2.5 Killing horizons

A null embedded hypersurface, invariant under the flow of a Killing vector k, which coincides with a connected component of the set
𝒩 [k ] := {g(k,k ) = 0, k ⁄= 0},
is called a Killing horizon associated to k. We will often write H [k] for 𝒩 [k], whenever 𝒩 [k] is a Killing horizon.

2.5.1 Bifurcate Killing horizons

The Schwarzschild black hole has an event horizon with a specific structure, which is captured by the following definition: A set is called a bifurcate Killing horizon if it is the union of a a smooth spacelike submanifold S of co-dimension two, called the bifurcation surface, on which a Killing vector field k vanishes, and of four smooth null embedded hypersurfaces obtained by following null geodesics in the four distinct null directions normal to S.

For example, the Killing vector x∂ + t∂ t x in Minkowski spacetime has a bifurcate Killing horizon, with the bifurcation surface {t = x = 0}. As already mentioned, another example is given by the set {r = 2m } in Schwarzschild–Kruskal–Szekeres spacetime with positive mass parameter m.

In the spirit of the previous definition, we will refer to the union of two null hypersurfaces, which intersect transversally on a 2-dimensional spacelike surface as a bifurcate null surface.

The reader is warned that a bifurcate Killing horizon is not a Killing horizon, as defined in Section 2.5, since the Killing vector vanishes on S. If one thinks of S as not being part of the bifurcate Killing horizon, then the resulting set is again not a Killing horizon, since it has more than one component.

2.5.2 Killing prehorizons

One of the key steps of the uniqueness theory, as described in Section 3, forces one to consider “horizon candidates” with local properties similar to those of a proper event horizon, but with global behavior possibly worse: A connected, not necessarily embedded, null hypersurface H0 ⊂ 𝒩 [k] to which k is tangent is called a Killing prehorizon. In this terminology, a Killing horizon is a Killing prehorizon, which forms a embedded hypersurface, which coincides with a connected component of 𝒩 [k]. The Minkowskian Killing vector ∂t − ∂x provides an example where 𝒩 is not a hypersurface, with every hyperplane t + x = const being a prehorizon.

The Killing vector k = ∂t + Y on n ℝ × π•‹, equipped with the flat metric, where n 𝕋 is an n-dimensional torus, and where Y is a unit Killing vector on 𝕋n with dense orbits, admits prehorizons, which are not embedded. This last example is globally hyperbolic, which shows that causality conditions are not sufficient to eliminate this kind of behavior.

Of crucial importance to the zeroth law of black-hole physics (to be discussed shortly) is the fact that the (k, k)-component of the Ricci tensor vanishes on horizons or prehorizons,

R (k,k ) = 0 on H [k]. (2.13 )
This is a simple consequence of the Raychaudhuri equation.

The following two properties of Killing horizons and prehorizons play a role in the theory of stationary black holes:

2.5.3 Surface gravity: degenerate, non-degenerate and mean-non-degenerate horizons

An immediate consequence of the definition of a Killing horizon or prehorizon is the proportionality of k and dN on H [k], where

N := g(k,k).
This follows, e.g., from g(k,dN ) = 0, since LkN = 0, and from the fact that two orthogonal null vectors are proportional. The observation motivates the definition of the surface gravity κ of a Killing horizon or prehorizon H [k], through the formula
d(g(k,k ))|H = − 2κk , (2.15 )
where we use the same symbol k for the covector ν μ gμνk dx appearing in the right-hand side as for the vector kμ∂μ.

The Killing equation implies dN = − 2∇kk; we see that the surface gravity measures the extent to which the parametrization of the geodesic congruence generated by k is not affine.

A fundamental property is that the surface gravity κ is constant over horizons or prehorizons in several situations of interest. This leads to the intriguing fact that the surface gravity plays a similar role in the theory of stationary black holes as the temperature does in ordinary thermodynamics. Since the latter is constant for a body in thermal equilibrium, the result

κ = constant on H [k ] (2.16 )
is usually called the zeroth law of black-hole physics [9Jump To The Next Citation Point].

The constancy of κ holds in vacuum, or for matter fields satisfying the dominant-energy condition, see, e.g., [151Jump To The Next Citation Point, Theorem 7.1]. The original proof of the zeroth law [9] proceeds as follows: First, Einstein’s equations and the fact that R (k,k) vanishes on the horizon imply that T (k,k) = 0 on H [k]. Hence, the vector field T (k) := Tμνk ν∂xμ is perpendicular to k and, therefore, space-like (possibly zero) or null on H [k]. On the other hand, the dominant energy condition requires that T(k) is zero, time-like or null. Thus, T (k) vanishes or is null on the horizon. Since two orthogonal null vectors are proportional, one has, using Einstein’s equations again, k ∧ R (k) = 0 on H [k], where R(k ) = R μνkμdx ν. The result that κ is constant over each horizon follows now from the general property (see, e.g., [314Jump To The Next Citation Point])

k ∧ dκ = − k ∧ R (k) on H [k ]. (2.17 )

The proof of (2.16View Equation) given in [314Jump To The Next Citation Point] generalizes to all spacetime dimensions n + 1 ≥ 4; the result also follows in all dimensions from the analysis in [165Jump To The Next Citation Point] when the horizon has compact spacelike sections.

By virtue of Eq. (2.17View Equation) and the identity d ω = ∗[k ∧ R(k)], the zeroth law follows if one can show that the twist one-form is closed on the horizon [270Jump To The Next Citation Point]:

[dω ]H [k] = 0 = ⇒ κ = constant on H [k ]. (2.18 )
While the original proof of the zeroth law takes advantage of Einstein’s equations and the dominant energy condition to conclude that the twist is closed, one may also achieve this by requiring that ω vanishes identically, which then proves the zeroth law under the second set of hypotheses listed below. This is obvious for static configurations, since then k has vanishing twist by definition.

Yet another situation of interest is a spacetime with two commuting Killing vector fields k and m, with a Killing horizon H [ξ] associated to a Killing vector ξ = k + Ωm. Such a spacetime is said to be circular if the distribution of planes spanned by k and m is hypersurface-orthogonal. Equivalently, the metric can be locally written in a 2+2 block-diagonal form, with one of the blocks defined by the orbits of k and m. In the circular case one shows that g(m, ω ) = g(ξ,ω ) = 0 ξ m implies dω = 0 ξ on the horizon generated by ξ; see [151Jump To The Next Citation Point], Chapter 7 for details.

A significant observation is that of Kay and Wald [184], that κ must be constant on bifurcate Killing horizons, regardless of the matter content. This is proven by showing that the derivative of the surface gravity in directions tangent to the bifurcation surface vanishes. Hence, κ cannot vary between the null-generators. But it is clear that κ is constant along the generators.

Summarizing, each of the following hypotheses is sufficient to prove that κ is constant over a Killing horizon defined by k:

The dominant energy condition holds;
the domain of outer communications is static;
the domain of outer communications is circular;
H [k] is a bifurcate Killing horizon.

See [270Jump To The Next Citation Point] for some further observations concerning (2.16View Equation).

A Killing horizon is called degenerate if κ vanishes, and non-degenerate otherwise.

As an example, in Minkowski spacetime, consider the Killing vector ξ = x ∂t + t∂x. We have

d(g(ξ,ξ)) = d(− x2 + t2) = 2(− xdx + tdt),
which equals twice β™­ μ ν ξ := gμνξ dx on each of the four Killing horizons
H (ξ )πœ–δ := {t = πœ–x, δt > 0}, πœ–,δ ∈ {±1 }.
On the other hand, for the Killing vector
k = y∂ + t∂ + x ∂ − y ∂ = y∂ + (t + x)∂ − y∂ (2.19 ) t y y x t y x
one obtains
d(g(k,k)) = 2(t + x)(dt + dx ),
which vanishes on each of the Killing horizons {t = − x,y ⁄= 0}. This shows that the same null surface can have zero or non-zero values of surface gravity, depending upon which Killing vector has been chosen to calculate κ.

A key theorem of Rácz and Wald [270] asserts that non-degenerate horizons (with a compact cross section and constant surface gravity) are “essentially bifurcate”, in the following sense: Given a spacetime with such a non-degenerate Killing horizon, one can find another spacetime, which is locally isometric to the original one in a one-sided neighborhood of a subset of the horizon, and which contains a bifurcate Killing horizon. The result can be made global under suitable conditions.

The notion of average surface gravity can be defined for null hypersurfaces, which are not necessarily Killing horizons: Following [238Jump To The Next Citation Point], near a smooth null hypersurface 𝒩 one can introduce Gaussian null coordinates, in which the metric takes the form

g = rφdv2 + 2dvdr + 2rh dxadv + h dxadxb. (2.20 ) a ab
The null hypersurface 𝒩 is given by the equation {r = 0}; when it corresponds to an event horizon, by replacing r by − r if necessary we can, without loss of generality, assume that r > 0 in the domain of outer communications. Assuming that 𝒩 admits a smooth compact cross-section S, the average surface gravity ⟨κ ⟩S is defined as
∫ ⟨κ⟩ = − 1-- φdμ , (2.21 ) S |S | S h
where dμh is the measure induced by the metric h on S, and |S | is the area of S. We emphasize that this is defined regardless of whether or not the hypersurface is a Killing horizon; but if it is with respect to a vector k, and if the surface gravity κ of k is constant on S, then ⟨κ⟩S equals κ.

A smooth null hypersurface, not necessarily a Killing horizon, with a smooth compact cross-section S such that ⟨κ⟩S ⁄= 0 is said to be mean non-degenerate.

Using general identities for Killing fields (see, e.g., [151Jump To The Next Citation Point], Chapter 2) one can derive the following explicit expressions for κ:

[ 1 ] [ 1 ] κ2 = − lim ---g(∇kk, ∇kk ) = − -ΔgN , (2.22 ) N→0 N 4 H[k]
where Δg denotes the Laplace–Beltrami operator of the metric g. Introducing the four velocity √ ---- u = kβˆ• − N for a time-like k, the first expression shows that the surface gravity is the limiting value of the force applied at infinity to keep a unit mass at H [k] in place: κ = lim (√ − N-|a|), where a = ∇ u u (see, e.g., [314]).
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