Hamilton-Jacobi-Einstein equation

Question

I have been looking at the Hamiltonian formalism of GR for some time and recently stumbled across the Hamilton-Jacobi-Einstein equation:

$$\frac{1}{\sqrt{g}} (\frac{1}{2}g_{pq}g_{rs} - g_{pr}g_{qs}) \frac{\delta S}{\delta g_{pq}} \frac{\delta S}{\delta g_{rs}} + \sqrt{g} R = 0$$

where $g_{ij}$ is the 3-metric dependent only on the spatial coordinates, i.e, $g_{ij} = g_{ij}$(r), and $g$ is the determinant of the 3-metric. ($R$ is the Ricci scalar of the 3D geometry). $S$ is the action dependent solely on the 3D metric $g_{ij}$.

Note that there's no time variable anywhere! Now, this is where I am confused at: Misner, Thorne, and Wheeler claim that "All the dynamic content of geometrodynamics is contained in this equation". But how does this equation govern the dynamics of the 3D metric in the 4D space, if there's no parameterization of any sort in this equation. Like how does this equation dictate the leaves $g_{ij}$ how to evolve if there's no reference to a time variable or any parameter?

Answer 1

1. The Hamilton-Jacobi-Einstein equation corresponds to the HJ equation $$ H[g_{ij},\frac{\delta W}{\delta g_{kl}}]~=~0$$ for Hamilton's characteristic functional$^1$ $W[g_{ij}]$ with vanishing energy $E=0$.

2. The Hamilton-Jacobi-Einstein equation is closely related to the Wheeler-DeWitt equation $$ \hat{H}[\hat{g}_{ij},\hat{\pi}^{kl}]|\Psi\rangle~=~0.$$

3. Since GR is reparametrization invariant, the Hamiltonian/energy vanishes, cf. e.g. this & this Phys.SE posts. Time evolution is just a gauge transformation, i.e. not physical. To obtain a unique evolution one must choose a gauge (and initial conditions).

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$^1$ In field theory Hamilton's characteristic function $W$ becomes a functional. It should not be conflated with the action $S$, cf. e.g. my Phys.SE answer here.