## 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 . 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
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 is three dimensional, the only possibilities are connected sums of
(and their quotients) and , 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 , the proof is immediate, see, e.g., [144, 152].

Theorem 3.1. Consider a spacetime containing a degenerate horizon with a compact cross section and assume the dominant energy condition holds. If then , except for the special case where the near-horizon geometry is flat (so ) and . If and the area of satisfies with equality if and only if the near-horizon geometry is , where is a compact quotient of hyperbolic space of genus .

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

where is the volume form of the horizon metric . Therefore, for and matter satisfying the dominant energy condition (see Eq. (29)), it follows that . Equality can only occur if , , : using Eqs. (17) and (18) this implies and , so the near-horizon geometry is the trivial flat solution . For the case the above argument fails and one finds no restriction on the topology of . Instead, for , one can derive a lower bound for the area of : 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 , , which implies and , so the near-horizon geometry is .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 in this case. An analogue of this invariant for dim is given by the Yamabe invariant . This is defined via the Yamabe constant associated to a given conformal class of metrics on . First consider the volume-normalised Einstein–Hilbert functional

where is a Riemannian metric on and is the associated volume form. As is well known, this functional is neither bounded from above or below. However, the restriction of to any conformal class of metrics is always bounded from below: the Yamabe constant for a given conformal class is then defined as the infimum of this functional. Parameterising the conformal class by , for smooth positive functions , we have , where The Yamabe invariant is defined by , 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 on compact , the functional 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 and assume the dominant energy condition holds. If , then either or the induced metric on the horizon is Ricci flat. If and the area of 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 for which the Yamabe constant then it follows that . Therefore, to establish that has positive Yamabe invariant, it is sufficient to show that for some the functional for all , since the solution to the Yamabe problem then tells us that for some .

For our Riemannian manifolds it is easy to show, except for one exceptional circumstance, that for all and thus . The proof is as follows. The horizon equation (17) can be used to establish the identity

for all , where we have defined the differential operator . It is worth noting that this identity relies crucially on the precise constants appearing in Eq. (17). This implies the following integral identity over : If , the dominant energy condition implies for all with equality only if , and . The exceptional case , and implies , which allows one to infer [91] that either or admits a metric of positive scalar curvature (and is thus positive Yamabe after all).As in four spacetime dimensions the above argument fails for , and thus provides no restriction of the topology of . Instead, assuming the dominant energy condition, Eq. (47) implies

for any , where the second inequality follows from Hölder’s inequality. Therefore, by the definition of , we deduce that . It follows that if , we get the stated non-trivial lower bound on the area of . We note that the lower bound can only be achieved if and , which implies , in which case necessarily minimises the functional in the conformal class so that . However, since , it need not be the case that the lower bound in Eq. (45) is saturated by such horizon metrics (in contrast to the 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 AdS_{2}-structure theorems

It is clear that a general near-horizon geometry, Eq. (7), possesses enhanced symmetry: in addition to the translation symmetry one also has a dilation symmetry where 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 is hypersurface orthogonal, i.e., everywhere.

Theorem 3.3 ([162]). Any static near-horizon geometry is locally a warped product of AdS_{2}, dS_{2} or
and . If is simply connected this statement is global. In this case if is compact
and the strong energy conditions holds it must be the AdS_{2} case or the direct product .

Proof : As a 1-form . A short calculation then reveals that if and only if

which are the staticity conditions for a near-horizon geometry. Locally they can be solved by where is a function on and is a constant. Substituting these into the near-horizon geometry and changing the affine parameter gives: The metric in the square bracket is a maximally symmetric space: AdS_{2}for , dS

_{2}for and for . If is simply connected then is globally defined on . Now consider Eq. (18), which in this case reduces to Assume is a globally-defined function. Integrating over shows that if the strong energy condition (28) holds then . The equality occurs if and only if , in which case is harmonic and hence a constant.

#### 3.2.2 Near-horizon geometries with rotational symmetries

We begin by considering near-horizon geometries with a rotational symmetry, whose orbits are generically cohomogeneity-1 on cross sections of the horizon . The orbit spaces have been classified and are homeomorphic to either the closed interval or a circle, see, e.g., [139]. The former corresponds to of topology times an appropriate dimensional torus, whereas the latter corresponds to . 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 for be the Killing vector fields generating the isometry. Define the 1-form , where is the volume form associated with the metric on . Note that is closed and invariant under the Killing fields and so defines a closed one-form on the orbit space. Hence there exists a globally-defined invariant function on such that

It follows that , where , so vanishes precisely at the endpoints of the closed interval where the matrix has rank . As a function on , has precisely one minimum and one maximum , which must occur at the endpoints of the orbit space. Hence .Introducing coordinates adapted to the Killing fields , we can use as a chart on everywhere except the endpoints of the orbit space. The metric for then reads

By standard results, the one-form may be decomposed globally on as where is co-closed. Since is invariant under the , it follows that and are as well. Further, periodicity of the orbits of the implies , i.e., . It is convenient to define the globally-defined positive function Next, writing and imposing that is co-closed, implies , where is a constant. It follows and since vanishes at the fixed points of the , this implies must vanish. Hence we may write where we define . It is worth noting that in the toroidal case one can introduce coordinates so that the horizon metric takes the same form, with now periodic and everywhere, although now the one-form may have an extra term since the constant need not vanish [131]. We are now ready to state the simplest of the AdS_{2} near-horizon symmetry enhancement
theorems:

Theorem 3.4 ([162]). Consider a -dimensional spacetime containing a degenerate horizon, invariant under an isometry group, and satisfying the Einstein equations . Then the near-horizon geometry has a global symmetry, where is either or the 2D Poincaré group. Furthermore, if 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 and components of the spacetime Einstein equations and changing the affine parameter , one can show

where and are constants. The metric in the square bracket is a maximally-symmetric space: AdS_{2}for , dS

_{2}for and for . Any isometry of these 2D base spaces transforms , for some function . Therefore, by simultaneously transforming , the full near-horizon geometry inherits the full isometry group of the 2D base, which for is and for 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 case.

The above result has a natural generalisation for Einstein–Maxwell theories. For the sake of generality, consider a general 2-derivative theory describing Einstein gravity coupled to Abelian vectors () and uncharged scalars () in dimensions, with action

where , is an arbitrary scalar potential (which allows for a cosmological constant), and or where 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 theory with symmetry. The near-horizon limit of this solution has a global symmetry, where is either or (the orientation-preserving subgroup of) the 2D Poincaré group. The Poincaré-symmetric case is excluded if and are positive definite, the scalar potential is non-positive, and the horizon topology is not .

Proof: The Maxwell fields and the scalar fields are invariant under the Killing fields , hence . By examining the and components of Einstein’s equations for the above general theory, and changing the affine parameter , one can show that the near-horizon metric is given by Eq. (58) and the Maxwell fields are given by

where are constants. Hence both the near-horizon metric and Maxwell fields are invariant under . Generic orbits of the symmetry group have the structure of fibred over a 2D maximally-symmetric space, i.e., AdS_{2}, dS

_{2}or . AdS

_{ 2}and dS

_{2}give symmetry, whereas gives Poincaré symmetry. The dS

_{ 2}and 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 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 spacetime with a degenerate horizon invariant under a
symmetry as in Theorem 3.4 and 3.5. The near-horizon geometry is static if it
is either a warped product of AdS_{2} and , or it is a warped product of locally AdS_{3} and a
-manifold.

The static conditions (49) for Eq. (58) occur if and only if: (i) for all , or (ii) and with . Case (i) gives a warped product of a 2D maximally-symmetric space and as in Theorem (3.3). For case (ii) one can introduce coordinates where , not necessarily periodic, so that and

The metric in the square brackets is locally isometric to AdS_{3}. If is a periodic coordinate the horizon topology is , where is some -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 and uncharged scalars with action

where is the 2-derivative action above, is a coupling constant, and 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 with near-horizon symmetry, and the near-horizon solution is analytic in . Then the near-horizon solution has symmetry to all orders in .

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 ).

So far, the results described all assume commuting rotational Killing fields. For this is the same number as the rank of the rotation group , so the above results are applicable to asymptotically-flat or globally-AdS black holes. For the rank of this rotation group is , which is smaller than , 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 commuting rotational isometries are assumed, in particular the case with 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 , whose generic orbit on is a cohomogeneity-1 -bundle over a -invariant homogeneous space . Furthermore, assume does not admit any -invariant one-forms. If Einstein’s equations hold, then the near-horizon geometry possesses a isometry group, where is either or the 2D Poincaré group. Furthermore, if 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 , with and , 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 dimensions, or all but one set equal in 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 . 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 , where , such that the surfaces orthogonal to the surfaces of transitivity are simply connected. Then the near-horizon geometry has an isometry group , where is either 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.

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

where as in the above cases we have rescaled the affine parameter . For the case, orthogonal transitivity follows from Einstein’s equations [72], which provides another proof of Theorem 3.4 and 3.5. For this result guarantees an AdS_{2}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.