Surface Stress Tensor and Junction Conditions on a Rotating Null Horizon

TL;DR

This paper derives the surface stress tensor and modifies junction conditions for rotating null horizons, enabling consistent matching of interior and exterior geometries at black hole horizons even with discontinuities.

Contribution

It introduces a modified set of junction conditions applicable to rotating null horizons, extending Israel's formalism to cases with discontinuous normals and non-propagating matter shells.

Findings

Derived surface stress tensor in terms of geometric invariants

Modified junction conditions valid for rotating null horizons

Applicable to matching black hole exteriors with interior geometries

Abstract

The general form of the surface stress tensor of an infinitesimally thin shell located on a rotating null horizon is derived, when different interior and exterior geometries are joined there. Although the induced metric on the surface must be the same approached from either side, the first derivatives of the metric need not be. Such discontinuities lead to a Dirac $\delta$-distribution in the Einstein tensor localized on the horizon. For a general stationary axisymmetric geometry the surface stress tensor can be expressed in terms of two geometric invariants that characterize the surface, namely the discontinuities $[\kappa]$ and $[\cal J]$ of the surface gravity $\kappa$ and angular momentum density $\cal J$. The Komar energy and angular momentum are given in coordinates adapted to the Killing symmetries, and the surface contributions to each determined in terms of $[\kappa]$ and $[\cal J]$. Guided by these, a simple modification of the original Israel junction conditions is verified directly from the Einstein tensor density to give the correct finite result for the surface stress, when the normal $\boldsymbol n$ to the surface is allowed to tend continuously to a null vector. The relation to Israel's original junction conditions, which fail on null surfaces, is given. The modified junction conditions are suitable to the matching of a rotating `black hole' exterior to any interior geometry joined at the Kerr null horizon surface, even when the surface normal is itself discontinuous and the Barrab\`es-Israel formalism is also inapplicable. This joining on a rotating null horizon is purely of the matter shell type and does not contain a propagating gravitational shock wave.