Phase Space GR 3+1 Form: Compact & Non-Compact

[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 $N^i$ 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

$$\{g_{ij},\pi^{ij},\gamma_{AB},p^{AB}\}$$

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

 

[Aaron Bex]

If so, then what would the dynamics of this extended phase space look like, and how would it be related to the dynamics of the original (closed) phase space?

 

[Entropia]

Yes, extending the spatial manifold \Sigma to include a boundary \partial\Sigma would require the phase space for GR to be extended as well. This is because the boundary term in the action for GR is necessary to ensure that the variational principle is well-defined, and therefore the boundary variables \gamma^{AB} and p^{AB} must be included in the phase space.

Furthermore, the inclusion of the boundary variables also reflects the fact that the boundary conditions at \partial\Sigma play an important role in the dynamics of the system. In particular, the boundary conditions can affect the evolution of the system and the behavior of the gravitational fields.

In summary, while the phase space for GR is usually taken to be the space of pairs (g_{ij},\pi^{ij}) over a closed spatial manifold \Sigma, extending \Sigma to include a boundary \partial\Sigma would require the phase space to be extended to include the boundary variables \gamma^{AB} and p^{AB}. This is necessary to properly account for the dynamics and boundary conditions of the system.