Focusing versus defocusing properties of truly naked black holes

TL;DR

This paper investigates the focusing and defocusing behaviors of null geodesics near truly naked horizons, revealing how energy conditions influence horizon properties and the validity of test particle approximations.

Contribution

It provides a detailed analysis of null geodesic congruences near TNHs, highlighting the effects of energy conditions and horizon extremality on geodesic behavior.

Findings

Outgoing rays' expansion tends to vanish near non-extremal horizons.

Divergence of ingoing rays occurs if NEC is satisfied outside the horizon.

Test particle approximation remains valid under certain conditions.

Abstract

We study the properties of the congruence of null geodesics propagating near the so-called truly naked horizons (TNH) - objects having finite Kretschmann scalar but with diverging tidal acceleration for freely falling observers. The expansion of outgoing rays near the future horizon always tends to vanish for the non-extremal case but may be non-zero for the distorted (ultra)extremal one. It tends to diverge for the ingoing ones if the the null energy condition (NEC) is satisfied in the vicinity of the horizon outside. However, it also tends to zero for NEC violating cases except the remote horizons. We also discuss the validity of test particle approximation for TNHs and find the sufficient condition for backreaction remaining small.