Light deflection in Observation: Angle differences between two null geodesics on the de Sitter spacetime with multi-lensing objects

TL;DR

This paper derives and analyzes the angle differences between null geodesics in de Sitter spacetime with multi-lensing objects, highlighting the impact of the cosmological constant and practical observability of black hole lensing effects.

Contribution

It provides a perturbative derivation of angle differences between light rays in de Sitter spacetime with multiple lenses, including the effects of the cosmological constant and realistic astrophysical scenarios.

Findings

Deflections by Sgr A* are about 10 microarcseconds, detectable with future observations.

Solar system lensing effects cancel out at first order, simplifying measurements.

Cosmological constant influences observed angle differences through source positions.

Abstract

We derive angle differences between two null geodesics, propagating from light sources to an observer, on the de Sitter spacetime with multi-lensing objects. Assuming the lensing objects are mass monopoles on the de Sitter background, we derive the metric tensor by solving the Einstein equation perturbatively. On that spacetime, we solve a null geodesic parametrized by the coordinate time. Using the null geodesics, we define the angle differences in a coordinate invariant way. We take in the relativistic effects up to the first order of perturbation and clarify the magnitude of approximation errors. We find that the rest observer, who sees the isotropic cosmic space, implicitly observes the effect of the cosmological constant on the angle differences through the positions of the light sources. As a practical application, we regard the massive black hole at our galactic center (Sgr A*) and the solar system as the lensing objects, further a star and a flare around Sgr A* as the light sources. We write the angle differences between these light sources using their spatial coordinates. We find that deflections by Sgr A* remain in the angle differences while deflections by the solar system cancel out up to the first order of perturbation. The deflections by Sgr A* amounts around 10 microarcseconds, which is detectable in the near future observations.