# Sympletic structure of General Relativity

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It made me wonder about symplectic structures in GR, specifically, is there something like a Louiville form? In my dilettante understanding, the existence of the ADM formulation essentially answers that for generic cases, but it is unclear to me how boundaries change this. Specifically, I know that if one has an interior boundary, then generally the evolution is not hamiltonian; on the other hand, if the interior boundary is an isolated horizon, then the it is hamiltonian iff the first law of blackhole thermodynamics is obeyed (see http://arxiv.org/abs/gr-qc/0407042).

The sharper form of the question is thus what happens cosmologically?

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retagged Mar 7, 2014
Wald's book on GR has a section on the hamiltonian formalism in General Relativity. It is an infinite-dimensional system, so you have to be a little careful when you talk about a symplectic structure. It certainly has a Poisson structure and it is constrained. The Poisson reduction gives you formally symplectic structure.

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Notice first that the phase space of any theory is nothing but the space of all its classical solutions. The traditional presentation of phase spaces by fields and their canonical momenta on a Cauchy surface is just a way of parameterizing all solutions by initial value data -- if possible. This is often possible, but comes with all the disadvantages that a choice of coordinates always comes with. The phase space itself exists independently of these choices and whether they exist in the first place. In order to emphasize this point one sometimes speaks of covariant phase space .

This is well known, even if it remains a bit hidden in many textbooks. For more details and an extensive and commented list of references on this see the $n$Lab entry phase space .

Then notice that the phase space of every field theory that comes from a local action functional (meaning that it is the integral of a Lagrangian which depends only on finitely many derivatives of the fields) comes canonically equipped with a canonical Liouville form and a canonical presymplectic form. The way this works is also discuss in detail at phase space . A good classical reference is Zuckerman, a more leisurely discussion is in Crncovic-Witten .

This canonical presymplectic form that exists on the phase space of every local theory becomes symplectic on the reduced phase space, which is the space obtained by quotienting out the gauge symmetries. This quotient is often very ill-behaved, but it always exists nicely as a "derived" quotient, and as such is modeled by the BV-BRST complex (as discussed there). The whole (Lagrangian) BV-BRST machinery is there to produce the canonical symplectic form existing on the reduced phase space of any local action functional.

Since the Einstein-Hilbert action and all of its usual variants with matter couplings etc. is a local action functional, all this applies to gravity. Recently Fredenhagen et al. have given careful discussions of the covariant phase space of gravity (and its Liouville form), see the references listed here .

It follows that the "dimension" of the covariant phase space of gravity does not depend on the "size of the universe", nor does it make much sense to ask this, in the first place. A given cosmology is one single point in this phase space (or rather it is so in the reduced phase space, after quotienting out symmetries).

However, you might be after some truncations or effective approximations or coarse graining to full covariant gravity. For these the story might be different.

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answered Oct 12, 2011 by (5,085 points)
Finally, there is nothing particularly mysterious about restricting yourself to a cosmological patch or to any other kind of patch of spacetime. Given any manifolds $X$ and $Y$, the space of solutions of Einstein equations, $\Gamma(X)$ or $\Gamma(Y)$, on either of them is infinite dimensional. Moreover, a diffeomorphism $X\to Y$ naturally induces the map $\Gamma(Y)\to \Gamma(X)$, by differential pullback. One may think of $X$ as smaller than $Y$ and hence $\Gamma(Y)=\Gamma(X)\times$(extra degrees of freedom). But $X$ and $Y$ could also be exchanged. That's life with diff-inv and inf-dim.

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Ron, the first line of the OP's question asks for the symplectic structure of the GR phase space and whether it has a Liouville form. My answer, after a lead-in paragraph on what the phase space actually is, discusses both of these structures on phase space.

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@Igor: The OP was just intimidated by all the incredibly trivial, but oh-so-math-y things being said, and decided that an up-voted +11 answer must be OK and complete. It isn't. It's essentially saying "phase-space==solution space", big duh. In your comment, I misread "diffeomorphic" for "isometric", of course you are right that manifolds can be diffeomorphic to submanifolds of themselves, this is why I dislike math-speak. Without a base metric, you can't find the horizon, or formulate the right notion of causal patch, so the correct physical notion of embedding requires boundary matching.

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@Urs: Yes, I was following the question from the physics stackexchange version, and the real issue for me was the question of how you define holographic dynamics classically in a small horizon patch. This answer answers the overly formal form of the question as it appears here, and I can't complain that you didn't answer a question that wasn't asked.

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@Igor: the restriction to a causal patch _is_ mysterious, and this is the whole point of the question. This answer does not answer anything, and the comments are saying trite/wrong things in overly formal language. Specifically: what do you mean to say "X and Y can be exchanged"? If X is diffeomorphic to a subset of Y, then Y is not diffeomorphic to a subset of X. A diffeomorphism from X to Y induces the _obvious inclusion_ map from all solutions on Y to all solutions on X, but not vice versa.

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For the record, Ashtekar is no slouch when it comes to the _covariant phase space_ cosntruction of the symplectic structure. If you look at the list of references on the nLab page Urs cited, you'll see the papers by Lee-Wald and Ashtekar-Bombelli-Reula, which are also often used as standard references on this topic. In fact, the $\Omega_V$ term Ashtekar writes down in section 7.2 of the paper you cited is constructed using precisely this method. I may say more about the boundary term $\Omega_S$, but I'll have to look at it in a bit more detail first.
A note of caution: these symplectic forms have to be obtained by integrating over a Cauchy surface. However, while the surface $M_1$ in Fig.6 of gr-qc/0407042 is Cauchy, neither $M$ nor $M_2$ are, because some inextensible timelike curves may intersect $M_1$ and $\Delta$, but neither of the other two surfaces. To obtain the correct symplectic form, integration over $M$ or $M_2$ must be supplemented by integration over the corresponding past-directed portion of $\Delta$. This has to be taken into account when computing the surface term $\Omega_S$.
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