Covariant global constants of motion in GR?

  1. tom.stoer

    tom.stoer 5,489
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    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 VF(t) covering M where t indicates a timelike direction (coordinate) perpendicular to VF(t). For each VF(t) one can define 3-integrals

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

    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

    [itex]\frac{dQ_F[\omega]}{dt} = 0[/itex]

    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 VF)?
  2. jcsd
  3. PAllen

    PAllen 5,891
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    Gold Member

  4. tom.stoer

    tom.stoer 5,489
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    :-) 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" ...
  5. PAllen

    PAllen 5,891
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    Gold Member

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

    tom.stoer 5,489
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    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 ...
  7. pervect

    pervect 8,157
    Staff Emeritus
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    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.
  8. tom.stoer

    tom.stoer 5,489
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    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.

  9. 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.
    Last edited: Sep 7, 2011
  10. WannabeNewton

    WannabeNewton 5,859
    Science Advisor

    What do you mean by that?
  11. 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?
  12. 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.
  13. WannabeNewton

    WannabeNewton 5,859
    Science Advisor

    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.
  14. tom.stoer

    tom.stoer 5,489
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    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?
  15. Sure, and therefore either you do the physically unjustified kv thing (see ) or I'd say you can't construct such thing in GR.

    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?
  16. tom.stoer

    tom.stoer 5,489
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    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.
  17. Quoting D. Hilbert: the Gauss-Bonnet theorem is just a 2-dimensional version of Einstein's field equation.
  18. tom.stoer

    tom.stoer 5,489
    Science Advisor

    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.
  19. 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.
  20. tom.stoer

    tom.stoer 5,489
    Science Advisor

    Something like Chern-Simons invariants or Pontryagin index?
  21. Have you heard something about TMG (topologically massive gravity) ? they use Chern-Simons forms too.It is related to Hořava–Lifgarbagez gravity.
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