How to go from a wave to a particle description of photon dynamics in curved spacetime?

Question

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?)

Answer 1

The movement of photons in GR is obtained from the Maxwell equations in the limit of geometric optics: Consider as an ansatz for the vector potential $$A_\mu = (a_\mu + \epsilon b_\mu )e^{i\psi/\epsilon} $$ with a small parameter $\epsilon$ and a phase $\psi$. The wave vector is defined as $k_\mu=\partial_\mu \psi$. The Maxwell equations for the gauge potential (in Lorenz gauge and for vacuum) read as: $$\nabla_\nu\nabla^\nu A_\mu -R^\mu _\nu A^\nu =0$$ If we insert the vector potential here and read off the leading order we find that $k_\nu k^\nu=0$. This of course implies $0=\nabla_\mu(k^\nu k_\nu)=2k^\nu\nabla_\mu k_\nu$. A short calculation shows: $\nabla_\mu k_\nu=\nabla_\nu k_\mu$. Therefore we have: $$k^\nu \nabla_\nu k^\mu=0$$This means that the wave vector follows a null geodesic, which is the same result one obtains from varying a suitable action. You can find a complete derivation for example in: Norbert Straumann, General Relativity, section 2.8