General Results

3 General Results

In this section we describe a number of general results concerning the topology and symmetry of near-horizon geometries under various assumptions.

3.1 Horizon topology theorem

Hawking’s horizon topology theorem is one of the fundamental ingredients of the classic four-dimensional black-hole uniqueness theorems [127 ]. It states that cross sections of the event horizon of an asymptotically-flat, stationary, black-hole solution to Einstein’s equations, satisfying the dominant energy condition, must be homeomorphic to $S2$ . The proof is an elegant variational argument that shows that any cross section with negative Euler characteristic can be deformed outside the event horizon such that its outward null geodesics converge. This means one has an outer trapped surface outside the event horizon, which is not allowed by general results on black holes.¹⁵

Galloway and Schoen have shown how to generalise Hawking’s horizon topology theorem to higher dimensional spacetimes [91 ]. Their theorem states if the dominant energy condition holds, a cross section $H$ of the horizon of a black hole, or more generally a marginally outer trapped surface, must have positive Yamabe invariant.¹⁶ The positivity of the Yamabe invariant, which we define below, is equivalent to the existence of a positive scalar curvature metric and is well known to impose restrictions on the topology, see, e.g., [89 ]. For example, when $H$ is three dimensional, the only possibilities are connected sums of $S3$ (and their quotients) and $S1 × S2$ , consistent with the known examples of black-hole solutions.

In the special case of degenerate horizons a simple proof of this topology theorem can be given directly from the near-horizon geometry [166 ]. This is essentially a specialisation of the simplified proof of the Galloway–Schoen theorem given in [189 ]. However, we note that since we only use properties of the near-horizon geometry, in particular only the horizon equation (17 ), we do not require the existence of a black hole.

For four dimensional spacetimes, so dim $H = 2$ , the proof is immediate, see, e.g., [144 , 152 ].

Theorem 3.1. Consider a spacetime containing a degenerate horizon with a compact cross section $H$ and assume the dominant energy condition holds. If $Λ ≥ 0$ then $H ∼= S2$ , except for the special case where the near-horizon geometry is flat (so $Λ = 0$ ) and $H ∼= T 2$ . If $Λ < 0$ and $χ (H ) < 0$ the area of $H$ satisfies $A ≥ 2 πΛ− 1χ(H ) H$ with equality if and only if the near-horizon geometry is $AdS2 × Σg$ , where $Σg$ is a compact quotient of hyperbolic space of genus $g$ .

The proof is elementary. The Euler characteristic of $H$ can be calculated by integrating the trace of Eq. (17 ) over $H$ to get

where $𝜖γ$ is the volume form of the horizon metric $γ$ . Therefore, for $Λ ≥ 0$ and matter satisfying the dominant energy condition $P γ ≥ 0$ (see Eq. (29

)), it follows that $χ (H ) ≥ 0$ . Equality can only occur if $Λ = 0$ , $Pγ ≡ 0$ , $ha ≡ 0$ : using Eqs. (17

) and (18

) this implies $Rab = 0$ and $F = 0$ , so the near-horizon geometry is the trivial flat solution $ℝ1,1 × T 2$ . For the $Λ < 0$ case the above argument fails and one finds no restriction on the topology of $H$ . Instead, for $χ(H ) < 0$ , one can derive a lower bound for the area of $H$ :

This agrees with the lower bounds found in [95 , 208 ] in the more general context of apparent horizons. The lower bound in Eq. (42

) is saturated if and only if $ha ≡ 0$ , $Pγ ≡ 0$ , which implies $Rab = − |Λ |γab$ and $F = Λ$ , so the near-horizon geometry is $AdS2 × Σg$ .

It is of interest to generalise these results to higher dimensions along the lines of Galloway and Schoen. As is well known, the total integral of the scalar curvature in itself does not constrain the topology of $H$ in this case. An analogue of this invariant for dim $H = n ≥ 3$ is given by the Yamabe invariant $σ(H )$ . This is defined via the Yamabe constant associated to a given conformal class of metrics $[γ]$ on $H$ . First consider the volume-normalised Einstein–Hilbert functional

where $γ′$ is a Riemannian metric on $H$ and $𝜖γ′$ is the associated volume form. As is well known, this functional is neither bounded from above or below. However, the restriction of $E$ to any conformal class of metrics is always bounded from below: the Yamabe constant $Y (H, [γ ])$ for a given conformal class is then defined as the infimum of this functional. Parameterising the conformal class by $-4- γ ′ = ϕ n−2γ$ , for smooth positive functions $ϕ$ , we have $Y(H, [γ]) ≡ infϕ>0 E γ[ϕ]$ , where

( ) ∫ 4(n−1)|∇ ϕ|2 + R ϕ2 𝜖 E [ϕ ] ≡--H---n−2-----------γ-----γ. (44 ) γ ( ∫ -2n- ) n−n-2 H ϕn−2𝜖γ

The Yamabe invariant $σ(H )$ is defined by $σ(H ) = sup Y(H, [γ]) [γ]$ , where the supremum is taken over all possible conformal classes. The solution to the Yamabe problem states the following remarkable fact: for every conformal class $[γ ]$ on compact $H$ , the functional $E γ[ϕ ]$ achieves its infimum and this occurs for a constant scalar curvature metric.

We are now ready to present the degenerate horizon topology theorem.

Theorem 3.2. Consider a spacetime containing a degenerate horizon with a compact cross section $H$ and assume the dominant energy condition holds. If $Λ ≥ 0$ , then either $σ (H ) > 0$ or the induced metric on the horizon is Ricci flat. If $Λ < 0$ and $σ(H ) < 0$ the area of $H$ satisfies

A simple proof exploits the solution to the Yamabe problem mentioned above [189 , 166 ]. First observe that if there exists a conformal class of metrics $[γ]$ for which the Yamabe constant $Y (H, [γ ]) > 0$ then it follows that $σ(H ) > 0$ . Therefore, to establish that $H$ has positive Yamabe invariant, it is sufficient to show that for some $γ ab$ the functional $E [ϕ] > 0 γ$ for all $ϕ > 0$ , since the solution to the Yamabe problem then tells us that $Y (H,[γ]) = E γ[ϕ0] > 0$ for some $ϕ0 > 0$ .

For our Riemannian manifolds $(H, γ)$ it is easy to show, except for one exceptional circumstance, that $E γ[ϕ] > 0$ for all $ϕ > 0$ and thus $Y(H, [γ]) > 0$ . The proof is as follows. The horizon equation (17 ) can be used to establish the identity

for all $ϕ$ , where we have defined the differential operator $Da ≡ ∇a + 12ha$ . It is worth noting that this identity relies crucially on the precise constants appearing in Eq. (17

). This implies the following integral identity over $H$ :

If $Λ ≥ 0$ , the dominant energy condition $P γ ≥ 0$ implies $Eγ[ϕ] ≥ 0$ for all $ϕ > 0$ with equality only if $ha ≡ 0$ , $P γ ≡ 0$ and $Λ = 0$ . The exceptional case $ha ≡ 0$ , $P γ ≡ 0$ and $Λ = 0$ implies $R γ = 0$ , which allows one to infer [91 ] that either $Rab = 0$ or $H$ admits a metric of positive scalar curvature (and is thus positive Yamabe after all).

As in four spacetime dimensions the above argument fails for $Λ < 0$ , and thus provides no restriction of the topology of $H$ . Instead, assuming the dominant energy condition, Eq. (47 ) implies

for any $ϕ > 0$ , where the second inequality follows from Hölder’s inequality. Therefore, by the definition of $Y (H, [γ])$ , we deduce that $2∕n Y (H,[γ]) ≥ − n|Λ|AH$ . It follows that if $σ (H ) < 0$ , we get the stated non-trivial lower bound on the area of $H$ . We note that the lower bound can only be achieved if $ha ≡ 0$ and $Pγ ≡ 0$ , which implies $Rγ = − n |Λ |$ , in which case $γ$ necessarily minimises the functional $E$ in the conformal class $[γ ]$ so that $Y(H, [γ]) = − n |Λ |A2∕n H$ . However, since $− Y (H, [γ]) ≥ |σ(H )|$ , it need not be the case that the lower bound in Eq. (45

) is saturated by such horizon metrics (in contrast to the $n = 2$ case above).

We note that the above topology theorems in fact only employ the scalar curvature of the horizon metric and not the full horizon equation (17 ). It would be interesting if one could use the non-trace part of the horizon equation to derive further topological restrictions.

3.2 AdS₂-structure theorems

It is clear that a general near-horizon geometry, Eq. (7 ), possesses enhanced symmetry: in addition to the translation symmetry $v → v + c$ one also has a dilation symmetry $(v,r) → (λv, λ−1r)$ where $λ ⁄= 0$ and together these form a two-dimensional non-Abelian isometry group. In this section we will discuss various near-horizon symmetry theorems that guarantee further enhanced symmetry.

3.2.1 Static near-horizon geometries

A static near-horizon geometry is one for which the normal Killing field $K$ is hypersurface orthogonal, i.e., $K ∧ dK ≡ 0$ everywhere.

Theorem 3.3 ([162 ]). Any static near-horizon geometry is locally a warped product of AdS₂, dS₂ or $ℝ1,1$ and $H$ . If $H$ is simply connected this statement is global. In this case if $H$ is compact and the strong energy conditions holds it must be the AdS₂ case or the direct product $1,1 ℝ × H$ .

Proof : As a 1-form $μ a 2 K = K μdx = dr + rhadx + r F dv$ . A short calculation then reveals that $K ∧ dK = 0$ if and only if

which are the staticity conditions for a near-horizon geometry. Locally they can be solved by

where $λ(x)$ is a function on $H$ and $A0$ is a constant. Substituting these into the near-horizon geometry and changing the affine parameter $r → e−λ(x)r$ gives:

The metric in the square bracket is a maximally symmetric space: AdS₂ for $A0 < 0$ , dS₂ for $A0 > 0$ and $ℝ1,1$ for $A = 0 0$ . If $H$ is simply connected then $λ$ is globally defined on $H$ . Now consider Eq. (18

), which in this case reduces to

Assume $λ$ is a globally-defined function. Integrating over $H$ shows that if the strong energy condition (28

) holds then $A0 ≤ 0$ . The equality $A0 = 0$ occurs if and only if $E = Λ$ , in which case $λ$ is harmonic and hence a constant.

3.2.2 Near-horizon geometries with rotational symmetries

We begin by considering near-horizon geometries with a $U(1)D −3$ rotational symmetry, whose orbits are generically cohomogeneity-1 on cross sections of the horizon $H$ . The orbit spaces $D −3 H ∕U (1 )$ have been classified and are homeomorphic to either the closed interval or a circle, see, e.g., [139 ]. The former corresponds to $H$ of topology $S2, S3,L(p,q )$ times an appropriate dimensional torus, whereas the latter corresponds to $H ∼= TD− 3$ . Unless otherwise stated we will assume non-toroidal topology.

It turns out to be convenient to work with a geometrically-defined set of coordinates as introduced in [152 ]. Let $mi$ for $i = 1 ...D − 3$ be the Killing vector fields generating the isometry. Define the 1-form $Σ = − im1 ⋅⋅⋅imD −3𝜖γ$ , where $𝜖γ$ is the volume form associated with the metric $γ$ on $H$ . Note that $Σ$ is closed and invariant under the Killing fields $mi$ and so defines a closed one-form on the orbit space. Hence there exists a globally-defined invariant function $x$ on $H$ such that

It follows that $|dx |2 = det B$ , where $Bij ≡ γ(mi, mj)$ , so $dx$ vanishes precisely at the endpoints of the closed interval where the matrix $Bij$ has rank $D − 4$ . As a function on $H$ , $x$ has precisely one minimum $x1$ and one maximum $x2$ , which must occur at the endpoints of the orbit space. Hence $D− 3 ∼ H ∕U (1) = [x1,x2]$ .

Introducing coordinates adapted to the Killing fields $mi = ∂ ∕∂ϕi$ , we can use $(x,ϕi)$ as a chart on $H$ everywhere except the endpoints of the orbit space. The metric for $x1 < x < x2$ then reads

By standard results, the one-form $h$ may be decomposed globally on $H$ as

where $β$ is co-closed. Since $h$ is invariant under the $mi$ , it follows that $β$ and $d λ$ are as well. Further, periodicity of the orbits of the $mi$ implies $mi ⋅ dλ = 0$ , i.e., $λ = λ (x)$ . It is convenient to define the globally-defined positive function

Next, writing $β = β dx + β dϕi x i$ and imposing that $β$ is co-closed, implies $βx (detB )1∕2 = c$ , where $c$ is a constant. It follows $iβΣ = c$ and since $Σ$ vanishes at the fixed points of the $mi$ , this implies $c$ must vanish. Hence we may write

where we define $ki(x) ≡ Γ hi$ . It is worth noting that in the toroidal case one can introduce coordinates $(x,ϕ )$ so that the horizon metric takes the same form, with $x$ now periodic and $det B > 0$ everywhere, although now the one-form $h$ may have an extra term since the constant $c$ need not vanish [131

We are now ready to state the simplest of the AdS₂ near-horizon symmetry enhancement theorems:

Theorem 3.4 ([162 ]). Consider a $D$ -dimensional spacetime containing a degenerate horizon, invariant under an $ℝ × U (1)D−3$ isometry group, and satisfying the Einstein equations $R μν = Λg μν$ . Then the near-horizon geometry has a global $G × U (1)D−3$ symmetry, where $G$ is either $O(2,1)$ or the 2D Poincaré group. Furthermore, if $Λ ≤ 0$ and the near-horizon geometry is non-static the Poincaré case is excluded.

Proof: For the non-toroidal case we use the above coordinates. By examining the $(xv )$ and $(xi)$ components of the spacetime Einstein equations and changing the affine parameter $r → Γ (x)r$ , one can show

where $A0$ and $i k$ are constants. The metric in the square bracket is a maximally-symmetric space: AdS₂ for $A0 < 0$ , dS₂ for $A0 > 0$ and $ℝ1,1$ for $A0 = 0$ . Any isometry of these 2D base spaces transforms $rdv → rdv + dψ$ , for some function $ψ(v,r )$ . Therefore, by simultaneously transforming $ϕi → ϕi − kiψ$ , the full near-horizon geometry inherits the full isometry group $G$ of the 2D base, which for $A0 ⁄= 0$ is $O (2,1)$ and for $A0 = 0$ is the 2D Poincaré group.

The toroidal case can in fact be excluded [131 ], although as remarked above the coordinate system needs to be developed differently.

In fact as we will see in Section 4 one can completely solve for the near-horizon geometries of the above form in the $Λ = 0$ case.

The above result has a natural generalisation for $D = 4,5$ Einstein–Maxwell theories. For the sake of generality, consider a general 2-derivative theory describing Einstein gravity coupled to Abelian vectors $I A$ ( $I = 1...N$ ) and uncharged scalars $A Φ$ ( $A = 1...M$ ) in $D = 4,5$ dimensions, with action

where $FI ≡ dAI$ , $V (Φ )$ is an arbitrary scalar potential (which allows for a cosmological constant), and

where $CIJK$ are constants. This encompasses many theories of interest, e.g., vacuum gravity with a cosmological constant, Einstein–Maxwell theory, and various (possibly gauged) supergravity theories arising from compactification from ten or eleven dimensions.

Theorem 3.5 ([162 ]). Consider an extremal–black-hole solution of the above $D = 4,5$ theory with $D −3 ℝ × U(1)$ symmetry. The near-horizon limit of this solution has a global $D−3 G × U (1)$ symmetry, where $G$ is either $SO (2,1)$ or (the orientation-preserving subgroup of) the 2D Poincaré group. The Poincaré-symmetric case is excluded if $f (Φ) AB$ and $g (Φ ) IJ$ are positive definite, the scalar potential is non-positive, and the horizon topology is not $D −2 T$ .

Proof: The Maxwell fields $F I$ and the scalar fields are invariant under the Killing fields $mi$ , hence $ΦI = ΦI(x)$ . By examining the $(xv)$ and $(xi)$ components of Einstein’s equations for the above general theory, and changing the affine parameter $r → Γ (x )r$ , one can show that the near-horizon metric is given by Eq. (58 ) and the Maxwell fields are given by

where $eI$ are constants. Hence both the near-horizon metric and Maxwell fields are invariant under $G$ . Generic orbits of the symmetry group have the structure of $T D− 3$ fibred over a 2D maximally-symmetric space, i.e., AdS₂, dS₂ or $1,1 ℝ$ . AdS₂ and dS₂ give $SO (2,1)$ symmetry, whereas $1,1 ℝ$ gives Poincaré symmetry. The dS₂ and $1,1 ℝ$ cases are excluded subject to the additional assumptions mentioned, which ensure that the theory obeys the strong energy condition.

Remarks:

In the original statement of this theorem asymptotically-flat or AdS boundary conditions were assumed [162 ]. These were only used at one point in the proof, where the property that the generator of each rotational symmetry must vanish somewhere in the asymptotic region (on the “axis” of the symmetry) was used to constrain the Maxwell fields. In fact, using the general form for the near-horizon limit of a Maxwell field (23 ) and the fact that for non-toroidal topology at least one of the rotational Killing fields must vanish somewhere, allows one to remove any assumptions on the asymptotics of the black-hole spacetime.
In the context of black holes, toroidal topology is excluded for $Λ = 0$ when the dominant energy condition holds by the black-hole–topology theorems.

An important corollary of the above theorems is:

Corollary 3.1. Consider a $D ≥ 5$ spacetime with a degenerate horizon invariant under a $ℝ × U(1)D −3$ symmetry as in Theorem 3.4 and 3.5. The near-horizon geometry is static if it is either a warped product of AdS₂ and $H$ , or it is a warped product of locally AdS₃ and a $(D − 3)$ -manifold.

The static conditions (49 ) for Eq. (58 ) occur if and only if: (i) $i k = 0$ for all $i = 1,...,D − 3$ , or (ii) $ki = constΓ$ and $kiki = ℓ−2Γ$ with $A0 = − ℓ−2 < 0$ . Case (i) gives a warped product of a 2D maximally-symmetric space and $H$ as in Theorem (3.3). For case (ii) one can introduce $U (1)D−3$ coordinates $(y1,yI)$ where $I = 2,...,D − 3$ , not necessarily periodic, so that $k = ℓ− 1∂ ∕∂y1$ and

The metric in the square brackets is locally isometric to AdS₃. If $y1$ is a periodic coordinate the horizon topology is $1 S × B$ , where $B$ is some $(D − 3)$ -dimensional manifold.

Theorem (3.5) can be extended to higher-derivative theories of gravity as follows. Consider a general theory of gravity coupled to Abelian vectors $AI$ and uncharged scalars $ΦA$ with action

where $S2$ is the 2-derivative action above, $λ$ is a coupling constant, and $ℒm$ is constructed by contracting (derivatives of) the Riemann tensor, volume form, scalar fields and Maxwell fields in such a way that the action is diffeomorphism and gauge-invariant.

Proposition 3.1 ([162 ]). Consider an extremal black-hole solution of the above higher-derivative theory, obeying the same assumptions as in Theorem 3.5. Assume there is a regular horizon when $λ = 0$ with $D− 3 SO (2, 1) × U (1)$ near-horizon symmetry, and the near-horizon solution is analytic in $λ$ . Then the near-horizon solution has $SO (2, 1) × U (1)D− 3$ symmetry to all orders in $λ$ .

Hence Theorem 3.5 is stable with respect to higher-derivative corrections. However, it does not apply to “small” black holes (i.e., if there is no regular black hole for $λ = 0$ ).

So far, the results described all assume $D − 3$ commuting rotational Killing fields. For $D = 4,5$ this is the same number as the rank of the rotation group $SO (D − 1)$ , so the above results are applicable to asymptotically-flat or globally-AdS black holes. For $D > 5$ the rank of this rotation group is $⌊D−1-⌋ 2$ , which is smaller than $D − 3$ , so the above theorems do not apply to asymptotically-flat or AdS black holes. An important open question is whether the above theorems generalise when fewer than $D − 3$ commuting rotational isometries are assumed, in particular the case with $⌊D−2-1⌋$ commuting rotational symmetries. To this end, partial results have been obtained assuming a certain non-Abelian cohomogeneity-1 rotational isometry.

Proposition 3.2 ([79 ]). Consider a near-horizon geometry with a rotational isometry group $U (1)m × K$ , whose generic orbit on $H$ is a cohomogeneity-1 $Tm$ -bundle over a $K$ -invariant homogeneous space $B$ . Furthermore, assume $B$ does not admit any $K$ -invariant one-forms. If Einstein’s equations $R μν = Λg μν$ hold, then the near-horizon geometry possesses a $G × U (1)m × K$ isometry group, where $G$ is either $O(2,1)$ or the 2D Poincaré group. Furthermore, if $Λ ≤ 0$ and the near-horizon geometry is non-static then the Poincaré group is excluded.

The assumptions in the above result reduce the Einstein equations for the near-horizon geometry to ODEs, which can be solved in the same way as in Theorem 3.4. The special case $K = SU (q)$ , $q−1 B = ℂ ℙ$ with $m = 1$ and $m = 2$ , gives a near-horizon geometry of the type that occurs for a Myers–Perry black hole with all the angular momenta of set equal in $2q + 2$ dimensions, or all but one set equal in $2q + 3$ dimensions, respectively.

The preceding results apply only to cohomogeneity-1 near-horizon geometries. As discussed above, this is too restrictive to capture the generic case for $D > 5$ . The following result for higher-cohomogeneity near-horizon geometries has been shown.

Theorem 3.6 ([166 ]). Consider a spacetime containing a degenerate horizon invariant under orthogonally transitive isometry group $ℝ × U (1)N$ , where $1 ≤ N ≤ D − 3$ , such that the surfaces orthogonal to the surfaces of transitivity are simply connected. Then the near-horizon geometry has an isometry group $N G × U (1)$ , where $G$ is either $SO (2,1)$ or the 2D Poincaré group. Furthermore, if the strong energy condition holds and the near-horizon geometry is non-static, the Poincaré case is excluded.

¹⁷

The near-horizon geometry in this case can be written as

where as in the above cases we have rescaled the affine parameter $r → Γ r$ . For the $N = D − 3$ case, orthogonal transitivity follows from Einstein’s equations [72 ], which provides another proof of Theorem 3.4 and 3.5. For $N < D − 3$ this result guarantees an AdS₂ symmetry for all known extremal–black-hole solutions, since all known explicit solutions possess orthogonally-transitive symmetry groups. In these higher cohomogeneity cases, the relation between Einstein’s equations and orthogonal transitivity is not understood. It would be interesting to investigate this further.

	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

3 General Results

3.1 Horizon topology theorem

3.2 AdS2-structure theorems

3.2.1 Static near-horizon geometries

3.2.2 Near-horizon geometries with rotational symmetries

3.2 AdS₂-structure theorems