Geometry near the apparent horizon

Abstract

Trapped regions bounded by horizons are the defining features of black holes. However, formation of a singularity-free apparent horizon in finite time of a distant observer is consistent only with special states of geometry and matter in its vicinity. In spherical symmetry such horizons exist only in two classes of solutions of the Einstein equations. Both violate the null energy condition (NEC) and allow for expanding and contracting trapped regions. However, an expanding trapped region leads to a firewall. The weighted time average of the energy density for an observer crossing this firewall is negative and exceeds the maximal NEC violation that quantum fields can produce. As a result, either black holes only can evaporate or the semiclassical physics breaks down already at the horizons. Geometry of a contracting trapped region approaches the ingoing Vaidya metric with decreasing mass. Only one class of solutions allows for a test particle to cross the apparent horizon, and for a thin shell to collapse into a black hole. These results significantly constrain the regular black hole models. Models with regular matter properties at the horizon can be realized only if significant departures from the semiclassical physics occur already at the horizon scale. The Hayward-Frolov model may describe only evaporation, but not formation of a regular black hole.