Killing horizon

A Killing horizon is a null hypersurface whose null generators are orbits of a Killing vector field $K$. On the horizon, $K^b{\nabla }_b{K}^a=\kappa K^a$ defines its surface gravity.

In an axisymmetric stationary spacetime, the horizon angular velocity is the constant coefficient in the null horizon generator $\chi =k+{\Omega }_{H}m$, where $k$ is the stationary Killing vector normalized at infinity and $m$ generates axial orbits with period $2\pi$. It measures rotation relative to the stationary frame at infinity. For a Kerr black hole, ${\Omega }_H=a/(r_{+}^2+a^2)$. It enters the rotating physical-process first law through the horizon flux of $E-{\Omega }_{H}J$.

A bifurcate Killing horizon consists of two intersecting null Killing horizons of the same Killing vector field, which vanishes on their spacelike intersection, the bifurcation surface. In regular Kruskal coordinates the horizons are $U=0$ and $V=0$, and their intersection is $U=V=0$. The displayed field is tangent and normal to each horizon branch and zero on the intersection.

A degenerate Killing horizon has zero surface gravity. In the radial quotient of an extremal black hole, a double horizon root produces a pole in the tortoise coordinate rather than the logarithm of a simple root. Such a horizon has no bifurcation surface of the usual nondegenerate type.

The null expansion, null shear and null twist of a congruence tangent to the generators of a Killing horizon all vanish there. If $k$ is its Killing generator and $\ell =fk$ is an affine generator, the screen-space projector kills every term proportional to $k$. The symmetric part of the projected derivative of $\ell$ therefore vanishes by the Killing equation. The antisymmetric part vanishes by Frobenius theorem, since $\ell$ is hypersurface-orthogonal. These conclusions extend through a regular bifurcation surface by continuity.

A cosmological horizon is a causal boundary beyond which events cannot communicate with a specified observer because of the global expansion or geometry of spacetime.

The surface gravity $\kappa$ measures the failure of a horizon-generating Killing vector field to be affinely parametrized. For a static metric $d{s}^2=-f(r)d{t}^2+f(r{)}^{-1}d{r}^2+\cdots$ with a simple horizon at $r=r_{+}$, $\kappa =|f^{\prime }(r_{+})|/2$.

For the future-directed central-time Killing vector field in a static patch of de Sitter spacetime, the outgoing null chart gives the future branch coefficient $+1/R$ in ${\nabla }_{k}k=\kappa k$. The ingoing chart gives $-1/R$ on the past branch. Both follow from $d(k^2)=-2\kappa k^{♭}$; the cross-term sign changes the horizon covector. The physical Hawking temperature uses $|\kappa |$, not a signed negative temperature.

For a static spacetime with $g_{tt}=-A(r)$, $g_{rr}=1/B(r)$ and simple simultaneous horizon zeros, the squared surface gravity of ${\partial }_t$ is $\lim B{A}^{\prime 2}/(4A)=A^{\prime }(r_{+})B^{\prime }(r_{+})/4$. Choose the positive value for the outer horizon. A rescaling of the time generator rescales its surface gravity.

The time translation is normalized at infinity. For the roots $r_{\pm }^2=M\pm \sqrt{M^2-Q^2}$, evaluate $V^{\prime }(r)/2$ to get the signed horizon values. The outer Hawking temperature in natural units is $\sqrt{M^2-Q^2}/(\pi r_{+}^3)$. The extremal limit has zero surface gravity; the neutral case has only the outer positive-radius horizon and temperature $1/(2\pi \sqrt{2M})$.

At a regular Killing horizon, the generator is hypersurface orthogonal and satisfies ${\nabla }_{K}K=\kappa K$. The Killing derivative contraction identity gives the displayed squared surface gravity, including through a bifurcation surface by continuity. The generator's normalization is part of this formula.

If a Killing horizon generator $\xi$ has constant nonzero surface gravity $\kappa$, write $\xi =F(\lambda )k$ with $k=d/d\lambda$ affine. The equation ${\nabla }_{\xi }\xi =\kappa \xi$ implies $dF/d\lambda =\kappa$, so shifting the affine origin gives $\xi =\kappa \lambda k$.

For a Schwarzschild black hole of mass $M$ in geometrized units, $f=1-2M/r$ gives $\kappa =f\prime (2M)/2=1/(4M)$. The Killing vector field is normalized at infinity; changing that normalization changes $\kappa$.

A bifurcation surface is a spacelike codimension-two cross-section on which the Killing generator of a bifurcate Killing horizon vanishes. Affine parameters on the future and past branches can be chosen to vanish there.