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

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 ${\mathcal{M}}^{\mathrm{opt}}$ 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 ${\mathcal{M}}^{\mathrm{opt}}$ or the curved $(r,\varphi )$ subspace ${\mathcal{M}}^{\mathrm{sub}}$ being a slice of constant time $t$ and (ii) the integral formula on the optical reference geometry ${\mathcal{M}}^{\mathrm{opt}}$ 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 $(r,\varphi )$ subspace ${\mathcal{M}}^{\mathrm{sub}}$. 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 $\mathcal{O}(\Lambda m)$ terms in addition to the Schwarzschild-like part, while order $\mathcal{O}(\Lambda )$ terms disappear.

23 pages, 7 figures

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