordinate) perpendicular to V_F(t). For each V_F(t) one can define 3-integrals

$Q_F[q] = \int_{V_F(T)} \,q$

using differential forms q?

Under which conditions do these Q represent "reasonable physical obervables" with a well-defined, covariant transformation law? Under which conditions can one find a conservation law

$\frac{dQ_F[\omega]}{dt} = 0$

Are there some physical relevant examples for q? How can q be constructed from the metric g (or a 3-bein e w.r.t to the 3-volume V_F)?

 

[PAllen]

The following is the best I know bridging the gap between purely local statements and statements at infinity:

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

 

[tom.stoer]

:-) I know this review article already; nevertheless - thanks a lot.

Do you have any idea regarding the more mathematical questions I am asking? I mean: the Q as defined above need not necessarily be energy (momentum, angular momentum), but "some global conserved quantity" ...

 

[PAllen]

:-) I know this review article already; nevertheless - thanks a lot.

Do you have any idea regarding the more mathematical questions I am asking? I mean: the Q as defined above need not necessarily be energy (momentum, angular momentum), but "some global conserved quantity" ...

Nope. I understand the question, but have no idea of the answer. I will follow this thread with interest.

 

[tom.stoer]

One idea is to consider topological invariants of 3-manifolds. One would have to study the effects of black hole formation, i.e. whether a 3-invariant is destroyed by a black hole singularity.

It would be interesting to study invariants which can be defined via integrals of local expressions. Then there is the question whether one can find a "locally conserved 4-vector current" which defines the invariant ...

 

[pervect]

Are you familiar with Noether's theorem? It may not do precisely what you want, but it does associate symmetries of the action with conserved quantities.

I think the correspondence works both ways (i.e. symmetries of the action imply the existence of conserved quantities,and vica-versa), though I'm not actually 100% sure on that point.

 

[tom.stoer]

I am familiar with Noether's theorem.

It allows one to construct locally conserved currents, but the problem in GR is to construct globally conserved charges (via integration); that does not work in GR in general; a famous example is the construction of a conserved energy.

 

[TrickyDicky]

One idea is to consider topological invariants of 3-manifolds. One would have to study the effects of black hole formation, i.e. whether a 3-invariant is destroyed by a black hole singularity.

It would be interesting to study invariants which can be defined via integrals of local expressions. Then there is the question whether one can find a "locally conserved 4-vector current" which defines the invariant ...

I also think it would be interesting, I've one question though, isn't something like a 4D current what is made to vanish in GR by imposing a covariantly divergence-less stress-energy tensor?
To get a globally conserved quantity, wouldn't you need an explicit choice of frame of reference? But this is exactly what you can't do in GR if you want to respect the general covariance of the 4-manifold. You would be imposing an artificial gauge.

 

[WannabeNewton]

To get a locally conserved quantity, wouldn't you need an explicit choice of frame of reference?

What do you mean by that?

 

[TrickyDicky]

Suppose we have a 4-dim. pseudo-Riemannian manifold M with spacelike foliations F, F', F'', ... For each F one can define a family of 3-volumes V_F(t) covering M where t indicates a timelike direction (coordinate) perpendicular to V_F(t). For each V_F(t) one can define 3-integrals

$Q_F[q] = \int_{V_F(T)} \,q$

using differential forms q?

Under which conditions do these Q represent "reasonable physical obervables" with a well-defined, covariant transformation law? Under which conditions can one find a conservation law

$\frac{dQ_F[\omega]}{dt} = 0$

Are there some physical relevant examples for q? How can q be constructed from the metric g (or a 3-bein e w.r.t to the 3-volume V_F)?

Under the same conditions that the energy quantity is obtained, I would guess? thru a KV field relevant for the quantity that one wants conserved like spacelike KV for momentum?
So they would have the same problem as with energy, don't you think?

 

[TrickyDicky]

What do you mean by that?

I,ve edited the post to change it, a little lapsus.
I understand what tom stoer looks for is the globally conserved quantity (integral), maybe I misinterpreted him.

 

[WannabeNewton]

I,ve edited the post to change it, a little lapsus.

Oh, yes then I do agree with you. The path dependence of parallel transport on curved manifolds will certainly change the result of the integral based on the reference frame.

 

[tom.stoer]

Constructing a globally conserved quantity from a conserved four-vector current density dj=0 works as usual by Stokes theorem. But constructing something like that based on a conserved tensor density DT=0 does not work b/c Stokes theorem does not apply to the Christoffel symbols appearing in the covariant derivative of T.

What I am looking for are some less restrictive conditions to construct a conserved entity Q as an integral over spacelike 3-manifolds and a "3-density" q (dj=0 is sufficient, but perhaps a weaker condition is available).

That's why I am asking for topological invariants of 3-manifolds. A simple example is the Gauss–Bonnet theorem for 2-manifolds. Is there something similar for 3-manifolds? Can the integrands (Gauss–Bonnet: curvature) be interpreted physically?

 

[TrickyDicky]

Constructing a globally conserved quantity from a conserved four-vector current density dj=0 works as usual by Stokes theorem. But constructing something like that based on a conserved tensor density DT=0 does not work b/c Stokes theorem does not apply to the Christoffel symbols appearing in the covariant derivative of T.

Sure, and therefore either you do the physically unjustified kv thing (see https://www.physicsforums.com/showpost.php?p=3438092&postcount=52 ) or I'd say you can't construct such thing in GR.

That's why I am asking for topological invariants of 3-manifolds. A simple example is the Gauss–Bonnet theorem for 2-manifolds. Is there something similar for 3-manifolds? Can the integrands (Gauss–Bonnet: curvature) be interpreted physically?

I've read something about 3-dim topological invariants in the context of QFT and quantum gravity but I'm not sure if that is what you are interested in or if they can be interpreted physically. What kind of physical property do you imagine?

 

[tom.stoer]

We agree on dj=0 and DT=0.

My question is if there could be other approaches (besides Noether currents) from which invariants as integrals over a spacelike 3-manifold can be constructed.

Regarding physically relevant examples: the Gauss-Bonnet theorem measures the (topologically constant) "total curvature". You could e.g. calculate a "mean curvature" by deviding by the volume. That's certainly interesting.

 

[TrickyDicky]

Regarding physically relevant examples: the Gauss-Bonnet theorem measures the (topologically constant) "total curvature". You could e.g. calculate a "mean curvature" by deviding by the volume. That's certainly interesting.

Quoting D. Hilbert: the Gauss-Bonnet theorem is just a 2-dimensional version of Einstein's field equation.

 

[tom.stoer]

Look at my first post.

Using the Gauss-Bonnet theorem in two dimensions my q would correspond to the Gaussian curvature. Are there examples for such a q on spacelike 3-manifold? Are there other approaches (besides Noether currents) from which invariants as integrals over a spacelike 3-manifold can be constructed?

Remember: I am not talking about arbitrary 3-manifolds, but about spacelike 3-manifolds derived from a foliation of 4-dim. spacetime.

 

[TrickyDicky]

Look at my first post.

Using the Gauss-Bonnet theorem in two dimensions my q would correspond to the Gaussian curvature. Are there examples for such a q on spacelike 3-manifold? Are there other approaches (besides Noether currents) from which invariants as integrals over a spacelike 3-manifold can be constructed?

Remember: I am not talking about arbitrary 3-manifolds, but about spacelike 3-manifolds derived from a foliation of 4-dim. spacetime.

Tom, I understand your question, and as I said, I'm not aware of any other approach and IMO probably there's no such. But I would sure be glad if someone came up with something like that.

 

[tom.stoer]

Something like Chern-Simons invariants or Pontryagin index?

 

[TrickyDicky]

Something like Chern-Simons invariants or Pontryagin index?

Have you heard something about TMG (topologically massive gravity) ? they use Chern-Simons forms too.It is related to Hořava–Lifgarbagez gravity.

 

[tom.stoer]

Have you heard something about TMG (topologically massive gravity) ?

I know that then approach exists, but I haven't studied one single paper. I only wanted to indicate that there may be some topological invariants related to local entities. Cherns-Simons theory is a bit strange as it relies on non-gravitational entities and is therefore not "physical" in the GR context. So something with pure metric (or tetrad) content is preferred. But I may be wrong ...

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EDIT: does this apply? remember: I am, not talking about 2+1 dim. field theory / gravity but about 3-manifolds induced by a spacelike (!) foliation of 3+1 dim. gravity.

 

[tom.stoer]

I think what I originally had in mind

Foliations F, F', F'', ... and 3-integrals over 3-volumes V_F(t) ...

... defining "reasonable physical obervables" Q

$Q_F[q] = \int_{V_F} \,q$

... which are conserved under time evolution

$\frac{dQ_F[q]}{dt} = 0$

... where q is constructed from the 3-metric g

does not really make sense.

Consider a non-compact, homogeneous 3-space of non-vanishing 3-curvature. Usually Q will diverge, therefore already in very simple cases no reasonable invariants Q defined as 3-integrals can exist.

 

[TrickyDicky]

I think what I originally had in mind



does not really make sense.

Consider a non-compact, homogeneous 3-space of non-vanishing 3-curvature. Usually Q will diverge, therefore already in very simple cases no reasonable invariants Q defined as 3-integrals can exist.

Yep, I think it will usually diverge. Hmm, except perhaps if the 3-curvature is positive.

 

[tom.stoer]

Yep, I think it will usually diverge. Hmm, except perhaps if the 3-curvature is positive.

Of course.

The problem is that it does not make sense to study invariants which are not defined for the universe we live in :-)

What I had in mind was the definition of globals invariants define as 3-integrals which are conserved under time evolution, either by Noether's theorem (which does not work for mass, energy, momentum, angular momentum, ... in general) or via topology (like Euler characteristic, ...). I thought that it would be interesting to study the relation between topology and dynamics (like Atiyah-Singer - anomalies - instanton numbers - ...) but based on the manifold structure of GR itself (metric, curvature) instead of fibre bundles.

Unfortunately in contradistinction to gauge field configurations you cannot control the manifold. You can define a gauge field on a non-compact manifold and study Chern–Simons forms, chern-classes etc. But you can't do that for the manifold itself.

Nevertheless, thanks for the discussion. At least I learned that it will not work :-)