Conservation laws in GR: a messy picture?

[bcrowell]

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.

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

 

[Mentz114]

There's much food for thought in your post and I'm not able to add anything useful except to note that you haven't mentioned the Landau-Lifgarbagez energy pseudo-tensor, which does have some relevance to conservation in GR.

My overall impression has been that we must be satisfied with local conservation laws, because global conservation can't be defined ( which is what you've said in effect, I think ). Global conservation would require action at a distance, which is specifically forbidden.

I still find it significant that gravity can appear as a gauge field if one applies local translational invariance of energy-momentum instead of the Newtonian global invariance. A naive interpretation is that gravity only exists to ensure local conservation laws, but that is putting the horse before the cart probably.

 

[bcrowell]

Hi, Lut --

Thanks very much for your comments! Good to know that at least one person knowledgeable about GR didn't think they came off as mad ravings, or were full of ridiculous mistakes :-)

A couple more things I would still like to understand better:

-If Gauss's theorem fails infinitely badly at a singularity, it would be interesting to be able to come up with a concrete example of the (presumably spectacular) consequences. Or maybe nature conspires to keep the consequences from being all that spectacular...? In the case of a black hole singularity, the observer who sees the consequences only has an infinitesimal time to contemplate them.

-I'm not clear on the role of topology in the swimming examples. It seems like you can have examples like this that are purely topological (like the swimmer reaching around a cylinder) and others that are purely about curvature.

- Ben

 

[pervect]

My perspective is this, I'm not sure if it will be helpful. What physicists call local conservation of mass-energy is mathematically stated by [itex]\del \dot T = 0. This type of conservation is actually built into Einstein's field equations. Global conservation of mass/energy/momentum requires special conditions, but these can be understood in a uniform framework as resulting from Noether's theorem, where time translation symmetry results in a conserved quantity which we call energy. Noether's theorem applies, as GR can be formulated as an "action" theory.

The most obvious sort of time translation symmetry is in a static system - this gives rise to the Komar mass. Asymptotically flat spacetimes have asymptotic symmetries at infinity - it gets messy here, but the infinite dimensonal symmetry groups have a preferred 4 dimensonal subgroup, which exhibits a time-translation symmetry. This gives rise to the ADM and Bondi masses.

I'm not quite sure where quasi-local masses fit into this framework - or if they do.

Conservation of charge is a bit simpler. Even in curved space-time, electromagnetism is a 2-form. So I think of the conservation of charge as being the result of our ability to represent a 2-form as a geometrical structure. Basically, I imagine space-time as being permeated by field-lines, which represent the 2-form structure of the electromagnetic field, and "counting the field lines" as giving us a conserved quantity. It's possible my understanding is flawed here, I should probably think about it, but that's the intuitive basis I currently use.

 

[Dmitry67]

Whats about some particular cases:
(non)conservation in CTL
And, as the most extreme example of CTL, Goedels Universe?

 

[bcrowell]

Whats about some particular cases:
(non)conservation in CTL
And, as the most extreme example of CTL, Goedels Universe?

What is CTL? Is it related to closed timelike curves (CTCs)?

It's not necessarily clear that CTCs would violate of conservation of energy:

https://www.physicsforums.com/showthread.php?p=819700#post819700
http://www.npl.washington.edu/AV/altvw69.html
http://golem.ph.utexas.edu/string/archives/000550.html

 

[Dmitry67]

Yes, CTCs, but your links are about wormholes.
I don't believe in wormholes, but I do believe in CTC inside Kerr BH

 

[Ben Niehoff]

You keep saying that Gauss's Law fails in curved spacetime, but I cannot imagine what you mean. Gauss's Law is simply (in appropriate units)

$$Q_{V} = \int_{\partial V} \star F.$$

This applies whatever the curvature of spacetime.

Perhaps what you mean is that the areas of spheres do not grow as r^2. But the electric field of a point charge always falls off as the areas of geodesic spheres, and so any curvature in the growth rate of spheres is exactly counterbalanced.

 

[pervect]

You keep saying that Gauss's Law fails in curved spacetime, but I cannot imagine what you mean. Gauss's Law is simply (in appropriate units)

$$Q_{V} = \int_{\partial V} \star F.$$

This applies whatever the curvature of spacetime.

Perhaps what you mean is that the areas of spheres do not grow as r^2. But the electric field of a point charge always falls off as the areas of geodesic spheres, and so any curvature in the growth rate of spheres is exactly counterbalanced.

This is related to what I was trying to say earlier - but I had to think about how to express it.

A 3-form and a 3-volume, taken together, "naturally" give a scalar - or in general, an n-form and an n-volume give a scalar, by a "natural" process of integration over a manifold.

Wald discusses this in appendex B in modern, formal language. MTW discusses it too, Wald also discusses the generalized version of Stoke's theorem, which we'll wind up needing. I tend to favor the geometric view, in which one thinks of the n-forms as geometric constructs, "field lines", and the integration process as "counting the field lines", but I'm not sure I can explain this very well - I think MTW does a decent job of this, though.

Anyway, given this background - that we know how to integrate an n-form over an n-volume to get a scalar.

How do we express charge in geometric language? Well, we take the one-form, J, which represents current density. And we take its dual, *J, which is a 3-form. We know how to integrate 3-forms over 3-boundaries to get a scalar. And this scalar, the integral of the 3-form, *J, over some 3 dimensional volume is just the charge enclosed in the volume.

And Maxwell's equations, in geometric form, say that d*F = 4 pi *J. How do we express the integral of the force normal to the surface over the two volume? Well, if we feed the Farady tensor F a unit time vector and a normal to the surface, we get a scalar, which is the component of force in that direction. But the vectors that compose the surface S are orthogonal to the normal vector and the time vector. So, we need the duality operation - we need *F, not F. And we can see that if we integrate *F, which is a 2-form, over some 2-surface S , we get the usual "force*area* integral.


The net result of all this is that we can draw a sphere around a black hole. And we integrate the normal force, measured by the clocks and rods of some physical observer on the sphere, times the area, measured in the same way. And this integral gives us the charge of the black hole in curved space-time, just as it does in non-curved space-time.

However, this sort of approach doesn't work to give us the mass of the black hole. It turns out in the static case we can integrate the "force at infinity" times the enclosing area to get the mass - see Wald starting at page 285 on energy. You might also see http://en.wikipedia.org/wiki/Komar_mass which is a rehash ov Wald's derivation that I posted to the Wiki.

But it'sreally not right to say that we don't have generalized versions of Stoke's theorem. We do. And for electromagnetism, they work just fine to define charge, in a manner that is almost the same as we are used to from flat space-time.

 

[bcrowell]

You keep saying that Gauss's Law fails in curved spacetime, but I cannot imagine what you mean.

Gauss's theorem fails in curved spacetime when the flux is a vector, which is what happens when you're talking about mass-energy. There's a good discussion here: http://www.phys.ncku.edu.tw/mirrors/physicsfaq/Relativity/GR/energy_gr.html I did say "we don't expect to have global conservation of any non-scalar quantity in GR," but it sounds like I caused confusion by dropping the "non-scalar" later in my post.

But it'sreally not right to say that we don't have generalized versions of Stoke's theorem. We do. And for electromagnetism, they work just fine to define charge, in a manner that is almost the same as we are used to from flat space-time.

Right, this is a case where the flux is a scalar.

-Ben