A generalized Damour-Navier-Stokes equation applied to trapping horizons

TL;DR

This paper derives a generalized Damour-Navier-Stokes equation applicable to various types of horizons in general relativity, extending previous null-horizon results to spacelike and trapping horizons, and introduces a new approach to horizon angular momentum.

Contribution

It generalizes the Damour-Navier-Stokes equation to non-null hypersurfaces and trapping horizons, incorporating a new normal fundamental form for broader applicability.

Findings

Derivation of a unified identity for different horizon types

Extension of the Damour-Navier-Stokes equation beyond null horizons

A new evolution equation for horizon angular momentum

Abstract

An identity is derived from Einstein equation for any hypersurface H which can be foliated by spacelike two-dimensional surfaces. In the case where the hypersurface is null, this identity coincides with the two-dimensional Navier-Stokes-like equation obtained by Damour in the membrane approach to a black hole event horizon. In the case where H is spacelike or null and the 2-surfaces are marginally trapped, this identity applies to Hayward's trapping horizons and to the related dynamical horizons recently introduced by Ashtekar and Krishnan. The identity involves a normal fundamental form (normal connection 1-form) of the 2-surface, which can be viewed as a generalization to non-null hypersurfaces of the Hajicek 1-form used by Damour. This 1-form is also used to define the angular momentum of the horizon. The generalized Damour-Navier-Stokes equation leads then to a simple evolution equation for the angular momentum.