The Isoperimetric Problem in Riemannian Optical Geometry

Abstract

In general relativity, spatial light rays of static spherically symmetric spacetimes are geodesics of surfaces in Riemannian optical geometry. In this paper, we apply results on the isoperimetric problem to show that length-minimizing curves subject to an area constraint are circles, and discuss implications for the photon spheres of Schwarzschild, Reissner-Nordstrom, as well as continuous mass models solving the Tolman-Oppenheimer-Volkoff equation. Moreover, we derive an isoperimetric inequality for gravitational lensing in Riemannian optical geometry, using curve shortening flow and the Gauss-Bonnet theorem.