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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.

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  • $\begingroup$ The Hamilton-Jacobi equation for what system? A free point particle? $\endgroup$
    – Ghoster
    Commented Oct 7, 2023 at 21:22
  • $\begingroup$ It's in terms of deriving the Johannsen metric, which is a continuous deviation from the Kerr metric. It is just postulated here so I'm quite clueless $\endgroup$
    – Geigercounter
    Commented Oct 7, 2023 at 21:27
  • $\begingroup$ Which paper? Which page? $\endgroup$
    – Qmechanic ♦
    Commented Oct 7, 2023 at 22:15

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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.

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