New Properties of the Hamilton-Jacobi Functional for General Relativity

It is shown that, analogously to the situation in the usual classical theories, the solution of the Hamilton-Jacobi equations of general relativity, $S$, satisfies the relation $S=\int \mathcal{L}{d}^{4}x$, where, however, $\mathcal{L}$ is the Lagrangian density of Dirac, rather than the more usual Einstein Lagrangian. This result is then used to prove that if we write $S=\int \mathcal{S}\left(g_{\mathrm{mn}}\right)d^{3}x$, the integrand $\mathcal{S}$ is a three-dimensional scalar density which is a homogeneous functional of $g_{\mathrm{mn}}$ of degree 1.