# Light deflection and Gauss–Bonnet theorem: definition of total deflection angle and its applications

Editor’s Choice (Research Article)

## 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 $$\varSigma ^4$$ on the optical reference geometry $${\mathscr {M}}^\mathrm{opt}$$ 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 $$\alpha$$; (1) the angular formula that uses four angles determined on the optical reference geometry $${\mathscr {M}}^\mathrm{opt}$$ or the curved $$(r, \phi )$$ subspace $${\mathscr {M}}^\mathrm{sub}$$ being a slice of constant time t and (2) the integral formula on the optical reference geometry $${\mathscr {M}}^\mathrm{opt}$$ which is the areal integral of the Gaussian curvature K in the area of a quadrilateral $$\varSigma ^4$$ and the line integral of the geodesic curvature $$\kappa _g$$ along the curve $$C_{\varGamma }$$. As the curve $$C_{\varGamma }$$, we introduce the unperturbed reference line that is the null geodesic $$\varGamma$$ on the background spacetime such as the Minkowski or the de Sitter spacetime, and is obtained by projecting $$\varGamma$$ vertically onto the curved $$(r, \phi )$$ subspace $${\mathscr {M}}^\mathrm{sub}$$. 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 $${\mathscr {O}}(\varLambda m)$$ terms in addition to the Schwarzschild-like part, while order $${\mathscr {O}}(\varLambda )$$ terms disappear.

## Keywords

Gravitation Cosmological constant Light deflection

## Notes

### Acknowledgements

This work was supported by JSPS KAKENHI Grant Number 15K05089.

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