Universal properties of the near-horizon optical geometry

Abstract

We make use of the fact that the optical geometry near a static non-degenerate Killing horizon is asymptotically hyperbolic to investigate universal features of black hole physics. We show how the Gauss-Bonnet theorem allows certain lensing scenarios to be ruled in or out. We find rates for the loss of scalar, vector and fermionic `hair' as objects fall quasi- statically towards the horizon. In the process we find the Lienard-Wiechert potential for hyperbolic space and calculate the force between electrons mediated by neutrinos, extending the flat space result of Feinberg and Sucher. We use the enhanced conformal symmetry of the Schwarzschild and Reissner-Nordstrom backgrounds to re-derive the electrostatic field due to a point charge in a simple fashion

The Jacobi metric for timelike geodesics in static spacetimes

“…In particular in [8] the Gaussian curvature of the Schwarzschild optical metric restricted to the equstorial plane was shown to be everywhere negative and to approach a constant value near the horizon r = 2M. In [9] this behaviour was found to be universal for the near horizons of non-extreme static black holes. One purpose of the present paper is to extend this work to the case of massive particles.…”

Standard static Finsler spacetimes

“…The conformal Riemannian metric on M , g 0 /Λ is called optical metric. It plays a fundamental role in the study of light rays of I × Λ M because its pregeodesics are the projections on M of the light rays in (I × M, g) (see [3], [21], [37]). Moreover, many of the causal properties of the spacetime I × Λ M are encoded in the geometry of the conformal manifold (M, g 0 /Λ).…”

Motion of charged particle in Reissner–Nordström spacetime: a Jacobi-metric approach

“…One of the early workers on this topic was Pin [13] who considered many body systems and in particular showed that the Gaussian curvature of the Jacobi metric has a sign opposite to the particle energy E (non-relativistic, without the rest energy). In later times Gibbons et al [14][15][16][17] have considered the optical metric in various physical situations which is a closely related concept for massless particles. Discussions of the Jacobi-metric approach in a modern perspective can be found in [18].…”