The optical geometry definition of the total deflection angle of a light ray in curved spacetime

Abstract

Assuming a static and spherically symmetric spacetime, we propose a novel concept of the total deflection angle of a light ray. The concept is defined by the difference between the sum of internal angles of two triangles; one of the triangles lies on curved spacetime distorted by a gravitating body and the other on its background. The triangle required to define the total deflection angle can be realized by setting three laser-beam baselines as in planned space missions such as LATOR, ASTROD-GW, and LISA. Accordingly, the new total deflection angle is, in principle, measurable by gauging the internal angles of the triangles. The new definition of the total deflection angle can provide a geometrically and intuitively clear interpretation. Two formulas are proposed to calculate the total deflection angle on the basis of the Gauss--Bonnet theorem. It is shown that in the case of the Schwarzschild spacetime, the expression for the total deflection angle $α_{\rm Sch}$ reduces to Epstein--Shapiro's formula when the source of a light ray and the observer are located in an asymptotically flat region. Additionally, in the case of the Schwarzschild--de Sitter spacetime, the expression for the total deflection angle $α_{\rm SdS}$ comprises the Schwarzschild-like parts and coupling terms of the central mass $m$ and the cosmological constant $Λ$ in the form of ${\cal O}(Λm)$ instead of ${\cal O}(Λ/m)$. Furthermore, $α_{\rm SdS}$ does not include the terms characterized only by the cosmological constant $Λ$.