Solving the Hamiltonian constraint equation in General Relativity

Question

The constraint equation of general relativity reads as follows \begin{align} \frac{(2\kappa)}{\sqrt{h}}\left(h_{ac} h_{bd} - \frac{1}{D-1} h_{ab} h_{cd} \right)p^{ab} p^{cd} - \frac{\sqrt{h}}{(2\kappa)}\left({}^{(D)}\mathscr{R} - 2 \Lambda\right) = 0, \end{align}

Where $h_{ab}$ are $D$ dimensional spatial metric and $p^{ab}$ are corresponding conjugate momenta. The above equation is algebraic, so it can be solved for the momenta. I find that the following ansatz solves the constraint

\begin{align} p^{ab} = \pm h^{ab} \frac{\sqrt{h}}{(2\kappa)}\sqrt{\frac{D-1}{D}\left(2\Lambda-{}^{(D)}\mathscr{R}\right)} \end{align}

Is this solution correct? Is this solution valid for all the $D$-hypersurface foliation or only on the initial slice? I am quite surprised to see that $p^{ab}$ do not depend on the other curvature tensors such as ${}^{(D)}\mathscr{R}_{ab}{}^{(D)}\mathscr{R}^{ab}$ etc. Any reason why?

Answer 1

I didn't check your math, but even if we accept that your equation for $p^{ab}$ in the middle of your question is correct, it's not really a "solution" of anything. The right-hand side of that equation still contains partial derivatives of $h_{ab}$ in it through the Ricci scalar term. So you started with a partial differential equation relating $p^{ab}$ and $h_{ab}$ and arrived at a different form of the same differential equation still relating those same tensor functions.

For your other questions, it's provable (Wald has a whole appendix on it in his book) that the constraints of the theory propagate, which is to say that $\partial_t H = 0$ and $\partial_t P^i = 0$ if those constraints are 0 on some initial hypersurface. A critical point is that there is not a single constraint, as the text of your question suggests, but 4 (more generally $D+1$) constraints via the scalar Hamiltonian constraint $H$ and the vector momentum constraint $P^i$.

As for why this depends only on the Ricci scalar (of the 3-metric, or in your case $D$-metric), I think that's not too surprising since the process of doing the ADM decomposition, giving the 3+1 formulation, starts with the regular Einstein-Hilbert action, which depends only on the Ricci scalar and metric determinant of the metric for the full spacetime. The result that the lower dimensional Hamiltonian constraint depends only on scalar curvature was essentially baked into the theory at the start.