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

TL;DR

This paper introduces a new geometric definition of the total deflection angle of light in curved spacetime, measurable via laser-baseline triangles, and derives formulas applicable to Schwarzschild and Schwarzschild-de Sitter spacetimes.

Contribution

It proposes a novel, measurable geometric definition of light deflection angles in curved spacetime using triangle internal angles, applicable to space missions and different spacetime models.

Findings

Defines total deflection angle via triangle internal angles on curved spacetime.

Derives formulas based on the Gauss--Bonnet theorem for Schwarzschild and Schwarzschild--de Sitter spacetimes.

Shows the new formula reduces to Epstein--Shapiro's in asymptotically flat regions.

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 $\alpha_{\rm Sch}$ 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 $\alpha_{\rm SdS}$ comprises the Schwarzschild-like parts and coupling terms of the central mass $m$ and the cosmological constant $\Lambda$ in the form of ${\cal O}(\Lambda m)$ instead of ${\cal O}(\Lambda/m)$. Furthermore, $\alpha_{\rm SdS}$ does not include the terms characterized only by the cosmological constant $\Lambda$.