Light Deflection and Gauss-Bonnet Theorem: Definition of Total Deflection Angle and its Applications

TL;DR

This paper introduces a new geometric approach using the Gauss-Bonnet theorem to define and calculate the total light deflection angle in Schwarzschild and Schwarzschild-de Sitter spacetimes, providing consistent formulas and new insights.

Contribution

It proposes a novel geometric definition of the total deflection angle using quadrilaterals on optical geometry and derives two equivalent formulas applicable to various spacetimes.

Findings

The two formulas for deflection angle are equivalent in Schwarzschild and Schwarzschild-de Sitter spacetimes.

The method recovers Epstein-Shapiro's formula at infinity for Schwarzschild spacetime.

Additional terms involving the cosmological constant appear in the Schwarzschild-de Sitter case.

Abstract

In this paper, we re-examine the light deflection in the Schwarzschild and the Schwarzschild-de Sitter spacetime. First, supposing a static and spherically symmetric spacetime, we propose the definition of the total deflection angle $\alpha$ of the light ray by constructing a quadrilateral $\Sigma^4$ on the optical reference geometry ${\cal M}^{\rm opt}$ determined by the optical metric $\bar{g}_{ij}$. On the basis of the definition of the total deflection angle $\alpha$ and the Gauss-Bonnet theorem, we derive two formulas to calculate the total deflection angle $\alpha$; (i) the angular formula that uses four angles determined on the optical reference geometry ${\cal M}^{\rm opt}$ or the curved $(r, \phi)$ subspace ${\cal M}^{\rm sub}$ being a slice of constant time $t$ and (ii) the integral formula on the optical reference geometry ${\cal M}^{\rm opt}$ which is the areal integral of the Gaussian curvature $K$ in the area of a quadrilateral $\Sigma^4$ and the line integral of the geodesic curvature $\kappa_g$ along the curve $C_{\Gamma}$. As the curve $C_{\Gamma}$, we introduce the unperturbed reference line that is the null geodesic $\Gamma$ on the background spacetime such as the Minkowski or the de Sitter spacetime, and is obtained by projecting $\Gamma$ vertically onto the curved $(r, \phi)$ subspace ${\cal M}^{\rm sub}$. We demonstrate that the two formulas give the same total deflection angle $\alpha$ for the Schwarzschild and the Schwarzschild--de Sitter spacetime. In particular, in the Schwarzschild case, the result coincides with Epstein--Shapiro's formula when the source $S$ and the receiver $R$ of the light ray are located at infinity. In addition, in the Schwarzschild--de Sitter case, there appear order ${\cal O}(\Lambda m)$ terms in addition to the Schwarzschild-like part, while order ${\cal O}(\Lambda)$ terms disappear.