Properties of space-time in the vicinity of trapped regions

We study the near-horizon geometry in axisymmetric space-times. The general axisymmetric metrics have seven different parameters depending on three coordinates, and consistency analysis of the Einstein equations on such space-times is a formidable task. After discussing some general aspects of axially symmetric metrics, we begin the investigations of the simplest representative of this class: the Kerr–Vaidya metrics. Kerr–Vaidya metrics can be derived from the Vaidya metric by the complex coordinate transformation suggested by Newman and Janis. We show that the energy-momentum tensor belongs to type III in the Segre-Hawking- Ellis classification but has a special form with all Lorentz-invariant eigenvalues belonging to zero. It is known that the apparent horizon of the outgoing Kerr–Vaidya metric coincides with that of the Kerr space-time and that it is not so for the ingoing metric. We find its location for quasi-stationary Kerr–Vaidya black holes. The energy-momentum tensor of the Kerr- Vaidya geometries violates the null energy condition. We show that similar to the spherically symmetric accreting black hole, energy density, pressure, and flux for an infalling observer are diverging in the outgoing Kerr–Vaidya metric. This firewall leads to the violation of a specific quantum energy inequality that bounds the violation of the null energy condition.