Nonexistence of local gravitational energy

[muppet]

I chanced upon an argument in Misner, Thorne and Wheeler to the effect that the energy/momentum of the gravitational field cannot classically be localised. Basic idea: you can make the Christoffel symbols vanish at any point, and hence the gravitational field at that point will vanish, taking any sensible conception of gravitational energy/momentum with it.

What does this mean for attempts to formulate gravity as a local quantum field theory? Can higher-order terms in some effective lagrangian contain derivatives of the Christoffel symbols that are non-vanishing, even when the Christoffel symbols themselves are?

Thanks in advance.

 

[atyy]

Maybe try Deser's http://arxiv.org/abs/gr-qc/0411023?

Usually, I the non-locality I read about seems to be the lack of local observables.
http://arxiv.org/abs/hep-th/0512200
http://arxiv.org/abs/1105.2036

Clearly QG can be formulated as local QFT at low energies. The problem is high energies. If gravity can be a local QFT even at high energies, it is presumably asymptotically safe. http://arxiv.org/abs/0709.3851

Some arguments against asymptotic safety are
http://arxiv.org/abs/0709.3555
http://arxiv.org/abs/1005.5408

As far as I can tell, the arguments against asymptotic safety are not regarded as conclusive, even by string theorists.

 

[muppet]

Thanks for the links- both asymptotic safety scenarios and Giddings' S-matrix approach are strewn across my desk as I type this! I've not read anyone comment on the energy distribution before; it seems to me that the "local energy density" would be the eigenvalues of the Hamiltonian density operator; following a few links on Wikipedia suggests that Misner has done plenty of work on this, and there's a subtlety in what energy actually means in this context.

 

[genneth]

Muppet: notice that your objection applies equally well to GR, and quantum mechanics does not suddenly make things worse. A Hamiltonian version of GR (see ADM formalism in MTW) is problematic even classically to deal with. It could simply mean that a Hamiltonian formalism for quantum gravity is not viable in all regimes. That is okay --- nothing about quantum mechanics fundamentally breaks without a preferred time evolution operator.

 

[muppet]

Thanks for the reply, Genneth; classically, I wasn't bothered by the non-localisability of gravitational energy because "gravity=geometry", and it makes sense to me that certain features of that can only be understood globally. In contrast, my understanding of the whole *point* of quantum field theory is the construction of local observables. Once I'd thought properly about what it would really mean in this context, I begun to feel slightly easier about it all (by which I really mean that I put the ADM formalism at the bottom of a very long to-do list and got on with some real work )

 

[Finbar]

Where did you pick up the understanding that the *point* of QFT is to define local observables?

This is probably a deep and common misconception. Physical observables are gauge independent and not always local for example wilson loops are non-local but gauge invariant.

I agree with genneth that the ADM formalism is actually the problem since one ends up with a bracket that is defined for objects which are not physical observables and then one has all the problems of constraints etc.


DeWitt has put forward the so called "global approach to QFT" (in his book of that name) which is based on the Peierls bracket, as supposed to the Poisson bracket, which has the advantage that it is only defined for observables.

 

[jonnyfish]

The last few times I've seen Shing-Tung Yau speak at math conferences, he's been advertising his recent work on quasilocal mass in general relativity. See for example http://arxiv.org/abs/0804.1174. He seems to have a completely general definition that is nonnegative and recovers other definitions of mass in various limit (eg asymptotically flat). He can prove rigorously that it has all sorts of desirable properties. I'm not sure if the physics community has caught on yet, but this is some really groundbreaking work that seems to be (at least partially) resolving some long-standing issues in GR. My secret hope is that this will lead to a better Hamiltonian formulation of GR (even classically) that could give hints towards the right way to quantize it.

 

[marcus]

The last few times I've seen Shing-Tung Yau speak at math conferences, he's been advertising his recent work on quasilocal mass in general relativity. See for example http://arxiv.org/abs/0804.1174. He seems to have a completely general definition that is nonnegative and recovers other definitions of mass in various limit (eg asymptotically flat). He can prove rigorously that it has all sorts of desirable properties. I'm not sure if the physics community has caught on yet, but this is some really groundbreaking work that seems to be (at least partially) resolving some long-standing issues in GR. My secret hope is that this will lead to a better Hamiltonian formulation of GR (even classically) that could give hints towards the right way to quantize it.

That is absolutely fascinating. I was not aware of it. I see there are 3 followup papers on this, for a total of 4

http://arxiv.org/abs/0805.1174
http://arxiv.org/abs/0804.1370
http://arxiv.org/abs/0906.0200
http://arxiv.org/abs/1002.0927

It would be great if in the GR context one could define the energy or mass within a certain baggie---inside some given 2-surface.
I gather the technique is to define admissible observers and take the INFIMUM, or greatest lower bound.

Since you have more familiarity with this, do me a favor if you would: indulge my curiosity and fantasize a bit about the consequences, if S-T Yau's definition finds acceptance.
Can you see how entropy might be defined? Can you see how geometric entropy might be defined---say within some baggie?

Heady stuff pray tell us more, oh Jonnyfish.

 

[jonnyfish]

I'm not sure that I can say too much--my expertise is not classical GR (though I do some differential geometry and PDE's). Better to hear from Yau and collaborators directly:

http://www.fields.utoronto.ca/audio/10-11/nonthematic_DLS/yau2/
http://pirsa.org/10050034/

There may be other recordings of talks as well but these are the ones I attended. I get the feeling that this might be overlooked--the definition and proofs of its properties might be a bit technical for many physicists, while on the other hand many of the mathematicians capable of really understanding it in detail don't particularly care about quantizing GR.

The fact that it is defined for surfaces suggests that there might be some kind of fundamental holographic principle in GR even at the classical level. I'm vaguely aware that there are stringy (and entropic?) arguments to suggest holography, so it seems like this could be more than just coincidence. More wild speculation on my part, but since UV divergences in QFT come from taking products of the field evaluated at the same spacetime point, it might be that a "quasilocal" quantized GR (as opposed to local QFT) might be exactly the thing that cures GR's problems. I would love to hear what QG researchers think of Yau's work.

 

[marcus]

Thanks! I've been listening to the Mu-Tao Wang talk some this afternoon, and looking over the papers by Yau and Wang.
EDIT: Eventually I found the talk by Yau himself the most helpful---it is audio+slides and it takes some guesswork to know when to advance to the next slide, keeping pace with the talk. Assuming that can be managed then of the two Jonnyfish mentioned I personally recommend this one
http://www.fields.utoronto.ca/audio/10-11/nonthematic_DLS/yau2/

 

[muppet]

Where did you pick up the understanding that the *point* of QFT is to define local observables?

This is probably a deep and common misconception. Physical observables are gauge independent and not always local for example wilson loops are non-local but gauge invariant.

This was based on the premise that fields are the building blocks of any local operator that commutes with itself at spacelike separations.

 

[genneth]

This was based on the premise that fields are the building blocks of any local operator that commutes with itself at spacelike separations.

To be fair, that is the impression one gets from most standard texts on QFT, which tend to be written with a bias towards high energy physics. In that context, one always has a flat background on which field live. If you like, the background is treated "classically", not as a part of the quantum system. Clearly, this treatment has issues when moving towards a quantum theory of geometry.

 

[tom.stoer]

I recommend the living-reviews article

http://relativity.livingreviews.org/Articles/lrr-2009-4/
Quasi-Local Energy-Momentum and Angular Momentum in General Relativity
László B. Szabados
Research Institute for Particle and Nuclear Physics of the Hungarian Academy of Sciences
lbszab@rmki.kfki.hu

Major update of lrr-2004-4 (see article history and change log for details.)

Abstract
The present status of the quasi-local mass, energy-momentum and angular-momentum constructions in general relativity is reviewed. First, the general ideas, concepts, and strategies, as well as the necessary tools to construct and analyze the quasi-local quantities, are recalled. Then, the various specific constructions and their properties (both successes and deficiencies are discussed. Finally, some of the (actual and potential) applications of the quasi-local concepts and specific constructions are briefly mentioned.

This review is based on talks given at the Erwin Schrödinger Institute, Vienna in July 1997, at the Universität Tübingen in May 1998, and at the National Center for Theoretical Sciences in Hsinchu, Taiwan and at the National Central University, Chungli, Taiwan, in July 2000.