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

Aug 14, 2017
34 pages
Published in:
  • Gen.Rel.Grav. 50 (2018) 5, 48
  • Published: Apr 5, 2018
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Abstract: (Springer)
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 α\alpha of the light ray by constructing a quadrilateral Σ4\varSigma ^4 on the optical reference geometry Mopt{\mathscr {M}}^\mathrm{opt} determined by the optical metric gˉij\bar{g}_{ij} . On the basis of the definition of the total deflection angle α\alpha and the Gauss–Bonnet theorem, we derive two formulas to calculate the total deflection angle α\alpha , (1) the angular formula that uses four angles determined on the optical reference geometry Mopt{\mathscr {M}}^\mathrm{opt} or the curved (r,ϕ)(r, \phi ) subspace Msub{\mathscr {M}}^\mathrm{sub} being a slice of constant time t and (2) the integral formula on the optical reference geometry Mopt{\mathscr {M}}^\mathrm{opt} which is the areal integral of the Gaussian curvature K in the area of a quadrilateral Σ4\varSigma ^4 and the line integral of the geodesic curvature κg\kappa _g along the curve CΓC_{\varGamma } . As the curve CΓC_{\varGamma } , we introduce the unperturbed reference line that is the null geodesic Γ\varGamma on the background spacetime such as the Minkowski or the de Sitter spacetime, and is obtained by projecting Γ\varGamma vertically onto the curved (r,ϕ)(r, \phi ) subspace Msub{\mathscr {M}}^\mathrm{sub} . We demonstrate that the two formulas give the same total deflection angle α\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 O(Λm){\mathscr {O}}(\varLambda m) terms in addition to the Schwarzschild-like part, while order O(Λ){\mathscr {O}}(\varLambda ) terms disappear.
Note:
  • 23 pages, 7 figures
  • Gravitation
  • Cosmological constant
  • Light deflection
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