Null geodesics in the C-metric with cosmological constant

TL;DR

This paper analyzes photon trajectories in the C-metric with a cosmological constant, providing exact solutions and classifying possible orbits using Hamilton-Jacobi separability and elliptic functions.

Contribution

It offers the first complete classification and explicit solutions for null geodesics in the C-metric with cosmological constant, including periodic orbits.

Findings

Exact solutions for photon trajectories using Jacobi elliptic functions

Classification of all possible null geodesics based on conserved quantities

Identification of periodic photon orbits on the photon surface

Abstract

In this paper we study the motion of photons or massless particles in the C-metric with cosmological constant. The Hamilton--Jacobi equations are known to be completely separable, giving a Carter-like quantity $Q$ which is a constant of motion. All possible trajectories are classified according to a two-dimensional parameter space representing the particle's angular momentum and energy scaled in units of $Q$. Exact solutions are given in the C-metric coordinates in terms of Jacobi elliptic functions. Using the exact solutions, we find examples of periodic orbits on the photon surface.