PHYSICAL REVIEW D, VOLUME 60, 104033 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 0556-2821/99/60͑10͒/104033͑5͒/$15.0060 104033-1 ©1999 The American Physical Society LIORM.BURKO PHYSICAL REVIEW D 60 104033 ⌽ 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␽ .