Notes on Spinoptics in a Stationary Spacetime

TL;DR

This paper derives covariant equations for spinoptics in stationary spacetimes, revealing non-conservation of certain quantities and differences for photon polarizations, with extensions to gravitational waves.

Contribution

It provides a covariant derivation of spinoptics equations in stationary spacetimes, highlighting new effects like non-conservation of inner products and polarization-dependent differences.

Findings

Null nature of modified photon trajectories is confirmed.

Inner product with stationary Killing vector is not conserved.

Differences between left and right polarized photons and gravitational waves are identified.

Abstract

In arXiv:1105.5629, equations of the modified geometrical optics for circularly polarized photon trajectories in a stationary spacetime are derived by using a (1+3)-decomposed form of Maxwell's equations. We derive the same results by using a four-dimensional covariant description. In our procedure, the null nature of the modified photon trajectory naturally appears and the energy flux is apparently null. We find that, in contrast to the standard geometrical optics, the inner product of the stationary Killing vector and the tangent null vector to the modified photon trajectory is no longer a conserved quantity along light paths. This quantity is furthermore different for left and right handed photon. A similar analysis is performed for gravitational waves and an additional factor of 2 appears in the modification due to the spin-2 nature of gravitational waves.