---

Geometry of the 2+1 Black Hole

arXiv:gr-qc/9302012v1 10 Feb 1993Geometry of the 2+1 Black HoleM´aximo Ba˜nados1,2,∗, Marc Henneaux1,3,#,Claudio Teitelboim1,2,4,∗and Jorge Zanelli1,2,∗1Centro de Estudios Cient´ıficos de Santiago Casilla 16443, Santiago 9, Chile2Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile.3 Facult´e des Sciences, Universit´e Libre de Bruxelles, Belgium.4 Institute for Advanced Study, Olden Lane, Princeton, New Jersey 08540, USA.Submitted to Phys. Rev.

D.November 19920

AbstractThe geometry of the spinning black holes of standard Einstein theoryin 2+1 dimensions, with a negative cosmological constant and without cou-plings to matter, is analyzed in detail. It is shown that the black hole arisesfrom identifications of points of anti-de Sitter space by a discrete subgroup ofSO(2, 2).

The generic black hole is a smooth manifold in the metric sense.The surface r = 0 is not a curvature singularity but, rather, a singularity inthe causal structure. Continuing past it would introduce closed timelike lines.However, simple examples show the regularity of the metric at r = 0 to beunstable: couplings to matter bring in a curvature singularity there.

Kruskalcoordinates and Penrose diagrams are exhibited. Special attention is given tothe limiting cases of (i) the spinless hole of zero mass, which differs from anti-de Sitter space and plays the role of the vacuum, and (ii) the spinning holeof maximal angular momentum .

A thorough classification of the elements ofthe Lie algebra of SO(2, 2) is given in an Appendix.PACS numbers 04.20 Jb, 97-60. Lf.1

1IntroductionThe black hole is one of the most fascinating structures that has ever emerged outof the theory of gravitation. And yet, it would seem fair to say, we are far from fullyunderstanding it.

It is therefore fortunate that full-fledged black holes have beenfound to exist[1] in the transparent setting of 2+1 standard Einstein gravity[2].The purpose of this article is to study in detail the geometry of the 2+1 blackhole without electric charge[3]. These results on the black hole geometry were onlyannounced and briefly summarized in[1].The plan of the article is the following: Section 2 deals with the action principleand its Hamiltonian version.

The Hamiltonian is specialized to the case of axiallysymmetric time independent fields and the equations of motion are solved. Theresulting metric has two integration constants which are next identified as the massand angular momentum.

This identification is achieved through an analysis of thesurface integrals at spacelike infinity that must be added to the Hamiltonian in orderto make it well defined. It is then shown that for a certain range of values of themass and angular momentum the solution is a black hole.

This black hole is shownto be quite similar to its 3+1 counterpart -the Kerr solution. It has an ergosphereand an upper bound in angular momentum for any given mass.The discussion of section 2 focuses on the physical properties of the black holeand ignores a question that must have been needling the geometer hiding withinevery theorist.

The spacetime geometry of the black hole is one of constant negativecurvature and therefore it is, locally, that of anti-de Sitter space. Thus, the blackhole can only differ from anti-de Sitter space in its global properties.

More precisely,as we shall see, the black hole arises from anti-de Sitter space through identificationsof points of the latter by means of a discrete subgroup of its symmetry group[4].Section 3 is devoted to this issue. The identifications are explicitly given and are,in particular, used to show that the black hole singularity at r = 0 is not one inthe metric, which is regular there, but rather a singularity in the causal structure.Continuing past r = 0 would bring in closed timelike lines.

When there is no angularmomentum an additional pathology appears at r = 0, a singularity in the manifoldstructure of the type present in the Taub-NUT space. This is dealt with in AppendixB.2

Once the identifications are geometrically understood, we pass, in Section 4, toexhibit special coordinate systems which reveal the causal structure. In particular,Kruskal coordinates are defined and the Penrose diagrams are drawn.

Special issuespertaining to the extreme rotating black hole with non-zero mass and to the zeromass limit of a non-rotating hole ( “vacuum”) are analyzed. Section 5 is devotedto some concluding remarks, showing the instability of the regularity of the metricat r2 = 0 in the presence of matter.

It is also briefly discussed how “chronology isprotected” in the 2+1 black hole.The classification of the elements of the Lie algebra of the symmetry groupSO(2, 2) is given in Appendix A.2Action Principle, Equations of Motionand their Solutions.2.1Action PrincipleThe action in lagrangian form may be taken to beI = 12πZ √−ghR + 2l−2id2xdt + B′,(2.1)where B′ is a surface term and the radius l is related to the cosmological constantby −Λ = l−2. [ Note that, for convenience in what follows, the numerical factor(16πG)−1 in front of the action is taken to be (2π)−1, i.e., we set the gravitationalconstant G, which has the dimensions of an inverse energy, equal to 18].Extremization of the action with respect to the spacetime metric gµν(x, t), yieldsthe Einstein field equationsRµν −12gµν(R + 2l−2) = 0(2.2)which, in a three dimensional spacetime, determine the full Riemann tensor asRµνλρ = −l−2(gµλgνρ −gνλgµρ)(2.3)describing a symmetric space of constant negative curvature.One may pass to the hamiltonian form of (2.1), which reads3

I =Z hπij ˙gij −N⊥H⊥−NiHiid2xdt + B(2.4)The surface term B will be discussed below. It differs from the B′ appearing inthe lagrangian form because the corresponding volume integrals differ by a surfaceterm.

The surface deformation generators H⊥,Hi are given byH⊥=2πg−1/2(πijπij −(πii)2) −(2π)−1g1/2(R + 2/l2)(2.5)Hi=−2πji/j(2.6)Extremizing the hamiltonian action with respect to the the lapse and shift func-tions N⊥, Ni, yields the constraint equations H⊥= 0 and Hi = 0 which are the⊥, ⊥and ⊥, i components of (2.2). Extremization with respect to the spatial metricgij and its conjugate momentum πij, yields the purely spatial part of the secondorder field equations (2.2), rewritten as a hamiltonian system of first order in time.2.2Axially symmetric stationary fieldOne may restrict the action principle to a class of fields that possess a rotationalKilling vector ∂/∂φ and a timelike Killing vector ∂/∂t.

If the radial coordinate isproperly adjusted, the line element may be written asds2=−(N⊥)2(r)dt2 + f −2(r)dr2 + r2(Nφ(r)dt + dφ)20 ≤φ < 2π,t1 ≤t ≤t2(2.7)The form of the momenta πij may be obtained from (2.7) through their relationπij = −(1/2π)g−1/2(Kij −Kgij) with the extrinsic curvature Kij, which, for a time-independent metric, simply reads 2N⊥Kij = (Ni|j + Nj|i). This gives as the onlycomponent of the momentum,πrφ = l2πp(r)(2.8)If expressions (2.7), (2.8) are introduced in the action, one findsI = −(t2 −t1)ZdrhN(r)H(r) + NφHφi+ B(2.9)4

withH≡2πf(r)H⊥= 2l2p2r3 + (f 2)′ −2 rl2(2.10)Hφ=−2lp′(2.11)N(r)=f −1N⊥(2.12)2.3SolutionsTo find solutions under the assumptions of time independence and axial symmetry,one must extremize the reduced action (2.9). Variation with respect to N and Nφ,yields that the generators H and Hφ must vanish.

These constraint equations arereadily solved to givep=−J2lf 2=−M +rl2+ J24r2(2.13)where M and J are two constants of integration, which will be identified below asthe mass and angular momentum, respectively.Variation of the action with respect to f 2 and p yields the equationsN′=0(Nφ)′ + 2lpr3 N=0(2.14)which determine N and Nφ asN=N(∞)Nφ=−J2r2N(∞) + Nφ(∞)(2.15)The constants of integration N(∞) and Nφ(∞) are part of the specification ofthe coordinate system, which is not fully fixed by the form of the line element (2.7)(see below).5

2.4Surface integrals at infinity2.4.1Quick analysisWe will be interested in including in the variational principle the class of fields thatapproach our solution (2.13), (2.15) at spacelike infinity. This means that the actionshould have an extremum under variations of gij and πij that for large r approachthe variations of the expressions (2.13), for any δM and δJ and for fixed N(∞),Nφ(∞) .

However, as seen most evidently from the reduced form (2.9) of the action,upon varying gij and πij one picks up a surface term. That is, one findsδI=(t2 −t1)[N(∞)δM −Nφ(∞)δJ] + δB+(Terms vanishing when the equations of motion hold)(2.16)Now, one must demand that when the equations of motion hold, the variationof the action should be zero[5].

Therefore, the boundary term B in the action mustbe adjusted so as to cancel the first two terms on the right side of (2.16). Thus, weputB = (t2 −t1)(−N(∞)M + Nφ(∞)J)(2.17)Equation (2.17) identifies M as the mass and J as the angular momentum.

Thisis because they appear as conjugates to the asymptotic displacements N(∞) andNφ(∞). (The minus sign in front of N(∞) appears because, conventionally, oneintroduces a minus sign in the generator when the displacement is along a timelikedirection.) That Nφ is the angular displacement is evident.

However, the fact thatthe rescaled lapse N given by (2.12) appears in (2.17) rather than the original N⊥,deserves explanation. The reason is the following.

The normal component of thedeformation that joins the surface of time t and that of time t + δt is δξ = nN⊥δt,where n is the unit normal. But the unit normal does not approach a Killing vectorat infinity.

If one multiplies it by f, one obtains, at infinity, a Killing vector K = nfwhose norm K · K = −f 2 is independent of N(∞).The displacement N(∞)δt(“Killing time”) is the component of the deformation δξ along K.6

2.4.2Detailed analysisThe preceding argument gives a quick way of obtaining the surface integrals thatmust be added to the action.It also puts in evidence the physical meaning ofM and J. However, a more careful analysis is needed.

One knows that in a gaugetheory such as General Relativity the conserved quantities are related to the asymp-totic symmetry group. This fact already emerged in the previous discussion where“displacements at infinity” played the key role.For 2+1 spacetime dimensionswith a negative cosmological constant, this asymptotic group is infinite dimensionaland contains SO(2, 2) as a subgroup.

The asymptotic Killing vectors ∂/∂φ andK = N(∞)−1∂t that appeared above are two of the generators in the Lie alge-bra of SO(2, 2). Thus, what we have called “Killing time displacements” are not“translations” but -rather- SO(2, 2) boosts.The general analysis of the asymptotic symmetry group of 2+1 gravity has beengiven in [6].

We briefly recall here its key aspects and apply them to the presenttreatment.One considers all metrics that for large r becomeds2 −→−rl2dt2 +rl2dr2 + r2dφ2,(2.18)[There is no loss of generality -in this context- in taking N(∞) = 1 and Nφ(∞) = 0.One must only remember that for any given spacetime the surface integrals are tobe calculated in a coordinate system obeying these conditions. ]The precise way in which ds2 approaches (2.18) for large r is obtained by actingon the solution (2.13), (2.15), with all possible anti-de Sitter group transformations.The rationale for this procedure is that one wants to have at least SO(2, 2) asan asymptotic symmetry group.

This is because the metric (2.18) coincides withthe asymptotic form of the anti-de Sitter metric, which has SO(2, 2) as its (exact)symmetry group.The remarkable feature is that the resulting class of allowedasymptotic metrics admits a much larger symmetry group.The asymptotic symmetry group turns out to be the conformal group. The con-formal group may be defined as the group of all transformations that leave invariantthe cylinder at infinity, up to a Weyl rescaling.

The conformal Killing vectors obey7

ξα;β + ξβ;α −12gαβξλ;λ = 0(2.19)The Lie algebra of the conformal group consists of two copies of the Virasoroalgebra. Therefore the conserved charges of 2+1 gravity are two sets Ln, Kn ofVirasoro generators (n = 0, ±1, ±2, ....).

Of these, the six SO(2, 2) generators areL0, L1, L−1, K0, K1, K−1, which form a subalgebra .The Ln and Kn obey the Virasoro algebra with a central charge proportional tothe radius of curvature. One has, in terms of non-quantum Poisson brackets,[Ln, Lm]=−i{(n −m)Ln+m + l · n(n2 −1)δn,−m}[Kn, Km]=−i{(n −m)Kn+m + l · n(n2 −1)δn,−m}(2.20)[Ln, Km]=0In the normalization for the central charge that has become standard in stringtheory, one hasc = 12l/¯h(2.21)The metric given by (2.13), (2.15) has only two charges which are non-zero(M = K0 + L0, J = K0 −L0).

However, by acting with the asymptotic group onecan endow it with other charges, much as by boosting a Schwarzschild solution onemay endow it with linear momentum.2.5The Black HoleThe lapse function N⊥vanishes for two values of r given byr± = lM21 ±s1 − JMl21/2. (2.22)whereas g00 vanishes atrerg = lM1/2(2.23)These three special values of r obey8

r−≤r+ ≤rerg(2.24)Just as it happens in 3+1 dimensions for the Kerr metric, r+ is the black holehorizon, rerg is the surface of infinite redshift and the region between r+ and rerg isthe ergosphere. In order for the solution to describe a black hole, one must haveM > 0,|J| ≤Ml.

(2.25)In the extreme case |J| = Ml, both roots of N2 = 0 coincide. Note that the radiusof curvature l = (−Λ)−1/2 provides the length scale necessary in order to have ahorizon in a theory in which the mass is dimensionless.

If one lets l grow very largethe black hole exterior is pushed away to infinity and one is left just with the inside.The vacuum state, namely what is to be regarded as empty space, is obtainedby making the black hole disappear. That is, by letting the horizon size go to zero.This amounts to letting M →0, which requires J →0 on account of (2.25).

Onethus obtains the line elementds2vac = −(r/l)2dt2 + (r/l)−2dr2 + r2dφ2. (2.26)As M grows negative one encounters the solutions studied previously in [7].

Theconical singularity that they possess is naked, just as the curvature singularity ofa negative mass black hole in 3 + 1 dimensions. Thus, they must, in the presentcontext, be excluded from the physical spectrum.

There is however an importantexceptional case. When one reaches M = −1 and J = 0 the singularity disappears.There is no horizon, but there is no singularity to hide either.

The configurationds2 = −(1 + (r/l)2)dt2 + (1 + (r/l)2)−1dr2 + r2dφ2(2.27)(anti-de Sitter space) is again permissible.Therefore, one sees that anti-de Sitter space emerges as a “bound state”, sepa-rated from the continuous black hole spectrum by a mass gap of one unit. This statecannot be deformed continuously into the vacuum (2.26), because the deformationwould require going through a sequence of naked singularities which are not includedin the configuration space.9

Note that the zero point of energy has been set so that the mass vanishes whenthe horizon size goes to zero.This is quite natural.It is what is done in 3+1dimensions. In the past, the zero of energy has been adjusted so that anti-de Sitterspace has zero mass instead.

Quite apart from this difference, the key point is thatthe black hole spectrum lies above the limiting case M = 0.We now pass, in the next section, to a detailed study of the geometry of theblack hole.3Black Hole as Anti-de Sitter Space Factored bya Subgroup of its Symmetry GroupWe will show in this section that the black hole arises from anti-de Sitter spacethrough identifications by means of a discrete subgroup of its isometry group SO(2, 2).This implies that the black hole is a solution of the source-free Einstein equationseverywhere, including r = 0. As we shall also see, the type of “singularity” thatis found at r = 0 is -generically- one in the causal structure and not in the curva-ture, which is everywhere finite (and constant).

It should be emphasized that thisstatement means that r = 0 is not a conical singularity.To proceed with the analysis we first review the properties of anti-de Sitter space.3.1Anti-de Sitter Space in 2+1 Dimensions3.1.1MetricAnti-de Sitter space can be defined in terms of its embedding in a four dimensionalflat space of signature (−−++)ds2 = −du2 −dv2 + dx2 + dy2(3.1)through the equation−v2 −u2 + x2 + y2 = −l2. (3.2)A system of coordinates covering the whole of the manifold may be introducedby setting10

u = l cosh µ sin λ,v = l cosh µ cos λ(3.3)with l sinh µ = √x2 + y2 and 0 ≤µ < ∞, 0 ≤λ < 2π. Inserting (3.3) into (3.1)givesds2 = l2 −cosh2 µ dλ2 + dx2 + dy2l2 + x2 + y2!

(3.4)an expression that can be further simplified by passing to polar coordinates in thex −y planex = l sinh µ cos θ,y = l sinh µ cos θ,(3.5)which yieldsds2 = l2 h−cosh2 µdλ2 + dµ2 + sinh2 µdθ2i(3.6)for the metric of anti-de Sitter space.Because λ is an angle, there are closed timelike curves in anti-de Sitter space (forinstance µ = µ0, θ = θ0). For this reason, one “unwraps” the λ coordinate, that is,one does not identify λ with λ+2π.

The space thus obtained is the universal coveringof anti-de Sitter space. It is this space which, by a common abuse of language, willbe called anti-de Sitter space in the sequel.

If the unwrapped λ is denoted by t/land if one sets r = l sinh µ, one obtainsds2 = ((r/l)2 + 1)dt2 + ((r/l)2 + 1)−1dr2 + r2dθ2(3.7)which is the metric (2.7) with M = −1, J = 0 (and φ replaced by θ).3.1.2IsometriesBy construction, the anti-de Sitter metric is invariant under SO(2, 2). The Killingvectors areJab = xb∂∂xa −xa∂∂xb(3.8)where xa = (v, u, x, y) or, in detail11

J01=v∂u −u∂vJ02=x∂v + v∂xJ03=y∂v + v∂yJ12=x∂u + u∂xJ13=y∂u + u∂yJ23=y∂x −x∂y(3.9)The vector J01 generates “time displacements” (J01 = ∂λ) whereas J23 generatesrotations in the x −y plane (J23 = ∂θ). The most general Killing vector is given by12ωabJab,ωab = −ωba(3.10)and is thus determined by an antisymmetric tensor in R4.3.1.3Poincar´e CoordinatesThe coordinates defined byz =lu + x,β =yu + x,γ =−vu + x.

(3.11)are called Poincar´e coordinates. They only cover part of the space, namely just oneof the infinitely many regions where u + x has a definite sign (see Fig.

1). Thesecoordinates are therefore not well adapted to the study of global properties.Interms of (z, β, γ) the anti-de Sitter line element readsds2 = l2"dz2 + dβ2 −dγ2z2#.

(3.12)For u + x > 0 one has z > 0 and for u + x < 0 one has z < 0. One can also findanalogous Poincar´e coordinates for each of the regions where u −x has a definitesign.3.2Identifications3.2.1Identification subgroup associated with a Killing vectorAny Killing vector ξ defines a one parameter subgroup of isometries of anti-de SitterspaceP →etξP.

(3.13)12

The mappings of (3.13) for which t is an integer multiple of a basic “step”, takenconventionally as 2π,P →etξP,t = 0, ±2π, ±4π, ....(3.14)define what we will call the identification subgroupSince the transformations (3.14) are isometries, the quotient space obtained byidentifying points that belong to a given orbit of the identification subgroup, inheritsfrom anti-de Sitter space a well defined metric which has constant negative curvature.The quotient space thus remains a solution of the Einstein equations.The identification process makes the curves joining two points of anti-de Sitterspace that are on the same orbit to be closed in the quotient space. In order forthe quotient space to have an admissible causal structure, these new closed curvesshould not be timelike or null.A necessary condition for the absence of closedtimelike lines is that the Killing vector ξ be spacelike,ξ · ξ > 0(3.15)This condition is not sufficient in general.

However, as it will be shown in Sec. 3.2.5,it turns out to be so for the particular Killing vectors employed in the identificationsleading to the black hole.3.2.2Singularities in the causal structureThere are some Killing vectors that do fulfill (3.15) everywhere in anti-de Sitterspace, for example∂∂θ, where θ is the angular coordinate appearing in (3.6).However, the Killing vectors appearing in the identifications that give rise to theblack hole are timelike or null in some regions.

These regions must be cut out fromanti-de Sitter space to make the identifications permissible.The resulting space-which we denote (adS)’- is invariant under (3.13) because the norm of a Killingvector is constant along its orbits. Hence, the quotient can still be taken.The space (adS)’ is geodesically incomplete since one can find geodesics that gofrom ξ · ξ > 0 to ξ · ξ < 0.

From the point of view of (adS)’ -i.e., prior to theidentifications- it is quite unnatural to remove the regions where ξ ·ξ is not positive.However, once the identifications are made, the frontier of the region ξ · ξ > 0, i.e.,13

the surface ξ · ξ = 0, appears as a singularity in the causal structure of spacetime,since continuing beyond it would produce closed timelike curves.For this reason, the region ξ · ξ = 0 may be regarded as a true singularity in thequotient space. If this point of view is taken, -as it is done here- the only incompletegeodesics are those that hit the singularity, just as in the 3+1 black hole.

It shouldbe stressed that the surface ξ·ξ = 0 is a singularity only in the causal structure. It isnot a conical curvature singularity of the type discussed in [7].

Indeed, the quotientspace is smooth [8]. Its curvature tensor is everywhere regular and given byRµνλρ = −l−2(gµλgνρ −gνλgµρ).

(3.16)The fundamental group of the quotient space is non trivial and isomorphic tothe identification subgroup. The orbits of the Killing vectors define closed curvesthat cannot be continuously shrunk to a point.

The “origin” ξ · ξ = 0 is neithera point nor a circle. It is a surface.

The topology of ξ · ξ = 0, and also that ofthe whole quotient space, can be inferred by inspection of the Penrose diagram inFig. 4c.

One finds that the black hole is topologically R2 × S1 and that the surfaceξ · ξ = 0 has infinitely many connected pieces, each of which is a cylinder whosecircular sections are null.3.2.3Explicit form of the identificationsWe claim that the black hole solutions are obtained by making identifications of thetype described above by the discrete group generated by the Killing vectorξ = r+l J12 −r−l J03 −J13 + J23(3.17)where the Jab are given by (3.8). The antisymmetric tensor ωab defined by (3.17)through ξ = 12ωabJab, is easily verified to possess real eigenvalues, namely, ±r+/land ±r−/l.

The corresponding Casimir invariants I1=ωabωab and I2 = 12ǫabcdωabωcdareI1 = −2l2(r2+ + r2−) = −2M,I2 = −4l2r+r−= −2|J|l(3.18)14

According to the classification given in Appendix A the Killing vector (3.17) isof type Ib when r+ ̸= r−, of type IIa when r+ = r−̸= 0 and of type III+ whenr+ = r−= 0.To prove that the identifications by e2πkξ yield the black hole metric, we startby considering the non - extreme case r2+ −r2−> 0. In that case, by performing anSO(2, 2) transformation, one can eliminate the last term in (3.17) and replace ξ bythe simpler expressionξ′ = r+l J12 −r−l J03(3.19)This follows from the analysis of Appendix A, where it is shown that any SO(2, 2)element with unequal real eigenvalues can be brought into the form (3.19) by anSO(2, 2) transformation.

Alternatively, one may rewrite (3.17) in Poincar´e coordi-nates as−ξ = r+l z ∂∂z + β ∂∂β + γ ∂∂γ!−r−l β ∂∂γ + γ ∂∂β!+ ∂∂β(3.20)and observe that the shiftsβ→β −r+r2+ −r2−(3.21)γ→γ −r−r2+ −r2−(3.22)-which are SO(2, 2) isometries- eliminate∂∂β in (3.20).The norm of ξ′ is given byξ′ · ξ′ = r2+l2 (u2 −x2) + r2−l2 (v2 −y2)(3.23)or, using (3.2),ξ′ · ξ′ = r2+ −r2−l2(u2 −x2) + r2−(3.24)Accordingly, the allowed region where ξ′ · ξ′ > 0 is−r2−l2r2+ −r2−< u2 −x2 < ∞. (3.25)15

The region ξ′ · ξ′ > 0 can be divided in an infinite number of regions of threedifferent types bounded by the null surfaces u2−x2 = 0 or v2−y2 = l2−(u2−x2) = 0.These regions are:Regions of type I: Smallest connected regions with u2 −x2 > l2 and y and u ofdefinite sign. These regions have no intersection with y = 0 since this would violateu2 −x2 = l2 + y2 −v2 > l2.

These regions are called “the outer regions”. The normof the Killing vector fulfills r2+ < ξ′ · ξ′ < +∞.Regions of type II: Smallest connected regions with 0 < u2 −x2 < l2 andu and v of definite sign.

These regions are called “the intermediate regions”. Thenorm of the Killing vector fulfills r2−< ξ′ · ξ′ < r2+.Regions of type III: Smallest connected regions with −r2−l2r2+−r2−< u2 −x2 < 0and x and v of definite sign.

These regions are called “the inner regions” and onlyexist for r−̸= 0. They do not intersect the x = 0 plane.

The norm of the Killingvector fulfills 0 < ξ′ · ξ′ < r2−The frontiers between the various regions are lightlike surfaces (the horizons! ).Each region of type I has one region of type II in its future and one in its past.For r−̸= 0, two situations are found for each region of type II: (i) it has one regionof type II and two regions of type I in its future as well as one region of type IIand two regions of type III in its past, or conversely (ii) the same description withI and III interchanged.

Finally, each region of type III has one region of type IIin its future and another one in its past. This is shown in Figures (2.a,b,c).

Letus now choose three contiguous regions of types I, II and III (one of each type).In these regions we introduce a (t, r, φ)- parametrization as follows (we assume fordefiniteness u, y > 0 in I, u, −v > 0 in II and x, −v > 0 in III).Region I. r+ < r:u=qA(r) cosh ˜φ(t, φ)x=qA(r) sinh ˜φ(t, φ)y=qB(r) cosh ˜t(t, φ)v=qB(r) sinh ˜t(t, φ)(3.26)Region II. r−< r < r+:u=qA(r) cosh ˜φ(t, φ)16

x=qA(r) sinh ˜φ(t, φ)y=−q−B(r) sinh ˜t(t, φ)v=−q−B(r) cosh ˜t(t, φ)(3.27)Region III. 0 < r < r−:u=q−A(r) sinh ˜φ(t, φ)x=q−A(r) cosh ˜φ(t, φ)y=−q−B(r) sinh ˜t(t, φ)v=−q−B(r) cosh ˜t(t, φ)(3.28)In (3.26), (3.27) and (3.28) we have setA(r)=l2r2−r2−r2+−r2−,B(r) = l2r2−r2+r2+−r2−˜t=(1/l) (r+t/l −r−φ),˜φ = (1/l) (−r−t/l + r+φ)(3.29)In the coordinates t, r, φ, the metric becomesds2 = −(N⊥)2dt2 + (N⊥)−2dr2 + r2(Nφdt + dφ)2(3.30)with −∞< t < ∞, −∞< φ < ∞i.e., it is the black hole metric but with φ anon-periodic coordinate.

The Killing vector ξ′ readsξ′ = ∂∂φ(3.31)By making the identificationφ →φ + 2kπ,(3.32)one gets the black hole spacetime as claimed above.It is clear from the construction that the coordinate system t, r, φ does notcover the domain ξ′ · ξ′ > 0 entirely, since it only covers one region of each type. Ifr−= 0 (in which case region III does not exist), this is only half of one connectedcomponent of the domain ξ′ · ξ′ > 0.If r−̸= 0, each of the regions I, II and17

III should be repeated an infinite number of times to completely cover the domainξ′ · ξ′ > 0 which is now connected. This infinite pattern follows from the fact thatone is dealing with the universal covering space of anti-de Sitter space and this willreappear in the Penrose diagrams given below.It is worthwhile emphasizing that it is the identification (3.32) that makes theblack hole.

If one does not say that φ is an angle, one simply has a portion of anti-deSitter space and the horizon is just that of an accelerated observer[9].3.2.4Extreme caseThe above derivation cannot be repeated in the extreme case r+ = r−. This isbecause the Killing vector (3.17) is now of a different type than (3.19).

Accordingto the classification given in the Appendix, when r+ = r−, (3.17) is of type IIa,while (3.19) is of type Ib with doubly degenerate roots. Hence, there is no SO(2, 2)transformation mapping one to the other.One can nevertheless argue that the identifications for anti-de Sitter space gener-ated by (3.17) yield the extreme black hole without exhibiting the precise coordinatetransformation that brings ξ into the form ∂/∂φ.

The argument runs as follows. Themetric (3.30) is regular even if one sets r2+ = r2−.

When φ is not identified, it de-scribes a portion of anti-de Sitter space for any value of r2+−r2−> 0, hence it does soalso in the limit r+ −r−→0. Similarly, ∂/∂φ is a Killing vector for any value of r−and r+.

By continuity, its Casimir invariants remain equal to I1 = −2(r2+ + r2−)/l2and I2 = −4r+r−/l2 in the limit r+ −r−→0. Hence, in the extreme case the vector∂/∂φ remains type Ib (with coincident roots) or becomes type IIa, since these arethe only two types compatible with the given I1, I2.

It is the latter alternative thatis realized. Indeed, type Ib may be excluded by noticing that the correspondingKilling vector has constant norm equal to r2+, whereas ∂/∂φ has a space-dependentnorm equal to r2.

Thus ∂/∂φ must be of type IIa and, thus, equal to (3.17) [up toa possible SO(2, 2) transformation that leaves the metric invariant].The preceding argument already establishes that the black hole is obtained fromanti-de Sitter space by an identification. However, for completeness we exhibit achange of coordinates in terms of which the identification just makes a coordinateperiodic.

The required coordinate transformation can be explicitly given in Poincar´ecoordinates. We start with the case M = r+ = r−= 0 (“the vacuum”), which is the18

more illuminating one.For M = 0, the region ξ · ξ > 0 splits into disjoint regions which are just thePoincar´e patches u + x > 0 or u + x < 0. Hence, to describe a connected domainwhere ξ·ξ > 0, one can just consider a single Poincar´e patch.

In Poincar´e coordinatesthe Killing vector ξ is −∂∂β and hence, the identificationsβ →β + 2kπ(3.33)in (3.12) lead to the black hole metric with M = 0 upon setting z = 1/r,β = φand γ = t.[Note that as depicted in Fig 1, the horizon-singularity r = 0 are the null surfacesu + x = 0 delimiting the Poincar´e region. Because the Killing vector ξ is againspacelike on the other side of u + x = 0, one can continue the solution with zeromass through r = 0 to negative values of r without encountering closed timelikecurves.

By doing so one includes, however, the closed lightlike curves that lie on thenull surface u + x = 0, as well as some singularities in the manifold structure of thetype discussed in Appendix B. ]The coordinate transformation bringing the anti-de Sitter metric to the extremecase with M ̸= 0 (and non-periodic in φ) is more complicated.

One needs in thatcase more than one Poincar´e patch to cover the black-hole spacetime. Actually aninfinite number of sets of patches is necessary, with each set containing one patchof each of the four types u + x > 0, u + x < 0, u −x > 0, u −x < 0.

We merelygive here that transformation in one of the patches u + x > 0, for r > r+.β=12 Tl + φ + e2r+φ −12r+! (3.34)γ=12 Tl + φ −e2r+φ +12r+!

(3.35)z=" 12r+(r2 −r2+)#−1/2er+φ(3.36)where T is given byT = 2t −l2r+r2 −r2+(3.37)19

and fulfills dT = 2dt+ 2r+l2rdr(r2−r2+)2. By substituting (3.34)-(3.36) in the Poincar´e metric,one gets the extreme black hole metric (with Nφ adjusted so that Nφ(r+) = 0).3.2.5Absence of Closed Timelike CurvesWe now complete the argument that there are no closed causal curves in the blackhole solution.

That is, we show that there is no non-spacelike, future-directed, curvelying in the region ξ · ξ > 0 of anti-de Sitter space and joining a point and its imagegenerated by exp2πξ.Since the surfaces r = r+ and r = r−are null, a causal curve which leaves anyone of the regions of types I, II or III through r = r+ or r = r−can never re-enterit. Furthermore, since the images of a point are all in the same region as that point,it is sufficient to consider each of these regions separately.In each of the regions of types I, II or III, the anti-de Sitter metric takes theformds2 = −(N⊥(r))2dt2 + (N⊥(r))−2dr2 + r2(Nφdt + dφ)2(3.38)where φ goes from −∞to +∞.

Consider a causal curve t(λ) r(λ) and φ(λ), wherethe parametrization is such that the tangent vector (dt/dλ, dr/dλ, dφ/dλ) does notvanish for any value of λ. The causal property of the curve reads(N⊥)2 dtdλ!2−(N⊥)−2 drdλ!2−r2 Nφ dtdλ + dφdλ!2≤0.

(3.39)In order to join the point (t0, r0, φ0) and (t0, r0, φ0 + 2kπ), the causal curvewould have to be such that dt/dλ = 0 for some value of λ, since t comes back to itsinitial value. But then, if (N⊥)2 > 0 it follows from (3.39) that dr/dλ = dφ/dλ = 0,leading to a contradiction.

Similarly, if (N⊥)2 < 0 (region II), the fact that dr/dλ =0 for some value of λ implies dt/dλ = dφ/dλ = 0, and the required contradiction. ✷It should be observed that if one were to admit the region ξ·ξ ≤0 in the solution,one could leave and re-enter the regions of type III through the surface ξ · ξ = 0,which is timelike for J ̸= 0.

(This is not possible when J = 0 because the surfaceξ · ξ = 0 is then null.) One would find that there are also closed timelike curvespassing through points in region III.

The boundary between the region where thereare no closed causal curves and the region in which there are is then the null surface20

r = r−.From the point of view of an outside observer staying at r > r+, theinclusion or non-inclusion of the region ξ · ξ ≤0 is irrelevant and cannot be probedsince the surface r = r+ remains in all cases an event horizon.3.2.6Black Hole has only two Killing vectorsThe black hole metric was obtained in Sec. 3.2.3 under the assumption of existenceof two commuting Killing vectors ∂/∂t and ∂/∂φ.

One may ask whether there areany other independent Killing vectors. The answer to this question is in the negativeas we now proceed to show.Before any identifications are made one has the six independent Killing vectorsJab of anti-de-Sitter space.

However, after the identifications, not all the correspond-ing vector fields will remain single valued in the quotient space.A necessary and sufficient condition for an adS vector field η to induce a welldefined vector field on the quotient space is that η be invariant under the transfor-mation of the identification subgroup,(exp2πξ)∗η = η(3.40)For a Killing vector, this condition becomes(exp2πξ)η(exp2πξ)−1 = η(3.41)i.e. [exp2πξ, η] = 0(3.42)where ξ and η are viewed as so(2, 2) matrices.Now, the matrix ξ can be decomposed asξ = s + n(3.43)where (i) s and n commute, (ii) s is semi-simple with real eigenvalues; and (iii)n is nilpotent (see Appendix A).

Accordingly, the semi-simple part of (exp2πξ) isexp2πs and its nilpotent part is (exp2πs)[(exp2πn) −1]. Any matrix commutingwith (exp2πξ) must thus separately commute with (exp2πs) and (exp2πn) (thesemi-simple and nilpotent parts of a matrix can be expressed polynomially in termsof that matrix).

This implies both21

[s, η] = 0(3.44)(because the eigenvalues of the matrix exp2πs are real and positive, any matrixcommuting with it must also commute with log(exp2πs) = 2πs) and[n, η] = 0(3.45)(the nilpotent matrix n can be expressed polynomially in terms of the nilpotentmatrix [(exp2πn) −1] and must thus commute with η). It follows from (3.44) and(3.45) that ξ and η commute,[ξ, η] = 0.

(3.46)The problem of finding all the Killing vectors of the black hole solution is thusequivalent to that of finding all the elements of the Lie algebra so(2, 2) that commutewith ξ.In order to solve equation (3.46) for η, we observe that so(2, 2) = so(2, 1) ⊕so(2, 1) and decompose accordingly ξ into its self-dual and anti-self-dual parts,ξ = ξ+ + ξ−(3.47)Similarly,η = η+ + η−(3.48)The Equation (3.46) is equivalent to[ξ+, η+] = 0,[ξ−, η−] = 0,(3.49)because self-dual and anti-self-dual elements automatically commute. Now, the onlyelements of so(2, 1) that commute with a given non-zero element of so(2, 1) are themultiples of that element.

Therefore, since ξ+ and ξ−are both non-zero for all valuesof the black hole parameters we conclude from (3.49)η+ = αξ+,η−= βξ−α, β ǫ R(3.50)this shows that the most general Killing vector is a linear combination of ∂/∂t and∂/∂φ.22

4Global StructureThe study of global properties of the 2+1 black hole reveals a strong coincidencewith the 3+1 case. The Penrose diagrams and maximal extensions are exactly thesame as those of a 3+1 black hole immersed in anti-de Sitter space.4.1Kruskal coordinatesWe follow the analysis of [10].

For the line elementds2 = −(N⊥)2dt2 + (N⊥)−2dr2 + r2(Nφdt + dφ)2(4.1)one may introduce a Kruskal coordinate patch around each of the roots of (N⊥)2 = 0to bring the metric to the formds2 = Ω2(du2 −dv2) + r2(Nφdt + dφ)2,(4.2)where t = t(u, v).If there is only one root (J = 0) then the Kruskal coordinates cover the wholespace. When two roots coincide, there are no Kruskal coordinates [11].For definiteness, we start with r+.

The Kruskal coordinates around r+ are definedbyPatch K+:r−< r ≤r+U+= −r+r+r+r+ r+r−r−r−r−/r+1/2sinh a+tV+=−r+r+r+r+ r+r−r−r−r−/r+1/2cosh a+t(a)r+ ≤r < ∞U+=r−r+r+r+ r+r−r−r−r−/r+1/2cosh a+tV+= r−r+r+r+ r+r−r−r−r−/r+1/2sinh a+t(b)(4.3)witha+ = r2+ −r2−l2r+,(4.4)The angular coordinate (denoted φ+) is chosen on K+ so that the constant ofintegration appearing in the solution of (2.14) is fixed to giveNφ(r+) = 0. (4.5)23

The metric takes the form (4.2), with the conformal factorΩ2(r) = (r2 −r2−)(r + r+)2a2+r2l2 r −r−r + r−!r−/r+r−< r < ∞. (4.6)With the choice of φ leading to (4.5), the term Nφdt in (4.2) remains regular atr+.Similarly, around r−, one definesPatch K−:0 < r ≤r−U−=−r+r−r+r− r+r+−r+r+r+/r−1/2cosh a−tV−=−r+r−r+r− r+r+−r+r+r+/r−1/2sinh a−t(a)r−≤r ≤r+U−=r−r−r+r− r+r+−r+r+r+/r−1/2sinh a−tV−=r−r−r+r− r+r+−r+r+r+/r−1/2cosh a−t(b)(4.7)witha−= r2−−r2+l2r−.

(4.8)This time, one chooses the angular coordinate φ−so that Nφ(r−) = 0. The metrictakes the form (4.2) withΩ2(r) = (r2+ −r2)(r + r−)2a2−r2l2 r+ −rr+ + r!r+/r−0 < r < r+.

(4.9)The overlap of the patches K+ and K−(r−< r < r+) will be called K. Justas in the 3+1 case one may maximally extend the geometry by glueing together aninfinite number of copies of patches K+, K−. We will not illustrate graphically thatextension in terms of Kruskal coordinates, but will rather go to the more economicalPenrose diagrams.4.2Penrose diagrams (r+ ̸= r−)The Penrose diagrams are obtained by the usual change of coordinatesU + V = tanp + q2U −V = tanp −q2.

(4.10)24

We define the inverse transformation by taking the usual determination of the inversetangent, namely the one that lies between −π/2 and +π/2.Consider first the case J = 0. From (4.10) and (4.3) (with r−= 0) it is easy toprove that, (i) r = ∞is mapped to the lines p = ±12π, (ii) the singularity r = 0is mapped to the lines q = ±12π and (iii) the horizon is mapped to p = ±q.

TheKruskal and Penrose diagrams associated with this geometry are shown in Fig.3.Next consider the case of the rotating black hole. By making the change of co-ordinates (4.10) in the two patches defined in Sec.

(4.1) we find one Penrose diagramfor each patch. These are shown in Figs.

(4a, b).The regions shown as K in parts (a) and (b) of Fig.4 are to be identified becausethey are the overlap. Now, the original black hole coordinates covered K and oneregion III in (4.a), and K and one region I in (4.b).

However, one wants to obtaina “maximal causal extension” (i.e., a maximal extension without closed timelikecurves). To this effect one must first include the other two regions in each diagramand then glue together an infinite sequence of them, as shown in Fig.

(4.c).4.3Extreme cases M = 0 and M = |J|/l4.3.1M = 0The metric isds2 = −(r/l)2dt2 + (r/l)−2dr2 + r2dφ2. (4.11)Defining the null dimensionless coordinatesu = tl −lr,v = −tl −lr(4.12)we findds2 = r2dudv + r2dφ2.

(4.13)and pass directly to Penrose coordinates byU = tan 12(p + q),V = tan 12(p −q). (4.14)The relation between the radial coordinate r and p, q is25

−r = lcos p + cos qsin p,(4.15)and the metric takes the formds2 = l2dp2 −dq2sin2 p+ r2dφ2. (4.16)From (4.15) it is easy to show that the origin is mapped to the segment of the linesp = π ± q running from p = 0 to p = π while spacelike infinity is mapped to thesegment of the p = π line that closes the triangle shown in Fig.

(5a).4.3.2M = |J|/lThe metric isds2 = −(r2 −r2+)2r2l2dt2 +r2l2(r2 −r2+)2dr2 + r2(Nφdt + dφ)2(4.17)where r = r+ = l(M/2)1/2 is the horizon. Introducing the null coordinates U = t+r∗and V = −t + r∗where r∗is the tortoise coordinater∗=Zdr(N⊥)2 =−rl22(r2 −r2+) + l24r+lnr −r+r + r+(4.18)and defining the Penrose coordinates p, q as in (4.14) we obtain the line elementds2 = 4(N⊥)2l2(dp2 −dq2)(cos p + cos q)2+ r2(Nφdt + dφ)2.

(4.19)Fromsin pcos q + cos p =−rl2(r2 −r2+) +l4r+lnr −r+r + r+ ,(4.20)one sees that the lines r = r+ are at ±45◦, whereas r = 0 is at p = (kπ)+ and r = ∞at p = (kπ)−. [By p = (kπ)+, we mean that r →0 as p →kπ from value greaterthan kπ, and similarly, r →∞as p →kπ from values smaller than kπ].

If we take forp the usual determination of the arc tangent in (4.14), so that the region 0 < r < r+is mapped on the triangle bounded by p = 0 (r = 0) and p = q = π, p −q = π, thenwe must take in the region r > r+ a different determination. Indeed, one must gluethe triangle corresponding to r > r+ to the triangle corresponding to 0 < r < r+along the sides r = r+ at 45◦, and not along the vertical sides (which are r = ∞in26

the region r > r+ and r = 0 in the region r < r+). For instance, one could mapr > r+ into the triangle bounded by p + q = π, p −q = −π and p = π.

Once this isdone, one can go safely across r = r+ because the zero of N⊥in (4.19) is cancelledby the zero in the denominator. To achieve the maximal extension one then needsto include an infinite sequence of triangles as shown in Fig.

(5.b) (the original blackhole geometry just included two adjacent triangles).5Instability of metric regularity at r2 = 0.Chronology ProtectionThe point of view taken in this article is that the region r2 < 0 must be cut outfrom the spacetime because it contains closed timelike lines (see Fig.6 for a Penrosediagram that includes the forbidden region). This is a consistent point of view andleads to a close analogy with the black hole in 3+1 dimensions.

There is, however, acompelling additional argument for considering the spacetime as ending at r = 0. Itis the fact that the introduction of matter produces a curvature singularity at r = 0.This can be easily seen in simple examples and we believe it to be a general feature(with the possible exception of very “fine-tuned” couplings).

The first example isthe collapse of a cloud of dust with J = 0 [12]. One can then verify that the matterwill reach infinite density at r=0.

In this case only the part of the surface r = 0 thatintersects the history of the dust becomes singular. This is due to the fact that thedust “probes” only part of the spacetime.

However, in the case of a field- such asthe electromagnetic field - which is our second example - all the spacetime is probed.As it was indicated in[1], the introduction of a Maxwell field that depends only onthe radial coordinate yields an electromagnetic field for which the gauge invariantscalar FµνF µν is proportional to r−2 and thus is singular at all points on the surfacer = 0.Therefore, in view of the curvature singularities that are brought in by mattercouplings, it seems not only reasonable, but also compulsory, to exclude the regionr2 < 0 from the spacetime.The collapsing dust is also interesting in that it may be regarded as a mechanismfor producing, without effort, closed timelike lines from a perfectly reasonable initialcondition ( with the help of a negative cosmological constant though!). However,27

one sees, first of all, that the closed timelike lines are hidden behind the horizon atr = r+ > 0 (Fig. 7).

But, moreover, if - say - an electromagnetic field is brought in,a barrier of infinite curvature is introduced at r = 0. This makes the closed timelikelines not reachable from r2 > 0.

In this sense we see that “chronology is protected”[13] in the 2+1 black-hole.28

AcknowledgementsInformative discussions with Steven Carlip, Frank Wilczek, and Edward Wit-ten are gratefully acknowledged. M. B. holds a Fundaci´on Andes Fellowship andM.H.

gratefully acknowledges the hospitality of the Institute for Advanced Studywhere the research reported in this paper was partially carried out.This workwas supported in part by grants 0862/91 and 0867/91 of FONDECYT (Chile),grant PG/082/92 of Departamento Postgrado y Post´ıtulo, Universidad de Chile,by research funds from F.N.R.S (Belgium), by a European Communities researchcontract, and by institutional support provided by SAREC (Sweden) and EmpresasCopec (Chile) to the Centro de Estudios Cient´ıficos de Santiago.29

Appendix A. One Parameter Subgroups of SO(2, 2)A.1Description of the problemThe purpose of this Appendix is to provide a complete classification of the in-equivalent one-parameter subgroups of SO(2, 2).Two one-parameter subgroups{g(t)} and {h(t)}, tǫR, are said to be equivalent if and only if they are conjugatein SO(2, 2), i.e.,g(t) = k−1h(t)k,kǫSO(2, 2)(A.1)By an SO(2, 2) rotation of the coordinate axes in R4, one can then map g(t) onh(t).

Since one-parameter subgroups are obtained by exponentiating infinitesimaltransformations, the task at hand amounts to classifying the elements of the Liealgebra so(2, 2) up to conjugation.Now, the elements of so(2, 2) are described by antisymmetric tensors ωab = −ωbain R4. If one conjugates the infinitesimal transformation Ra b = δa b + εωa b byk ǫSO(2, 2), (kTηk = η,η = diag(−−++)), one finds that the antisymmetricmatrix ω ≡(ωab) transforms asω →ω′ = kTωk, kǫSO(2, 2)(A.2)Hence we have to classify antisymmetric tensors under the equivalence relation(A.2).A.2StrategyAny linear operator M can be uniquely decomposed as the sum of a semi-simple(diagonalizable over the complex numbers) linear operator S and a nilpotent oper-ator N that commute,M = S + N,(A.3)[S, N] = 0(A.4)withNq = 0for some q(A.5)30

andS = L−1(diagonal matrix)L, for some L(A.6)(Jordan - Chevalley decomposition of M).The eigenvalues of S coincide with those of M and provide an intrinsic character-ization of S. When the eigenvalues of S are non-degenerate, the nilpotent operatorN is identically zero and M is thus completely characterized (up to similarity) byits eigenvalues. However, if some eigenvalues are repeated, N may be non-zero andM cannot be reconstructed from the knowledge of its eigenvalues: one needs alsoinformation about its nilpotent part (the dimensions of the irreducible invariantsubspaces).We shall construct the sought-for invariant classification of elements of so(2, 2) bymeans of the Jordan - Chevalley decomposition of the operator ωa b.Since ηab ̸= δabfor SO(2, 2), the operator iωa b is, in general, not hermitian.

Accordingly, it maypossess a non-trivial nilpotent part when its eigenvalues are degenerate. The classi-fication of the possible ωa b is analogous to the invariant classification of the electro-magnetic field in Minkowski space and is also reminiscent of the Petrov classificationof the Weyl tensor in General Relativity.Because the matrix ωab is real and antisymmetric, there are restrictions on itseigenvalues.

These constraints are contained in the following elementary Lemmas.Lemma 1: If λ is an eigenvalue of ωab, then −λ is also an eigenvalue of ωab.Proof: From(ωab −ληab)lb = 0(A.7)one infers the characteristic equationdet(ω −λη) = 0(A.8)But then 0 = det(ω −λη)T = det(−ω −λη) = det(ω + λη), i.e., −λ is also a rootof the characteristic equation.✷Lemma 2: If λ is an eigenvalue, then λ* is also an eigenvalue.31

Proof: This is a consequence of the reality of ωab, which implies that the char-acteristic equation (A.8) has real coefficients.✷A.2.1Types of eigenvaluesIt follows from these theorems that the four eigenvalues of ω are of the followingfour possible types:1. λ, −λ, λ∗, −λ∗,λ = a + ib,a ̸= 0 ̸= b2. λ1 = λ∗1, −λ1, λ2 = λ∗2, −λ2,(λ1 and λ2 real)3. λ1, −λ1 = λ∗1, λ2, −λ2 = λ∗,(λ1 and λ2 imaginary)4. λ1 = λ∗1, −λ1, λ2, −λ2 = λ∗2,(λ1 real, λ2 imaginary)In each case, the eigenvalues involve only two independent real numbers, whoseknowledge is equivalent to knowing the two Casimir invariants.I1 = ωabωab,I2 = 12ǫabcdωabωcd(A.9)[If one replaces SO(2, 2) by SO(4), iωa b is hermitian and therefore diagonalizable.Hence there is no nilpotent part and iωa b is completely characterized by its eigen-values and thus by I1 and I2.

]Multiple roots can occur only in the following circumstances:• Cases (2) and (3), when λ1 = λ2 (or −λ2). If λ1 ̸= 0, then λ1 and −λ1 aredistinct roots.

If λ1 = 0, then 0 is a quadruple root; or• Cases (2),(3) or (4), when one of the roots vanishes.A.2.2Types of antisymmetric tensorsFor simple roots, one can give a unique canonical form to which any matrix ωabwith a given set of eigenvalues can be brought to by an SO(2, 2) transformation.This is the form of ωab in the basis where ωa b is diagonal. In the presence of multipleroots, there are inequivalent canonical forms because ωa b may contain a non-trivial32

nilpotent part N. But for each possible type of N, there is a unique canonical form.These canonical forms are all derived in the next subsections.We shall say that the matrix ωab is of type k if its nilpotent part is of orderk, Nk = 0. The types I and II can be further classified according to the realityproperties of the roots.

We thus define:Type I (N = 0)Ia: 4 complex roots λ, −λ, λ∗, −λ∗(λ ̸= ±λ∗).Ib: 4 real roots λ1, −λ1, λ2, −λ2.Ic: 4 imaginary roots λ1, −λ1, λ2, −λ2.Id: 2 real (λ1 and −λ1), and two imaginary roots (λ2 and −λ2).Type II (N ̸= 0, N2 = 0)IIa: 2 real double roots, λ and −λ.IIb: 2 imaginary double roots, λ and −λ.IIc: 1 double root (0) and 2 simple roots (λ and −λ, with λ real or imaginary. )Type III (N2 ̸= 0, N3 = 0): one quadruple root, zero.Type IV (N3 ̸= 0, N4 = 0): one quadruple root, zero.We shall write in all casesλ = a + ib(A.10)We close this section by proving the following useful Lemma.Lemma 3: Let va and ua be eigenvectors of ωa b with respective eigenvalues λand µ,ωabvb = λva,ωabub = µua.

(A.11)Then vaua = 0 unless λ + µ = 0. In particular, if λ ̸= 0, then va is a null vector.Proof: One has uaωa bvb = λuava = −µuava, and thus (λ + µ)uava = 0.

✷We now proceed to the explicit determination of the canonical forms.A.3Type IaOne has by definition of type Ia,ωablb=λla(A.12a)33

ωabmb=−λmb(A.12b)ωabl∗b=λ∗l∗a(A.12c)ωabm∗b=−λ∗m∗a(A.12d)where the eigenvectors la, l∗a, ma, ma∗are complex and linearly independent. Theonly scalar products that can be different from zero are lama and la∗m∗a.

They cannotvanish since the metric would then be degenerate. By scaling ma if necessary onecan assume lama = 1.

One then has also la∗m∗a = 1. The metric is given byηab = lamb + l∗am∗b + [a ↔b](A.13)since(ηab −lamb −l∗am∗b −[a ↔b]) ubis zero whenever ua equals la, ma, la∗, mb∗.

The tensor ωab is given byωab = λ(lamb −lbma) + λ∗(l∗am∗b −l∗bm∗a)(A.14)because this reproduces (A.12a)-(A.12d).Our goal is to achieve a canonical expression for ωa b over the real numbers.Therefore we decompose the vectors la and ma into their real and imaginary com-ponentsla = ua + iva, ma = na + iqa,(A.15)(the transformation la, ma, l∗a, m∗a →ua, va, na, qa is invertible and so, the vectorsua, va, na, and qa form a basis). This givesηab=2(uanb −vaqb) + [a ↔b](A.16)ωab=2a(uanb −vaqb) −2b(uaqb + vanb) −[a ↔b](A.17)In the orthonormal basis where the vectors ua, va, na, qa, have components ua =(0, 12, 12, 0), na = (0, −12, 12, 0), va = ( 12, 0, 0, 12),and qa = ( 12, 0, 0, −12), ωab take theform34

ωab=0b0a−b0a00−a0b−a0−b0(A.18)Eq. (A.18) is the canonical form of an antisymmetric tensor of type Ia.

TheCasimir invariants are found from (A.9) to beI1=4(b2 −a2))(A.19a)I2=4(b2 + a2))(A.19b)A.4Type IbOne has, by definition of type Ib,ωablb=λ1la, ωabmb = −λ1ma(A.20a)ωabnb=λ2na, ωabub = −λ2ua(A.20b)The vectors la, ma, na, and ua are real and linearly independent, and the non-vanishing scalar products are l · m and n · u. Straightforward steps yield then, in anorthonormal basis, the canonical formωab=000−λ200−λ100λ100λ2000(A.21)The Casimir invariants are given by35

I1=−2(λ21 + λ22),(A.22a)I2=4λ1λ2. (A.22b)A.5Type IcOne has, by definition of type Icωablb=ib1la, ωablb∗= −ib1l∗a(A.23a)ωabmb=ib1ma, ωabmb∗= −ib1m∗a(A.23b)The only non-vanishing scalar products are lala∗and mama∗.

One can rescale laand ma so that l · l∗= ±1, m · m∗= ∓1. If l · l∗= 1, then m · m∗= −1 and viceversa.

[Through la =1√2(ua + iva), one associates to a vector la obeying lala∗= 1,two real vectors ua, va , such that uaua = 1 = vava,uava = 0. So, if lala∗= 1,one must have mama∗= −1 in order to agree with the signature (−−++) of themetric.

]One obtains the final canonical formωab=0b100−b1000000b200−b20(A.24)for ωab in a real orthonormal basis.The Casimir invariants are found to beI1=2(b21 + b22),(A.25a)I2=4b1b2. (A.25b)36

A.6Type IdType Id does not exist. Indeed, the real eigenvalue brings a block of signature(+−), while the imaginary eigenvalue brings a block of signature (++) or (−−).This is inconsistent with signature (−−++).A.7Role of the Casimir invariants for type IIf one compares (A.19a), (A.19b), (A.22a), (A.22b) and (A.25a), (A.25b), onesees that the Casimir invariants completely characterize the matrices ωab of typeI.

If I1±I2 are both positive, the type is type Ic. If I1±I2 are both negative, thetype is type Ib.

Otherwise, the type is Ia. Furthermore, the eigenvalues can bereconstructed from I1 and I2.

The roots are degenerate when I1+I2 or I1−I2 vanish.It is easy to see that I1±I2 are the Casimir invariants of the two algebras so(2, 1)contained in so(2, 2, ) = so(2, 1) ⊕so(2, 1). The self-dual and anti-self-dual (real)matrices ω±ab = ωab ± 12ǫab cdωcd define irreducible representations of so(2, 2) (ω+abtransforms as a vector under the first so(2, 1), while ω−ab transforms as a vectorunder the second.) One has 2I1 = ω+abω+ab and 2I2 = ω−abω−ab.

There is however, noparticular advantage in working with the self-dual and anti-self-dual components ofωab in the subsequent discussion. For that reason, we shall not perform the split.A.8Type IIaBy definition of type IIa, there are two doubly degenerate, non-zero, real eigen-values λ and −λ.

Each eigenvalue has at least one eigenvector, thus one can find laand ma such thatωablb=λla(A.26a)ωabmb=−λma(A.26b)Within each invariant subspace we can introduce an additional vector to completel, m to a basis. Since ωa b has a nilpotent part, at least one of the additional vectorswill not be an eigenvector.

We can thus write, without loss of generality,37

ωabub=λua + la(A.27a)ωabsb=−λsa + αma(A.27b)It follows from (A.26a), (A.26b) and (A.27a), (A.27b) that l·l = l·m = l·u = 0.Hence, since the metric is non-degenerate we must have l · s ̸= 0. This implies inturn that α must be different from zero since (A.27a), (A.27b) gives l·s+αm·u = 0.By a rescaling of m we can set α = 1, so one hasωabsb = −λsa + ma(A.28)The remaining scalar products are evaluated as follows.First, one can takeuasa = 0 since one can redefine ua →ua + ρla without changing any of the previousrelations.

Second, by multiplying (A.27a) with ua, one gets, using uala = 0, thatuaua = 0. One then finds from (A.28) uama = −1 as the only remaining non-vanishing scalar product.The metric and antisymmetric tensor ωab readηab=lasb −maub + [a ↔b](A.29a)ωab=λ(lasb −uamb) −lamb −[a ↔b](A.29b)In a suitable orthonormal frame, this givesωab=011λ−10λ1−1−λ01−λ−1−10(A.30)When λ ̸= 0, a simpler, equivalent canonical form, can be achieved by replacingma by m′a + la/2λ and sa by s′a + ua/2λ.

This leaves ηab unchangedηab = las′b −m′aub + [a ↔b],(A.31)38

but modifies ωab toωab = λ(las′b −uam′b) + la(ub −m′b) −[a ↔b],(A.32)which, in an appropriate orthonormal frame, yieldsωab=000λ00λ10−λ01−λ−1−10(A.33)The forms (A.30) and (A.33) are not equivalent when λ = 0. It is only (A.30) thatis if type IIa in that case, since (A.33) with λ = 0 possesses a non trivial nilpotentpart of order 3 and is thus of type III.

The Casimir invariants are found to beI1=−4λ2,(A.34a)I2=4λ2. (A.34b)i.e., they are exactly the same as those of (A.22a), (A.22b) with λ1 = λ2.

However,the canonical forms (A.30) or (A.33) are not equivalent to (A.21) with λ1 = λ2 sincethey possess a non trivial nilpotent part , while (A.21) does not for any value of λ1,λ2.A.9Types IIb and IIcThe analysis of type IIb proceeds as for type IIa.We only quote the finalcanonical form in an orthonormal basisωab=0b −1−10−b + 1001100b + 10−1−b −10(A.35)39

and the Casimir invariantsI1 = 4b2,I2 = 4b2. (A.36)Type IIc is incompatible with a non-degenerate metric and so it does not exist.Indeed, the equations ωablb = 0, ωabmb = la (0 is a double root and ωab is a nontrivial nilpotent matrix in the corresponding invariant eigenspace), together withωabub = λua, ωabvb = −λva imply l · l = l · ω · m = −(lω) · m = 0, l · m = ω · m = 0,l · u = λ−1l · ω · u = 0, l · v = −λ−1l · v = 0.

So la would be a non zero vectororthogonal to any vector and the metric would be degenerate.A.10Types III and IVIn type III, zero is a quadruple root of the characteristic equation. Since ωa b isnilpotent of order 3, one can find a basis such thatωablb = 0(A.37a)ωabmb = 0,ωabub = ma,ωabtb = ua.

(A.37b)The scalar product of la with ua vanishes from (A.37b). Similarly, m · m =m · u = 0.

Hence m · t cannot vanish, say m · t = ±1. Then, by a redefinition ofla, la →la + ρma, one can assume l · t = 0.

It follows that l · l ̸= 0 since otherwisethe metric would be degenerate. We set l · l = −ε, ε = ±1.

By making appropriateredefinitions of ta if necessary and using the fact that the metric is of signature(−−++), one finally obtainsηab=ε(−lalb −matb −tbma + uaub)(A.38a)ωab=ε(maub −uamb)(A.38b)This yields in an appropriate orthonormal basisType III+ (ε = +1).40

ωab=0000000100010−1−10(A.39)Type III−(ε = −1)ωab=0−1−10100010000000(A.40)The two Casimir invariants vanish for type III and yet the matrix ωab is not zero.Type IV does not exist. Indeed for the case of nilpotency of order 4, one hasωablb = 0, ωabmb = la, ωabub = ma and ωabtb = ua.

By taking the scalar product ofthe equation with la, one finds l · l = l · m = l · u = 0. So l · t ̸= 0, say l · t = k.But then u · m = m · ωt = −l · t ̸= 0 (from the last relations), while the equationsωabub = ma and the antisymmetry of ωabub imply u · m = 0.

This contradictionshows that type IV is inconsistent.41

A.11Summary of ResultsWe summarize our results by giving for each type the canonical form of theKilling vector (1/2)ωabJab and the corresponding Casimir invariants in a table.TypeKilling vector14I114I2Iab(J01 + J23) −a(J03 + J12)b2 −a2b2 + a2Ibλ1J12 + λ2J03−12(λ21 + λ22)λ1λ2Icb1J01 + b2J2312(b21 + b22)b1b2IIaλ(J03 + J12) + J01 −J02 −J13 + J23−λ2λ2orλ(−J03 + J12) −J13 + J23 (λ ̸= 0)−λ2−λ2IIb(b −1)J01 + (b −1)J23 + J02 −J13b2b2III+−J13 + J2300III−−J01 + J0200Table 1. Classification of one-parameter subgroups of SO(2, 2)Note that for the second canonical form of type IIa, valid when λ ̸= 0, we havereplaced J03 by −J03 to comply with the form given in the text.

This amounts toreplace λ2 by −λ2, and can be acheived by redefinining ξ0 as −ξ0. This is why thesecond Casimir invariant, which is not parity-invariant, changes its sign.The cases of interest for the black hole are Ib, IIa and III+, for which theeingenvalues of ωab, namely ±r+/l and ±r−/l are all real.

(These cases exist onlybecause the signature of the metric is (−−++)). Type Ib (with λ1 ̸= λ2) describea general black hole with |J| < Ml, type IIa describes an extreme black hole withnon-zero mass, while type III+ describes the ground state with M = 0.

The typebecomes more and more special [from four distinct real roots to one single real root(zero)] as one goes from the general black hole to the ground state.It is interesting to notice that if one expresses r+ and r−as functions of J and Mand goes beyond the extreme limit |J| = Ml, the roots r+ and r−become complexconjugates. This strongly suggests that type Ia describes the spacetime whose metricis obtained by setting |J| > Ml in the black hole line element.

On the other hand,if one keeps |J| < Ml and takes M < 0, the roots r+ and r−become two differentpurely imaginary numbers. This strongly suggests that there is a close relationshipbetween type Ic and the negative mass solutions of [5].42

Finally, on an even more parenthetical note, we mention that for the Euclideanblack hole the group SO(2, 2) is replaced by SO(3, 1). In that case the eigenvaluesof ωa b are of the form (a, −a, ib, −ib) with real a and b.

This form may be obtainedfrom that of type Ib above by setting MEuc = M, JEuc = −iJ in the formula (2.22),expressing the eigenvalues in terms of M and J. This is just the prescription forthe (real) Euclidean continuation of the Minkowskian signature black hole [see, forexample[1].Appendix B. Smoothness of the Black HoleGeometryThis Appendix addresses the question of whether the smoothness of anti-de Sitterspace subsists after the identifications leading to the black hole are made.

That is,we ask whether the quotient spaces we deal with are Hausdorffmanifolds.Theconclusion is that this is so when J ̸= 0, but when J = 0 the Hausdorffmanifoldstructure is destroyed at r = 0.As discussed by Hawking and Ellis [9],the quotient spaces are Hausdorffmanifoldsif and only if the action of the identification subgroup H = {exp2πkξ, kǫZ} isproperly discontinuous, namely, if the following properties hold,(i) Each point Qǫ adS has a neighbourhood U such that (exp2πkξ)(U) ∩U = φ forall kǫZ, k ̸= 0; and(ii) If P, Q ǫ adS do not belong to the same orbit of H (i.e., there is no k ǫZ suchthat (exp2πkξ)(P) = Q), then there are neighborhoods B and B′ of P and Qrespectively such that (exp2πkξ)(B) ∩B′ = φ for all k ǫZ.To proceed with the analysis we introduce the Euclidean norm[(u′ −u)2 + (v′ −v)2 + (x′ −x)2 + (y′ −y)2]1/2(B.1)on R4. The norm of the Killing vectorξ = r+l u ∂∂x + x ∂∂u!−r−l v ∂∂y + y ∂∂v!

(B.2)43

is bounded from below by r−> 0,∥ξ · ξ ∥E="r2+l2 (u2 + x2) + r2−l2 (v2 + y2)#="r2+ −r2−l2(u2 + x2) + r2−l2 (u2 + x2 + v2 + y2)#1/2≥r−> 0(on u2 + v2 = x2 + y2 + l2)(B.3)Let Q0 be a point of anti-de Sitter space with coordinates (u0, v0, x0, y0) satisfyingu20 + v20 −x20 −y20 = l2. Its successive images Qn are given byun=(cosh α)u0 + (sinh α)x0(B.4)xn=(sinh α)u0 + (cosh α)x0(B.5)vn=(cosh β)v0 −(sinh β)y0(B.6)yn=−(sinh β)v0 + (cosh β)v0(B.7)(B.8)with n ǫ Z, α = 2πr+/l, β = 2πr−/l.

The Euclidean distance dE(Q0, Qn), (n ̸= 0)between Q0 and Qn is bounded from below bydE(Q0, Qn) ≥lq2(cosh β −1) > 0, (n ̸= 0). (B.9)Indeed, one has(un −u0)2 + (xn −x0)2 + (vn −v0)2 + (yn −y0)2(B.10)≥|(un −u0)2 −(xn −x0)2| + |(vn −v0)2 −(yn −y0)2|=2(cosh nα −1)|u20 −x20| + 2(cosh nβ −1)|v20 −y20|≥2(cosh β −1)[|u20 −x20| + |v20 −y20|]≥2(cosh β −1)|u20 −x20 + v20 −y20|=2(cosh β −1)l2(B.11)The bound (B.9) is uniform, i.e., it does not depend on Q0.Let P0 be another point of anti-de Sitter space with coordinates (¯u0, ¯v0, ¯x0, ¯y0).It is easy to see, by using formulas analogous to (B.3) for P0, that the distancedE(Pn, Q0) between Q0 and the images of P0 goes to infinity as n →±∞.

Hence,44

there is a minimum “distance of approach” of the orbit of P0 to Q0 (which may bezero if Q0 = Pk for some k). That minimum distance of approach varies continuouslyif one varies P0 continuously.Let U be the open ball centered at Q0 with radius r

Theimage of any point of this ball by exp2πkξ, (k ̸= 0) cannot be in U. Otherwise thebound (B.3) would be violated.

This proves (i).Now, turn to (ii). Let P0 be a point that is not mapped on Q0 by any power ofexp2πξ.

In the open ball U, there can be at most one image of P0. If there werenone, by continuity, the points sufficiently close to P0 will have no image in U andthus (ii) would hold.

So let us assume that there is one image of P0 in U, say Pn.Let ˜B be an open ball centered at Pn and entirely contained in U. All the imagesof the points in ˜B lie outside U, i.e., (exp2πkξ)( ˜B) ∩U = φ.

Let B” be an openball centered at Q0 such that B ∩˜B = φ. Then B = (exp −2πnξ)( ˜B) and B” fulfillcondition (ii).

[For simplicity we have used in this analysis the simpler form of the Killing vectoronly appropriate for |J| < Ml. One can easily check that for |J| = Ml there are nofixed points and that all the orbits go to infinity, just as for |J| < Ml.

It then easilyfollows that the results for |J| < Ml remain valid for |J| = Ml. The details are leftto the reader]The above argument breaks down when there is no angular momentum becausethe Killing vector ξ = r+l (u ∂∂x + x ∂∂u) vanishes in that case along the line u = x = 0,which is thus a line of fixed points.

This makes the bound (B.3) empty. Furthermore,each fixed point is an accumulation point for the orbits of the points obeying u±x = 0and having the same values of v and y.

Hence, both (i) and (ii) are violated ifone takes for Q one of the fixed points. The action of the group is not properlydiscontinuous.

This leads to a singularity in the manifold structure of the Taub-NUT type. [This kind of singularity has been discussed in [14].

Another example of it hasbeen found in [15]. For an analysis see [9], where a discussion of identificationsunder boosts in two-dimensional Minkowski space is given.

To make contact withthat analysis observe that near r = 0 one can neglect the cosmological constant.The SO(2, 2) group goes then over to the Poincar´e group in three dimensions. Theidentification Killing vector (3.17) becomes then a boost plus a translation in a45

transverse direction. It is the presence, in our case, of this additional transversedirection which is responsible for the smooth behavior when J ̸= 0: the combinationof a boost and a transverse translation does not have fixed points.

]46

References[*]e-mail: cecsnet@uchcecvm[#] e-mail: henneaux@ulb.ac.be[1] M. Ba˜nados, C. Teitelboim and J. Zanelli, Phys. Rev.

Lett., 69, 1849 (1992)[2] The black hole considered here is a solution of the standard Einstein equationswith a negative cosmological constant and without matter. Its properties arein very close correspondence with its counterpart in four spacetime dimensions.Models for black holes in two spacetime dimensions can also be constructed [foran early attempt see J.D.

Brown, M. Henneaux and C. Teitelboim, Phys. Rev.D33, 319 (1986)] and have been the object of considerable interest recentlyin connection with string theory [see E. Witten, Phys.

Rev. D44 314 (1991),C-G. Callan, S.B.

Giddings, and J.A. Harvey and A. Strominger, Phys.

Rev.D45, 1005 (1992). The action is then not the one of Einstein’s theory and itincludes, in the string case, a dilaton field as an essential ingredient.

A relatedstring-inspired model in three spacetime dimensions (the “black string”) hasalso been considered [J.H. Horne and G.T.

Horowitz, Nucl. Phys.

B368, 444(1992)]. [3] The charged 2+1 black hole was briefly discussed in [1].

It is not a spacetime ofconstant curvature. This makes the analysis given below, in terms of identifica-tions in anti de Sitter space, not applicable.

For an independent discussion ofsolutions of 2+1 gravity coupled to electromagnetism, without a cosmologicalconstant, see I.I. Kogan, Mod.

Phys. Lett.

A7, 2341 (1992). [4] For signature (+++) there is a theorem stating that any geodesically completespace of constant negative curvature is a quotient of “Euclidean anti-de Sit-ter” (i.e.

the three-dimensional Lobachevsky plane) by a discrete subgroup ofO(3, 1). [W.P.

Thurston, Geometry and Topology on Three-Manifolds, Prince-ton Lecture Notes, 1979 (unpublished)]. S. Carlip informs us that an equallyclear-cut theorem does not seem to be available for signature (- ++) [see, in thiscontext G. Mess, Lorentz Spacetimes of Constant Curvature, IHES/M/90/28preprint, unpublished].

However - as persuasively argued by E. Witten (private47

communication) - it was natural to expect the black hole geometry to also bea quotient and this indeed turned out to be the case. [5] T. Regge and C. Teitelboim, Ann.

of Phys. (NY) 88, 286 (1974).

[6] J.D. Brown and M.Henneaux, Commun.

Math. Phys.

104, 207 (1986)[7] S. Deser, R. Jackiw and G.’t Hooft, Ann. Phys.

(NY) 152, 220 (1984). S. Deserand R. Jackiw, Ann.

Phys. (NY) 153, 405 (1984).

A. Staruszkiewicz, Acta Phys.Pol. 24, 734 (1963).

J. Gott and M. Alpert, Gen. Rel. & Grav.

16, 3 (1984). [8] The condition for the quotient space to be smooth is that the action of theisometry group generated by ξ be “properly discontinuous”, see S. Hawking andG.F.R.

Ellis, The Large Scale Structure of Spacetime, (Cambridge UniversityPress, Cambridge, 1973). As discussed in Appendix B, in the black hole case thiscondition is met when the angular momentum is different from zero.

However,when J=0, the group action is not properly discontinuous and the singularityis of the type present in Taub-NUT space. [9] One may regard 2+1 gravity as a Chern-Simon theory for SO(2.2) [E. Wit-ten, Nucl.

Phys. B 311, 46 (1988)].

The holonomies of that approach are thenexp2πnξ with the Killing vector ξ used in the identification regarded as an ele-ment of the Lie algebra of SO(2.2). This correspondence appears to hold underrather general conditions on the manifold [see e.g., the work by Mess[4] andalso S. Carlip, Class.

and Quant. Gravity 8, 5 (1991)].

For the black geometry,the Chern-Simons holonomies have been evaluated directly by D. Cangemi, M.Leblanc and R. B. Mann (Gauge Formulation of the Spinning Black Hole in(2+1) - Dimensional Anti-de Sitter Space, MIT Preprint CTP #2162, 1992,unpublished). Their results agree with our expressions (3.19) (applicable when|J| < Ml).

However, since ξ · ξ is a smooth surface (recall sec. 3.2.2) there isno source there for the curvature tensor (or anywhere else as a matter of fact).One gets a non-trivial holonomy -that can be thought of as a flux integral-because of the non-trivial topology (“charge without charge”.

See J.A. WheelerGeometrodynamics, Academic Press, New York 1992.

)48

[10] R.H. Boyer and R.W. Lindquist, J.

Math. Phys.

8, 265 (1967), B. Carter, Phys.Rev. 141, 1242, (1966); 174, 1559 (1968), J. Graves and D. Brill, Phys.

Rev.120, 1507 (1960). [11] B. Carter, Phys.

Lett. 21, 423 (1966).

[12] S.F. Ross and R.B.

Mann, “Gravitationally Collapsing Dust in (2+1) dimen-sions”. Preprint WATPHYSTH 92/07, unpublished.

[13] S.W. Hawking, Phys.

Rev. D. 46, 603 (1992).

[14] C.W. Misner, in Relativity Theory and Astrophysics I: Relativity and Cosmol-ogy, J. Ehlers, ed., Lectures in Applied Mathematics, 8 (American MathematicalSociety), p.

160. [15] C. Nappi and E. Witten, Phys.

Lett. 293 B, 309 (1992).49

FIGURE CAPTIONSFigure 1.Poincar´e Patches(a) Section with surface y = 0. The solid lines have u + x = 0,y = 0.

Thesecurves are lightlike and asymptotic to λ = (k + 1/2)π. The pattern is periodicin λ.

(b) Section with surface x = 0. The solid lines (including the axis λ = 0) haveu + x = 0, x = 0 in anti-de Sitter space.

The pattern is again periodic in λ.As one lets the angle θ approach ±π/2, the lines u + x = 0 become more andmore horizontal until they reach the configuration shown.Figure 2.Regions determined by the norm of ξ′. (a) Section with surface y = 0 when r−̸= 0.The solid lines are the curvesξ′ · ξ′ = 0, y = 0.

They are timelike. The dotted lines are the lines ξ′ · ξ′ = r2−(u2 −x2 = 0), bounding regions II and III.

The lines formed by dots andsegments have ξ′ · ξ′ = r2+, y = 0. (b) Section with surface x = 0 when r−̸= 0.

The surface x = 0 has ξ′ · ξ′ > 0everywhere when r−̸= 0. The horizontal solid lines are the lines ξ′ · ξ′ = r2−,x = 0.

The lightlike lines formed by dots and segments have ξ′ · ξ′ = r2+. Theregion ξ′ · ξ′ > 0, x = 0 splits into disconnected components separated by thehorizontal lines and containing two regions I and two regions II.

(c) Section with surface y = 0 where r−= 0. The solid lines have ξ′·ξ′ = 0, y = 0.The lines formed by dots and segments have ξ′ · ξ′ = r2+.

The region ξ′ · ξ′ > 0splits into disconnected components separated by the horizontal lines witheach component consisting of two regions II (and two regions I, not seen inthis figure since they have no intersection with y = 0).Regions III havedisappeared. Note that the Killing vector ξ′ is now tangent to the lightlikecurves u2 −x2 = 0, y = 0.Figure 3.Spacetime diagrams for J = 0(a) Kruskal diagram, (b) Penrose diagram.50

Figure 4 . Penrose diagrams for J ̸= 0.

(a) Patch K−, (b) Patch K+, (c) Completediagram obtained by joining an infinite sequence of patches K−, K+ on theoverlap KFigure 5.Penrose Diagrams for the extreme cases (a) M = 0 = J, (b) M = |J/l| ̸= 0.Figure 6.Penrose diagrams for the maximally extended non-extremal spinning blackhole (Ml > |J| > 0), showing also the regions beyond the singularity wherethe Killing vector ξ is timelike. Regions III’ are defined by −∞< ξ · ξ < r2−and contain regions III(0 < ξ · ξ < r2−).

The metric in these regions isisomorphic to the metric in regions I but with the roles of t and φ exchanged.The singularity r = 0 in III corresponds then to the stationary surface inI. There are closed timelike curves through each point in regions III’.

Theseclosed timelike curves cross ξ · ξ = 0.Figure 7.Penrose diagram for a collapsing body in the case J = 0.51

---

출처: arXiv:9302.012 • 원문 보기