Register to reply 
Covariant global constants of motion in GR? 
Share this thread: 
#1
Sep511, 04:06 PM

Sci Advisor
P: 5,366

We know that in GR it is not possible for arbitrary spacetimes to define a conserved energy by using a 3integral. There are some obstacles like
Now let's forget about this specific case and ask the following more general questions: Suppose we have a 4dim. pseudoRiemannian manifold M with spacelike foliations F, F', F'', ... For each F one can define a family of 3volumes 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 3integrals [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 welldefined, 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 3bein e w.r.t to the 3volume V_{F})? 


#2
Sep511, 05:23 PM

Sci Advisor
PF Gold
P: 5,027

The following is the best I know bridging the gap between purely local statements and statements at infinity:
http://relativity.livingreviews.org/...es/lrr20094/ 


#3
Sep511, 05:45 PM

Sci Advisor
P: 5,366

:) 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" ... 


#4
Sep511, 09:47 PM

Sci Advisor
PF Gold
P: 5,027

Covariant global constants of motion in GR?



#5
Sep611, 07:20 AM

Sci Advisor
P: 5,366

One idea is to consider topological invariants of 3manifolds. One would have to study the effects of black hole formation, i.e. whether a 3invariant 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 4vector current" which defines the invariant ... 


#6
Sep611, 06:25 PM

Emeritus
Sci Advisor
P: 7,596

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 vicaversa), though I'm not actually 100% sure on that point. 


#7
Sep711, 12:11 AM

Sci Advisor
P: 5,366

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. 


#8
Sep711, 07:20 AM

P: 3,006

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 divergenceless stressenergy 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 4manifold. You would be imposing an artificial gauge. 


#10
Sep711, 07:33 AM

P: 3,006

So they would have the same problem as with energy, don't you think? 


#11
Sep711, 07:38 AM

P: 3,006

I understand what tom stoer looks for is the globally conserved quantity (integral), maybe I misinterpreted him. 


#12
Sep711, 07:55 AM

C. Spirit
Sci Advisor
Thanks
P: 5,427




#13
Sep711, 08:24 AM

Sci Advisor
P: 5,366

Constructing a globally conserved quantity from a conserved fourvector 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 3manifolds and a "3density" q (dj=0 is sufficient, but perhaps a weaker condition is available). That's why I am asking for topological invariants of 3manifolds. A simple example is the Gauss–Bonnet theorem for 2manifolds. Is there something similar for 3manifolds? Can the integrands (Gauss–Bonnet: curvature) be interpreted physically? 


#14
Sep711, 12:52 PM

P: 3,006




#15
Sep711, 04:34 PM

Sci Advisor
P: 5,366

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 3manifold can be constructed. Regarding physically relevant examples: the GaussBonnet theorem measures the (topologically constant) "total curvature". You could e.g. calculate a "mean curvature" by deviding by the volume. That's certainly interesting. 


#16
Sep711, 05:09 PM

P: 3,006




#17
Sep711, 05:29 PM

Sci Advisor
P: 5,366

Look at my first post.
Using the GaussBonnet theorem in two dimensions my q would correspond to the Gaussian curvature. Are there examples for such a q on spacelike 3manifold? Are there other approaches (besides Noether currents) from which invariants as integrals over a spacelike 3manifold can be constructed? Remember: I am not talking about arbitrary 3manifolds, but about spacelike 3manifolds derived from a foliation of 4dim. spacetime. 


#18
Sep711, 05:40 PM

P: 3,006




Register to reply 
Related Discussions  
Local constants of motion?  Differential Equations  1  
Independent constants of motion  Advanced Physics Homework  0  
Isometries and constants of motion  Differential Geometry  0  
Local vs Global Constants of Motion  General Physics  3  
Constants Of Motion  Advanced Physics Homework  4 