Hamiltonian formulation of general relativity

Using the lapse function $N$, shift vector $N^i$ and spatial metric $h_{ij}$, the Legendre transform in mechanics of the Einstein-Hilbert action gives, in units $16\pi G=1$,Here $\pi =h_{ij}{\pi }^{ij}$ and ${\pi }^{ij}$ is the canonical momentum of the spatial metric. Varying lapse and shift imposes the Hamiltonian constraint and momentum constraint. An asymptotic time translation requires an Arnowitt-Deser-Misner energy boundary term. The bulk constraints generate gauge evolution; the nonzero asymptotic energy comes from the boundary.

On a compact spatial Cauchy hypersurface without boundary, the gravitational Hamiltonian consists entirely of constraints and vanishes on solutions. There is no asymptotic time-translation boundary charge to add. This is the canonical statement that a closed universe has zero total energy; it does not assert that local matter energy density or local curvature vanishes.

The momentum constraint is the equation obtained by varying the shift vector in the Hamiltonian formulation of general relativity. It expresses the tangential-normal projection of the Einstein field equations and generates spatial coordinate transformations. Writing $p^{ij}={\pi }^{ij}/\sqrt{h}$ avoids ambiguities about covariant derivatives of tensor densities.

With the convention $K_{ij}=({\dot{h}}_{ij}-D_i{N}_j-D_j{N}_i)/(2N)$ for extrinsic curvature, differentiation of the gravitational Lagrangian gives ${\pi }^{ij}=\sqrt{h}(K^{ij}-K{h}^{ij})$ in units $16\pi G=1$. It is a weight-one tensor density. In three spatial dimensions, $\pi =-2\sqrt{h}K$, soThe lapse function and shift vector have no time derivatives in this action and consequently have vanishing conjugate momenta. Changing the sign convention for extrinsic curvature changes the corresponding momentum relation; it must not be silently combined with this formula.