[tex]g_{ij} = \textrm{metric on a spatial surface}[/tex]

[tex]\pi^{ij} = \textrm{momentum conjugate to }g_{ij}[/tex]

[tex]N^i = \textrm{shift vector}[/tex]

[tex]N = \textrm{lapse function}[/tex]

Both [itex]N[/itex] and [tex]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 [itex](g_{ij},\pi^{ij})[/itex] over a given spatial manifold [itex]\Sigma[/itex]. (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 [itex]\Sigma[/itex] is taken to be closed, i.e., compact and without boundary. However, if we extend these ideas by setting [itex]\partial\Sigma\ne0[/itex] 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}\}[/tex]

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