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 [179Jump To The Next Citation Point, 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

1 ∇L (L ⋅ N ) = L ⋅ (∇LN ) = L ⋅ (∇N L ) =-∇N (L ⋅ L) = 0 (4 ) 2
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

( a 1 ) a b g = 2dv dr + rhadx + 2rfdv + γabdx dx , (5 )
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. (5View Equation) 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

( a 1 2 ) a b g = 2dv dr + rha (r,x )dx + 2r F (r,x)dv + γab(r, x)dx dx . (6 )

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. (6View Equation) 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

( ) gNH = 2dv dr + rha(x )dxa + 1r2F (x)dv + γab(x)dxadxb, (7 ) 2
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 (7View Equation), denoted by (eA), where A = (+, − ,a ), a = 1,...,n and

e+ = dv, e− = dr + rha ˆea + 1r2Fdv, ea = ˆea, (8 ) 2
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
1 2 ˆ e+ = ∂v − 2Fr ∂r, e− = ∂r, ea = ∂a − rha∂r, (9 )
where ˆ∂a denote the dual vectors to ˆea. The connection 1-forms satisfy deA = − ωA ∧ eB B and are given by
ω+− = rF e+ + 1haea, ω+a = 1r2(∂ˆaF − F ha)e+ − 1hae − + rˆ∇ [ahb]eb, 1 + 2 2 + 2 ω −a = − 2hae , ωab = ˆωab − r ˆ∇[ahb]e , (10 )
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:
S++ = − 12(∇ˆa − 2ha)S+a, (13 ) ˆ b 1 c b S+a = − ∇ [Rba − 2γba(R c + 2R+ − )] + h Rba − haR+ − , (14 )
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. (7View Equation) 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:

1- ρσ R μν = Λgμν + Tμν − n g Tρσgμν, (15 )
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. (6View Equation) 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

( ) T = 2dv T+ − dr + r(βa + T+ − ha)dxa + 1r2(α + T+− F)dv + Tabdxadxb, (16 ) 2
where T+− ,α are functions on H and βa is a 1-form on H. Working in the vielbein frame (8View Equation), 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:
1 Rab = 2hahb − ∇ (ahb) + Λγab + Pab, (17 ) F = 1h2 − 1∇aha + Λ − E, (18 ) 2 2
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. (17View Equation), (18View Equation) 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:

a a b b α = − 12(∇ − 2h )βa, βa = − (∇ − h )Tab − haT+ − , (21 )
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. (12View Equation). The first equation in (21View Equation) and the identity (13View Equation) imply that the ++ equation is satisfied as a consequence of the +a equation. Finally, substituting Eqs. (17View Equation) and (18View Equation) into the identity (14View Equation), and using the second equation in (21View Equation), implies the +a equation. Alternatively, a tedious calculation shows that the +a equation follows from Eqs. (17View Equation) and (18View Equation) using the contracted Bianchi identity for Eq. (17View Equation), together with the second equation in (21View Equation).

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

( ρ 2 ) Tμν = 2 ℱμρℱ ν − 14ℱ gμν , (22 )
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
1 a b ℱNH = d (rΔ (x)dv) + 2Bab(x )dx ∧ dx , (23 )
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

( ) ---2---- ρ1...ρp−1 -1- 2 Tμν = (p − 1)! ℱμρ1...ρp−1ℱ ν − 2p ℱ gμν , (26 )
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
ℱNH = d (Y ∧ rdv) + X, (27 )
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. (17View Equation) and (18View Equation) 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 (17View Equation) and (18View Equation) remain valid in the more general context of extremal isolated horizons [163Jump To The Next Citation Point, 209, 28] and Kundt metrics [144Jump To The Next Citation Point].

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

E ≥ Λ. (28 )
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
ab Pγ ≡ Pabγ ≥ 0. (29 )
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. (28View Equation) and (29View Equation). 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 (17View Equation), 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 [123Jump To The Next Citation Point, 79Jump To The Next Citation Point, 154Jump To The Next Citation Point].

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

∫ AH = 𝜖γ, (30 ) H
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

∫ -1-- J = 16π ⋆dm. (31 ) S∞
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:
∫ --1- J = 16 π H(h ⋅ m )𝜖γ. (32 )
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 (23View Equation) one can show that, in the gauge ℒmA = 0,
1 ∫ J = ---- (h ⋅ m + 4(m ⋅ A )Δ )𝜖γ, (33 ) 16 π H
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

d ⋆ ℱ + 2√ξ-ℱ ∧ ℱ = 0, (34 ) 3
where ξ is the CS coupling constant. The angular momentum in this case can also be written purely as an integral over H:
1 ∫ J = ---- (h ⋅ m + 4(m ⋅ A )Δ ) 𝜖γ +-1√6-ξ(m ⋅ A )A ∧ B. (35 ) 16 π H 3 3
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:

1 ∫ Qe = --- ⋆ℱ . (36 ) 4 π S∞
By applying Stokes’ Theorem to a spacelike hypersurface Σ as above, and using the Maxwell equation, one easily finds
1 ∫ Qe = --- Δ ðœ–γ. (37 ) 4π H
For D = 5 Einstein–Maxwell–CS theory one instead gets
∫ -1- √2- Qe = 4π H Δ ðœ–γ + 3ξA ∧ B. (38 )
For D = 4 one also has a conserved magnetic charge ∫ Qm = 1- ℱ 4π S∞. Using the Bianchi identity this can be written as
∫ -1- Qm = 4π B. (39 ) H
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
∫ ∫ -1- 1-- 𝒟 = 2π 2 ℱ = 2π 2 B, (40 ) S S
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 (35View Equation) and (38View Equation), see [123, 155Jump To The Next Citation Point].

  Go to previous page Scroll to top Go to next page