# Phase space for GR

1. Nov 29, 2006

### shoehorn

In the 3+1 formulation of GR we have the following basic variables:

$$g_{ij} = \textrm{metric on a spatial surface}$$
$$\pi^{ij} = \textrm{momentum conjugate to }g_{ij}$$
$$N^i = \textrm{shift vector}$$
$$N = \textrm{lapse function}$$

Both $N$ and $$N^i[/itex] are purely gauge variables, so are essentially unimportant. This means that the phase space for GR is essentially the space of pairs $(g_{ij},\pi^{ij})$ over a given spatial manifold $\Sigma$. (I know this isn't really the phase space since GR has constraints, but that's irrelevant for my current question.) The point is that, in most treatments of which I am aware, the spatial manifold $\Sigma$ is taken to be closed, i.e., compact and without boundary. However, if we extend these ideas by setting $\partial\Sigma\ne0$ then we know that the action for GR also requires a boundary term. Does this then imply that the phase space for GR should be extended to the set [tex]\{g_{ij},\pi^{ij},\gamma_{AB},p^{AB}\}$$

where $\gamma^{AB}$ is a two-dimensional metric on $\partial\Sigma$ and $p^{AB}$ is its conjugate momentum?

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