Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 311 / 1

The action contains neither $\dot{N}$ nor ${\dot{N}}^i$, so their conjugate momentum vanishes. They impose constraints rather than independent evolution equations: after the Legendre transform in mechanics, the lapse function and shift vector are Lagrange multipliers in the Hamiltonian formulation of general relativity.

Use precisely the paper's sign convention for extrinsic curvature, $K_{ij}=({\dot{h}}_{ij}-D_i{N}_j-D_j{N}_i)/(2N)$. At fixed $h_{ij},N,N^i$,Consequently the canonical momentum of the spatial metric isThis is a weight-one tensor density, with indices raised and lowered using $h_{ij}$. Its trace and inverse relation areThe sign of $K_{ij}$ used here is opposite to the convention in extrinsic curvature of a spatial hypersurface; mixing the two conventions would reverse the momentum formula.

In the paper's convention, ${\dot{h}}_{ij}=2N{K}_{ij}+D_i{N}_j+D_j{N}_i$. The Legendre transform in mechanics therefore givesIndeed ${\pi }^{ij}{K}_{ij}=\sqrt{h}(K_{ij}{K}^{ij}-K^2)$, and inversion of the canonical momentum of the spatial metric givesTo integrate the shift term without ambiguity about tensor densities, put $p^{ij}={\pi }^{ij}/\sqrt{h}$. Metric compatibility of the spatial covariant derivative gives, up to a boundary term,Thus the Hamiltonian formulation of general relativity hasVariation of the lapse function imposes the Hamiltonian constraint $\mathcal{H}=0$, while variation of the shift vector imposes the momentum constraint ${\mathcal{H}}_i=0$.

Choose the asymptotic time translation: $N=1+O(r^{-1})$, $N^i=O(r^{-1})$, with their derivatives $O(r^{-2})$, in asymptotically Cartesian coordinates. Use the usual differentiable falloff $h_{ij}={\delta }_{ij}+O(r^{-1})$, ${\partial }_k{h}_{ij}=O(r^{-2})$, ${\pi }^{ij}=O(r^{-2})$, and analogous falloff for allowed variations; impose the standard additional conditions needed for finite bulk integrals. These variations need not fix the Arnowitt-Deser-Misner energy.

Only the spatial curvature term produces the leading nonvanishing surface variation. Applying the variation of the Ricci scalar and applying integration by parts to its two derivative terms yieldsThe terms involving derivatives of the asymptotic lapse function, and the shift boundary terms, vanish under these falloff conditions. The displayed surface integral need not vanish: its integrand is $O(r^{-2})$ and the area is $O(r^2)$.

Hamilton's equations require a Hamiltonian with a variation expressed by the bulk functional derivatives alone for the allowed variations. The unwanted term is exactly $-\delta E_{\mathrm{ADM}}$, so differentiability of the gravitational Hamiltonian at spatial infinity requiresIts variation cancels the surface term. On the constraint surface, $H_{\mathrm{total}}=E_{\mathrm{ADM}}$. Restoring units inserts $1/(16\pi G)$ in the surface integral; the paper has set this factor to one. The original PDF has the standard equation ${\partial }_t{h}_{ij}=\delta H/\delta {\pi }^{ij}$; the extra $t$ in the extracted TeX is a transcription error.

A closed spatial Cauchy hypersurface is compact and has no boundary. Thus there is no asymptotic surface charge, and the complete gravitational Hamiltonian is the integral of the Hamiltonian constraint and momentum constraint derived above. Both constraints vanish on solutions, givingThis is the vanishing canonical Hamiltonian of a closed universe: canonical evolution is generated by constraints and there is no asymptotic time translation whose boundary charge could define an Arnowitt-Deser-Misner energy. It does not require local matter energy densities or local curvature to vanish. With matter included, the corresponding total constraints still vanish.