We present some new derivations of the effect of a plane gravitational wave
on a light ray. A simple interpretation of the results is that a gravitational
wave causes a phase modulation of electromagnetic waves. We arrive at this
picture from two contrasting directions, namely null geodesics and Maxwell's
equations, or, geometric and wave optics. Under geometric optics, we express
the geodesic equations in Hamiltonian form and solve perturbatively for the
effect of gravitational waves. We find that the well-known time-delay formula
for light generalizes trivially to massive particles. We also recover, by way
of a Hamilton-Jacobi equation, the phase modulation obtained under wave optics.
Turning then to wave optics - rather than solving Maxwell's equations directly
for the fields, as in most previous approaches - we derive a perturbed wave
equation (perturbed by the gravitational wave) for the electromagnetic
four-potential. From this wave equation it follows that the four-potential and
the electric and magnetic fields all experience the same phase modulation.
Applying such a phase modulation to a superposition of plane waves
corresponding to a Gaussian wave packet leads to time delays.Comment: Accepted for publication in Physical Review D, matches published
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