Strength of the Null Singularity Inside Black Holes

Strength of the null singularity inside black holes

Lior M. Burko Theoretical Astrophysics, California Institute of Technology, Pasadena, California 91125 ͑Received 3 May 1999; published 26 October 1999͒ We study analytically the Cauchy horizon singularity inside spherically symmetric charged black holes, coupled to a spherical, self-gravitating, minimally coupled, massless scalar ﬁeld. We show that all causal geodesics terminate at the Cauchy horizon at a null singularity, which is weak according to the Tipler classi- ﬁcation. The singularity is also deformationally weak in the sense of Ori. Our results are valid at arbitrary points along the null singularity, in particular at late retarded times, when nonlinear effects are crucial. ͓S0556-2821͑99͒04920-6͔ PACS number͑s͒: 04.70.Bw, 04.20.Dw

I. INTRODUCTION spherical charged black hole, the main features of the CH singularity were first found analytically for simplified models The issue of spacetime singularities—which are known to based on null fluids ͓10–12͔, and later confirmed numeri- inevitably occur inside black holes under very plausible as- cally for a model with a self-gravitating scalar field ͓13,14͔. sumptions ͓1͔—is an intriguing puzzle of physics. The laws Expressions for the divergence rate of the blueshift factors of physics, as we presently understand them ͑e.g., classical for that model, which are valid everywhere along the CH, general relativity͒, are presumably invalid at singularities. were found analytically in Ref. ͓15͔. Those expressions are Instead, some other theories ͑e.g., quantum gravity͒, as yet exact on the CH as functions of retarded time. However, they unknown, are expected to take over from general relativity are only asymptotic expressions as functions of advanced and control the spacetime structure. The general relativistic time ͑see below͒. predictions are nevertheless of the greatest importance, as The strength of the null singularity is of crucial impor- they reveal the spacetime structure under extreme conditions tance for the question of the hypothetical possibility of hy- in the strong-field regime. Of particular interest is the possi- perspace travel through the CH of black holes. A necessary bility that there are two distinct ways in which general rela- condition for this possibility to be realized is that physical tivity can fail at different types of singularities: for one type objects would traverse the CH peacefully. Because the CH is of singularity the failure is through infinite destructive ef- known to be a curvature singularity, it is necessary that the fects on physical objects, whereas for the other type the fail- singularity would be weak according to the Tipler classifica- ure is through the breakdown of predictability. tion of singularity strengths. For the toy model of a spherical Until recently, the only known generic singularity in gen- charged black hole, which we shall study here, the properties eral relativity was the Belinsky-Khalatnikov-Lifshitz ͑BKL͒ of the CH singularity which have been found in Refs. ͓11– singularity ͓2͔. According to the BKL picture, spacetime de- 15͔ are all consistent with the picture of a Tipler weak sin- velops a succession of Kasner epochs in which the axes of gularity. However, the weakness of the singularity was dem- contraction and expansion change directions chaotically. onstrated only for the simplified Ori model ͓12͔ and at This succession ends at unbounded oscillations at a spacelike asymptotically early times for spinning black holes ͓5͔, singularity, which is unavoidably destructive for any physi- where there are still no strong nonlinear effects, such as fo- cal object—a strong singularity. In the last several years, cusing of the null generators of the CH, which are crucial at however, evidence has been accumulating that the BKL sin- later times. In the context of spherical charged black holes gularity is not the only type of singularity which may evolve and a self-gravitating scalar field, several important features in general relativity from generic initial data. of the spacetime structure have been found in fully nonlinear The new type of singularity forms at the Cauchy horizon numerical simulations. Specifically, it was shown that for ͑CH͒ of spinning or charged black holes. ͑For a recent re- any point along the CH singularity there existed coordinates view see Ref. ͓3͔.͒ The features of this singularity are mark- for which the metric coefficients were finite and the metric edly different from those of the BKL singularity: ͑i͒ It is null determinant was nondegenerate in an open neighborhood to ͑rather than spacelike͒, ͑ii͒ it is weak ͑according to Tipler’s the past ͓13,14͔. However, despite previous claims ͓5,14,16͔, classification ͓4͔͒, specifically, the tidal deformations which this still does not guarantee that the singularity is weak in the an extended physical object suffers upon approaching the Tipler sense ͓17͔. singularity are bounded. In the case of a spinning black hole, It is the purpose of this paper to present an analytical the evidence for the null and weak singularity has emerged demonstration of the weakness of the singularity for the from analytical perturbative ͓5,6͔ and nonperturbative ͓7͔ model of a spherical charged black hole with a self- analyses. In addition, the local existence and genericity of a gravitating, minimally coupled, massless, real scalar field. null and weak singularity in solutions of the vacuum Einstein Our results are valid at arbitrary points along the CH singu- equations was demonstrated in Ref. ͓8͔. This was more re- larity, in particular at late times, where strong nonlinear ef- cently demonstrated also in the framework of plane- fects ͑focusing of the null generators of the CH and growth symmetric spacetimes in Ref. ͓9͔. For the toy model of a of the blueshift factors͒ are crucial. In fact, our results are

⌽ valid everywhere along the CH singularity, down to the ,V are exactly inversely proportional to r(u), in the follow- event where the generators of the CH are completely fo- ing sense. Consider two outgoing null rays, and let one ray ϭ ϭ cused, and the singularity becomes spacelike and Tipler be at u u1, say, and the other at u u2. The ratios ͓ ͔ ⌽ ⌽ strong 18 . We emphasize that although our discussion here r,V(2)/r,V(1) and ,V(2)/ ,V(1) approach r(u1)/r(u2)as → is analytical, we do make assumptions which are based on V 0. Taking now u1 to be in the asymptotically early parts Ϸ ⌽ results obtained by numerical simulations. of the CH, where r(u1) rϪ , we find that both r,V and ,V are inversely proportional to r(u). As r(u) is monotonically II. STRENGTH OF THE SINGULARITY decreasing as a function of retarded time along the CH, we ⌽ find that r,V and ,V grow monotonically along the CH. This We write the general spherically symmetric line element growth is a nonlinear effect which indicates the strengthen- in the form ing of the singularity along the CH ͑although the singularity is still weak according to the Tipler classification; see be- ds2ϭϪf ͑u,v͒dudvϩr2͑u,v͒d⍀2, ͑1͒ low͒. where d⍀2ϭd␽2ϩsin2 ␽d␸2 is the line element on the unit All the nonzero components of the Riemann-Christoffel curvature tensor R␮␯␳␴ are given completely in terms of the two sphere. The coordinates u,v are any outgoing and ingo- ⌽ ing null coordinates, correspondingly. ͑Below, we shall spe- divergent blueshift factors r,V , ,V , and the finite quantities ⌽ cialize to a specific choice of gauge, and define a particular r,r,u , ,u , and F. Interestingly, R␮␯␳␴ does not depend on choice of an ingoing null coordinate.͒ We consider the class gradients of F. This can be understood from the following of scalar field perturbations which is inherent to any gravita- consideration. The tensor R␮␯␳␴ can be written as the sum of tional collapse process. These are the perturbations which the Weyl tensor, and another tensor which is built from the ͑ result from the evolution of nonvanishing multipole mo- Ricci and the metric tensors but not involving their deriva- ͒ ments during the collapse. When these perturbations propa- tives . In spherical symmetry the Weyl tensor is given com- gate outwards, they are partially reflected off the spacetime pletely in terms of the mass function, which is defined by ,␮ϭ Ϫ ϩ 2 2 curvature and captured by the black hole. This process re- r,␮r 1 2M(u,v)/r q /r , q being the charge of the sults in a scalar field, which at late advanced times decays black hole. In Kruskal-like coordinates the mass function ϭ ϩ ϩ 2 along the event horizon according to an inverse power of M(u,V) (r/2)(1 4r,ur,V /F) q /(2r), which depends ͑ advanced time. Specifically, we assume that the scalar field only on r,r,V ,r,u , and F. The divergence of the mass func- behaves along the event horizon at late times according to tion at the singularity, and consequently also the divergence Ϫ ⌽EHϰ(␬v ) n ͓19–21͔, where n is a positive integer which of curvature, is evident from the divergence of r,V and the e ͒ is related to the multipole moment under consideration. ͑We finiteness of r, r,u , and F. The Ricci tensor R␮␯ ϭ ⌽ ⌽ do not consider, however, other possible sources of pertur- 2 ,␮ ,␯ , and consequently R␮␯␳␴ is independent of gra- ͓ ͔ ͒ dients of F. bations 22 . By ve we denote the usual advanced time in the Eddington gauge, and rϮ ,␬ are the outer and inner ho- We find that the components of R␮␯␳␴ which have the ͑ rizons and the surface gravity of the latter, respectively, for a strongest divergence near the CH are RV␸V␸ and RV␽V␽ . In Reissner-Nordstro¨m black hole having the same external pa- the Appendix we list all the nonzero independent compo- ͒ rameters as the black hole we consider has at late times. We nents of the Riemann-Christoffel tensor. It can be readily define the dimensionless ingoing Kruskal-like coordinate by shown that VϵϪexp(Ϫ␬v ). In the Kruskal gauge we denote the metric e ϭϪ 2͑⌽ ͒2ϭ Ϫ2 ␽ ͑ ͒ Ϫ RV␽V␽ r ,V sin RV␸V␸ . 4 function guV by F/2. For this model, it can be shown analytically that at arbitrary points along the CH the following Ϫ We denote these two components schematically and collec- relations are satisfied, to the leading orders in ͓Ϫln(ϪV)͔ 1 tively by R. We find that, to the leading orders in V and in ͓15͔ ͓Ϫln(ϪV)͔Ϫ1, ͒2 ͑nrϪA Ϫ Ϫ Ϫ r ϭ ͓Ϫln͑ϪV͔͒Ϫ2nϪ2͕1ϩb ͓Ϫln͑ϪV͔͒Ϫ1 R͑V͒ϰV 2͓Ϫln͑ϪV͔͒ 2n 2. ͑5͒ ,V rV 1

ϩ Ϫ Ϫ ͒ Ϫ2ϩ Ϫ Ϫ ͒ Ϫ3 ͑ ͒ ͑The other divergent components of R␮␯␳␴ , i.e., b2͓ ln͑ V ͔ O͓ ln͑ V ͔ ͖, 2 RuVuV , R␽␸␽␸ , Ru␽V␽ , and Ru␸V␸ are proportional to the leading order in V to VϪ1 times a logarithmic factor.͒ Be- ͑nrϪA͒ ⌽ ϭ Ϫ Ϫ ͒ ϪnϪ1 ϩ Ϫ Ϫ ͒ Ϫ1 ,V ͓ ln͑ V ͔ ͕1 c1͓ ln͑ V ͔ cause both metric functions r and F have finite values at the rV CH ͑which is known from the numerical simulations of Refs. ϩ Ϫ Ϫ ͒ Ϫ2ϩ Ϫ Ϫ ͒ Ϫ3 ͑ ͒ ͓13,14͔͒, it is easy to show that the dependence of R(V)on c2͓ ln͑ V ͔ O͓ ln͑ V ͔ ͖. 3 V does not change when we transform to a parallel- Here, Aϭ͓rϩ /(2rϪ)͔(rϩ /rϪϩrϪ /rϩ), and the expansion propagated frame. ␶ coefficients bi and ci are functions of retarded time only. We next find V( ) as a function of affine parameter Note that in the limit V→0 these are exact expressions as ͑proper time͒ ␶ along a general null ͑timelike͒ geodesic. For functions of retarded time. That is, to the leading order in general causal geodesics, the geodesic equations are ͓Ϫln(ϪV)͔Ϫ1 there is implicit dependence on retarded time ϭ 2 ˙ 2 through r r(u), and along the CH singularity both r,V and u˙ v˙ ϭ͑mr ⍀ Ϫp͒, ͑6͒

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͑ ͓Ϫ Ϫ ͔Ϫ1 v¨ ϩ͑ f / f ͒v˙ 2ϩ2mr ͑ fu˙ v˙ ϩp͒/͑rf͒ϭ0, ͑7͒ Higher order terms in ln( V) are functions of re- ,v ,u ͒ tarded time. We note that only b1 and c1 are constrained. The coefficients of higher-order terms in ͓Ϫln(ϪV)͔Ϫ1 are u¨ ϩ͑ f / f ͒u˙ 2ϩ2mr ͑ fu˙ v˙ ϩp͒/͑rf͒ϭ0. ͑8͒ ,u ,v immaterial near the CH for our determination of the strength → → Here, mϭ0(1) for radial ͑nonradial͒ geodesics, and p of the singularity. Note that F F0(u)asV 0, and that F ϭ0(Ϫ1) for null ͑timelike͒ geodesics. A dot denotes differ- is not analytic in V. In fact, this is an important property of entiation with respect to affine parameter ͑proper time͒, and the CH singularity: In a Kruskal-like gauge the metric functions r and F are finite at the singularity, but their gradients ⍀˙ 2ϭ␽˙ 2ϩ 2 ␽␸˙ 2 sin . The geodesic equations can be solved to in the outgoing direction diverge. The finiteness of r and F at ͓ Ϫ ͔Ϫ1 the leading order in ln( V) for all causal geodesics. This the CH also implies that the metric determinant is nondegen- ͓ ͔ is done by using the field equation 14 erate. This expression for F is similar to the behavior of the ͓ ͔ ϭ ϩ ͑⌽ ͒2 ͑ ͒ guV metric function found for the simplified Ori model 12 . F,V /F r,VV /r,V r ,V /r,V 9 We stress that although this expression for F is exactly valid everywhere along the CH singularity, it still does not allow to find F /F explicitly. Substituting Eqs. ͑2͒ and ͑3͒ in Eq. ,V us to find the variation of F with retarded time, as we do not ͑9͒ we find know the form of F0(u)orr(u) along the CH. We note that 1 1 near the CH the metric function F is monotonic in V. This ͒ ϭ Ϫ ͒ ϩ ϩ Ϫ ͒ Ϫ Ϫ ͒ Ϫ1 ͓ ͔ ͑ln F ,V ͓ln͑ r,V ͔,V ͑2c1 b1 ͓ ln͑ V ͔ result is in accord with the numerical results of Ref. 14 . V V ͑Notice, however, the disagreement with the numerical re- ͓ ͔ 1 sults of Ref. 16 . It is reasonable to expect the behavior of ϩ͑b2Ϫb Ϫ2b c ϩc2ϩ2c ͒ ͓Ϫln͑ϪV͔͒Ϫ2 the metric functions near the CH to be similar for both cases 1 2 1 1 1 2 V of real and complex scalar fields. The lower panels of Figs. 3 of Ref. ͓16͔ imply, however, a nonmonotonic behavior of F. 1 Ϫ ϩOͭ ͓Ϫln͑ϪV͔͒ 3ͮ . ͑10͒ That kind of behavior can be obtained from a numerical code V with a specific choice of parameters if the latter is far from ͒ Integration yields convergence near the CH. Let us consider first radial geodesics. ͑The case of nonradial geodesics will be treated next.͒ In the null case (mϭ0 ln Fϭln ˜F ϩln͑Vr ͒Ϫ͑2c Ϫb ͕͒ln͓Ϫln͑ϪV͔͖͒ 0 ,V 1 1 and pϭ0) it is easy to solve the geodesic equations ͑6͒–͑8͒. ϩ͑ 2Ϫ Ϫ ϩ 2ϩ ͓͒Ϫ ͑Ϫ ͔͒Ϫ1 b1 b2 2b1c1 c1 2c2 ln V For outgoing geodesics one readily finds that the solution is uϭconst and V˙ ϭconst/F. The metric function F can be ex- ϩ ͕͓Ϫ ͑Ϫ ͔͒Ϫ2͖ ͑ ͒ O ln V . 11 panded in ͓Ϫln(ϪV)͔Ϫ1, despite its nonanalyticity in V.To the leading orders in ͓Ϫln(ϪV)͔Ϫ1 we find that ˜ Here, ln F0 is an integration constant, which can be a func- ͑ ͒ ϭ ͒ ϩ Ϫ Ϫ ͒ Ϫ1 ͑ ͒ tion of u. Exponentiating both sides, and substituting Eq. 2 F F0͑u ͕1 B͓ ln͑ V ͔ ͖, 15 for r,V we find such that ͑ ͒2 nrϪA Ϫ Ϫ ϩ Ϫ Fϭ˜F ͓Ϫln͑ϪV͔͒ 2n 2 b1 2c1 const 0 r V˙ ϭ ͕1ϪB͓Ϫln͑ϪV͔͒Ϫ1͖. ͑16͒ F0 ϫ ϩ Ϫ Ϫ ͒ Ϫ1ϩ Ϫ Ϫ ͒ Ϫ2 ͕1 b1͓ ln͑ V ͔ O͓ ln͑ V ͔ ͖ The solution for V(␶) is then given asymptotically close to ϫ ͕͑ 2Ϫ Ϫ ϩ 2ϩ ͓͒Ϫ ͑Ϫ ͔͒Ϫ1 Ϫ1 exp b1 b2 2b1c1 c1 2c2 ln V the CH, to the leading orders in ͓Ϫln(ϪV)͔ by Ϫ2 ϩO͓Ϫln͑ϪV͔͒ ͖. ͑12͒ V͑␶͒ϭ␶͕1ϪB͓Ϫln͑Ϫ␶͔͒Ϫ1͖. ͑17͒

From the numerical results of Refs. ͓13,14͔ it is known that ͑Recall that the affine parameter is given up to a linear trans- as V→0, F approaches a finite value. Consequently, in order formation.͒ To the leading order in ͓Ϫln(Ϫ␶)͔Ϫ1 we can thus to have consistency with the numerical results we require approximate V(␶)Ϸ␶. Note that although asymptotically Ϫ ϭ ϩ that b1 2c1 2n 2, which implies that V(␶) and V˙ (␶) behave as ␶ and ␶˙ ϵ1, correspondingly, V¨ (␶) ␶¨ ϵ ¨ ␶ ϭ ͑ ͕͒ ϩ ͓Ϫ ͑Ϫ ͔͒Ϫ1ϩ ͓Ϫ ͑Ϫ ͔͒Ϫ2͖ behaves very differently from 0. In fact, V( ) diverges as F F0 u 1 B ln V O ln V , ␶→ ␶ Ϸ␶ ͑13͒ 0. Therefore, one can approximate V( ) only if one is interested in V(␶) itself, or at the most in V˙ (␶). This ϭ ϩ Ϫ ϩ 2ϩ where B (2n 3)b1 b2 c1 2c2, and F0(u) approximation is invalid for V¨ (␶) or higher derivatives. For ϭ˜ 2 ϭ ϭϪ F0(nrϪA) /r. For the logarithmic derivative of F we find, radial timelike geodesics (m 0 and p 1) one uses the ͓Ϫ Ϫ ͔Ϫ1 ˙ to the leading order in ln( V) , that finiteness of F,u to find approximately that again V Ϸconst/F, and consequently one finds the same result for 1 Ϫ V(␶). In the case of nonradial geodesics (mϭ1) one can F /FϭB ͓Ϫln͑ϪV͔͒ 2. ͑14͒ ,V V consider a specific value of the retarded time at which the

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␶ geodesic hits the CH singularity. Than, r,r,u , and F can be Jacobi field J( ) for which at least one parallel-propagated approximated by their values at the singularity. When this is tetrad component is unbounded at the limit ␶→0 ͓24͔. This done, the equations for null nonradial (mϭ1 and pϭ0͒ geo- definition is more physically motivated than Tipler’s defini- desics become inhomogeneous linear equations. The corre- tion, because it classifies a singularity as strong not only sponding homogeneous equations are nothing but the equa- when the volume element vanishes, but also when the vol- tions for the radial geodesics, which we already solved. ume element diverges to infinity, or there is infinite compres- Particular solutions for the inhomogeneous equations are sion in one direction, and infinite stretching in a different easy to generate, and one finds that again V(␶) is given direction, such that the volume element remains bounded. In asymptotically as before. The last case is the case of nonra- fact, it can be shown that the failure of the necessary condi- dial timelike (mϭ1 and pϭϪ1) geodesics. In this case the tion for the singularity to be Tipler strong implies not only geodesic equation becomes ͑under similar assumptions͒ an the boundedness of the volume element, but also the bound- inhomogeneous non-linear equation. Although this equation edness of the Jacobi fields themselves ͓24͔, such that objects is hard to solve directly, it can be checked that the same are not expected to be destroyed also because of distortions leading order proportionality of V(␶) and ␶ is the solution which preserve the volume element or divergence to infinity also for this case. We thus find that for all causal geodesics, of the volume element. Consequently, the singularity we are to the leading order in ͓Ϫln(Ϫ␶)͔Ϫ1, V(␶) is proportional to studying here is weak also in the sense of Ori ͑deformation- ␶. ally weak͒. ͑Ori’s definition does not include null geodesics. We next reexpress R as a function of affine parameter However, extended physical objects move along timelike ͑proper time͒ along radial or nonradial null ͑timelike͒ geode- geodesics, such that this deficiency does not restrict our dis- sics. To the leading order in ͓Ϫln(Ϫ␶)͔Ϫ1 we find that in a cussion. It is conceivable that both Ori’s definition and theo- parallel-propagated frame rem for the necessary condition for the singularity to be Ori strong can be generalized to all causal geodesics.͒ R͑␶͒ϰ␶Ϫ2͓Ϫln͑Ϫ␶͔͒Ϫ2nϪ2. ͑18͒ We note that according to the Kro´lak classification of singularities ͓25͔ this is a strong singularity. Specifically, if we A necessary condition for a singularity to be strong in the integrate over the divergent components of the Riemann- Tipler sense is given by the following theorem ͓23͔: For null Christoffel tensor only once, the integral does not converge ͑timelike͒ geodesics, if the singularity is strong in the Tipler on the singularity. This means that the expansion diverges sense, then for at least one component of the Riemann- ͑negatively͒ on the singularity ͑Kro´lak strong͒, but still the Christoffel curvature tensor in a parallel-propagated frame, volume element ͑and the distortion in general͒ remains finite the twice integrated component with respect to affine param- ͑Tipler and Ori weak͒. One might be worried that even if eter ͑proper time͒ does not converge at the singularity. Spe- spacetime were classically extendible beyond the CH, this cifically, the necessary condition for the singularity to be infinite negative expansion would inevitably result in un- Tipler strong is that avoidable destruction of any extended physical object subse- quent to its traversing of the CH ͓26͔. Of course, any classi- ␶ ␶Ј cal extension of geometry beyond the CH is not unique. We I͑␶͒ϭ ͵ d␶Ј ͵ d␶Љ͉R͑␶Љ͉͒ ͑19͒ can, however, consider an extension with a continuous (C0) metric and a unique C1 timelike geodesic, and assume that does not converge as ␶→0. It can be readily shown that the object follows this geodesic ͓27͔. Any extension of clas- when R(␶) is integrated twice with respect to ␶, I(␶) con- sical geometry beyond the CH ͑which can be modeled as a verges in the limit ␶→0. Consequently, we find that a nec- thin layer wherein the geometry is inherently quantum͒ re- essary condition for any causal geodesic to terminate at a quires an infinite flux of negative energy traveling along the Tipler strong singularity is not satisfied. Hence, all causal contracting CH. This negative energy flux may then act to geodesics terminate at a Tipler weak singularity, namely, the regularize the expansion, such that the deformation rate of singularity is Tipler weak. The physical content of this result physical objects beyond the CH would be bounded ͓27͔. is that the volume element of physical objects remains ͑The infinite expansion is likely not to destroy physical ob- bounded at the singularity. We emphasize that this result is jects up to the CH ͓27͔, in contrast with Ref. ͓26͔.͒ Indeed, a valid everywhere along the singularity, in particular at late simplified two-dimensional quantum model shows an infinite retarded times where the nonlinear effects ͑focusing of the ingoing flux of negative energy along the CH ͓28,29͔. More generators of the CH and the growth of the blueshift factors͒ recent semiclassical toy models of a quantum field on a are crucial. mass-inflation background are not inconsistent with this picture ͓30,31͔. One should not take these quantum results too III. CONCLUDING REMARKS seriously, however, because in these models the semiclassical contributions are dominated by the regime where curva- Recently, Ori suggested to define a deformationally strong ture is Planckian, such that the semiclassical approximation singularity in the following way. Let ␭(␶) be a timelike geo- is not expected to be valid anymore. Instead, a full quantum desic with proper time ␶ along it. The geodesic ␭(␶) termi- theory of gravity is of need. Of course, in the absence of a nates at a deformationally-strong singularity at ␶ϭ0ifat valid theory of quantum gravity it is difficult to make pre- least one of the following two conditions holds: ͑i͒␭(␶) dictions on the detailed interaction of the thin layer of the terminates at a Tipler strong singularity or ͑ii͒ there exists a CH with physical objects, but the evidence we currently have

104033-4 STRENGTH OF THE NULL SINGULARITY INSIDE... PHYSICAL REVIEW D 60 104033 do not preclude the possibility of objects traversing the CH f q2 singularity peacefully. ␽ ␽ϭϪ Ϫ ͩ Ϫ ͪ Ru v r,ur,v 1 , 4 r2 ACKNOWLEDGMENTS ϭϪ 2⌽2 I have beneﬁted from useful discussions with Patrick Rv␽v␽ r ,v , Brady and Amos Ori. This work was supported by NSF grant AST-9731698. f q2 2 ␸ ␸ϭϪͫ ϩ ͩ Ϫ ͪͬ ␽ Ru v r,ur,v 1 sin , APPENDIX 4 r2 The independent components of the Riemann-Christoffel curvature tensor which do not vanish identically are ϭϪ 2⌽2 2 ␽ Ru␸u␸ r ,u sin , 1 f q2 R ϭϪ ͫ 4r r ϩ f ͩ 1Ϫ2 ͪͬϩ f ⌽ ⌽ , uvuv 4 2 ,u ,v 2 ,u ,v ϭϪ 2⌽2 r r Ru␽u␽ r ,u ,

r2 ϭ 2 ␽ R␽␸␽␸ 4 r,ur,v sin , ϭϪ 2⌽2 2 ␽ f Rv␸v␸ r ,v sin .

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