Spacetime and Geometry · Problem 7.3

Cosmology — Sean Carroll

Physics — General Relativity · fermats-principle-gravitational-lensing

Problem

Fermat's principle states that a light ray moves along a path of least time. For a medium with refractive index

n(\mathbf{x})

, this is equivalent to extremizing the time

t = \int n(\mathbf{x})[\delta_{ij}dx^i dx^j]^{1/2} \tag{7.204}

along the path. Show that Fermat's principle, with the refractive index given by

n = 1 - 2\Phi

, leads to the correct equation of motion for a photon in a spacetime perturbed by a Newtonian potential.