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

TL;DR

The paper extends Damour's Navier-Stokes-type description of black-hole horizons to general hypersurfaces foliated by spacelike 2-surfaces, deriving a generalized DNS identity from the Einstein equations. By introducing the normal fundamental form $\Omega^{(ell)}$ and a normal evolution vector $h$, the authors obtain a DNS-like balance that remains valid for null, spacelike, or timelike horizons and for trapping or dynamical horizons. The work also develops a quasilocal angular-momentum flux law $dJ(\varphi)/dt$ in terms of matter flux and viscous terms, and shows consistency with established results by Ashtekar-Krishnan and Booth-Fairhurst in appropriate limits. The formalism provides a local, covariant framework for horizon dynamics and angular-momentum transport with potential applications in numerical relativity and black-hole thermodynamics.

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.

Figures (4)

• Figure 1: Foliation of a hypersurface ${\mathcal{H}}$ by a family $({\mathcal{S}}_t)_{t\in\mathbb{R}}$ of spacelike 2-surfaces, and the associated evolution vector $\bm{h}$.
• Figure 2: Vector plane ${\mathcal{T}}_p({\mathcal{S}}_t)^\perp$ normal to ${\mathcal{S}}_t$ at a given point $p$, with some orthonormal frame $(\bm{n},\bm{s})$ and some null frame $(\bm{\ell},\bm{k})$. The directions of $\bm{\ell}$ and $\bm{k}$ are uniquely defined as the intersections of ${\mathcal{T}}_p({\mathcal{S}}_t)^\perp$ with the light cone emanating from $p$, whereas the directions of $\bm{n}$ and $\bm{s}$ can be changed by an arbitrary boost in a direction normal to ${\mathcal{S}}_t$.
• Figure 3: Null vectors $(\bm{\ell},\bm{k})$ associated with the evolution vector $\bm{h}$ by $\bm{h} = \bm{\ell} - C \bm{k}$; the plane of the figure is the plane ${\mathcal{T}}_p({\mathcal{S}}_t)^\perp$, so that ${\mathcal{S}}_t$ is reduced to a point.
• Figure 4: Same as Fig. \ref{['f:coupe_normal']} but with in addition the vector $\bm{m}$ normal to the hypersurface ${\mathcal{H}}$.