Gravitational birefringence of light in Schwarzschild spacetime

Abstract

We compute the gravitational birefringence of light as it undergoes gravitational lensing. To this end we re-derive the Souriau-Saturnini equations in the Schwarzschild metric and solve them numerically and perturbatively. Our main result is an offset between the trajectories of the photons of opposite polarisations, which grows with time. We also find an intriguing instability of the spin component transverse to the momentum.To the memory of Christian Duval 1 supported by the OCEVU Labex (ANR-11-LABX-0060) funded by the "Investissements d'Avenir"French government program

Gravitational spin Hall effect of light

“…A massless limit of these equations was derived by Souriau and Saturnini [28,29], and particular examples adapted to certain spacetimes have been discussed in Refs. [30][31][32]. Another commonly used method is the Wentzel-Kramers-Brillouin (WKB) approximation for various field equations on curved spacetimes.…”

How does the photon’s spin affect gravitational wave measurements?

“…Let us add, finally, that gravitational birefringence had already been considered experimentally in 1974 [45], resulting in an upper bound for this effect in gravitational lensing, but the results were somewhat inconclusive, since the effect can actually be expected to be much weaker than the experiment precision [38,39]. Thanks to the high sensitivity of LIGO and Virgo, experimental bounds can also be found for birefringence predicted by other theories, for example those violating the Lorentz invariance [46,47].…”

“…In specific examples, the treatment of the problem with the help of a Berry phase and the treatment with the MPD equations with the Tulczyjew SSC, or their symplectic description, agree with each other. See, for instance [36,37] for the treatment of chiral fermions, and [38,39] for birefringence of a photon in a Schwarzschild spacetime (note that there is a typo in the very last formula in [38]: their anomalous velocity is indeed transverse to the geodesic plane, just as in [39]). Still another support for the Tulczyjew SSC was provided by Souriau who showed [40] that geometric quantization of the symplectic system which derives the MPD equations with this SSC, when considered with a flat background, leads to the Maxwell equations.…”

“…Note that if causality is not violated over distances larger than the wavelength of the photon, it should not imply any problem, since the pole-dipole approximation as such only holds if the length scales tied to the particle (here wavelength of the photon) are much smaller than the curvature length scale. Indeed, in papers where the Tulczyjew SSC was employed, e.g., to study photons in the Schwarzschild, de Sitter or FLRW backgrounds [23,39,41,42], causality has not been found to be violated over meaningful distances.…”

“…The Souriau-Saturnini equations describe the trajectory of a massless photon with spin in a gravitational field. While they work rather well in a Robertson-Walker background [41], or in the proximity of a star [23,39], they break down when the curvature of the gravitational background vanishes. This is due to the lonely term R(S)(S) in the denominator of (3.1).…”