Geometric optics in general relativity
Dr. Rahulkumar Solanki
Visiting Professor, Harvey Mudd College
How gravity affects light propagation can be explained by the principle that one cannot locally distinguish gravitational field from uniform acceleration. This is a consequence of the equivalence of inertial and gravitational mass. In an accelerated frame, the rectilinear motion of light appears curvilinear. In general, this uniform acceleration will be different at different points. In other words, unlike special relativity, the proper time and length measured by observers at rest in the same coordinate system but at different locations will differ. In static gravitational fields (not changing with time and produced by a non-rotating source), light rays follow Fermat’s variational principle of least (stationary) time. According to this principle, light ray trajectories are the geodesics of the associated Fermat metric. For curved space, the metric gives the distance between two infinitesimally separated points, and geodesics is the generalization of straight lines in flat space. Additionally, if the Fermat metric can be written in isotropic coordinates, light propagation can be mimicked by an optical medium in ordinary optics with an appropriate index of refraction. If two static gravitational fields share identical light rays, then the associated Fermat metrics are called projectively equivalent. This property, however, does not imply that the lengths and angles measured by observers in these fields will be identical. I will show that the projective equivalence between two Fermat metrics, when viewed in isotropic coordinates, corresponds to the refractive indices, which are proportional. This is analogous to ordinary geometric optics: Snell’s law remains invariant if refractive indices are rescaled by the same constant. I will discuss its implications for quasi-Newtonian approximations applied to cosmological observations. I will also discuss part of Inq Soncharoen’s thesis work on finding cylindrically symmetric static gravitational fields that share identical light rays.