Degenerate Horizons and Near-Horizon Geometry

2 Degenerate Horizons and Near-Horizon Geometry

2.1 Coordinate systems and near-horizon limit

In this section we will introduce a general notion of a near-horizon geometry. This requires us to first introduce some preliminary constructions. Let $𝒩$ be a smooth⁹ codimension-1 null hypersurface in a $D$ dimensional spacetime $(M, g)$ . In a neighbourhood of any such hypersurface there exists an adapted coordinate chart called Gaussian null coordinates that we now recall [179 , 86 ].

Let $N$ be a vector field normal to $𝒩$ whose integral curves are future-directed null geodesic generators of $𝒩$ . In general these will be non-affinely parameterised so on $𝒩$ we have $∇N N = κN$ for some function $κ$ . Now let $H$ denote a smooth $(D − 2)$ -dimensional spacelike submanifold of $𝒩$ , such that each integral curve of $N$ crosses $H$ exactly once: we term $H$ a cross section of $𝒩$ and assume such submanifolds exist. On $H$ choose arbitrary local coordinates $(xa)$ , for $a = 1,...,D − 2$ , containing some point $p ∈ H$ . Starting from $p ∈ H$ , consider the point in $𝒩$ a parameter value $v$ along the integral curve of $N$ . Now assign coordinates $(v,xa)$ to such a point, i.e., we extend the functions $a x$ into $𝒩$ by keeping them constant along such a curve. This defines a set of coordinates $a (v,x )$ in a tubular neighbourhood of the integral curves of $N$ through $p ∈ H$ , such that $N = ∂ ∕∂v$ . Since $N$ is normal to $𝒩$ we have $N ⋅ N = gvv = 0$ and $N ⋅ ∂∕∂xa = gva = 0$ on $𝒩$ .

We now extend these coordinates into a neighbourhood of $𝒩$ in $M$ as follows. For any point $q ∈ 𝒩$ contained in the above coordinates $a (v,x )$ , let $L$ be the unique past-directed null vector satisfying $L ⋅ N = 1$ and $a L ⋅ ∂∕∂x = 0$ . Now starting at $q$ , consider the point in $M$ an affine parameter value $r$ along the null geodesic with tangent vector $L$ . Define the coordinates of such a point in $M$ by $(v,r,xa)$ , i.e., the functions $v,xa$ are extended into $M$ by requiring them to be constant along such null geodesics. This provides coordinates $(v, r,xa)$ defined in a neighbourhood of $𝒩$ in $M$ , as required.

We extend the definitions of $N$ and $L$ into $M$ by $N = ∂∕∂v$ and $L = ∂ ∕∂r$ . By construction the integral curves of $L = ∂∕∂r$ are null geodesics and hence $grr = 0$ everywhere in the neighbourhood of $𝒩$ in $M$ in question. Furthermore, using the fact that $N$ and $L$ commute (they are coordinate vector fields), we have

and therefore $L ⋅ N = gvr = 1$ for all $r$ . A similar argument shows $a L ⋅ ∂∕∂x = gra = 0$ for all $r$ .

These considerations show that, in a neighbourhood of $𝒩$ in $M$ , the spacetime metric $g$ written in Gaussian null coordinates $(v,r,xa)$ is of the form

where $𝒩$ is the hypersurface ${r = 0}$ , the metric components $f,ha,γab$ are smooth functions of all the coordinates, and $γ ab$ is an invertible $(D − 2) × (D − 2)$ matrix. This coordinate chart is unique up to choice of cross section $H$ and choice of coordinates $a (x )$ on $H$ . Upon a change of coordinates on $H$ the quantities $f, ha,γab$ transform as a function, 1-form and non-degenerate metric, respectively. Hence they may be thought of as components of a globally-defined function, 1-form and Riemannian metric on $H$ .

The coordinates developed above are valid in the neighbourhood of any smooth null hypersurface $𝒩$ . In this work we will in fact be concerned with smooth Killing horizons. These are null hypersurfaces that possess a normal that is a Killing field $K$ in $M$ . Hence we may set $N = K$ in the above construction. Since $K = ∂∕ ∂v$ we deduce that in the neighbourhood of a Killing horizon $𝒩$ , the metric can be written as Eq. (5 ) where the functions $f,ha,γab$ are all independent of the coordinate $v$ . Using the Killing property one can rewrite $∇ K = κK K$ as $d(K ⋅ K ) = − 2 κK$ on $𝒩$ , where $κ$ is now the usual surface gravity of a Killing horizon.

We may now introduce the main objects we will study in this work: degenerate Killing horizons. These are defined as Killing horizons $𝒩$ such that the normal Killing field $K$ is tangent to affinely parameterised null geodesics on $𝒩$ , i.e., $κ ≡ 0$ . Therefore, $d(K ⋅ K )|𝒩 = 0$ , which implies that in Gaussian null coordinates $(∂rgvv)|r=0 = 0$ . It follows that $2 gvv = r F$ for some smooth function $F$ . Therefore, in the neighbourhood of any smooth degenerate Killing horizon the metric in Gaussian null coordinates reads

We are now ready to define the near-horizon geometry of a $D$ -dimensional spacetime $(M, g)$ containing such a degenerate horizon. Given any $𝜖 > 0$ , consider the diffeomorphism defined by $v → v ∕𝜖$ and $r → 𝜖r$ . The metric in Gaussian null coordinates transforms $g → g 𝜖$ where $g𝜖$ is given by Eq. (6 ) with the replacements $F(r,x) → F (𝜖r,x )$ , $ha(r,x) → ha(𝜖r,x)$ and $γab(r,x) → γab(𝜖r,x)$ . The near-horizon limit is then defined as the $𝜖 → 0$ limit of $g 𝜖$ . It is clear this limit always exists since all metric functions are smooth at $r = 0$ . The resulting metric is called the near-horizon geometry and is given by

where $F(x ) = F (0,x),ha (x ) = ha(0,x)$ and $γab(x) = γab(0,x)$ . Notice that the $r$ dependence of the metric is completely fixed. In fact the near-horizon geometry is completely specified by the following geometric data on the $(D − 2)$ -dimensional cross section $H$ : a smooth function $F$ , 1-form $ha$ and Riemannian metric $γ ab$ . We will often refer to the triple of data $(F,h ,γ ) a ab$ on $H$ as the near-horizon data.

Intuitively, the near-horizon limit is a scaling limit that focuses on the spacetime near the horizon $𝒩$ . We emphasise that the degenerate assumption $gvv = O(r2)$ is crucial for defining this limit and such a general notion of a near-horizon limit does not exist for a non-degenerate Killing horizon.

2.2 Curvature of near-horizon geometry

As we will see, geometric equations (such as Einstein’s equations) for a near-horizon geometry can be equivalently written as geometric equations defined purely on a $(D − 2)$ -dimensional cross section manifold $H$ of the horizon. In this section we will write down general formulae relating the curvature of a near-horizon geometry to the curvature of the horizon $H$ . For convenience we will denote the dimension of $H$ by $n = D − 2$ .

It is convenient to introduce a null-orthonormal frame for the near-horizon metric (7 ), denoted by $(eA)$ , where $A = (+, − ,a )$ , $a = 1,...,n$ and

so that $A B + − a a g = ηABe e = 2e e + e e$ , where $a ˆe$ are vielbeins for the horizon metric $a a γ = ˆe ˆe$ .¹⁰ The dual basis vectors are

where $ˆ∂a$ denote the dual vectors to $ˆea$ . The connection 1-forms satisfy $deA = − ωA ∧ eB B$ and are given by

where $ˆωab$ and $ˆ ∇a$ are the connection 1-forms and Levi-Civita connection of the metric $γab$ on $H$ respectively. The curvature two-forms defined by $C ΩAB = d ωAB + ωAC ∧ ω B$ give the Riemann tensor in this basis using $ΩAB = 12RABCDeC ∧ eD$ . The curvature two forms are:

( ) Ω = Ωˆ + e+ ∧ e− ˆ∇ h + re+ ∧ ed − h ˆ∇ h + ˆ∇ ∇ˆ h + 1h ˆ∇ h − 1h ∇ˆ h , ab ab [a b] ( d [a b] d [a b] )2 a [b d] 2 b [a d] (1- ) + − b + ˆ 1 ˆ 1ˆ a b Ω+ − = 4haha − F e ∧ e + re ∧ e ∂bF − F hb + 2ha∇ [ahb] + 2∇ [ahb]e ∧ e , 2 + d [( 1ˆ ) ˆ 1 ˆ ˆ ˆ 1 ˆ ] Ω+a = r e ∧ e − 2∇d + hd (∂aF − F ha) + 2F ∇ [ahd] + ∇[cha]∇ [chd] + 2h [a∇d ]F + − ( 1 ) 1 − b( 1 ) + re ∧ e haF − ∂ˆaF − 2hb∇ˆ[bha ] + 2 e ∧ e ˆ∇ahb − 2hahb ( ) + reb ∧ ed − ∇ˆd ˆ∇ [ahb] + 12ha∇ˆ[dhb] − 12hb∇ˆ[ahd ] , ( ) Ω−a = 12e+ ∧ eb ∇ˆbha − 12hahb , (11 )

where $ˆΩab$ is the curvature of $ωˆab$ on $H$ . The non-vanishing components of the Ricci tensor are thus given by:

R+ − = F − 12haha + 12 ˆ∇aha, ˆ ˆ 1 Rab = Rab[ + ∇ (ahb) − 2hahb, ] R = r2 − 1ˆ∇2F + 3haˆ∇ F + 1F∇ˆah − F h h + ∇ˆ h ˆ∇ h ≡ r2S , ++ [ 2 2 a 2 a ]a a [c a] [c a] ++ R = r ∇ˆ F − Fh − 2h ˆ∇ h + ∇ˆ ˆ∇ h ≡ rS , (12 ) +a a a b [a b] b [a b] +a

where $ˆR ab$ is the Ricci tensor of the metric $γ ab$ on $H$ . The spacetime contracted Bianchi identity implies the following identities on $H$ :

which may also be verified directly from the above expressions.

It is worth noting that the following components of the Weyl tensor automatically vanish: $C −a−b = 0$ and $C −abc = 0$ . This means that $e− = ∂r$ is a multiple Weyl aligned null direction and hence any near-horizon geometry is at least algebraically special of type II within the classification of [47 ]. In fact, it can be checked that the null geodesic vector field $∂r$ has vanishing expansion, shear and twist and therefore any near-horizon geometry is a Kundt spacetime.¹¹ Indeed, by inspection of Eq. (7 ) it is clear that near-horizon geometries are a subclass of the degenerate Kundt spacetimes,¹² which are all algebraically special of at least type II [184 ].

Henceforth, we will drop the “hats” on all horizon quantities, so $Rab$ and $∇a$ refer to the Ricci tensor and Levi-Civita connection of $γab$ on $H$ .

2.3 Einstein equations and energy conditions

We will consider spacetimes that are solutions to Einstein’s equations:

where $Tμν$ is the energy-momentum and $Λ$ is the cosmological constant of our spacetime. We will be interested in a variety of possible energy momentum tensors and thus in this section we will keep the discussion general.

An important fact is that if a spacetime containing a degenerate horizon satisfies Einstein’s equations then so does its near-horizon geometry. This is easy to see as follows. If the metric $g$ in Eq. (6 ) satisfies Einstein’s equations, then so will the 1-parameter family of diffeomorphic metrics $g𝜖$ for any $𝜖 > 0$ . Hence the limiting metric $𝜖 → 0$ , which by definition is the near-horizon geometry, must also satisfy the Einstein equations.

The near-horizon limit of the energy momentum tensor thus must also exist and takes the form

where $T+− ,α$ are functions on $H$ and $βa$ is a 1-form on $H$ . Working in the vielbein frame (8

), it is then straightforward to verify that the $ab$ and $+−$ components of the Einstein equations for the near-horizon geometry give the following equations on the cross section $H$ :

where we have defined

Pab ≡ Tab − 1-(γcdTcd + 2T+ − )γab, (19 ) ( n ) n-−--2 1- ab E ≡ − n T+ − + nγ Tab. (20 )

It may be shown that the rest of the Einstein equations are automatically satisfied as a consequence of Eqs. (17

), (18

) and the matter field equations, as follows.

The matter field equations must imply the spacetime conservation equation $∇ μT = 0 μν$ . This is equivalent to the following equations on $H$ :

which thus determine the components of the energy momentum tensor $α,βa$ in terms of $T+− ,Tab$ . The $++$ and $+a$ components of the Einstein equations are $S++ = α$ and $S+a = βa$ respectively, where $S ++$ and $S +a$ are defined in Eq. (12

). The first equation in (21

) and the identity (13

) imply that the $++$ equation is satisfied as a consequence of the $+a$ equation. Finally, substituting Eqs. (17

) and (18

) into the identity (14

), and using the second equation in (21

), implies the $+a$ equation. Alternatively, a tedious calculation shows that the $+a$ equation follows from Eqs. (17

) and (18

) using the contracted Bianchi identity for Eq. (17

), together with the second equation in (21

Although the energy momentum tensor must have a near-horizon limit, it is not obvious that the matter fields themselves must. Thus, consider the full spacetime before taking the near-horizon limit. Recall that for any Killing horizon $R μνK μK ν|𝒩 = 0$ and therefore $TμνK μK ν|𝒩 = 0$ . This imposes a constraint on the matter fields. We will illustrate this for Einstein–Maxwell theory whose energy-momentum tensor is

where $ℱ$ is the Maxwell 2-form, which must satisfy the Bianchi identity $dℱ = 0$ . It can be checked that in Gaussian null coordinates $μ ν ab TμνK K |𝒩 = 2 ℱvaℱvbγ |r=0$ and hence we deduce that $ℱva|r=0 = 0$ . Thus, smoothness requires $ℱva = 𝒪 (r)$ , which implies the near-horizon limit of $ℱ$ in fact exists. Furthermore, imposing the Bianchi identity to the near-horizon limit of the Maxwell field relates $ℱvr$ and $ℱ va$ , allowing one to write

where $Δ$ is a function on $H$ and $B$ is a closed 2-form on $H$ . The 2-form $B$ is the Maxwell field induced on $H$ and locally can be written as $B = dA$ for some 1-form potential $A$ on $H$ . It can be checked that for the near-horizon limit

2(n − 1) 2 1 ab E = --------Δ + -BabB , (24 ) n ( n cd) P = 2B B c+ 2-Δ2 − BcdB--- γ . (25 ) ab ac b n n ab

We will present the Maxwell equations in a variety of dimensions in Section 6.

It is worth remarking that the above naturally generalises to $p$ -form electrodynamics, with $p ≥ 2$ , for which the energy momentum tensor is

where $ℱ$ is a $p$ -form field strength satisfying the Bianchi identity $dℱ = 0$ . It is then easily checked that $TμνK μK ν|𝒩 = 0$ implies $ℱva1...ap−1ℱva1...ap−1|r=0 = 0$ and hence $ℱva1...ap−1|r=0 = 0$ . Thus, smoothness requires $ℱ = 𝒪 (r) va1...ap−1$ , which implies that the near-horizon limit of the $p$ -form exists. The Bianchi identity then implies that the most general form for the near-horizon limit is

where $Y$ is a $(p − 2 )$ -form on $H$ and $X$ is a closed $p$ -form on $H$ .

The Einstein equations for a near-horizon geometry can also be interpreted as geometrical equations arising from the restriction of the Einstein equations for the full spacetime to a degenerate horizon, without taking the near-horizon limit, as follows. The near-horizon limit can be thought of as the $𝜖 → 0$ limit of the “boost” transformation $(K, L) → (𝜖K, 𝜖−1L )$ . This implies that restricting the boost-invariant components of the Einstein equations for the full spacetime to a degenerate horizon is equivalent to the boost invariant components of the Einstein equations for the near-horizon geometry. The boost-invariant components are $+ −$ and $ab$ and hence we see that Eqs. (17 ) and (18 ) are also valid for the full spacetime quantities restricted to the horizon. We deduce that the restriction of these components of the Einstein equations depends only on data intrinsic to $H$ : this special feature only arises for degenerate horizons.¹³ It is worth noting that the horizon equations (17 ) and (18 ) remain valid in the more general context of extremal isolated horizons [163 , 209 , 28 ] and Kundt metrics [144 ].

The positivity of $E$ and $Pab$ can be related to standard energy conditions. For a near-horizon geometry $R μν(K − L )μ(K − L)ν|𝒩 = − 2R μνK μLν|𝒩$ . Since $K − L$ is timelike on the horizon, the strong energy condition implies $RμνK μL ν|𝒩 ≤ 0$ . Hence, noting that $R μνK μLν|𝒩 = R+ − = − E + Λ$ we deduce that the strong energy condition implies

On the other hand the dominant energy condition implies $μ ν TμνK L |𝒩 ≤ 0$ . One can show $T μνK μLν|𝒩 = − 12 Pabγab$ . Therefore, the dominant energy condition implies

Since $P = − 2T γ +−$ , if $n ≥ 2$ the dominant energy condition implies $E ≥ 0$ : hence, if $Λ ≤ 0$ the dominant energy condition implies both Eqs. (28

) and (29

). Observe that Einstein–Maxwell theory with $Λ ≤ 0$ satisfies both of these conditions.

In this review, we describe the current understanding of the space of solutions to the basic horizon equation (17 ), together with the appropriate horizon matter field equations, in a variety of dimensions and theories.

2.4 Physical charges

So far we have considered near-horizon geometries independently of any extremal–black-hole solutions. In this section we will assume that the near-horizon geometry arises from a near-horizon limit of an extremal black hole. This limit discards the asymptotic data of the parent–black-hole solution. As a result, only a subset of the physical properties of a black hole can be calculated from the near-horizon geometry alone. In particular, information about the asymptotic stationary Killing vector field is lost and hence one cannot compute the mass from a Komar integral, nor can one compute the angular velocity of the horizon with respect to infinity. Below we discuss physical properties that can be computed purely from the near-horizon geometry [123 , 79 , 154 ].

Area. The area of cross sections of the horizon $H$ is defined by

where $𝜖γ$ is the volume form associated to the induced Riemannian metric $γab$ on $H$ .

For definiteness we now assume the parent black hole is asymptotically flat.

Angular momentum. The conserved angular momentum associated with a rotational symmetry, generated by a Killing vector $m$ , is given by a Komar integral on a sphere at spacelike infinity $S ∞$ :¹⁴

This expression can be rewritten as an integral of the near-horizon data over $H$ , by applying Stokes’ theorem to a spacelike hypersurface $Σ$ with boundary $S∞ ∪ H$ . The field equations can be used to evaluate the volume integral that is of the form $∫ ⋆R (m ) Σ$ , where $R(m )μ = R μνm ν$ . In particular, for vacuum gravity one simply has:

For Einstein–Maxwell theories the integral $∫ Σ⋆R (m )$ can also be written as an integral over $H$ , giving extra terms that correspond to the contribution of the matter fields to the angular momentum. For example, consider pure Einstein–Maxwell theory in any dimension so the Maxwell equation is $d ⋆ ℱ = 0$ . Parameterising the near-horizon Maxwell field by (23

) one can show that, in the gauge $ℒmA = 0$ ,

so the angular momentum is indeed determined by the near-horizon data.

In five spacetime dimensions it is natural to couple Einstein–Maxwell theory to a Chern–Simons (CS) term. While the Einstein equations are unchanged, the Maxwell equation now becomes

where $ξ$ is the CS coupling constant. The angular momentum in this case can also be written purely as an integral over $H$ :

Of particular interest is the theory defined by CS coupling $ξ = 1$ , since this corresponds to the bosonic sector of minimal supergravity.

Gauge charges. For Einstein–Maxwell theories there are also electric, and possibly magnetic, charges. For example, in pure Einstein–Maxwell theory in any dimension, the electric charge is written as an integral over spatial infinity:

By applying Stokes’ Theorem to a spacelike hypersurface $Σ$ as above, and using the Maxwell equation, one easily finds

For $D = 5$ Einstein–Maxwell–CS theory one instead gets

For $D = 4$ one also has a conserved magnetic charge $∫ Qm = 1- ℱ 4π S∞$ . Using the Bianchi identity this can be written as

For $D > 4$ asymptotically-flat black holes there is no conserved magnetic charge. However, for $D = 5$ black rings $H ∼= S1 × S2$ , one can define a quasi-local dipole charge over the $S2$

where in the second equality we have expressed it in terms of the horizon Maxwell field.

Note that in general the gauge field $A$ will not be globally defined on $H$ , so care must be taken to evaluate expressions such as (35 ) and (38 ), see [123 , 155 ].

	Abstract
1	Introduction
	1.1	Black holes in string theory
	1.2	Gauge/gravity duality
	1.3	Black hole classification
	1.4	This review
2	Degenerate Horizons and Near-Horizon Geometry
	2.1	Coordinate systems and near-horizon limit
	2.2	Curvature of near-horizon geometry
	2.3	Einstein equations and energy conditions
	2.4	Physical charges
3	General Results
	3.1	Horizon topology theorem
	3.2	AdS₂-structure theorems
4	Vacuum Solutions
	4.1	Static: all dimensions
	4.2	Three dimensions
	4.3	Four dimensions
	4.4	Five dimensions
	4.5	Higher dimensions
5	Supersymmetric Solutions
	5.1	Four dimensions
	5.2	Five dimensions
	5.3	Six dimensions
	5.4	Ten dimensions
	5.5	Eleven dimensions
6	Solutions with Gauge Fields
	6.1	Three dimensional Einstein–Maxwell–Chern–Simons theory
	6.2	Four dimensional Einstein–Maxwell theory
	6.3	Five dimensional Einstein–Maxwell–Chern–Simons theory
	6.4	Theories with hidden symmetry
	6.5	Non-Abelian gauge fields
7	Applications and Related Topics
	7.1	Black-hole uniqueness theorems
	7.2	Stability of near-horizon geometries and extremal black holes
	7.3	Geometric inequalities
	7.4	Analytic continuation
	7.5	Extremal branes
	Acknowledgements
	References
	Footnotes