Topological censorship and higher genus black holes

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Topological censorship and higher genus black holes

Transcript Of Topological censorship and higher genus black holes

PHYSICAL REVIEW D, VOLUME 60, 104039

Topological censorship and higher genus black holes

G. J. Galloway,1,* K. Schleich,2,† D. M. Witt,2,‡ and E. Woolgar3,§
1Department of Mathematics and Computer Science, University of Miami, Coral Gables, Florida 33124 2Department of Physics and Astronomy, University of British Columbia, 6224 Agriculture Road, Vancouver, BC, Canada V6T 1Z1 3Department of Mathematical Sciences and Theoretical Physics Institute, University of Alberta, Edmonton, AB, Canada T6G 2G1
͑Received 22 February 1999; published 27 October 1999͒
Motivated by recent interest in black holes whose asymptotic geometry approaches that of anti–de Sitter spacetime, we give a proof of topological censorship applicable to spacetimes with such asymptotic behavior. Employing a useful rephrasing of topological censorship as a property of homotopies of arbitrary loops, we then explore the consequences of topological censorship for the horizon topology of black holes. We find that the genera of horizons are controled by the genus of the space at infinity. Our results make it clear that there is no conflict between topological censorship and the nonspherical horizon topologies of locally anti–de Sitter black holes. More specifically, let D be the domain of outer communications of a boundary at infinity ‘‘scri.’’ We show that the principle of topological censorship ͑PTC͒, which is that every causal curve in D having end points on scri can be deformed to scri, holds under reasonable conditions for timelike scri, as it is known to do for a simply connected null scri. We then show that the PTC implies that the fundamental group of scri maps, via inclusion, onto the fundamental group of D: i.e., every loop in D is homotopic to a loop in scri. We use this to determine the integral homology of preferred spacelike hypersurfaces ͑Cauchy surfaces or analogues thereof͒ in the domain of outer communications of any four-dimensional spacetime obeying the PTC. From this, we establish that the sum of the genera of the cross sections in which such a hypersurface meets black hole horizons is bounded above by the genus of the cut of infinity defined by the hypersurface. Our results generalize familiar theorems valid for asymptotically flat spacetimes requiring simple connectivity of the domain of outer communications and spherical topology for stationary and evolving black holes. ͓S0556-2821͑99͒08218-1͔

PACS number͑s͒: 04.70.Bw, 04.20.Gz

I. INTRODUCTION
It is generally a matter of course that the gross features of shape, i.e., the topology, of composite objects, from molecules to stars, are determined by their internal structure. Yet black holes are an exception. A black hole has little internal structure, but the topology of its horizon is nonetheless strongly constrained, seemingly by the external structure of spacetime. This was made apparent in the early work of Hawking ͓1͔, who established via a beautiful variational argument the spherical topology of stationary horizons. Hawking’s proof was predicated on the global causal theoretic result that no outer trapped surfaces can exist outside the black hole region, unless energy conditions or cosmic censorship are violated. As argued by Hawking in ͓1͔, the possibility of toroidal topology, which arises as a borderline case in his argument, can be eliminated by consideration of a certain characteristic initial value problem and the assumption of analyticity; see ͓2͔ for further discussion of this issue.
In recent years an entirely different approach to the study of black hole topology has developed, based on the notion of topological censorship. In 1994, Chrus´ciel and Wald ͓3͔, improving in the stationary setting the result on black hole topology considered in ͓2͔ and ͓4͔, were able to remove the analyticity assumption in Hawking’s theorem by making use
*Email address: [email protected] †Email address: [email protected] ‡Email address: [email protected] §Email address: [email protected]

of the active topological censorship theorem of Friedman, Schleich, and Witt ͓5͔ ͑FSW͒. This latter result states that in a globally hyperbolic, asymptotically flat ͑AF͒ spacetime obeying an averaged null energy condition ͑ultimately, in the modified form used below—see ͓6͔͒, every causal curve beginning and ending on the boundary at infinity could be ho-
motopically deformed to that boundary. When topological censorship holds, the domain of outer communications ͑DOC—the region exterior to black and white holes͒ of an AF spacetime must be simply connected ͓6͔.
Jacobson and Venkataramani ͓7͔, also using the topological censorship theorem of FSW, were able to extend the result of Chrus´ciel and Wald on black hole topology beyond the stationary case. The principle behind their arguments was that any horizon topology other than spherical would allow certain causal curves outside the horizon to link with it, and so such curves would not be deformable to infinity, which would contradict FSW.
In the early 1990s, new solutions with nonspherical black hole horizons were discovered in locally anti–de Sitter ͑AdS͒ spacetimes ͓8–13͔; for a recent review, see ͓14͔. The original topological censorship theorem did not apply to these spacetimes since they were not AF and not globally hyperbolic. However, two improvements to the proof of topological censorship indicated that these differences ought not matter. Galloway ͓15͔ was able to produce a ‘‘finite infinity’’ version of topological censorship that replaced the usual asymptotic conditions on the geometry with a mild geometrical condition on a finitely distant boundary, and Galloway and Woolgar ͓16͔ were able to replace the assumption of global hyperbolicity by weak cosmic censorship.

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PHYSICAL REVIEW D 60 104039

Moreover, it was soon observed ͓17͔ that topological censorship in the sense of FSW held true for each of the newly discovered black hole constructions in locally AdS spacetime, although no general proof was known in this setting and obviously the aforementioned corollary implying spherical horizon topology could not hold. This is not paradoxical—the topology of the locally AdS black hole spacetimes is such that no causal curve links with a nonspherical horizon in such a way as to preclude a homotopy deforming that curve to infinity.1 In fact, we will see below that the corollary implying the spherical topology of black holes in AF spacetime is merely a special case of a more general corollary of topological censorship which gives a relationship between the topology of the black hole horizons and the ‘‘topology of infinity.’’ In this sense, the topology of the black hole horizons is governed by a structure that is as ‘‘external’’ as possible, being entirely at infinity.
In this paper, we will consider spacetimes that obey the Principle of Topological Censorship ͑PTC͒. Let M be a spacetime with metric gab . Suppose this spacetime can be conformally included into a spacetime with boundary MЈϭMഫI, with metric gaЈb whose restriction to M obeys gaЈbϭ⍀2gab , and where I is the ⍀ϭ0 surface, which defines I as the boundary at infinity. For a connected boundary component I,2 the domain of outer communication D is defined by

DªIϩ͑ I͒പIϪ͑ I͒.

͑1͒

The PTC is the following condition on D: Principle of Topological Censorship. Every causal curve
whose initial and final end points belong to I is fixed end point homotopic to a curve on I.3
The PTC has already been established for general, physically reasonable AF spacetime; cf. ͓5͔, ͓16͔. In the next section we present a proof of the PTC in a setting that includes many asymptotically locally anti–de Sitter black hole spacetimes. This generalization exploits the fact that PTC proofs generally follow from a condition on double-null components of the Ricci tensor. These components can be related to the double-null components of the stress-energy tensor through the double-null components of the Einstein equations. This relation involves no trace terms and so clearly is insensitive to the cosmological constant. This corrects the impression that the PTC is invalid in the presence of a negative cosmological constant ͓13,18͔, an impression that, if it were true, would imply that the PTC in this case would im-

1That topological censorship proofs should generalize to space-
times having the asymptotic structure of these new black holes was first suggested in ͓7͔.
2For the purpose of discussion, in the AF case we assume for simplicity that i0 is included in I.
3FSW’s original statement of topological censorship required every causal curve beginning and ending on I to be homotopic to one
in a ‘‘simply connected neighborhood of infinity.’’ Our phrasing above accommodates spacetimes for which I has no simply con-
nected neighborhoods.

pose no constraints at all on the topology of black hole horizons. Hawking’s gϭ0 restriction ͓1,19,20͔ on the horizon topology, by way of contrast, does assume non-negative energy density on the horizon and so is not in force in the presence of a negative cosmological constant in vacuum, but as we will see, the PTC remains valid and imposes constraints on the topology.
In Sec. III, we will prove that the mapping from the fundamental group of I to that of the domain of outer communications is a surjection for spacetimes satisfying the PTC. This theorem generalizes previous results on the DOC being simply connected in asymptotically flat spacetimes. This characterization allows us to shift attention from causal curves to arbitrary loops in further study of the consequences of the PTC.
In Sec. IV, by considering loops restricted to certain spacelike surfaces and using arguments from algebraic topology, we give a very direct derivation of the topology of black hole horizons and the integral homology of hypersurfaces in exterior regions of four-dimensional spacetimes. More precisely, we consider the topology of the closure of the Cauchy surfaces or analogues thereof for the DOC whose intersections with the horizons are closed two-manifolds ͑‘‘good cuts’’͒. We find that
k
͚ giрgo ,
iϭ1
where the gi are the genera of the cuts of the black hole horizons and go is the genus of a cut of I by the surface. Thus the topology of black hole horizons is constrained by the topology at infinity. This result also pertains to any subdomain of the DOC that lies to the future of a cut of I and whose Cauchy surfaces—or analogues thereof—meet the horizons at closed two-manifolds. Therefore, it applies to the topology of the black hole horizons in the presence of black hole formation or evolution at times for which the appropriate subdomain and surfaces can be found. Finally, we demonstrate that the integral homology of these surfaces is torsion free and consequently completely determined by the Betti numbers. Furthermore, these are completely fixed in terms of the genera of the boundary.
Section V contains a discussion concerning these results and their applicability to the case of nonstationary black holes.
II. VALIDITY OF THE PTC
The aim of this section is to present a version of the PTC applicable to spacetimes which are asymptotically locally anti–de Sitter. Hence we consider a spacetime M, with metric gab , which can be conformally included into a spacetime with boundary MЈϭMഫI, with metric gaЈb , such that ‫ץ‬MЈϭI is timelike ͑i.e., is a Lorentzian hypersurface in the induced metric͒ and MϭMЈ‫گ‬I. We permit I to have multiple components. With regard to the conformal factor ⍀෈C1(MЈ), we make the standard assumptions that ͑a͒ ⍀Ͼ0 and gaЈbϭ⍀2gab on M, and ͑b͒ ⍀ϭ0 and d⍀ 0 pointwise on I.

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Just as in the case of spacetimes without a boundary, we say that a spacetime with boundary MЈ is globally hyperbolic provided MЈ is strongly causal and the sets Jϩ( p,MЈ)പJϪ(q,MЈ) are compact for all p,q෈MЈ. Note that when I is timelike, as in the situation considered here, M can never be globally hyperbolic. However, in many ex-
amples of interest, such as anti–de Sitter space and the do-
main of outer communications of various locally AdS spaces, MЈ is.
For later convenience, we define a Cauchy surface for MЈ to be a subset VЈʚMЈ which is met once and only once by each inextendible causal curve in MЈ. Then VЈ will be a spacelike hypersurface which, as a manifold with boundary, has a boundary on I. It can be shown, as in the standard case, that a spacetime with timelike boundary DЈ which is globally hyperbolic admits a Cauchy surface VЈ and is homeomorphic to RϫVЈ. ͑This can be shown by directly modifying the proof of proposition 6.6.8 in ͓19͔. Many of the locally anti–de Sitter and related models ͓8–11,13,14͔ which have
been constructed have DOCs which admit Cauchy surfaces
of this sort.
The proof of the PTC is a consequence of the following
basic result.

Theorem 2.1. Let MʚMЈ be as described above, and as-
sume the following conditions hold. ͑i͒ MЈϭMഫI is globally hyperbolic. ͑ii͒ There is a component I0 of I which admits a compact
spacelike cut. ͑iii͒ For each point p in M near I0 and any future complete null geodesic s→␩(s) in M starting at p, ͐ 0ϱ Ric( ␩ Ј , ␩ Ј ) d s у 0. Then I0 cannot communicate with any other component of I, i.e., Jϩ(I0)പ(I‫گ‬I0)ϭл.

Condition ͑iii͒ is a modified form of the average null en-

ergy condition ͑ANEC͒. This term usually refers to a condi-

tion of the same form as ͑iii͒ except that the integral is taken

over geodesics complete to both past and future. Note that if

one assumes that the Einstein equations with cosmological

constant

R

a

b

Ϫ

1 2

R

g

a

b

ϩ



g

a

b

ϭ

8



T

a

b

hold,

then

for

any

null vector X, Ric(X,X)ϭRabXaXbϭ8␲TabXaXb. Then the

integrand Ric(␩Ј,␩Ј) in ͑iii͒ could be replaced by

T(␩Ј,␩Ј). Clearly, the presence and sign of the cosmologi-

cal constant are irrelevant to whether or not a spacetime sat-

isfying the Einstein equations will satisfy condition ͑iii͒. For the next theorem, if ‫ץ‬MЈ is not connected, let I de-
note a single component of ‫ץ‬MЈ. Let DϭIϩ(I)പIϪ(I) be the domain of outer communications of M with respect to I. Assume that D does not meet any other components of ‫ץ‬MЈ. Note that if the conditions of theorem 2.1 hold, this

latter assumption is automatically satisfied. Using in addition the fact that I is timelike, it follows that D is connected and the closure of D in MЈ contains I. Then DЈªDഫI is a connected spacetime with boundary, with ‫ץ‬DЈϭI and Dϭ DЈ ‫ گ‬I.

We now state the following topological censorship theo-

rem, applicable to asymptotically locally AdS spacetimes.

Theorem 2.2. Let D be the domain of outer communications with respect to I as described above, and assume the follow-
ing conditions hold. ͑i͒ DЈϭDഫI is globally hyperbolic. ͑ii͒ I admits a compact spacelike cut. ͑iii͒ For each point p in M near I and any future complete null geodesic s→␩(s) in D starting at p, ͐0ϱRic(␩Ј,␩Ј)ds у 0. Then the PTC holds on D.
Remark. Let K be a cut of I, and let IK be the portion of I to the future of K, IKϭIപIϩ(K). Let DK be the domain of outer of communications with respect to IK , DKϭIϩ(IK)പIϪ(IK)ϭIϩ(K)പIϪ(I). Theorems 2.1 and 2.2 apply equally well to DK . This procedure, first discussed by Jacobson and Venkataramani ͓7͔, allows one, by the
methods of this paper, to study the topology of cuts on the future event horizon Hϭ‫ץ‬IϪ(I) of the form ‫ץ‬Iϩ(IK)പH. By taking K sufficiently far to the future, this procedure enables one to consider cuts on H well to the future of the
initial formation of the black hole, where one has a greater expectation that the intersection of H with ‫ץ‬Iϩ(IK) will be reasonable ͑i.e., a surface͒. See ͓7͔ and Secs. IV and V below
for further discussion of this point. In what follows, it is worth keeping in mind that I may refer to the portion of scri
to the future of a cut. Proof of theorem 2.1. The global hyperbolicity of MЈ implies that I0 is strongly causal as a spacetime in its own right and that the sets Jϩ(x,I0)പJϪ(y,I0) ͓ʚJϩ(x,MЈ)പJϪ(y,MЈ)͔ for x,y෈I0 have compact closure in I0 . This is sufficient to imply that I0 is globally hyperbolic as a spacetime in its own right. Assumption ͑ii͒ then implies that I0 is foliated by compact Cauchy surfaces.
Now suppose that Jϩ(I0) meets some other component I1 of I: i.e., suppose there exists a future-directed causal curve from a point p෈I0 to a point q෈I1 . Let ⌺0 be a Cauchy surface for I0 passing through p. Push ⌺0 slightly in the normal direction to I0 to obtain a compact spacelike surface ⌺ contained in M. Let V be a compact spacelike
hypersurface with boundary spanning ⌺0 and ⌺. By properties of the conformal factor, we are assured, for suitable
pushes, that ⌺ is null mean convex. By this we mean that the future-directed null geodesics issuing orthogonally from ⌺ which are ‘‘inward pointing’’ with respect to I0 ͑i.e., which point away from V͒ have negative divergence.
Under the present supposition, Jϩ(V) meets I1 . At the same time, Jϩ(V) cannot contain all of I1 . If it did, there would exist a past inextendible causal curve in I1 starting at q෈I1 contained in the compact set Jϩ(V)പJϪ(q), contradicting the strong causality of MЈ. Hence ‫ץ‬Jϩ(V) meets I1 ; let q0 be a point in ‫ץ‬Jϩ(V)പI1 . Since MЈ is globally hyperbolic, it is causally simple; cf. proposition 6.6.1 in ͓19͔,
which remains valid in the present setting. Hence ‫ץ‬Jϩ(V)ϭJϩ(V)‫گ‬Iϩ(V), which implies that there exists a future-directed null curve ␩ʚ‫ץ‬Jϩ(V) that extends from a point on V to q0 . ͑Alternatively, one can prove the existence of this curve from results of ͓21͔, which are also valid in the present setting.͒ It is possible that ␩ meets I several times before reaching q0 . Consider only the portion ␩0 of ␩ which

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PHYSICAL REVIEW D 60 104039

extends from the initial point of ␩ on V up to, but not including, the first point at which ␩ meets I.

By properties of achronal boundaries and the conformal factor, ␩0 is a future complete null geodesic in M emanating from a point on V. Since ␩0 cannot enter Iϩ(V), it follows that ͑1͒ ␩0 actually meets V at a point p0 of ⌺, ͑2͒ ␩0 meets ⌺ orthogonally at p0 , and ͑3͒ ␩0 is inward pointing with respect to I0 ͑i.e., ␩0 points away from V͒. Since ␩0 is
future complete, the energy condition ͑iii͒ and null mean

convexity of ⌺ imply that there is a null focal point to ⌺

along ␩0 . But beyond the focal point, ␩0 must enter

Iϩ(⌺)͓ʚIϩ(V)͔, contradicting ␩0ʚ‫ץ‬Jϩ(V).

᭿

Proof of theorem 2.2 The proof is an application of theorem
2.1, together with a covering space argument. Fix p෈I. The inclusion map i:I→DЈ induces a homomorphism of fundamental groups i* :⌸1(I, p)→⌸1(DЈ, p). The image G ϭi*„⌸1(I, p)… is a subgroup of ⌸1(DЈ, p). Basic covering space theory guarantees that there exists an essentially

unique covering space D˜ Ј of DЈ such that ␲*„⌸1(D˜ Ј,˜p)… ϭGϭi*„⌸1(I, p)…, where ␲:D˜ Ј→DЈ is the covering map and ␲(˜p)ϭp. Equip D˜ Ј with the pullback metric ␲*(gЈ) so

that ␲:D˜ Ј→DЈ is a local isometry. We note that when I is

simply connected, D˜ Ј is the universal cover of DЈ, but in general it will not be. In a somewhat different context ͑i.e., when DЈ is a spacetime without boundary and I is a space-

like hypersurface͒, D˜ Ј is known as the Hawking covering spacetime: cf. ͓19,22͔.

The covering spacetime D˜ Ј has two basic properties: ͑a͒

The component ˜I of ␲Ϫ1(I) which passes through ˜p is a

copy of I: i.e., ␲*͉˜I :˜I→I is an isometry, and ͑b͒ i*„⌸1(˜I,˜p)…ϭ⌸1(D˜ Ј,˜p), where i:˜ID˜ Ј.
Property ͑b͒ says that any loop in D˜ Ј based at ˜p can be

deformed through loops based at ˜p to a loop in ˜I based at ˜p.

This property is an easy consequence of the defining prop-

erty ␲*„⌸1(D˜ Ј,˜p)…ϭi*„⌸1(I, p)… and the homotopy lifting property. In turn, property ͑b͒ easily implies that any curve

in D˜ Ј with end points on ˜I is fixed end point homotopic to a

curve in ˜I. Now let ␥ be a future-directed causal curve in DЈ with
end points on I. Assume ␥ extends from x෈I to y෈I. Let ˜␥

be the lift of ␥ into D˜ Ј starting at ˜x෈˜I. Note that the hy-

potheses of theorem 2.1, with MϭD˜ ª␲Ϫ1(D), MЈϭD˜ Ј,

and I0ϭ˜I, are satisfied. Hence the future end point ˜y of ˜␥

must also lie on ˜I. Then we know that ˜␥ is fixed end point

homotopic to a curve in ˜I. Projecting this homotopy down to

DЈ, it follows that ␥ is fixed end point homotopic to a curve

in I, as desired.

᭿

Remarks. Since many examples of locally AdS spacetimes—

for example, those constructed by identifications of AdS

͓9–11,13,14͔—do not obey the generic condition, our aim

was to present a version of the PTC which does not require

it. However, if one replaces the energy condition ͑iii͒ by the generic condition and the ANEC ͐Ϫϱ ϱRic(␩Ј,␩Ј)dsу0 along any complete null geodesic s→␩(s) in D, then theo-

rems 2.1 and 2.2 still hold, and moreover they do not require the compactness condition ͑ii͒. The proof of theorem 2.1 under these new assumptions involves the construction of a complete null line ͑globally achronal null geodesic͒ which is incompatible with the energy conditions. The results in this setting are rather general, and in particular the proofs do not use in any essential way that I is timelike. We also mention that the global hyperbolicity assumption in theorem 2.2 can be weakened in a fashion similar to what was done in the AF case in ͓16͔.
We now turn our attention to a characterization of topological censorship which will allow us to more easily explore the consequences of topological censorship for black hole topology.

III. ALGEBRAIC CHARACTERIZATION OF TOPOLOGICAL CENSORSHIP
The main result of this section is a restatement of the properties of a spacetime satisfying the PTC in a language amenable to algebraic topological considerations.
Let D and I be as in Sec. I. Then DЈϭDഫI is a spacetime with boundary, with ‫ץ‬DЈϭI. The inclusion map i:I→DЈ induces a homomorphism of fundamental groups i* :⌸1(I)→⌸1(DЈ). Then we have the following.
Proposition 3.1. If the PTC holds for DЈ, then the group homomorphism i* :⌸1(I)→⌸1(DЈ) induced by inclusion is surjective.

Remark. Note that the fundamental groups of D and DЈ are

trivially isomorphic, ⌸1(D)Х⌸1(DЈ). Hence proposition

3.1 says roughly that every loop in D is deformable to a loop

in I. Moreover, it implies that ⌸1(D) is isomorphic to the

factor group nected, then

⌸ so

1 ( I) / is D

ker .

i

*

.

In

particular,

if

I

is

simply

con-

Our proof relies on the following straightforward lemma

in topology

Lemma 3.2. Let N be a manifold and S an embedded submanifold with inclusion mapping i:S→N. If in the universal
covering space of N the inverse image of S by the covering map is connected, then i* :⌸1(S)→⌸1(N) is surjective.

Proof. Strictly speaking, we are dealing with pointed spaces

(S,p) and (N,p), for some fixed point p෈S, and we want to show that i* :⌸1(S, p)→⌸1(N, p) is onto. Let ͓c0͔ be an element of ⌸1(N, p): i.e., let c0 be a loop in N based at p.

Let N˜ be the universal covering space of N, with covering

map ␲:N˜ →N. Choose ˜p෈N˜ such that ␲(˜p)ϭ p. Let ˜c0 be

the lift of c0 starting at ˜p; then, ˜c0 is a curve in N˜ extending from ˜p to a point ˜q with ␲(˜q)ϭ p. Since ˜p,˜q෈␲Ϫ1(S) and ␲Ϫ1(S) is path connected, there exists a curve ˜c1 in ␲Ϫ1(S) from ˜p to ˜q. The projected curve c1ϭ␲(˜c1) is a loop in S

based at p, i.e., ͓c1͔෈⌸1(S, p). Since N˜ is simply con-

nected, ˜c0 is fixed end point homotopic to ˜c1 . But this im-

plies that c0 is homotopic to c1 through loops based at p, i.e.,

i*(͓c1͔)ϭ͓c0͔, as desired.

᭿

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Proof of proposition 3.1. We let N˜ be the universal covering

spacetime of NªDЈ with projection ␲:N˜ →N. Then N˜

ϭD˜ ഫ˜I, where D˜ ϭ␲Ϫ1(D) is the universal covering space-

time of D and ˜Iϭ␲Ϫ1(I) is the boundary of N˜ .

Every point in D˜ belongs to the inverse image by ␲ of some point in D, and every point in D lies on some causal

curve beginning and ending on I, so every point in D˜ lies on

some causal curve beginning and ending on ˜I. By the PTC

and basic lifting properties, no such curve can end on a dif-

ferent connected component of ˜I than it began on. Therefore,

if we label these connected components by ␣, the open sets

Iϩ(˜I␣)പIϪ(˜I␣) are a disjoint open cover of D˜ . But D˜ is

connected, so ␣ can take only one value, whence ˜I is con-

nected. It follows that DЈ satisfies the conditions of lemma

3.2 with SϭI and therefore i* :⌸1(I)→⌸1(DЈ) is surjec-

tive. Thus i* :⌸1(I)→⌸1(D) is surjective.

᭿

All known locally anti–de Sitter black holes and related

spacetimes are in accordance with proposition 3.1. We con-

clude this section with the following simple corollary to

proposition 3.1.

Corollary 3.3. If the PTC holds for DЈ, then D is orientable if I is.

Proof. In fact, DЈ is orientable, for if DЈ were not orientable and I were, then DЈ would possess an orientable double cover containing two copies I1 and I2 of I. Then a curve from p1෈I1 to p2෈I2 , where ␲( p1)ϭ␲( p2)ϭ p෈I, would project to a loop in DЈ not deformable to I, contrary

to the surjectivity of i* .

᭿

IV. APPLICATION TO BLACK HOLE TOPOLOGY

The boundary of the region of spacetime visible to observers at I by future-directed causal curves is referred to as the event horizon. This horizon is a set of one or more null surfaces, also called black hole horizons, generated by null geodesics that have no future end points, but possibly have past end points. The topology of these black hole horizons is constrained in spacetimes obeying the PTC because, as was seen in Sec. III, the topology of the domain of outer communications is constrained by the PTC—intuitively, causal curves that can communicate with observers at I cannot link with these horizons in a nontrivial way: rather, they only carry information about the nontriviality of curves on I.
Useful though this description is, it does not characterize the topology of these horizons themselves, but rather the hole that their excision leaves in the spacetime. However, one can obtain certain information about these horizons if one considers the topology of the intersection of certain spacelike hypersurfaces with the horizons in fourdimensional spacetimes, those whose intersection with the horizons are closed spacelike two-manifolds ͑good cuts of the horizons͒. For example, if one has a single horizon of product form, as is the case for a stationary black hole, then the horizon topology is determined by that of the twomanifold. Of course, black hole horizons are generally not of product form; however, the topology of any good cut is still

closely related to that of the horizons. For example, if each horizon has a region with product topology, this topology is determined by that of a good cut passing through this region. As demonstrated by Jacobson and Venkataramani ͓7͔ for asymptotically flat spacetimes, the PTC constrains the topology of good cuts of the horizons by the closure of a Cauchy surface for the domain of outer communications to be twospheres.
Now it is possible that a black hole horizon admits two good cuts by spacelike hypersurfaces, one entirely to the future of the other, such that the cuts are not homeomorphic. This can only happen if null generators enter the horizon between the two cuts. Physically, such a situation corresponds to a black hole with transient behavior, such as that induced by formation from collapse, collision of black holes, or absorption of matter. Jacobson and Venkataramani observed that their theorem could also be applied to such situations for certain spacelike hypersurfaces that cut the horizons sufficiently far away from regions of black hole formation, collision, or matter absorption. Precisely, when the domain of outer communications is globally hyperbolic to the future of a cut of I and if the PTC holds on this subdomain, then the PTC will constrain the topology of this subdomain, of its Cauchy surface, and ultimately of good cuts of the horizons. Thus, though the PTC does not determine the topology of arbitrary embedded hypersurfaces or the cuts they make on the horizons, it does do so for hypersurfaces homeomorphic to Cauchy surfaces for these subdomains that make good cuts of the horizons. Such subdomains can be found for black hole horizons that settle down at late times.
Below we provide a generalization of these results applicable to a more general set of spacetimes satisfying the PTC than asymptotically flat spacetimes. This generalization is based on the observation that if one can continuously push any loop in the DOC down into an appropriate spacelike surface that cuts the horizons in spacelike two-manifolds, it follows from proposition 3.1 that the fundamental group of the spacelike surface is related to that of I. In the interests of a clear presentation that carefully treats all technical details, we first prove the following results for globally hyperbolic four-dimensional spacetimes with timelike4 I and, we emphasize, for any globally hyperbolic subdomain corresponding to the future of a cut of I. We then provide the results for the case of globally hyperbolic, asymptotically flat spacetimes. From the method of proof of these two cases, it is manifestly apparent that these theorems can be easily generalized to a wider class of spacetimes that satisfy the PTC. We conclude with a remark about how to provide the correct technical statement and proofs for these cases. We will also comment further on the transient behavior associated with black hole formation in the Discussion section.
Let M be a spacetime with timelike infinity I and domain of outer communications D. Assume I is connected and ori-
4We remind the reader that the notions of global hyperbolicity and Cauchy surfaces for spacetimes with timelike I are reviewed at the beginning of Sec. II.

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entable. Let K be a spacelike cut of I, and let IK be the portion of I to the future of K, IKϭIപIϩ(K). Let DK be the DOC with respect to IK , DKϭIϩ(IK)പIϪ(IK) ϭIϩ(K)പIϪ(I). Note that DKЈ ªDKഫIK is a connected spacetime with timelike boundary, with ‫ץ‬DKЈ ϭIK and DKϭDKЈ ‫گ‬IK . In the following we will assume that DKЈ is globally hyperbolic and has a Cauchy surface VЈ as in Sec. II. The following theorem is the main result pertaining to the topology of black holes.
Theorem 4.1. Let DK be the domain of outer communications to the future of the cut K on I as described above. Assume that DKЈ is globally hyperbolic and satisfies the PTC. Suppose VЈ is a Cauchy surface for DKЈ such that its closure V ϭ¯VЈ in MЈ is a compact topological three-manifold with boundary whose boundary ‫ץ‬V (corresponding to the edge of VЈ in MЈ) consists of a disjoint union of compact twosurfaces,

k

‫ץ‬Vϭ ⌺i ,

͑2͒

iϭ0

where ⌺0 is on I and the ⌺i , iϭ1, . . . ,k, are on the event horizon. Then

k

͚ giрgo ,

͑3͒

iϭ1

where g jϭthe genus of ⌺ j , jϭ0,1,...,k. In particular, if ⌺0 is a two-sphere, then so is each ⌺i , iϭ1,...,k.
Remark 1. Many known examples of locally AdS black hole spacetimes have black hole horizons with genus equal of that of scri ͓9–11,13͔. In fact, for these examples, V is a product space. Remark 2. Theorem 4.1 has been stated for spacetimes with timelike I. An analogous version for asymptotically flat spacetimes holds as well, but differs slightly in technical details. The AF case will be considered later in the section.
Theorem 4.1 is established in a series of lemmas. Lemma 4.2 connects the fundamental group of the Cauchy surface to that of I. It is the only lemma that uses the conditions that the spacetime is globally hyperbolic and satisfies the PTC. The remaining lemmas are based purely on algebraic topology and are in fact applicable to any three-manifold with compact boundary.

Lemma 4.2. Let the setting be as in theorem 4.1. Then the group homomorphism i* :⌸1( ͚0)→⌸1(V) induced by inclusion i: ͚0→V is onto.
Proof. Let Za be a timelike vector field on DKЈ tangent to IK . Let r:DKЈ →VЈ be the continuous projection map sending each point p෈DKЈ to the unique point in VЈ determined by the integral curve of Za through p. Note that r(IK)ϭ ͚0 .
Fix p෈ ͚0 . All loops considered are based at p. Let c0 be a loop in V. By deforming c0 slightly we can assume c0 is in VЈ. Since DKЈ satisfies the hypotheses of theorem 2.2, the

PTC holds for DKЈ . Then, by proposition 3.1, c0 can be continuously deformed through loops in DKЈ to a loop c1 in IK . It follows by composition with r that c0ϭr‫ؠ‬c0 can be continuously deformed through loops in V to the loop
c2ϭr‫ؠ‬c1 in ͚0 , and hence i*(͓c2͔)ϭ͓c0͔, as desired. ᭿ By corollary 3.3 and the assumptions of theorem 4.1, V is
a connected, orientable, compact three-manifold with bound-
ary whose boundary consists of kϩ1 compact surfaces ͚i , iϭ0,...,k. For the following results, which are purely topo-
logical, we assume V is any such manifold.
All homology and cohomology below are taken over the integers. The jth homology group of a manifold P will be denoted by H j(P), b j(P)ª the rank of the free part of H j(P) will denote the jth Betti number of P, and ␹͑P͒ will denote the Euler characteristic ͑the alternating sum of the
Betti numbers͒. In the case of b j(V) ͓ϭb j(V0)͔, we will simply write b j .
In general, for V as assumed above, b0ϭ1, b3ϭ0, and b1 and b2 satisfy the following inequalities.

Lemma 4.3. For V as above, then ͑a͒ b1у ͚ikϭ0 gi and ͑b͒ b2уk.

Proof. The boundary surfaces ͚1 , ͚2 , . . . , ͚k clearly deter-

mine k linearly independent two-cycles in V, and hence

͑b͒ holds. To prove ͑a͒ we use the formula, ␹(V)

ϭ

1 2



(

‫ץ‬

V

)

,

valid

for

any

compact,

orientable,

odd-

dimensional manifold. This formula, together with the ex-

pressions ␹(V)ϭ1Ϫb1ϩb2 and ␹‫ץ)‬V)ϭ ͚ikϭ0 ␹( ͚i)ϭ 2(kϩ1)Ϫ2 ͚ikϭ0 gi , implies the equation

k

͚ b1ϭb2ϩ giϪk.

͑4͒

iϭ0

The inequality ͑a͒ now follows immediately from ͑b͒. ᭿

Lemma 4.4. If the →⌸ (V) induced by

group homomorphism i* :⌸1( ͚0) inclusion is onto, then the inequality

1

͑3͒,

͚

k iϭ

1

g

i

р

g

0

,

holds.

In

particular,

if

͚0

is

a

two-sphere,

then so is each ͚i , iϭ1, . . . ,k.

Proof. We use the fact that the first integral homology group
of a space is isomorphic to the fundamental group modded
out by its commutator subgroup. Hence modding out by the commutator subgroups of ⌸1( ͚0) and ⌸1(V), respectively, induces from i* a surjective homomorphism from H1( ͚0) to H1(V). It follows that the rank of the free part of H1(V) cannot be greater than that of H1( ͚0), i.e.,

b1рb1͑⌺0͒ϭ2g0 .

͑5͒

Combining this inequality with the inequality ͑a͒ in lemma

4.3 yields the inequality ͑3͒. Since V is orientable, so are its

boundary components. If ͚0 is a two-sphere, then each ͚i ,

iϭ1,...,k, is forced by ͑3͒ to have genus zero and, hence, is a

two-sphere.

᭿

Proof of theorem 4.1. The proof follows immediately from

lemmas 4.2 and 4.4.

᭿

Next we show that the condition on i* in lemma 4.4 completely determines the homology of V.

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Proposition 4.5. If i* :⌸1( ͚0)→⌸1(V) is onto, then the integral homology H*(V,Z) is torsion free and, hence, is completely determined by the Betti numbers. Furthermore,
the inequalities in lemma 4.3 become equalities: ͑a͒ b1ϭ ͚ikϭ0 gi and ͑b͒ b2ϭk.

Proof. We first prove that H*(V,Z) is torsion free. Since V has a boundary, H3(V)ϭ0. Also, H0(V) is one copy of Z as
V is connected. Thus we need to show that H1(V) and
H2(V) are free.
Claim 1. H2(V) is free. To prove claim 1 we recall the classic result that
HnϪ1(Nn) is free for an orientable closed n-manifold Nn. To make use of this, let VЈ be a compact orientable threemanifold without boundary containing V ͑e.g., take VЈ to be the double of V͒, and let BϭVЈ‫گ‬V.
Assume that W is a nontrivial torsion element in H2(V). Now view W as an element in H2(VЈ). Suppose Wϭ0 in H2(VЈ). Then Wϭ0 in H2(VЈ,B). By excision, H2(VЈ,B) ϭH2(V,‫ץ‬V), where ‫ץ‬V is the manifold boundary of V. Hence Wϭ0 in H2(V,‫ץ‬V). This means that Wϭa sum of
boundary components in H2(V). But a sum of boundary
components cannot be a torsion element. Thus W 0 in
H2(VЈ). Moreover, if nWϭ0 in H2(V), then nWϭ0 in H2(VЈ). It follows that W is a nontrivial torsion element in H2(VЈ), a contradiction. Hence H2(V) is free.
Claim 2. H1(V) is free. To prove claim 2 we first consider the relative homology
sequence for the pair Vʛ ͚0 :





‫ץ‬

¯→H1͑ ⌺0͒→H1͑ V ͒→H1͑ V,⌺0͒→H˜ 0͑ ⌺0͒ϭ0. ͑6͒

͓Here H˜ 0( ͚0) is the reduced zeroth-dimensional homology group.͔ Since, as discussed in lemma 4.4, ␣ is onto, we
have ker ␤ϭIm ␣ϭH1(V), which implies ␤ϵ0. Hence ker‫ץ‬ϭIm ␤ϭ0, and thus ‫ ץ‬is injective. This implies that
H 1 ( V , ⌺ 0 ) ϭ 0. Now consider the relative homology sequence for the
triple Vʛ‫ץ‬Vʛ ͚0 :

¯→H1͑‫ץ‬V,⌺0͒→H1͑V,⌺0͒ϭ0→H1͑V,‫ץ‬V͒

‫ץ‬

→H0͑‫ץ‬V,⌺0͒→¯ .

͑7͒

Since H0‫ץ)‬V, ͚0) is torsion free and ‫ ץ‬is injective, H1(V,‫ץ‬V) is torsion free. Next, Poincare´-Lefschetz duality gives H2(V)ХH1(V,‫ץ‬V). Hence H2(V) is torsion free. The
universal coefficient theorem implies that

H2͑ V ͒ХHom„H2͑ V ͒,Z… Ext„H1͑ V ͒,Z….

͑8͒

The functor Ext͑Ϫ, Ϫ͒ is bilinear in the first argument with respect to direct sums and Ext(Zk ,Z)ϭZk . Hence H2(V)
cannot be torsion free unless H1(V) is. This completes the
proof of claim 2 and the proof that H*(V) is torsion free. It remains to show that the inequalities in lemma 4.3 be-
come equalities. We prove b2ϭk; the equation b1ϭ ͚ikϭ0 gi then follows from Eq. ͑4͒. In view of lemma 4.3, it is sufficient to show that b2рk. Since H2(V) is finitely gener-

ated and torsion free, we have H2(V)ХH2(V)

ХH1(V,‫ץ‬V), where we have again made use of Poincare´-

Lefschetz duality. Hence b2ϭrank H1(V,‫ץ‬V). To show that

rank H1(V,‫ץ‬V)рk, we refer again to the long exact se-

quence ͑7͒. By excision, H0‫ץ)‬V, ͚0)ХH0‫ץ)‬V‫͚ گ‬0 ,л)

ϭH0‫ץ)‬V‫͚ گ‬0). Hence, by the injectivity of ‫,ץ‬

rank H1(V,‫ץ‬V)рrank H0‫ץ)‬V, ͚0)ϭthe number of compo-

nents of ‫ץ‬V‫͚ گ‬0ϭk. This completes the proof of proposition

4.5.

᭿

The conclusion of proposition 4.5 applies to the spacelike

three-surface with boundary V of theorem 4.1. Thus we have

completely determined the homology of the Cauchy surfaces

of DKЈ .

We now consider the asymptotically flat case with null infinity IϭIϩഫIϪ. For this case, let K be a spacelike cut of IϪ, and let IK be the portion of I to the future of K, IKϭIപJϩ(K). Let DK be the domain of outer of communications with respect to IK , DKϭIϩ(IK)പIϪ(IK) ϭIϩ(K)പIϪ(Iϩ). Here DKЈ ªDKഫIϩ is a connected spacetime with boundary, with ‫ץ‬DKЈ ϭIϩ and DKϭDKЈ ‫گ‬Iϩ.

We then have the following analogue of theorem 4.1.

Theorem 4.1Ј. Let DK be the domain of outer communications to the future of the cut K on IϪ of an asymptotically flat spacetime M as described above. Assume that DKЈ is globally hyperbolic and satisfies the PTC. Suppose V0 is a Cauchy surface for DK such that its closure Vϭ¯V0 in M is a topological three-manifold with boundary, compact outside a small neighborhood of i0, with boundary components con-
sisting of a disjoint union of compact two-surfaces:

k
‫ץ‬Vϭ ⌺i ,
iϭ1
where the ͚i , iϭ1, . . . ,k, are on the event horizon. Then all ͚i , iϭ1, . . . ,k, are two-spheres. Moreover, V0 has the topology of a homotopy three-sphere minus kϩ1 closed three-balls.

Remark 1. In the AF case the asymptotic topology is spheri-
cal, which corresponds to g0ϭ0 in inequality ͑3͒. But since giϭ0, iϭ1, . . . ,k, inequality ͑3͒ is satisfied in the AF case, as well. Again, the topology of the event horizon is con-
strained by the topology at infinity.
Remark 2. This theorem is a slightly strengthened version of the main theorem in ͓7͔; it does not assume orientability of
V0 , and we conclude a stronger topology for this Cauchy surface. Proof of theorem 4.1Ј. The arguments used to prove theorem
4.1 can be easily adapted, with only minor technical changes
involved, to prove in the AF case that each ͚i is a twosphere. Alternatively, one may argue as follows. By known results on topological censorship in the AF case ͓5,6͔, DK is simply connected and, hence, so is V0 . It follows that ˜V ϭVഫ͕i0͖ is a compact simply connected three-manifold with boundary, with boundary components ͚i , iϭ1, . . . ,k. Then, according to lemma 4.9, p. 47 in Hempel ͓23͔, each ͚i is a two-sphere. By attaching three-cells to

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PHYSICAL REVIEW D 60 104039

each ͚i , we obtain a closed simply connected threemanifold, which by well-known results ͑see ͓23͔͒ is a homo-

topy three-sphere. Removing the attached three-cells and i0 ,

we obtain that V0 is a homotopy three-sphere minus kϩ1

closed balls.

᭿

Remark. Although the above results were proved assuming

global hyperbolicity, it is clear that the same results will hold

for a more general set of spacetimes that satisfy the PTC and

for which a version of lemma 4.2 can be proved. Spacetimes

that are not globally hyperbolic but satisfy a weaker condi-

tion such as weak cosmic censorship can still admit a pro-

jection onto a preferred spacelike surface. In particular, one

can generalize the projection given by the integral flow of a

timelike vector field on the domain of outer communications

used to push loops into the Cauchy surface to be a retract.

Recall a retract of X onto a subspace A is a continuous map r:X→A such that r͉Aϭid. Thus, if V0 is a regular retract of D, that is, if there exists a retract r:DഫI→V0ഫ⌺0 such that r(I)ʕ⌺0 , then one can again establish lemma 4.2.

V. DISCUSSION
We wish to emphasize that the results concerning black hole topology in four-dimensional spacetimes obtained in Sec. IV in no way contradict the numerical findings of ͓24͔ concerning the existence in principle of temporarily toroidal black holes in asymptotically flat spacetimes. The consistency of topological censorship with asymptotically flat models containing temporarily toroidal black hole horizons has been clearly elucidated in ͓25͔. The acausal nature of crossover sets, expected to be present in the early formation of the event horizon, permits slicings of the event horizon in asymptotically flat black hole spacetimes with exotic ͑i.e., nonspherical͒ topologies. See the recent papers in Refs. ͓26, 27͔ for further discussion. As described in Sec. IV, the method of topological censorship for exploring the topology of black hole horizons makes use of specific time slices, namely, Cauchy surfaces for the DOC or for the subregion of the DOC to the future of a cut on I. Surfaces exhibiting temporarily toroidal black hole horizons are not such surfaces. Moreover, the method requires such a slice to have a nonempty edge which meets the horizons in C0 compact surfaces. We elaborate further on these points below.
It is important to keep in mind that not all Cauchy sur-
faces V0 for the DOC are interiors of orientable manifolds with boundary V corresponding to the intersection of a
spacelike slice with the black hole horizons. Consider the t ϭ0 slice of the R P3 geon. As discussed in ͓5͔, this spacetime is constructed from the tϭ0 slice of Schwarzschild spacetime by identifying antipodal points at the throat r ϭ2M . The maximal evolution of this slice is a spacetime with spatial topology R P3Ϫ pt. Its universal covering space is the maximally extended Schwarzschild spacetime.
The tϭ0 slice of the R P3 geon contains a nonorientable R P2 with zero expansion. This R P2 is not a trapped surface as it does not separate the slice into two regions. It is not part of the DOC as any radially outward directed null geodesic from this surface does not intersect I; thus, it is clearly part of the horizon. The intersection of the DOC with the tϭ0

slice produces a simply connected V0 . The intersection of the horizon with this slice is R P2. However, we cannot at-
tach this surface to V0 to produce a manifold with boundary V by the inclusion map; instead, this map reproduces the original tϭ0 slice which has no interior boundary. Note, however, that any spacelike slice that does not pass through this R P2 will intersect the horizon at an S2. In fact, this will be the generic situation. Moreover, the intersection of such a
slice with the DOC will produce a simply connected V0 which is the interior of a closed connected orientable V with an S2 interior boundary.
Clearly, this example does not contradict any results of
Sec. IV, which assumes an orientable V with two or more boundaries. However, it does yield the important lesson that
one must construct V to apply the theorem, not V0 . It also gives an example of a badly behaved cut of the horizon, again illustrating the usefulness of taking slices to the future of a cut of I.
For our second example, we construct a toy model of a black hole spacetime that mimics a special case of topology change, namely, that of black hole formation from a single collapse. This model illustrates several features: how a choice of a hypersurface can affect the description of horizon topology and how some cuts of I give rise to a Cauchy surface whose edge on the horizon is not a two-manifold.
We begin with a three-dimensional model; later a fourdimensional example will be constructed by treating the three-dimensional model as a hyperplane through an axis of symmetry in the larger spacetime. Our spacetime can have either anti–de Sitter or flat geometry; as both are conformal to regions of the Einstein static cylinder, we use as coordinates in the construction below those of the conformally related flat metric ͑cf. ͓19͔, Secs. 5.1 and 5.2͒. We depict I as timelike in the accompanying figures, but it can equally well be null.
We begin in three dimensions with a line segment L defined by tϭyϭ0, ͉x͉рl. The future Iϩ(L) of this line segment is a sort of elongated cone, whose traces in hyperplanes of constant t have the shape of rectangles with semicircular caps attached to the two short sides. We foliate the spacetime by hyperboloids:

t ϭ a ϩ ͱr 2 ϩ b ,

͑9͒

where r2ϭx2ϩy2 and a,b are conveniently chosen parameters. These hyperboloids cut I in a circle. We now remove from spacetime all points of Jϩ(L) above a hyperplane t
ϭd intersecting it to the future. What remains is a black hole
spacetime and has a globally hyperbolic domain of outer communications D ͑again, cf. Sec. II͒. The black hole is the set of remaining points of Iϩ(L). The horizon ‫ץ‬IϪ(Iϩ) is
generated by null geodesics that all begin on L. The Cauchy surface for D will have topology R2 and does
not cross the horizon. Thus, to probe the topology of the
horizon, one needs to consider spacetimes corresponding to the future of a cut of I. However, not every cut of I will
produce a spacetime with a Cauchy slicing with the correct

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FIG. 1. A bad cut of the horizon. The surface VK is the Cauchy surface for the domain of outer communications of the spacetime to the future of the cut K of I. The inner boundary of the causal future
of K intersects the horizon at a spacelike line segment. Consequently the closure of VK has topology R2.

FIG. 3. Two slicings of the horizon by hyperboloids. Both illustrations concentrate on the region near the horizon. The top illustration is of a hyperboloid that intersects the horizon at two topological circles. The bottom illustration is of a hyperboloid that lies to the future of the first. It intersects the horizon at one topological circle.

FIG. 2. A good cut of the horizon. The surface VK is the Cauchy
surface for the domain of outer communications of the spacetime to the future of the cut K of I. The intersection of the inner boundary of the causal future of K with the horizon is now S1. Consequently, the closure of VK intersects the horizon at S1.

FIG. 4. Entanglement and nonentanglement of curves on black
hole horizons. Each illustration displays a cross section of a cut of I. The cutaway reveals a cut of a black hole horizon inside. In the top illustration, the cut of I has genus 0, that of the horizon has genus 1, and, as illustrated, there are curves not deformable to I. In the middle illustration, the genus of the cut of I and that of the
horizon are both 2, and they are linked in such a manner that every curve is deformable to I. In the illustration at bottom, the genus of the cut of I exceeds that of the horizon, and again every curve is deformable to I.

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GALLOWAY, SCHLEICH, WITT, AND WOOLGAR
properties. Such a bad cut of I is illustrated in Fig. 1. The boundary of the causal future of this cut intersects the horizon at a segment I of L. The topology of a Cauchy slice for its DOC is R2‫گ‬I. Its closure intersects the horizon at I; thus, as in the R P3 geon above, the closure of this slice has no inner boundary, being in this case R2.
This 3D spacetime corresponds to a 4D axisymmetric spacetime. The correspondence between axisymmetric spatial hypersurfaces and the xy planes of the spacetime is generated by rotating each xy plane about the y axis. After this rotation, one sees that the line segment L becomes a disk in the 4D axisymmetric spacetime. The Cauchy surface in question meets the horizon in a disk ͑a closed two-ball͒ and has topology R3 minus that disk. The closure of the Cauchy surface is R3 and again has no internal boundary. Thus the results of Sec. IV do not apply to this Cauchy slice.
A good cut of I is illustrated in Fig. 2. This cut intersects the horizon at a sphere. The topology of a Cauchy surface for its DOC is R2‫گ‬B2, and the intersection of its closure is S1. The Cauchy surface in the corresponding four-dimensional model has topology R3‫گ‬B3 with internal boundary S2. The results of Sec. IV clearly apply in the latter case.
Of course, not all spacelike surfaces need be Cauchy surfaces of the spacetime to the future of a cut of I. The family of hyperboloids ͑9͒, two of which are illustrated in Fig. 3, provides an example of such surfaces. As recognized in ͓25͔ in a similar model, this family exhibits the formation of a temporarily toroidal black hole horizon as the parameter a in Eq. ͑9͒ increases; these surfaces intersect the horizon in a pair of topological circles, which by axial symmetry correspond to a toroidal horizon in the four-dimensional spacetime. The circles increase in size and eventually meet, whence the horizon topology changes. After this point, these surfaces meet the horizon at a circle, corresponding to a sphere by axial symmetry.
In contrast, with respect to constant-t surfaces, the horizon forms completely at the tϭ0 instant. For every tϾ0 hypersurface, the black hole has spherical topology, and inequality

PHYSICAL REVIEW D 60 104039
͑3͒ holds. Any tϭt0Ͼ0 hypersurface is a Cauchy surface for the region of D that lies in the future of an appropriate cut of I. The apparent change of horizon topology from toroidal to spherical was an effect entirely dependent on the choice of hypersurface. The only unambiguous description of this black hole is that no causal curve was able to link with the horizon, i.e., that the PTC was not violated.
In the Introduction, we offered the view that the topology of the boundary at infinity constrained that of the horizons, but one could equally well reverse this picture. Let us contemplate a black hole considered as a stationary, causally well-behaved, isolated system cut off from the Universe by a sufficiently distant boundary I. Then we have shown that topological censorship requires the genus of the horizon to be a lower bound for that of the boundary. As remarked above, this seems an intuitive result; as illustrated in Fig. 4, when one visualizes placing a genus g1 surface within a genus g2 ‘‘box’’ with no possibility of entangling curves, it seems clear that g2уg1 . Yet, as is often the case with such things, the powerful machinery of algebraic topology was required to prove it. An advantage of using this powerful tool is that we were able to completely specify the homology of well-behaved exterior regions of black holes and, in virtue of proposition 4.5, to say that all interesting homology of these exteriors, save that which is reflected in the topology of scri, is directly attributable to the presence of horizons.
ACKNOWLEDGMENTS
E.W. wishes to acknowledge conversations with R. B. Mann and W. Smith concerning locally anti–de Sitter spacetimes. G.J.G. and E.W. thank Piotr Chrus´ciel for conversations concerning the validity of topological censorship for these spacetimes. This work was partially supported by the Natural Sciences and Engineering Research Council of Canada and by the U.S. National Science Foundation, Grant No. DMS-9803566.

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