Computing Lagrangian for Light Propagation in Spacetime

[atoasttoall]

Does anyone know the Lagrangian for the propagation of light in curved spacetime? I'm disappointed to discover that I don't actually know how to compute the action for a given null curve.

 

[Matterwave]

It's the same as for a massive object except instead of the proper time being your parameterization, you have to use just some random affine parameter.