Bilocal geodesic operators in static spherically-symmetric spacetimes

TL;DR

This paper develops a method to compute exact optical observables in static spherically symmetric spacetimes using bilocal geodesic operators, applicable to various models and configurations, demonstrated with Schwarzschild spacetime.

Contribution

It introduces a novel approach to derive exact expressions for optical observables in static spherically symmetric spacetimes using the bilocal geodesic operator formalism.

Findings

Derived general expressions for optical observables in static spherically symmetric spacetimes.

Applied the method to Schwarzschild spacetime to analyze distance measures.

Showed the behavior of angular diameter, parallax, and distance slip outside the photon sphere.

Abstract

We present a method to compute exact expressions for optical observables for static spherically symmetric spacetimes in the framework of the bilocal geodesic operator formalism. The expressions are obtained by solving the linear geodesic deviation equations for null geodesics, using the spacetime symmetries and the associated conserved quantities. We solve the equations in two different ways: by varying the geodesics with respect to their initial data and by directly integrating the equation for the geodesic deviation. The results are very general and can be applied to a variety of spacetime models and configurations of the emitter and the observer. We illustrate some of the aspects with an example of Schwarzschild spacetime, focusing on the behaviour of the angular diameter distance, the parallax distance, and the distance slip between the observer and the emitter outside the photon sphere.