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A Global Uniqueness Theorem for Stationary Black Holes

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A Global Uniqueness Theorem for Stationary Black Holes G′a bor Etesi Institute for Theoretical Physics,E¨o tv¨o s University Puskin u.5-7,Budapest,H-1088Hungary e-mail:etesi@poe.elte.hu Abstract A global uniqueness theorem for stationary black holes is proved as a direct consequence of the Topological Censorship Theorem and the topo-logical classi?cation of compact,simply connected four-manifolds.1Introduction There is a remarkable interplay between di?erential geometry,the theory of di?erential equations and the physics of gravitation in the famous proof of the uniqueness of stationary black holes.The ?rst proof was given in a series of papers by Carter,Hawking,Israel and Robinson (for a survey see [8],[12]).In the eighties a very elegant shorter proof was discovered by Mazur [10]who found a hidden symmetry of the electromagnetic and gravitational ?elds.These very deep and di?cult investigations all were devoted to the uniqueness problem of the metric on a suitable four-manifold carrying a Lorentzian asymptotically ?at structure in the spirit of Penrose’s description of in?nity of space-times.More recently physicist’s e?ort is addressed to the topology of the event horizons of general (i.e.non-stationary)black holes.The ?rst theorems were proven by Hawking [8],[9]and later by Gannon,Galloway [6],[7]and others.

Based on the celebrated Topological Censorship Theorem of Friedman,Schleich and Witt [5]and using energy conditions Chru′s ciel and Wald gave a short proof that the event horizon of a stationary black hole in a “moment”is always a sphere.

The question naturally arises what can one say about the topology of the space-time itself in this case.

On the other hand the ?nal step in the understanding of four-manifolds mak-ing use “classical”(i.e.non-physical)methods was done by Freedman in 1981who gave a complete (topological)classi?cation of compact simply connected four-manifolds.

G.Etesi:Black Hole Uniqueness2

In this paper,referring to the results of Chru′s ciel–Wald and Freedman,we prove that a global uniqueness also holds for stationary black holes and more generally stationary space-times i.e.not only the metric but even the topology of the space-time in question is unique.Our method is based on a natural compacti?cation of the space-time manifold and a careful study of a vector?eld extended to this compact manifold.

Truely speaking,this is not a very surprising result in light of the local uniqueness.However,it demonstrates the power of the theorems mentioned above.

2Vector Fields

First we de?ne the precise notion of a stationary,asymptotically?at space-time containing a black hole collecting the standard de?nitions.Let us summarize the properties of an asymptotically?at and empty space-time that we need;for the whole de?nitions see[12]and the notion of an asymptotically empty and ?at space-time can be found in[8].

Let(M,g)be a space-time manifold and x∈M.then J±(x)is called the causal future and past of x respectively.If a space-time manifold(M,g) is asymptotically?at and empty then there exists a conformal inclusion i: (M,g)→(?M,?g)such that

??i(M)={i0}∪I+∪I?,where i0is the space-like in?nity and the future and past null-in?nities I±satisfy I±:=?J±(i0)\{i0}=S2×R;

??M\i(M)=J?(i0);

?There exists a function?:?M→R+which is smooth everywhere(except possibly at i0)satisfying?g|i(M)=?2 i?1 ?g and?|?i(M)=0;

?Every null geodesic on(?M,?g)has future and past end-points on I±re-spectively.

De?nition.Let(M,g)be asymptotically?at.(M,g)is called strongly asymptotically predictable if there exists an open region?V??M such that?V?

G.Etesi:Black Hole Uniqueness3

Remarks.Moreover we require(M,g)to be stationary i.e.there exists a future-directed time-like Killing?eld K on(M,g).In this case H is a three-dimensional null-surface in M,hence for each t∈R,H t:=H∩S t(S t is a Cauchy-surface)is a two-dimensional surface in M.We shall assume that H t (the event horizon in a“moment”)is a two dimensional embedded,orientable, smooth,compact surface in M without boundary.This is the requirement the event horizon to be“regular”.This condition is satis?ed by physically relevant black hole solutions of Einstein’s equations but there is no a priori reason to assume it.

Let(M,g)be a maximally extended space-time manifold as above.In the following considerations we shall focus on one outer,asymptotically?at region of it i.e.a part of(M,g)whose boundary at in?nity in?M is connected.To get such an(incomplete)manifold we simply cut up(M,g)along one connected component of its event horizon.We shall continue to denote this separated part also by(M,g)(g is the original metric restricted to our domain).We can see that under the conformal inclusion i the manifold(M,g)has boundary ?i(M)=?H∪I+∪{i0}∪I?where?H=i(H)and?H,I±are connected now.

Proposition1.Let(M,g)be a space-time manifold.Then K|H∈Γ(T H).

Proof.Assuming the existence of a point p∈H such that K p/∈T p H,let γ:(?1,1)→M be a smooth integral curve of K satisfyingγ(0)=p and ˙γ(0)=K p.Hence there is anε∈(?1,1)such thatγ(?ε)∈B andγ(ε)/∈B. But this means thatγ(?ε)∈J?(I+),since it can be connected by an integral curve of K toγ(ε)∈J?(I+).Hence this assumption led us to a contradiction. 3

Corollary.H is invariant under the?ow generated by K on M and,being H a null-surface,K|H is a null vector?eld.3

https://www.wendangku.net/doc/188728896.html,ing a heuristic argument here we can identify H t up to homeo-morphism as follows.Since the boundary of(M,g)at in?nity is homeomorphic to S2×R we may assume due to the stationarity that H t is homeomorphic to S2.According to recent articles one can prove that this is indeed the case for a stationary black hole[6],[7],[8],[9].For our purposes it is more important to refer to a stronger result of Chru′s ciel and Wald[3].

They prove that an outer asymptotically?at region of a stationary space-time manifold satisfying the null energy condition is simply connected as a con-sequence of the Topological Censorship Theorem[5].Under suitable additional hypothesis(e.g.the compactness of H t)it follows that H t is homeomorphic to a sphere.

Hence,with the aid of these results we get that H is homeomorphic to S2×R.

Now let us study the behaviour of the Killing?eld near the in?nity!Let ?K:=i

?K induced by the inclusion i.

G.Etesi:Black Hole Uniqueness4

Proposition2.?K becomes a null vector?eld at in?nity.

Proof.Let?γ:R→?M be an inextendible integral curve of?K!We may write:

?K?γ(t)|i(M) 2=?g|i(M)(?K?γ(t),?K?γ(t))=?2(?γ(t)) i?1 ?g(i?Kγ(t),i?Kγ(t))=

=?2(?γ(t))g(Kγ(t),Kγ(t)).

We have used the third property of asymptotic?atness.However,using the fact that K is a Killing?eld on M,we can write

?K?γ(t)|i(M) 2=a?2(?γ(t)),

where a:=g(Kγ(t

0),Kγ(t

)is a constant for an arbitrary t0∈R.But

lim

t→±∞

?2(?γ(t))=0

because of the asymptotic?atness.3

In the light of Proposition1.and2.we can see that K approaches a null vector?eld near the boundary of M.Hence it is straightforward to study the behaviour of null vector?elds on M and?M.Applying a smooth deformation to K on(M,g)we can produce a smooth,nowhere vanishing(but highly non-unique!)null vector?eld K0on(M,g)whose integral curves are inextendible geodesics.Denoting by?K0the image of this?eld under i,i.e.?K0=i?K0,?K0 has future and past end-points on I±respectively by the fourth property of asymptotically?at space-times.

Remark.Of course,we could have started with this null vector?eld instead of the Killing?eld K.The reason for dealing with the naturally given Killing ?eld was the attempt to exploit as much as possible the structure of a stationary, asymptotically?at space-time manifold.

Now let?X be an inextendible null vector?eld on(?M,?g)i.e.its integral curves are inextendible.We would like to study the extension of this?eld to the null in?nities hence?rst we have to extend the domain of its integral curves, which is R in this moment.Let us suppose that this extended domain is the circle S1.

Proposition3.Let?X be a null vector?eld on(?M,?g)with extended domain whose integral curves are geodesics.Then?X can be extended to I±if and only if?X|I±=0.

Proof.Let i(S)∪{i0}=:?S??M be a space-like hypersurface and let x∈?S\(?S∩?B)(x=i0).Then there exists an integral curve?γ:R→?M such that?γ(0)=x.So we can?nd a point q∈I+possessing the property

q=lim

t→+∞

?γ(t).

G.Etesi:Black Hole Uniqueness5 Let us de?ne a mapφ:?S\(?S∩?B)→I+byφ(x)=q.

Let we assume that we have extended?X to I+in a smooth manner and there is a q∈I+such that?X q=0!Hence there is a smooth curve?β:(?ε,ε)→I+ for a suitable small positive numberεsatisfying

?β(0)=q,˙?β(0)=?X

Obviously we can?nd aδ<εsuch that for q=q′∈U q

?β(δ)=q′,˙?β(δ)=?X

=0.

q′

But in this case there is an x′∈U x(U x denotes a small neighbourhood of x) and an integral curve?γ′of?X such thatφ(x′)=q′.

In other words q′satis?es

q′∈im?β,q′∈im?γ′,

and

q∈im?β,q/∈im?γ′.

This means that q′is a branching point of an integral curve of?X.But in this case even if?X q′is well de?ned??X q′is not and this is a contradiction.

Similar argument holds for I?.3

Hence we are naturally forced to?nd a null vector?eld tending to zero on the null in?nities.However note that on(?M,?g)there is a natural cut-o?function, namely?.Certainly there exists a k∈N such that the vector?eld de?ned by

:=?k?K0

is a zero vector?eld restricted to I±because of the third property of asymptotic ?atness.Note that this new vector?eld can be extended as zero to i0as well: Surrounding i0by a small neighbourhood U one can see(since U∩(I+∪I?)=?)?X

is arbitrary small in U.

It is straightforward that?X0is not transversal to I±in the sense of[1] since it approaches the null-in?nities as a tangential?eld.This would cause some di?culties later on in our construction.Fortunately one can overcome this non-transversality phenomenon by a general method.Due to standard transversality arguments[1]applying a generic small perturbation to?X0in a suitable neighbourhood of I±we can achieve that the perturbed?eld?Xεwill be transversal to the submanifolds of null-in?nities(and even remains zero on them,of course).

3The Compacti?cation Procedure

Now let N1be a smooth four-manifold.We call a subset C?N1a domain of N1if it is di?eomorphic to a closed four-ball B4.Let V,U±?N1be domains and j:?M→N1a smooth embedding satisfying the following conditions:

G.Etesi:Black Hole Uniqueness6

?N1\j(?M)=int V;

?There exists a point p0∈N1and domains U±satisfying U+∩U?={p0} such that j(i0)=p0and j(I±)??U±and j(

Z(q,t):=(0,0,0,f?(t));

where for the smooth cut-o?function f?the following holds:

f?(t)= 0if t≤?1

1if t≥?1

2,1]

Z(q,t):=(0,0,0,f+(t));

where

f+(t)= 1if t≤1

G.Etesi:Black Hole Uniqueness7 Clearly Z is a smooth vector?eld in W and Z|A±=0.Now take a smooth cut-o?functionρ:R4→R+satisfyingρ|V=1and being zero on the complement of W.De?ne the extension of Y1by

ρZ+(1?ρ)Y1.

It is obvious that the extended?eld(also denoted by Y1)is a smooth vector ?eld on N1due to the transversality of the original?eld to I±(more precisely it has not been de?ned in U±yet)and satis?es Y1|?U±=0.

As a?nal step let us apply a smooth homotopy for N1contracting the four-balls U±to the point p0.In this way we get a smooth compact mani-fold N0without boundaries and,due to the transversality conditions on Y1,a well-de?ned smooth vector?eld Y0on it.It is clear that Y0has only one(de-generated)isolated singular point,namely p0.Its index is+2as easy to see due to the fourth property of asymptotic?atness.

Theorem.N0is homeomorphic to the four-sphere S4.

Proof.First it is not di?cult to see that N0is simply connected.Note that N0is a union of an outer region of the original stationary,asymptotically ?at space-time M and a solid torus-like space T homeomorphic to B3×S1 with one B3pinched into a point(namely which corresponds to p0).Choosing p0as a base point consider the loop l in N0representing the generator of the fundamental groupπ1(T)=Z.Clearly this loop can be deformed continuously into M∪{p0}.But referring again to the theorem of Chru′s ciel and Wald[3] the deformed loop l is homotopically trivial since M is simply connected.It is obvious that every other loops in N0are contractible provingπ1(N0)=0.

Secondly,we have constructed a smooth vector?eld Y0on N0having one isolated singular point p0with index+2.Taking into account that N0is a smooth,compact manifold without boundaries this means that

χ(N0)=2

according to the classical Poincar′e–Hopf Theorem[2][10].

The Euler characteristic of a simply connected compact four-manifold S always has the formχ(S)=2+b2where b2denotes the rank of its intersection form(or second Betti number).In our case this rank is zero,hence our form is of even type.By the uniqueness of the trivial zero rank matrix representing this even form and referring to the deep theorem of Freedman[4]which gives a full classi?cation of simply connected compact topological four-manifolds in terms of their intersection matrices we deduce that N0is homeomorphic to the four-sphere since its even intersection form is given by the same zero-matrix.3 Corollary.The uniqueness of N0implies the uniqueness of the original space-time manifold(more precisely one connected piece of its outer region) M since we simply have to remove a singular solid torus T from N0and this

G.Etesi:Black Hole Uniqueness8 can be done in a unique way being N0simply connected.Hence,taking into account the explicit example of the Kerr-solution,this outer region is always homeomorphic to S2×R2.3

4Conclusion

We have proved that topological uniqueness holds for space-times carrying a stationary black hole.Note that we could prove only a topological equivalence although our constructed manifold N0carries a smooth structure,too.It would be interesting to know if this smooth structure was identical to the standard one in light of the unsolved problem of the four dimensional Poincar′e-conjecture in the smooth category.

From the physical point of view this uniqueness is important if we are inter-ested in problems concerning the whole structure of such space-time manifolds.

For example one may deal with the description of the vacuum structure of Yang-Mills?elds on the background of a gravitational con?guration containing a black hole or some other singularity(taking the singularity theorems into consideration this question is very general and natural).One would expect that a black hole may have a strong in?uence on these structures and using our theorem presented here we can study this problem e?ectively in our following paper.

Moreover we hope that our construction works for non-stationary i.e.higher genus black holes as well giving an insight into the structure of such more general space-time manifolds.

5Acknowledegment

The work was partially supported by the Hungarian Ministry of Culture and Education(FKFP0125/1997).

6References

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G.Etesi:Black Hole Uniqueness9

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