# Sympletic structure of General Relativity

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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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. –  José Figueroa-O'Farrill Oct 12 '11 at 12:19
–  Ben Crowell Aug 24 '13 at 4:52

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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Nice answer, the point about phase space being a covariant object should be more widely appreciated. –  user566 Oct 12 '11 at 15:33
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. –  Igor Khavkine Oct 12 '11 at 15:48
I think most of the interesting physics is in the last sentence: the full solution extended beyond the cosmological horizon defines a point in this much too large phase space, the space of all Einsteinian metrics, but the original question was about the reduction of the phase space to describe the dynamics of a cosmological patch. This reduction should give that there are more effective degrees of freedom as the universe expands, and the reduction process is mysterious. I think that the spirit of the question is: can you make sense of a causal-patch reduction? –  Ron Maimon Oct 12 '11 at 17:46
The boundary term $\Omega_S$ in Ashtekar's paper, it seems, has everything to do with a Chern-Simons boundary term added to the action of GR while studying "black hole entropy". See, arxiv.org/abs/gr-qc/9710007 . With no need to pay attention to motivation for this extra term, the covariant phase space formalism described by Urs gives you both terms $\Omega_V$ and $\Omega_S$. –  Igor Khavkine Oct 13 '11 at 9:38
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. –  Igor Khavkine Oct 13 '11 at 9:50