# 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 \(\varSigma ^4\) 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.

## Supplementary material

## References

- 1.Dyson, F.W., Eddington, A.S., Davidson, C.: A determination of the deflection of light by the sun’s gravitational field, from observations made at the total eclipse of May 29, 1919. Philos. Trans. R. Soc. Lond. Ser. A
**220**, 291–333 (1920)ADSCrossRefGoogle Scholar - 2.Will, C.M.: The confrontation between general relativity and experiment. Living Rev. Relat.
**17**, 4 (2014)ADSCrossRefzbMATHGoogle Scholar - 3.Schneider, P., Ehlers, J., Falco, E.E.: Gravitational Lenses. Springer, Berlin, Heidelberg, New York (1999)Google Scholar
- 4.Schneider, P., Kochanek, C., Wambsganss, J.: Gravitational Lensing: Strong, Weak and Micro. Springer, Berlin, Heidelberg, New York (2006)Google Scholar
- 5.Islam, J.N.: The cosmological constant and classical tests of general relativity. Phys. Lett. A
**97**, 239–241 (1983)ADSMathSciNetCrossRefGoogle Scholar - 6.Rindler, W., Ishak, M.: Contribution of the cosmological constant to the relativistic bending of light revisited. Phys. Rev. D
**76**, 043006 (2007)ADSCrossRefGoogle Scholar - 7.Ishak, M., Rindler, W.: The relevance of the cosmological constant for lensing. Gen. Relativ. Gravit.
**42**, 2247–2268 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 8.Lake, K.: Bending of light and the cosmological constant. Phys. Rev. D
**65**, 087301 (2002)ADSMathSciNetCrossRefGoogle Scholar - 9.Park, M.: Rigorous approach to gravitational lensing. Phys. Rev. D
**78**, 023014 (2008)ADSMathSciNetCrossRefGoogle Scholar - 10.Khriplovich, I.B., Pomeransky, A.A.: Does the cosmological term influence gravitational lensing? Int. J. Mod. Phys. D
**17**, 2255–2259 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 11.Simpson, F., Peacock, J.A., Heavens, A.F.: On lensing by a cosmological constant, on lensing by a cosmological constant. MNRAS
**402**, 2009–2016 (2010)ADSCrossRefGoogle Scholar - 12.Bhadra, A., Biswas, S., Sarkar, K.: Gravitational deflection of light in the Schwarzschild–de Sitter space-time. Phys. Rev. D
**82**, 063003 (2010)ADSCrossRefGoogle Scholar - 13.Miraghaei, H., Nouri-Zonoz, M.: Classical tests of general relativity in the Newtonian limit of the Schwarzschild–de Sitter spacetime. Gen. Relativ. Gravit.
**42**, 2947–2956 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 14.Biressa, T., de Freitas Pacheco, J.A.: The cosmological constant and the gravitational light bending. Gen. Relativ. Gravit.
**43**, 2649–2659 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 15.Arakida, H., Kasai, M.: Effect of the cosmological constant on the bending of light and the cosmological lens equation. Phys. Rev. D
**85**, 023006 (2012)ADSCrossRefGoogle Scholar - 16.Hammad, F.: A note on the effect of the cosmological constant on the bending of light. Mod. Phys. Lett. A
**28**, 1350181 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 17.Lebedev, D., Lake, K.: On the influence of the cosmological constant on trajectories of light and associated measurements in Schwarzschild de Sitter space. arXiv:1308.4931 (2013)
- 18.Batic, D., Nelson, S., Nowakowski, M.: Light on curved backgrounds. Phys. Rev. D
**91**, 104015 (2015)ADSMathSciNetCrossRefGoogle Scholar - 19.Arakida, H.: Effect of the cosmological constant on light deflection: time transfer function approach. Universe
**2**, 5 (2016)ADSCrossRefGoogle Scholar - 20.Gibbons, G.W., Werner, M.C.: Applications of the Gauss Bonnet theorem to gravitational lensing. Class. Quantum Gravity
**25**, 235009 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 21.Gibbons, G.W., Warnick, C.M., Werner, M.C.: Light bending in Schwarzschild de Sitter: projective geometry of the optical metric. Class. Quantum Gravity
**25**, 245009 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 22.Ishihara, A., Suzuki, Y., Ono, T., Kitamura, T., Asada, H.: Gravitational bending angle of light for finite distance and the Gauss–Bonnet theorem. Phys. Rev. D
**94**, 084015 (2016)ADSMathSciNetCrossRefGoogle Scholar - 23.Ishihara, A., Suzuki, Y., Ono, T., Asada, H.: Finite-distance corrections to the gravitational bending angle of light in the strong deflection limit. Phys. Rev. D
**95**, 044017 (2017)ADSCrossRefGoogle Scholar - 24.Weinberg, S.: The cosmological constant problem. Rev. Mod. Phys.
**61**, 1–23 (1989)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 25.Carroll, S.: The cosmological constant. Living Rev. Relativ.
**4**, 1 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 26.Abramowicz, M.A., Carter, B., Lasota, J.P.: Optical reference geometry for stationary and static dynamics. Gen. Relativ. Gravit.
**20**, 1173–1183 (1988)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 27.Carroll, S.: Spacetime and Geometry: An Introduction to General Relativity. Addison Wesley, San Francisco (2004)zbMATHGoogle Scholar
- 28.Klingenberg, W.: A Course in Differential Geometry. Springer, New York (1978)CrossRefzbMATHGoogle Scholar
- 29.Kreyszig, E.: Differential Geometry. Dover Publications, New York (1991)zbMATHGoogle Scholar
- 30.do Carmo, M.P.: Differential Geometry of Curves and Surfaces, 2nd edn. Dover Publications, Mineola, New York (2016)zbMATHGoogle Scholar
- 31.Rindler, W.: Relativity: Special, General, and Cosmological, 2nd edn. Oxford University Press, New York (2006)zbMATHGoogle Scholar
- 32.Epstein, R., Shapiro, I.I.: Post-post-Newtonian deflection of light by the Sun. Phys. Rev. D
**22**, 2947–2949 (1980)ADSCrossRefGoogle Scholar - 33.Kottler, F.: Über die physikalischen Grundlagen der Einsteinschen Gravitationstheorie. Annalen. Phys.
**361**, 401–462 (1918)ADSCrossRefzbMATHGoogle Scholar - 34.Flamm, L.: Beiträge zur Einsteinschen Gravitationstheorie. Phys. Z.
**17**, 448–454 (1916)zbMATHGoogle Scholar