I've been trying to organize my thoughts about conservation laws in GR, and so far I'm not having as much success as I'd like in bringing order to the whole topic. Maybe this is just the way GR is -- conservation laws don't play their usual central role, and their behavior varies on a case-by-case basis -- but I wonder if I'm missing some more general insights that would allow me to bring more order to the picture.(adsbygoogle = window.adsbygoogle || []).push({});

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.

Here's a summary of my understanding at this point:

Because vectors at different locations in spacetime can only be compared subject to the ambiguities of path-dependent parallel transport, we don't expect to have global conservation of any non-scalar quantity in GR; GR doesn't even have the language needed to state such a conservation law. Even if such a quantity has a local conservation law, that doesn't imply a corresponding global conservation law, since Gauss's theorem fails in curved spacetime. The non-scalar conserved quantities we have in nonrelativistic mechanics are momentum and angular momentum. In GR we have local conservation of energy-momentum, but there is no corresponding global conservation law. A good example of this is the relativistic swimmer.[Gueron 2005] The swimmer obeys local conservation of momentum at every point within its body, but violates global conservation of momentum.

We can say more about scalars like charge on a global scale, but what we can say is still weaker than the corresponding conservation laws in flat spacetime. For example, in a closed universe we can make a 3-surface around the "equator," and we expect that if we don't see any net current across the equator, then the total flux across the equator should be constant. But if the flux did change, we wouldn't be able to tell in which hemisphere the violation of conservation of charge was happening. (MTW has a discussion of this example on p. 457.)

In classical GR, the existence of horizons makes it difficult to relate a global conservation law to observables. If we sweep an electron behind a horizon, it becomes impossible to verify later on whether or not its lepton number continued to be conserved. In the case of a black hole's horizon, the electrovac no-hair theorem makes it seem like there is no possible observable that would test for nonconservation of any quantity other than mass, charge, and angular momentum. Wald discusses on p. 413 how this issue becomes even more acute when you start talking about quantum gravity; once a black hole has evaporated completely, you clearly have nonconservation of lepton number, etc. However, all of this gets a little muddy for me because the generalized no-hair conjecture fails.[Heusler 1998] Black hole evaporation could only ever be observed by humans in the case of black holes that started off microscopic, e.g., black holes created at the LHC if there are large extra dimensions, and in that context, I would be suspicious of any argument that assumed special status for the electromagnetic interaction as opposed to the other fundamental forces.

Another interesting point that comes up when you deal with singularities is that when Gauss's theorem fails in curved spacetime, the bound on the error is proportional to the curvature. Therefore you could theoretically have arbitrarily bad nonconservation of a non-scalar quantity as you get closer and closer to a black hole or Big Bang singularity. In popular writing, you often hear statements like, "The laws of physics break down at a singularity." In the past, I'd construed this to mean that (1) classical GR no longer gives well-defined predictions about initial value problems (Earman's famous "lost socks and green slime"), and (2) classical GR should give way to quantum gravity at the Planck scale. But it seems like there is also (3) a total breakdown of non-scalar conservation laws in classical GR.

First off I'd like to hear whether the above analysis holds up to inspection. If it does, I'd also be interested in whether anyone sees a simpler or more systematic perspective. It seems like every conservation law is a special case unto itself.

"'Swimming' versus 'swinging' in spacetime", Gueron, Maia, and Matsas, http://arxiv.org/abs/gr-qc/0510054

Markus Heusler, "Stationary Black Holes: Uniqueness and Beyond." http://www.livingreviews.org/lrr-1998-6

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# Conservation laws in GR: a messy picture?

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