Kastor-Traschen Black Holes, Null Geodesics and Conformal Circles

TL;DR

This paper analyzes the null geodesic structure of Kastor-Traschen black hole solutions, revealing how projected light rays relate to conformal circles and providing methods to derive explicit geodesic expressions.

Contribution

It introduces a new differential system for projected null geodesics and links these to conformal circles, expanding understanding of Kastor-Traschen spacetime geometry.

Findings

Projected null geodesics are integral curves of a third order ODE system.

A new system matches the projection of null curves from magnetic flows.

Analytic expressions for geodesics are derived via a known diffeomorphism.

Abstract

The Kastor-Traschen metric is a time-dependent solution of the Einstein-Maxwell equations with positive cosmological constant $\Lambda$ which can be used to describe an arbitrary number of charged dynamical black holes. In this paper, we consider the null geodesic structure of this solution, in particular, focusing on the projection to the space of orbits of the timelike conformal retraction. It is found that these projected light rays arise as integral curves of a system of third order ordinary differential equations. This system is not uniquely defined, however, and we use the inherent freedom to construct a new system whose integral curves coincide with the projection of distinguished null curves of Kastor-Traschen arising from a magnetic flow. We discuss our results in the one-centre case and demonstrate a link to conformal circles in the limit $\Lambda \rightarrow 0$. We also show how to construct analytic expressions for the projected null geodesics of this metric by exploiting a well-known diffeomorphism between the K-T metric and extremal Reissner-Nordstrom deSitter. We make some remarks about the two-centre solution and demonstrate a link with the one-centre case.