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=∫Ld⁴x, where, however, 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=∫S(gₘₙ)d³x, the integrand S is a three-dimensional scalar density which is a homogeneous functional of gₘₙ of degree 1.