Bifurcate Killing horizons, rotation and normalization

Work in a smooth time-oriented Lorentzian four-manifold $(M,g)$ of signature $(-,+,+,+)$, with metric-compatible Levi–Civita connection $\mathrm{\nabla }$, and let $K$ be a smooth Killing field, meaning ${\mathcal{L}}_{K}g=0$. A smooth embedded hypersurface $H$ is null if the restriction of $g$ to each tangent space $T_{p}H$ has a nonzero radical: the vectors in $T_{p}H$ orthogonal to all of $T_{p}H$. It is a Killing horizon for $K$ when $H_{\ast }:=\{p\in H:K_p\mathrm{\ne }0\}$ is nonempty and dense in $H$ and, on $H_{\ast }$, $K$ is tangent and normal to $H$. The null normal line is the radical of the induced form. An integral curve of this line field, or horizon generator, means here a smooth curve $\gamma :I\to H$, where $I\subset \mathbb{R}$ is a nonempty open interval, such that $\dot{\gamma }(s)\mathrm{\ne }0$ and $\dot{\gamma }(s)$ lies in the null normal line at $\gamma (s)$ for every $s\in I$. Curves related by a smooth diffeomorphic reparameterization are identified; restrictions to open subintervals are allowed generator segments. No maximality, completeness, affine parametrization or preferred parameter scale is implicit.

The signed surface gravity of this specified $K$ on $H_{\ast }$ is the unique smooth scalar ${\kappa }_H$ satisfying the ambient covector equation $d(g(K,K))=-2{\kappa }_H{K}^{\mathrm{♭}}$, where $K^{\mathrm{♭}}=g(K,\cdot )$. Equivalently, ${\mathrm{\nabla }}_{K}K={\kappa }_{H}K$ on $H_{\ast }$. A chosen future branch is a specified branch with its future direction; neither the sign nor constancy of ${\kappa }_H$ follows from the terminology.

A bifurcate Killing horizon here consists of two distinct specified smooth embedded null hypersurfaces $H_{+}$ and $H_{-}$ in a common neighborhood of a smooth spacelike embedded two-surface $S$, each a Killing horizon for the same $K$, such that $K=0$ on $S$, their intersection near each point of $S$ is $S$, and $T_p{H}_{+}\cap T_p{H}_{-}=T_{p}S$. Spacelike means that the induced metric on $S$ is positive definite. The surface $S$ is the bifurcation surface. A surface-gravity value at $p\in S$ is specified by the common value only when the two branch scalars extend smoothly to $p$ along their respective branches and agree there. The equation at $K=0$ alone fixes no value.

For a specified stationary Killing field $T$, meaning a Killing field timelike on a specified nonempty open region, a horizon is non-rotating relative to $T$ if $T_p$ lies in the null normal line at every point of the horizon; it is rotating relative to $T$ if that tangency fails somewhere. Zero vectors satisfy the tangency condition. A normalization specifies the scale of $K$; replacing $K$ by $aK$ for a nonzero constant $a$ replaces ${\kappa }_H$ by $a{\kappa }_H$. Thus a numerical surface gravity, or boost rate when that term is used for this normalized scalar, requires the normalization to be stated. No existence, constancy, nonzero-value, field-equation or global-extension assertion is included.

Killing fields of a Lorentzian metric

Metric compatibility, torsion freedom and the metric-selected connection class

Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor

Causal solutions, Killing horizons, escape and metric extensions

The horizon supplier establishes that the scalar in the ambient covector equation exists uniquely and smoothly on $H_{\ast }$; density makes a continuous extension unique if it exists. By metric compatibility and the Killing equation, ${\mathrm{\nabla }}_a(g(K,K))=2K^b{\mathrm{\nabla }}_a{K}_b=-2K^b{\mathrm{\nabla }}_b{K}_a$. Nondegeneracy of $g$ therefore makes the covector equation equivalent to ${\mathrm{\nabla }}_{K}K={\kappa }_{H}K$. Rescaling $K$ by a nonzero constant multiplies the left side of the covector equation by $a^2$ and $K^{\mathrm{♭}}$ by $a$, giving the stated rescaling of ${\kappa }_H$. The null normal line and tangency condition are pointwise conditions on the specified data; the definition does not assert that suitable data exist.

The horizon and bifurcation definitions, including existence and smoothness of the signed scalar on $H_{\ast }$, are supplied by Causal solutions, Killing horizons, escape and metric extensions under its Definition (b) hypotheses and Proof 2.1. Reall, Section 5.3, equations (5.21)–(5.22), independently states the Killing-equation characterization used in [F2]. General Relativity, printed p. 56 This lecture-note citation is limited to that identity; the horizon, density and branch-extension conventions come from the declared mathematical supplier.