Rotation, surface gravity and horizon degeneracy

PDF p. 2: §1.1, rotation and surface gravity; PDF p. 5: §1.3, degeneracy and multiple black holes

Let $(M,g)$ be a smooth time-oriented Lorentzian four-manifold with Levi–Civita connection $\mathrm{\nabla }$, and let $T$ be a stationary Killing field: ${\mathcal{L}}_{T}g=0$ and $T$ is timelike on a specified nonempty open region. Let $H$ be a smooth null hypersurface. Its null normal line at $p$ is the radical of $g{\mathrm{\mid }}_{T_{p}H}$, namely the vectors in $T_{p}H$ orthogonal to all of $T_{p}H$; its integral curves, where specified, are the null generators. The horizon is non-rotating relative to $T$ if $T_p$ lies in this line at every $p\in H$, and rotating relative to $T$ if this condition fails somewhere. The zero vector belongs to the line.

For a Killing horizon of a specified Killing field $K$, put $H_{\ast }=\{p\in H:K_p\mathrm{\ne }0\}$ and suppose it is nonempty and dense, with $K$ tangent and normal to $H$ on $H_{\ast }$. Its signed surface-gravity function is the unique smooth scalar ${\kappa }_H$ on $H_{\ast }$ satisfying $d(g(K,K))=-2{\kappa }_H{K}^{\mathrm{♭}}$, where $K^{\mathrm{♭}}=g(K,\cdot )$; equivalently ${\mathrm{\nabla }}_{K}K={\kappa }_{H}K$. Fixing $K$ fixes its scale. Under a constant rescaling $K\mapsto aK$, $a\mathrm{\ne }0$, the signed function becomes $a{\kappa }_H$.

In the article's convention, suppose the chosen future-horizon generator has a constant coefficient ${\kappa }_H=\kappa \ge 0$. This number is its surface gravity: the horizon is non-degenerate if $\kappa >0$ and degenerate if $\kappa =0$. Constancy, nonnegativity and the existence of the particular generator are additional properties, not consequences of these definitions. At a point where $K=0$, the equation ${\mathrm{\nabla }}_{K}K=\kappa K$ by itself determines no value of $\kappa$.

A bifurcate Killing horizon consists locally of two distinct smooth null Killing horizons $H_{+}$ and $H_{-}$ of the same field $K$, intersecting transversely in a smooth spacelike two-surface $S$ on which $K=0$; spacelike means that the induced metric on $S$ is positive definite. If $S$ is diffeomorphic to the two-sphere, call it a bifurcation sphere. A surface-gravity value on $S$ is specified by the common value only when the branch functions extend smoothly there and agree. In the article's terminology, multiple black holes means that the horizon has several connected components.

Bifurcate Killing horizons, rotation and normalization

Given: The data and hypotheses stated in the Definition.

Goal: Show that the terminology and the stated numerical conventions are well-defined under those hypotheses.

1.1 The null normal line and the pointwise membership condition give complementary rotation alternatives. On the nonzero part of the specified Killing horizon, the horizon interface supplies a unique smooth signed coefficient and identifies its covector and acceleration equations. Its constant-rescaling rule shows why the numerical coefficient depends on the chosen scale of $K$. [given, F1]

2.1 When the coefficient is assumed constant and nonnegative, exactly one of $\kappa =0$ and $\kappa >0$ holds, so the two degeneracy labels are unambiguous. At a zero of $K$, both sides of the acceleration equation vanish, so that equation alone selects no coefficient. The bifurcation and extension conditions give the stated intersection terminology without asserting existence of an extension; the sphere and connected-component conditions are additional classifications of the supplied sets. [given, F2, step 1.1] ∎

The article attributes the existence of its horizon generator and the constant nonnegative coefficient to rigidity results. The present definition records the convention conditional on those properties; it does not prove them.