Evolution of the electric field along null rays for arbitrary observers and spacetimes

Abstract

We derive, in a manifestly covariant and electromagnetic gauge independent way, the evolution law of the electric field $E^{\alpha }=E{e}^{\alpha }(e^{\alpha }{e}_{\alpha }=1)$, relative to an arbitrary set of instantaneous observers along a null geodesic ray, for an arbitrary Lorentzian spacetime, in the geometrical optics limit of Maxwell's equations in vacuum. We show that, in general, neither the magnitude $E$ nor the direction $e^{\alpha }$ of the electric field (here interpreted as the observed polarization of light) are parallel transported along the ray. For an extended reference frame around the given light ray, we express the evolution of the direction $e^{\alpha }$ in terms of the frame's kinematics, proving thereby that its expansion never spoils parallel transport, which bears on the unbiased inference of intrinsic properties of cosmological sources, such as, for instance, the polarization field of the cosmic microwave background (CMB). As an application of the newly derived laws, we consider a simple setup for a gravitational wave (GW) interferometer, showing that, despite the (kinematic) shear induced by the GW, the change in the final interference pattern is negligible since it turns out to be of the order of the ratio of the GW and laser frequencies.

arXiv comment6 pages, 3 figures

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• Influence of gravitational waves upon light in the Minkowski background: from null geodesics to interferometry
• The Response of Laser Interferometric Gravitational Wave Detectors Beyond the Eikonal Equation
• Influence of gravitational waves upon light. Part II. Electric field propagation and interference pattern in a gravitational wave detector
• Influence of gravitational waves upon light. Part I. Null geodesics, radar distance and frequency shift