Brown-York Energy and Radial Geodesics

TL;DR

This paper explores the relationship between Brown-York and Misner-Sharp quasilocal energies in spherically symmetric spacetimes, revealing their analogy to geodesic energies and explaining phenomena like repulsive geodesics in charged black holes.

Contribution

It establishes a precise analogy between quasilocal energies and geodesic energies, and generalizes the connection between Brown-York energy and geodesic behavior in various metrics.

Findings

Brown-York energy relates to geodesic effective potential.

Negative Brown-York energy correlates with repulsive geodesics.

The relationship extends inside horizons in certain spacetimes.

Abstract

We compare the Brown-York (BY) and the standard Misner-Sharp (MS) quasilocal energies for round spheres in spherically symmetric space-times from the point of view of radial geodesics. In particular, we show that the relation between the BY and MS energies is precisely analogous to that between the (relativistic) energy E of a geodesic and the effective (Newtonian) energy E_{eff} appearing in the geodesic equation, thus shedding some light on the relation between the two. Moreover, for Schwarzschild-like metrics we establish a general relationship between the BY energy and the geodesic effective potential which explains and generalises the recently observed connection between negative BY energy and the repulsive behaviour of geodesics in the Reissner-Nordstrom metric. We also comment on the extension of this connection between geodesics and the quasilocal BY energy to regions inside a horizon.