Hamilton-Jacobi equation in context of GR

Question

I'm reading a paper in which the Hamilton-Jacobi equation is stated as $$-\frac{\partial S}{\partial \tau} = \frac{1}{2}g^{\alpha\beta}\partial_\alpha S \partial_\beta S,\tag{1}$$ where $\tau$ is the proper time. This is the first time I've come across the Hamilton-Jacobi equation in the context of GR.

I understand the Hamilton-Jacobi equation in the classical sense (see e.g. wikipedia https://en.wikipedia.org/wiki/Hamilton%E2%80%93Jacobi_equation) but I can't seem to generalize this to the expression above in GR-terms.

Answer 1

OP's eq. (1) is evidently the Hamilton-Jacobi (HJ) equation for the Hamiltonian $H=\frac{1}{2}g^{\alpha\beta}p_{\alpha}p_{\beta}$ of a relativistic point particle in the $e=1$ gauge, cf. e.g. this related Phys.SE post.