The general interaction of EM fields with gravity is given by,

$$ S = \int d^4x \sqrt{-g} F_{\mu \nu} F^{\mu \nu}$$

for EM field tensor $ F_{\mu \nu}$ and $g$ is the determinant of the metric tensor. Through the variational principle, this equation governs the motion of an EM wave in a gravitational field. We can think of this as the 'macrophysical' description.

Now, in GR the motion of a particle (e.g. a photon) is governed by the equations which result from the Hamiltonian,

$$ H = \frac{1}{2} g^{\mu \nu} p_{\mu} p_{\nu}$$

$$\dot{x^{\mu}} = \frac{\partial H}{\partial p_{\mu}} \, \, \, ; \, \, \, \dot{p}_{\mu} = -\frac{\partial H}{\partial x^{\mu}} \ , $$

which $=0$ for a photon, and $p_{\mu}$ is the photon 4-momentum. This is the 'microphysical' description.

In some limit, it seems appropriate that the macrophysical and microphysical (i.e. the wave/particle) descriptions are equivalent.

Is it possible to go from equation 1 (re EM action), to equations 2/3 (Hamiltonian approach?)