How Generic Are Null Spacetime Singularities?
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View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server How generic are null spacetime singularities? Amos Ori Department of Physics, Technion - Israel Institute of Technology, 32000 Haifa, Israel. Eanna E. Flanagan EnricoFermi Institute, University of Chicago, Chicago, IL 60637-1433. The spacetime singularities inside realistic black holes are sometimes thought to b e spacelike and strong, since there is a generic class of solutions (BKL) to Einsteins equations with these prop erties. We show that null, weak singularities are also generic, in the following sense: there is a class of vacuum solutions containing null, weak singularities, dep ending on 8 arbitrary (up to some inequalities) analytic initial functions of 3 spatial co ordinates. Since 8 arbitrary functions are needed (in the gauge used here) to span the generic solution, this class can b e regarded as generic. 1 typ e of generic singularity, in the last two decades it has One of the most fascinating outcomes of general rela- b een widely b elieved that the nal state of a realistic tivity is the observation that the most fundamental con- gravitational collapse must b e the strong, spacelike, os- cept in physics | the fabric of space and time | may cillatory, BKL singularity. b ecome singular in certain circumstances. A series of sin- Recently, there havebeenavariety of indications that gularity theorems [1] imply that spacetime singularities a spacetime singularity of a completely di erenttyp e ac- are exp ected to develop inside black holes. The observa- tually forms inside realistic (rotating) black holes. In tional evidence at present is that black holes do exist in particular, this singularityisnul l and weak, rather than the Universe. The formation of spacetime singularities spacelike and strong. The rst evidence for this new pic- in the real world is thus almost inevitable. However, the ture came from the mass-in ation mo del [8,3] | a toy- singularity theorems tell us almost nothing ab out the na- mo del in which the Kerr background is mo deled by the ture and lo cation of these singularities. Despite a variety spherically-symmetric Reissner-Nordstrom solution, and of investigations, there is to day still no consensus on the the gravitational p erturbations are mo deled in terms of structure of singularities inside realistic black holes. two cross owing null uids. More direct evidence came At issue are the following features of singularities: their from a non-linear p erturbation analysis of the inner struc- lo cation, casual character (spacelike, timelikeornull), ture of rotating black holes [9]. Both the mass-in ation and, most imp ortantly, their strength. The strength of mo dels and the nonlinear p erturbation analysis of Kerr singularities has far-reaching physical consequences. We strongly suggest that a null, weak, scalar-curvature sin- will call a singularity weak if the metric tensor has a well- gularity develops at the inner horizon of the background de ned and regular limit at the singular hyp ersurface, geometry. (See also an earlier mo del by Hisco ck [10].) and strong otherwise [2]. A physical ob ject whichmoves Despite the ab ove comp elling evidence, there still is a towards a strong curvature singularity will b e completely debate concerning the nature of generic black-hole singu- torn apart by the diverging tidal force, which causes un- larities. It is sometimes argued that the Einstein equa- b ounded tidal distortion. On the other hand, if the sin- tions, due to their non-linearity, do not allow generic so- gularityisweak, the total tidal distortion is nite, and lutions with null curvature singularities [11]. According may b e arbitrarily small, so that phyical observers may to this argument, the non-linearity, combined with the p ossibly not b e destroyed by the singularity [2,3]. diverging curvature, immediately catalyzes the transfor- The main diculty in determining the structure of mation of the null curvature singularityinto a strong black hole singularities is that the celebrated exact black spacelike one | presumably the BKL singularity. (Such hole solutions (the Kerr-Newman family) do not givea a phenonemon o ccurs in the example of two colliding realistic description of the geometry inside the horizon, plane waves [11,12]). A similar argumentwas also given, although they do describ e well the region outside. This a long time ago, by Chandrasekhar and Hartle [4]. Ac- is b ecause the well known no-hair prop erty of black holes cording to this p oint of view, the results of the non-linear | that arbitrary initial p erturbations are harmlessly ra- p erturbation analysis of Kerr are to b e interpreted as diated away and do not qualitatively change the space- an artifact of the p erturbative approach used [14] (and time structure | only applies to the exterior geometry. the mass-in ation mo del is a toy-mo del, after all). This The geometry inside the black hole (near the singularity ob jection clearly marks the need for a more rigorous, and/or the Cauchy horizon) is unstable to initial small non-p erturbative, mathematical analysis, to show that a p erturbations [4,5], and consequently wemust go b eyond generic null weak singularity is consistent with Einsteins the classic exact solutions to understand realistic black equations. hole interiors. To determine the structure of generic sin- Recently, Brady and Chamb ers showed that a null sin- gularities, it is necessary to take initial data corresp ond- gularity could b e consistent with the constraint section ing to the classic black hole solutions, make generic small of Einsteins equations formulated on null hyp ersurfaces p erturbations to the initial data, and evolve forward in [15]. However, their result do es not completely resolve time to determine the nature of the resulting singularity. the ab ove issue. The hyp othesis raised in Ref. [11], ac- For this purp ose a linear evolution of the p erturbations cording to which nonlinear e ects will immediately trans- may b e insucient | the real question is what happ ens form the singular initial data into a spacelike singular- in full nonlinear general relativity. ity, is not necessarily inconsistent with the analysis of The simplest black-hole solution, the Schwarzschild so- Ref. [15]. It is p ossible that a spacelike singularity could lution, contains a central singularity which is spacelike form just at the intersection p oint of the twocharac- and strong. For manyyears, this Schwarzschild singular- teristic null hyp ersurfaces considered in Ref. [15]. It is itywas regarded as the archetyp e for a spacetime singu- primarily the evolution equations which will determine larity. Although as mentioned ab ove this particular typ e whether singular initial data will evolveinto a null sin- of singularityisknown to day to b e unstable to deviations gularityorinto a spacelike one. from spherical symmetry (and hence unrealistic) [6], an- The purp ose of the present pap er is to present a new other typ e of a strong spacelike singularity, the so-called mathematical analysis which addresses the ab ove ques- BKL singularity [7], is b elieved to b e generic (b elowwe tion. Our analysis shows that (i) the vacuum Einstein shall further explain and discuss the concept of generic- equations (b oth the constraint and evolution equations) ity). Since the BKL singularity is so far the only known 2 0 C . admit solutions with a null weak singularity, and (ii) the Our construction is local in the sense that the manifolds class of such singular solutions is so large that it de- we construct are extendible (in directions away from the p ends on the maximum p ossible numb er of indep endent null singularity); roughly sp eaking they can b e thought functional degrees of freedom. We will call such classes of as op en regions in a more complete spacetime, part of of solutions functional ly generic (see b elow). There- whose b oundary consists of the singular null hyp ersur- fore any attempt to argue, on lo cal grounds, that a face. Wedonot prove that null weak singularities arise null weak singularity is necessarily inconsistent with the in the maximal Cauchyevolution of any asymptotically non-linearities of Einsteins equations, must b e false. In at, smo oth initial data set. On the other hand, our the present letter we outline this analysis and present + spacetimes are of the form D (), where is an op en the main results; a full account of this work is given in region in an analytic initial data set. However, the cur- Ref. [16]. vature singularity is already present on the b oundary of Let us rst explain what we mean by \degrees of free- in the initial data. dom" and \functionally generic". Supp ose that is some We shall rst demonstrate the main idea b ehind our eld on a 3+1 dimensional spacetime, whichmaybea mathematical construction by applying it to a simpler multi-comp onent eld. Supp ose that initial data for problem | a scalar eld. Consider, as an example, a are sp eci ed on some spacelikehyp ersurface S .We shall real scalar eld in at spacetime, satisfying the non- say that has k \degrees of freedom" if k is the number linear eld equation of initial functions (i.e. functions of the 3 spacelike co- ordinates parameterizing S ) which need to b e sp eci ed ; + = V () (1) ; on S in order to uniquely determine inside D (S ) the solution to the eld equations satis ed by [17]. The where V () is some non-linear analytic function. (We number k dep ends on the typ e of eld, and also p ossibly add this non-linear piece in order to obtain a closer on the gauge condition used if there is gauge freedom.