on the topology of black hole event horizons in higher...

Introduction and MotivationsHawking’s Theorem RevisitedFive-Dimensional Black HolesSix-Dimensional Black Holes

Comments on Higher DimensionsOpen Questions and Outlook

On The Topology of Black Hole Event Horizonsin Higher Dimensions

Yaron Oz (Tel-Aviv University)

General Relativity In Higher Dimensions,Hebrew University, February 2007

Yaron Oz On The Topology of Black Hole Event Horizons in Higher Dimensions



Outline

1 Introduction and Motivations

2 Hawking’s Theorem Revisited

3 Five-Dimensional Black Holes

4 Six-Dimensional Black Holes

5 Comments on Higher Dimensions

6 Open Questions and Outlook




Introduction and Motivations

In four dimensions the topology of the event horizon of anasymptotically flat stationary black hole is uniquelydetermined to be the two-sphere S2 (Hawking).Hawking’s theorem: The integrated Ricci scalar curvatureR with respect to the induced metric h on the event horizonMH , is positive ∫

MH

RdS > 0 (1)

This condition applied to two-dimensional manifoldsdetermines uniquely the topology.





Another way to determine the topology of the event horizonis via the so called topological censorship (Friedman,Schleich, Witt:1993): the region of space-time outside theblack hole should have a simple topology.Mathematically it requires the horizon to be cobordant to asphere via a simply connected oriented cobordism.For a two-dimensional horizon it means that there is asimply connected three-dimensional oriented manifoldwhose boundary is the oriented disjoint union of thehorizon and the two-sphere.Topological censorship implies that the topology of theevent horizon is that of the two-sphere S2 (Chrusciel,Wald:1994).





The classification of the topology of the event horizons inhigher dimensions is more complicated.For instance, for five-dimensional asymptotically flatstationary black holes, in addition to the known S3 topologyof event horizons, stationary black hole solutions withevent horizons of S2 × S1 topology (Black Rings) havebeen constructed (Emparan, Reall:2001).





We will consider the topology of event horizons indimensions higher than four.First, we reconsider Hawking’s theorem and show that itcontinues to hold in higher dimensions.Using this and Thurston’s geometric types classification ofthree-manifolds, we find that the only possible geometrictypes of event horizons in five dimensions are S3 andS2 × S1.





In six dimensions we use the requirement that the horizonis cobordant to a four-sphere, Friedman’s classification oftopological four-manifolds and Donaldson’s results onsmooth four-manifolds, and show that simply connectedevent horizons are homeomorphic to S4 or S2 × S2.We find allowed non-simply connected event horizonsS3 × S1 and S2 × Σg (Σg is a genus g Riemann surface),and event horizons with finite non-abelian first homotopygroup, whose universal cover is S4.We will make some comments on the classification indimensions higher than six.





This talk is based on the paper: ”On the Topology of BlackHole Event Horizons in Higher Dimensions”, with C.Helfgott and Y. Yanay, JHEP 0602 (2006) 025.





Galloway and Schoen (2006) have shown that genericallythe event horizon admits a metric of positive scalarcurvature (positive Yamabe type).This, together with the fact that closed orientable3-manifolds of positive Yamabe type are connected sumsof S3 and S2 × S1, specifies the topologies of eventhorizons in five dimensions.The theorem does not hold for Ricci flat horizons, and forflat horizons in four and five dimensions.





Fields Medalists that will cast in this talk:

Thom (1958): CobordismSmale (1966): h-cobordismThurston (1982): Topology of 3-manifoldsDonaldson (1986): Smooth 4-manifoldsFreedman (1986): Topological 4-manifoldsPerelman (2006, declined): Geometrization conjecture




Hawking’s Theorem Revisited

Let us reconsider Hawking’s theorem. We will find that alsofor asymptotically flat stationary black holes in dimensionshigher than four the integrated Ricci scalar curvature Rwith respect to the induced metric h on the event horizonMh, is positive.Idea of the proof: Consider a closed 2-surface Σ, which isan intersection of the event horizon and a space-likehypersurface. Hawking shows that if Σ is not S2, then itcan be deformed to an outer trapped surface, theexpansion θ < 0 outside the black hole.However, any closed trapped surface must lie inside theblack hole.





Consider a stationary n-dimensional space-time M with ametric g. M is required to be regular predictable, i.e. itsfuture is predictable from a Cauchy surface.Denote by (Y1, Y2) two future-directed null vectorsorthogonal to MH , normalized as

Y a1 Y2a = −1 (2)

We take Y1 to be the future-directed null vector pointing outof the horizon, and Y2 to be the vector pointing into thehorizon.We now deform the event horizon by moving each point onit a parameter distance ω along an orthogonal nullgeodesic with tangent vector Y a

2 .





Following the same steps as in the original proof, onederives the equation

d θ

dω= pb;d hbd −RacY a

1 Y c2 +RadcbY d

1 Y c2 Y a

2 Y b1 +

papa − Y a1 ;c hc

dY d2 ;bhb

a (3)

where θ ≡ Y a1 ;bhb

a and pa = −habY2c;bY c1 .

The (positive definite) induced metric on the horizon reads

hab = gab + Y1aY2b + Y2aY1b (4)





Using a rescaling:

Y1 → Y ′1 = eyY1, Y2 → Y ′

2 = e−yY2 (5)

pa → p′a = pa + haby;b, we get

d θ′

dω′

∣∣∣∣∣ω=0

= pb;d hbd + y;bd hbd −RacY a1 Y c

2 +

RadcbY d1 Y c

2 Y a2 Y b

1 + p′ap′a (6)





Using the Gauss-Godazzi equations, evaluated on thehorizon yields

R = Rijkl hik hjl = R− 2RijklY i1Y j

2Y k1 Y l

2 + 4RijY i1Y j

2 (7)

where R is the Ricci scalar associated to the inducedmetric, and the unhatted quantities are the curvaturetensors of the full metric.Also, we use Einstein’s equations and the dominant energycondition:

12R+RabY a

1 Y b2 = 8πTabY a

1 Y b2 ≥ 0 (8)





We get:

d θ′

dω′

∣∣∣∣∣ω=0

=

∫MH

(−12

R + 8πTabY a1 Y b

2 )dS + p′ap′a (9)





Suppose d θ′

dω′

∣∣∣ω=0

is positive. We then take ω′ to be a smallnegative value, thereby looking at a surface slightly outsidethe horizon on which θ′ is now negative.Such a surface is an outer trapped surface, which isforbidden in a stationary regular predictable space-timesatisfying the dominant energy condition.




Five-Dimensional Black Holes

The horizons are three-manifolds. Thurston introducedeight geometric types in the classification ofthree-manifolds.There are eight basic homogeneous geometries, up to anequivalence relation, called geometric types. Out of thesetypes one constructs geometric structures, which arespaces that admit a complete locally homogeneous metric.Geometrization conjecture, proven by Perelman: Anycompact and oriented three-manifold has a decompositionas a connected sum of these basic geometric types.





Consider an orientable, connected, complete and simplyconnected Riemannian three-manifold X which ishomogeneous with respect to an orientation preservinggroup of isometries G. The eight geometric types classify(X , G). The equivalence relation (X , G) ∼ (X ′, G′) holdswhen there is a diffeomorphism of X onto X ′, which takesthe action of G onto the action of G′.Out of these types one constructs spaces (geometricstructures) M ' X/Γ where Γ is a subgroup of G. Here theaction of Γ is discontinuous, discrete and free. M is locallyhomogeneous with respect to the metric on (X , G). It isisometric to the quotient of X by Γ.





The first three types in the classification are based on thethree constant curvature spaces, the 3-sphere S3

(Spherical geometry), which has a positive scalar curvatureand isometry group G = SO(4), the Euclidean space R3

(Euclidean geometry) with R = 0 and isometry groupG = R3 × SO(3) and the hyperbolic space H3 (Hyperbolicgeometry) with R < 0 and isometry group G = PSL(2, C).Of these three geometric types, only the S3 type satisfiesthe curvature condition and is allowed as an horizon.





The next two types are based on S2 × R and H2 × R. Ofthese two geometric types, only the S2 × R type satisfiesthe curvature condition and is allowed as an horizon. Inthis allowed class we have S2 × S1.





The last three geometric types are Nil geometry, Solgeometry and the universal cover of the Lie groupSL(2, R).





The Nil geometry: this is the geometry of thethree-dimensional Lie group of 3× 3 real upper triangularmatrices of the form 1 x z

0 1 y0 0 1

(10)

under matrix multiplication (Heisenberg group).





We can think about Nil as (x , y , z) ∈ R3 with themultiplication

(x , y , z) · (x ′, y ′, z ′) = (x + x ′, y + y ′, z + z ′ + xy ′) (11)

The Nil metric (the left-invariant metric on R3) is given by

ds2 = dx2 + dy2 + (dz − xdy)2 (12)

and has R = −12 . The Nil geometry type does not satisfy

the curvature condition and is not allowed as an horizon.





The Sol geometry: this is the geometry of the Lie groupobtained by the semidirect product of R with R2. We canthink about Sol as (x , y , z) ∈ R3 with the multiplication

(x , y , z) · (x ′, y ′, z ′) = (x + e−zx ′, y + ezy ′, z + z ′) (13)

The left-invariant Sol metric is given by

ds2 = e2zdx2 + e−2zdy2 + dz2 (14)

and has R = −2. The Sol geometry type does not satisfythe curvature condition and is not allowed as an horizon.





The ˜Sl(2, R) geometry: this is the geometry of theuniversal covering of the three-dimensional Lie group of all2× 2 real matrices with determinant one Sl(2, R). The˜Sl(2, R) geometry type has negative R and is not allowed

as an horizon.





Summary: We find that only two geometric types areallowed horizons in five dimensions: the S3 geometric typeand the S2 × R geometric type.




Six-Dimensional Black Holes

Consider now six-dimensional stationary black holes inasymptotically flat space-times.We will use topological censorship together withFriedman’s classification of topological four-manifolds andDonaldson’s results on smooth four-manifolds, in order toclassify possible event horizons.





First we note that oriented cobordism from the eventhorizon to a four-sphere S4 exists if and only if the horizonmanifold Mh has vanishing Pontrjagin and Steifel-Whitneynumbers.In general, two smooth closed n-dimensional manifolds arecobordant iff all their corresponding Steifel-Whitneynumbers are equal (Thom).If in addition we require the cobordism to be oriented then(when n = 4k ) their corresponding Pontrjagin numbers areequal.





We start by considering simply connected event horizons,that is

Π1(Mh) = 0 (15)

This, in particular, implies that the cohomology groupsH1(MH) (which is the abelianization of the first homotopygroup), and H3(MH) (by Hodge duality) vanish.





One can use the second cohomology group H2(MH) todefine an intersection form

Q(α, β) =

∫MH

α ∧ β (16)

where α, β ∈ H2(MH). Q is the basic topological invariantof a compact four-manifold.Q is symmetric, non-degenerate withrank(Q) = b2 ≡ dim H2(MH), and can be diagonalizedover R.





Since the four-sphere S4 has zero second cohomologygroup, all its intersection numbers vanish and

Q(S4) = (0) (17)





The signature σ of a four-manifold is defined by thedifference of positive and negative eigenvalues of Q.It can be expressed using the Hirzebruch signaturetheorem as

σ(MH) =13

∫MH

p1 (18)

where p1 is the first Pontrjagin class

p1 = − 18π2 TrR ∧ R (19)





Since p1(S4) vanishes, topological censorship implies thenthat the signature of Q(MH) vanishes

σ(Q(MH)) = 0 (20)

Mathematically, the signature is cobordant invariant.





Consider next the Stiefel-Whitney classes

ωi ∈ H i(MH , Z2) (21)

For a compact, simply-connected orientable manifoldω1 = ω3 = 0.ω2 is the obstruction to a spin-structure.





Although ω2(S4) = 0, oriented cobordism does not implythat the second Stiefel-Whitney class of MH is zero. Inother words, MH is not necessarily a spin manifold.The reason is that we require the Stiefel-Whitney numbersand not the classes to vanish.Here it means that

〈ω22, µ〉 = 0 (22)

where µ ∈ H4(M, Z2) is the so called fundamentalhomology class.





The intersection form Q is actually defined on the latticeH2(MH , Z ) and is a unimodular (det(Q) = ±1) symmetricbilinear form over the integers.One says that Q is of even type if

Q(α, α) ∈ 2Z (23)

for all α ∈ H2(MH , Z ).





Fact: If ω2 = 0 then Q is even.Summary: event horizons which are spin manifolds arecharacterized as topological four-manifolds by anintersection form Q(MH), which has vanishing signatureand is of even type.





When the event horizons are not spin manifolds, ω2 6= 0and Q(MH) is odd.In this case there are two topological four-manifolds MHfor a given intersection form.They are distinguished by the Kirby-Siebenmann invariantα(MH), which is zero if MH × S1 is smooth and one ifMH × S1 is not smooth.





In the classification of possible intersection forms of MHwe distinguish two cases:

(i) Q(MH) is positive definite.

(ii) Q(MH) is indefinite.





Consider first the case when Q(MH) is positive definite.Theorem (Donaldson): If Q(MH) is even then MH ishomeomorphic to the four-sphere S4.If Q(MH) is odd then MH is homeomorphic to aconnected sum of positively oriented CP2’s.H2(CP2, Z ) has one generator and Q(CP2) = (1). Thus,the signature of the connected sum in nonzero, and this isnot an allowed horizon.





If Q(MH) is indefinite, then if it is even it can be written as

Q(MH) = aE8 + bH, a, b ∈ Z b 6= 0 (24)

where E8 is the Cartan matrix of the Lie algebra E8 and

H =

(0 11 0

)(25)

is the intersection form of S2 × S2.H2(S2 × S2, Z ) is generated by a = S2 × pt andb = pt × S2. The intersection matrix Q(S2 × S2) in thebasis (a, b) is H.





Since the signature σ(E8) = 8 and we require thatσ(Q(MH)) = 0 this implies that a = 0.The basic case is b = 1 and MH is S2 × S2. If we takeb > 1 we will get a connected sum of S2 × S2.





We should note, however, that by using connected sumsthere is a way to construct other event horizons whoseintersection form has vanishing signature and is of eventype.





Consider for instance a K 3. Its intersection form isQ(K 3) = −2E8 + 3H. Since its signature is nonzero, K 3 isnot an allowed horizon.However, if we take a connected sum of the K 3 and −K 3,where −K 3 has an opposite orientation we get an evenintersection form with vanishing signature, sinceQ(−K 3) = −Q(K 3).





Theorem (Freedman):If Q(MH) is indefinite and odd then MH is a connectedsum of ±CP2’s, where −CP2 has the opposite orientationof CP2 and Q(−CP2) = −Q(CP2) = (−1).





Summary: We found that if the horizon is simply connectedthen it is homeomorphic, up to a connected sum operation,to S4 or to S2 × S2. Note that both S4 and S2 × S2 arespin manifolds.





Consider next the case when MH is not simply connected.When Q(MH) is positive definite, one can relax thecondition that Π1(MH) = 0 by requiring only that there areno non-trivial homomorphisms of Π1(MH) into SU(2). Thisimplies that every flat SU(2) bundle over MH is trivial andthat H1(MH) being the abelianization of Π1(MH) vanishes.This allows Π1(MH) to be any finite simple nonabeliangroup. With this relaxed condition we get four-manifoldevent horizons, whose universal cover is S4.





There are three other non-simply connected cases that wewould like to explore.Consider T 4. Its intersection form is Q(T 4) = 3H and it isnot excluded by the previous discussion from being anevent horizon.However, it does not satisfy our curvature condition (atleast with the flat metric).





Next consider S3 × S1. It is not ruled out by our analysissince it has vanishing Pontrjagin and Steifel-Whitneynumbers. Also, it satisfies the curvature condition.The last examples are Σg × Σh, where Σg and Σh areRiemann surfaces of genus g and h respectively, and wehave assumed that the induced metric decomposes as adirect (unwarped) sum.





Assuming a product metric, the curvature condition reads

(g − 1)Vol(Σh) + (h − 1)Vol(Σg) < 0 (26)

This can be satisfied by Σh = S2 and

g < 1 +Vol(Σg)

Vol(S2)(27)

We cannot exclude, a priori, the possibility that the ratio ofvolumes can be as large as we want and therefore allgenera g are allowed. We encountered above the caseg = 0, namely the horizon S2 × S2. The intersection formQ(S2 × Σg) = H and it is not ruled out by topologicalcensorship.





Summary: We found that if the event horizon has vanishingfirst homotopy group then it is homeomorphic to S4 orS2 × S2.If the event horizon has finite simple nonabelian firsthomotopy group and positive intersection form, then itsuniversal cover is homeomorphic to S4.We found other allowed non-simply connected casesS3 × S1 with first homotopy group Z and S2 × Σg with firsthomotopy group1 Π1(Σg).

1Π1(Σg) is generated by a1, b1, ..., ag , bg with the relationa1b1a−1

1 b−11 ...agbga−1

g b−1g = 1.




Comments on Higher Dimensions

The event horizons MH are now closed differentiablen-manifolds with dimension n higher than four, cobordantto the n-sphere Sn.An important concept in differential topology is that ofh-cobordism. Two cobordant n-manifolds M1, M2 areh-cobordant if their inclusion map in the n + 1-dimensionalmanifold W are homotopy equivalent:M1 → W ∼ M2 → W .





The h-cobordism theorem (Smale) implies that if thehorizon manifold MH is h-cobordant to Sn then it isdiffeomorphic to Sn.Note, however, that h-cobordism is a stronger requirementthan what is implied by topological censorship.





Let us introduce the concept of spin cobordism.Mathematically, it requires in our context the existence ofan (n + 1)-dimensional compact spin manifold, whoseboundary is the oriented disjoint union of Sn and MH . Inparticular, MH is a spin manifold whose spin structure isinduced from that of the (n + 1)-dimensional manifold.





The concept of spin cobordism may be relevant, since weare mainly interested in higher-dimensional black holessolutions to supergravity equations as the low energyeffective description of the superstring equations.Therefore, we would like the geometry to accommodatefermions.Note, that in the classification of the previous section, spincobordism would have implied that the intersection form isof even type.





n = 5 : For five-dimensional manifolds with vanishingsecond Steifel-Whitney class , there exists a classificationof all possible closed simply connected manifolds. Themanifolds are in 1-1 correspondence with finitely generatedabelian groups.n = 6 : Six-dimensional closed manifolds with vanishingfirst and second homotopy groups Π1 = 0 and Π2 = 0(2-connected) are homeomorphic to S6 or connected sumof copies of S3 × S3.





n = 2k : There are general results which enumerate the(k − 1)-connected 2k -manifolds.n ≥ 5: If MH is n-dimensional compact, simply connectedand spin cobordant to Sn, it is obtained from Sn by doingsurgery on spheres of codimension greater than two.




Open Questions and Outlook

In order to have a complete classification of event horizons inhigher dimensions, additional physical requirements areneeded. This can come, for instance, from:

Refinement of topological censorshipConsiderations of stabilitySupersymmetryAnd, how to find solutions in six dimensions with S2 × S2

or S3 × S1 horizon topology ?


on the topology of black hole event horizons in higher...

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