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.

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黎曼光学几何中的等周问题

在广义相对论中，静态球对称时空的空间光线是黎曼光学几何中表面的测地线。在本文中，我们应用等周问题的结果来证明受面积约束的长度最小曲线是圆，并讨论对 Schwarzschild、Reissner-Nordstrom 的光子球以及求解 Tolman-奥本海默-沃尔科夫方程。此外，我们使用曲线缩短流和 Gauss-Bonnet 定理推导出了黎曼光学几何中引力透镜的等周不等式。