# The Isoperimetric Problem in Riemannian Optical Geometry

## TL;DR

This paper explores the isoperimetric problem in Riemannian optical geometry related to general relativity, showing that length-minimizing curves are circles and deriving inequalities relevant to gravitational lensing.

## Contribution

It applies isoperimetric results to Riemannian optical geometry, linking geometric properties to photon spheres and gravitational lensing in static spherically symmetric spacetimes.

## Key findings

- Length-minimizing curves are circles under area constraints.
- Derived an isoperimetric inequality for gravitational lensing.
- Discussed implications for photon spheres in various spacetime models.

## 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.