How Generic Are Null Spacetime Singularities?

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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 singularities, 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

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.

analogy with the non-linear gravitational case.) In or-

For example, for a scalar eld k = 2, b ecause one needs

der to show that this eld admits a functionally generic

to sp ecify b oth and on S .For the gravitational eld,

null singularity,we pro ceed as follows: Let x; y ; u; v be

it is well known that there are 2 2=4inherent degrees

the standard, double-null, Minkowski co ordinates (i.e.

of freedom. The actual number k,however, is 4 plus the

2 2 2

ds = dx + dy dudv ). Equation (1) reads

numb er of un xed gauge degrees of freedom, which de-

p ends on the sp eci c gauge conditions used. In the gauge

= V () (2)

;uv

we use, we nd that k = 8 (see b elow).

where here and b elow the indices a; b; ::: run over the

We shall say that a class of solutions to the eld equa-

co ordinates x and y .Wenow de ne

tions is functional ly generic, if this class dep ends on k

arbitrary functions of three indep endentvariables [18].

1=n

w v (3)

This concept of genericity is basically the same as that

used by BKL [7]. The motivation b ehind this de ni-

for some o dd number n 3. We also de ne

tion is obvious: Supp ose that a given particular solution

t w + u; zwu: (4)

admits some sp eci c feature (e.g. a singularity of some

typ e). Obviously, in order for this feature to b e stable to

Re-expressing the eld equation (2) in terms of t and z ,

small (but generic) p erturbations in the initial data, it is

we obtain

necessary that the class of solutions satisfying this fea-

ture should dep end on k arbitrary functions. Functional

= + n [(z + t)=2] V () : (5)

;tt ;z z

genericityisthus a necessary condition for stability, and

Let M denote some compact neighb orho o d of the ori- is also necessary in order that there b e an op en set in

gin (x = y = z = t = 0), and let S b e the inter- the space of solutions with the desired feature, in any

section of the hyp ersurface t=0 with M (see Fig. 1). reasonable top ology on the space of solutions [19].

Let f (x; y ; z ) and f (x; y ; z )betwo analytic functions As we mentioned ab ove, the goal of this pro ject is to

1 2

of their arguments, de ned on S . For any such pair demonstrate, mathematically, the existence of a function-

+ 0

of functions, there exists a neighborhood M M of ally generic null weak singularity. More sp eci call y,we

S , and a unique analytic solution (x; y ; z ; t)tothe prove that there exists a class of solutions (M; g)tothe

+ +

eld equation (5) in M , such that on S , = f vacuum Einstein equations, which all admit a (weak) cur-

and d=dt = f . This follows directly from the Cauchy- vature singularity onanull hyp ersurface, and which de-

Kowalewski theorem [20], in view of the form of Eq. (5). p end on k = 8 (see b elow) arbitrary analytic functions of

Let us denote the intersecti on of M with the null hy- three indep endentvariables. (In Ref. [16] we shall give

+ +

p ersurface v =0byN . Recall that N includes a a more precise formulation of this statement.) The sin-

neighb orho o d of the origin in the hyp ersurface v =0. gularities may also b e characterized by the fact that the

Returning now to the original indep endentvariables manifold may b e extended through the null surface to an

0 0 0

(u; v ), we nd that (x; y ; u; v ) is continuous through- analytic manifold (M ;g ) where the metric g is analytic

out M .Wenow fo cus attention on the section v<0, everywhere except on the null surface where it is only 3

change of indep endentvariables in the di erential equa- t 0ofM , whichwe denote by M . Since the

tions R =0;thus, the unknowns in Eq. (7) b elow are transformation from (z; t)to(u; v ) is analytic as long

still the six metric functions g , which corresp ond to the as v 6=0,we nd that (x; y ; u; v ) is analytic through-

co ordinates (x; y ; u; v ).] By taking certain linear combi- out M .However, will generally fail to b e smo oth at

1=n1

nations of the equations R = 0, it is p ossible to rewrite v =0: =(1=n)v will diverge at v =0as

i ;v ;w

the evolution equations in the schematic form long as 6= 0 there. We assume that at the origin,

df =dz 6= f . This ensures that at least in some neigh-

1 2

g = f (g ;g ;g ;g ;g ;z;t): (7)

i;tt i j j;t j;A j;AB j;At

b orho o d of the origin, b oth and are nonzero.

;w ;u

Let N b e the intersecti on of that neighb orho o d with the

Here, the indices A; B run over the \spatial" variables

section t 0ofN .We nd that diverges on N .

x; y ; z . If we imp ose certain inequalities on the ini-

Moreover, the invariant diverges on N to o [it is

;

tial data [which ensure that in the region of interest

1+1=n

dominated by(2=n)v ]. N is thus a singu-

;u ;w

det (g ) 1], then the functions f are analytic in all

lar null hyp ersurface.

their arguments. [The gauge conditions (6) are crucial in

We conclude that there exists a class of solutions to

deriving Eq. (7).]

Eq. (1), which dep ends on two analytic functions of

Wenow consider the evolution of initial data under the

(x; y ; z )(f and f ) that can b e chosen arbitrarily (apart

1 2

system (7). As b efore, we take the initial hyp ersurface

from the ab ove inequality), and which contains a singu-

to b e t = 0. Equation (7) requires twelve initial func-

larityonanull hyp ersurface. In other words, the scalar

tions to b e sp eci ed on this hyp ersurface: the six func-

eld admits a functionally generic null singularity. (Note

tions h (x; y ; z ) g (x; y ; z ; t = 0), and the six functions

i i

that hasawell- de ned limit on the singular hyp ersur-

p (x; y ; z ) g (x; y ; z ; t = 0). The form of Eq. (7) is

i i;t

face; this is the scalar- eld analog of the notion of weak

suitable for an application of the Cauchy- Kowalewski

singularity.)

+ +

theorem. Thus, de ning S , M and M as b efore, and

We turn now to generalize this construction to the

following the arguments ab ove, we arrive at the following

gravitational eld. As b efore, our co ordinates are de-

conclusion: For anychoice of the ab ovetwelve analytic

noted (x; y ; u; v ). We adopt the gauge

functions h (x; y ; z ) and p (x; y ; z ) on the section S of

i i

g = g = g = g =0; (6)

ux uy uu vv

t=0 (sub ject to certain inequalities), there exists an an-

alytic solution g (x; y ; z ; t) to Eq. (7) in M . Again, re-

which in turn implies that g = 0. This ensures that the

turning from the variables (z; t) to the original indep en-

co ordinate v is null (that is, the hyp ersurfaces v = const

dentvariables (u; v ), we nd that the metric functions

are null). There are six non-trivial metric functions,

g (x; y ; u; v ) are continuous throughout M (and in par-

whichwe denote by g (i =1;6), where here and b elow

ticular at v = 0) and, moreover, are analytic throughout

the indices i;j;::: represent the six pairs of spacetime

M .However, at the hyp ersurface v =0,g typically

i;v

co ordinates (xx; xy; yy; vx; vy; uv).

1+1=n

diverge like v . As a consequence, the Riemann

In this gauge, the number k of arbitrary functions in a

comp onents R generically diverge there [16]. More-

av bv

general solution is k = 8. This can b e seen as follows. De-

over, it can b e shown that the scalar K R R

ne the new variables T v + u; Z vu. Then to de-

2+1=n

also generically diverges at v = 0 (like v ). Thus,

termine a solution of the evolution equations, twelve ini-

fo cusing attention on the physical region M ,we nd that

tial functions need to b e sp eci ed on the spacelikehyp er-

the solutions constructed in that way are absolutely regu-

surface T = const, namely g (x; y ; Z ) and g (x; y ; Z ),

i i;T

lar inside the region M , but develop a null, weak, scalar-

1 i 6. However, these 12 functions must satisfy 4

curvature singularity on the p ortion v = 0 of its b ound-

constraint equations, as is always the case in general rel-

ary.

ativity, so that the numb er of indep endently sp eci able

The twelve initial functions h (x; y ; z ) and p (x; y ; z )

functions is k = 8. This conclusion can also b e reached

i i

are sub ject to four constraint equations. It is there-

by adding the conventional number of intrinsic degrees

fore natural to exp ect that eight of these 12 initial func-

of freedom of the vacuum gravitational eld (2 2= 4)

tions can b e chosen arbitrarily. This is not trivial to

to the numb er of un xed gauge degrees of freedom in the

prove mathematically,however, esp ecially b ecause the

gauge (6), whichwe show in Ref. [16] to b e 4.

constraint equations (expressed in the variables x; y ; z )

We shall now outline the generalization of the ab ove

are somewhat pathological at z =0. After some ef-

scalar- eld construction to the gravitational eld. First,

fort, we found a mathematical construction which proves

one writes the Einstein equations R = 0 in the gauge

the ab ove statement. More sp eci cally, in our mathe-

(6). These equations can b e naturally divided into six

matical scheme one is free to cho ose the six h (x; y ; z ),

evolution equations and four constraint equations. At

p (x; y ; z ), and one other function p(x; y ; z ). We can

this stage we fo cus attention on the evolution equations,

then show (using the Cauchy-Kowalewski theorem) the

which can b e taken to b e R = 0. Next, we de ne w ,

existence of a solution of the constraint equations (in a

t and z as b efore [Eqs. (3),(4)], and transform the eld

neighb orho o d of z = 0). The remaining initial functions

equations from the indep endentvariables (u; v )to(z; t).

p (x; y ; z ) are then determined from that solution. The

[Toavoid confusion, we emphasize that what we are do-

ab ove eight analytic functions can b e chosen arbitrarily,

ing here is not a co ordinate transformation: it is just a 4

up to some inequalities. [6] A.G. Doroshkevich, Ya. B. Zel'dovich and I.D. Novikov,

Sov. Phys. {JETP 22 122 (1965).

To summarize, our mathematical construction shows

[7] V. A. Belinsky and I. M. Khalatnikov, Zh. Eksp. and

the existence of a class of solutions to the vacuum Ein-

Teor. Fiz. 57, 2163 (1969)[ Sov. Phys.-JEPT 30, 1174

stein equations, which contains a weak scalar-curvature

(1970)]; I. M. Khalatnikov and E. M. Lifshitz, Phys. Rev.

singularity at the null hyp ersurface v = 0, and which

Lett. 24, 76 (1970); V. A. Belinsky, I. M. Khalatnikov and

dep ends on k = 8 analytic functions of (x; y ; z ). Our

E. M. Lifshitz, Usp. Fiz. Nauk. 102, 463 (1970) [Advances

construction therefore demonstrates the existence of a

in Physics 19, 525 (1970)].

functionally generic null, weak, scalar-curvature singu-

[8] E. Poisson and W. Israel, Phys. Rev. D 41, 1796 (1990).

larity.

[9] A. Ori, Phys. Rev. Lett. 68, 2117 (1992).

The main limitation of our construction is its restric-

[10] W. A. Hisco ck, Phys. Lett. 83A 110 (1981).

tion to analytic initial functions. We b elieve that this is

[11] U. Yurtsever, Class. Quantum Grav. 10, L17 (1993)

merely a technical limitation of the mathematical con-

[12] Note, however, that despite what was argued in Ref. [11],

struction used, and the same physical situation (a null

the existence of a generic null weak singularity at the

weak singularity) will evolveeven if the initial functions

inner horizon is absolutely consistent with the mo del of

are not analytic (provided they are suciently smo oth

two colliding plane waves [13].

at v<0). Atany rate, it would b e worthwhile to com-

[13] A. Ori, to b e published.

pare the mathematical status of our generic null weak

[14] P. R. Brady, D. Nunez and S. Sinha, Phys. Rev. D 47

singularity to that of the BKL singularity.To the b est of

4239 (1993).

the authors' knowledge, the existence of even a single in-

[15] P. R. Brady and C. M. Chamb ers, Phys. Rev. D 51 4177

homogeneous singular vacuum solution of the BKL typ e

(1995)

has not yet b een proved mathematically - let alone the

[16] E. E. Flanagan and A. Ori, to b e published.

generality of this class of singular solutions. On the other

[17] This is a slight abuse of the more usual terminology, ac-

cording to which the numb er of degrees of freedom is said hand wehave demonstrated rigorously the existence of a

to b e k=2, rather than k .

huge class of exact solutions containing null weak singu-

[18] Strictly sp eaking, in order for this de nition to b e useful,

larities.

the nature of the dep endence of the solutions on the k

Another imp ortant limitation of the present construc-

functions must b e restricted in some (very natural) ways;

tion is that it cannot demonstrate the evolution of a sin-

see Ref. [16] for further discussion.

gularity from regular initial data. All wehave shown is

[19] See Ref. [16] for more detailed discussions of genericity,

the lo cal consistency of null weak singularities with Ein-

stability and top ologies on the space of solutions.

steins equations, despite the non-linearity of the latter.

[20] See, e.g., Theorem 10.1.1. in R. M. Wald, General Rela-

The imp ortant problem of the onset of the singularity

tivity, (University of Chicago Press, Chicago, 1984).

from regular initial data (e.g. in gravitational collapse)

still remains op en; although the nonlinear p erturbation

analysis of Ref. [9] suggests a mechanism for this onset, it

FIG. 1. Spacetime diagram in z t co ordinates, illustrating

is still to b e proved rigorously that a null singularity can

the mathematical construction used.

form starting from an asymptotically at, regular initial

data.

This researchwas supp orted in part by the Israel Sci-

ence Foundation administrated by the Israel Academyof

Sciences and Humanities, by the Fund for Promotion of

Research in the Technion, and by the US National Sci-

ence Foundation grants AST 9114925 and PHY 9220644.

[1] See, e.g., Ref. [20], Sec.9.5.

[2] F. J. Tipler, Phys. Lett. 64A, 8 (1977).

[3] A. Ori, Phys. Rev. Lett. 67, 789 (1991)

[4] S. Chandrasekhar and J.B. Hartle, Pro c. R. So c. Lond.

A 384 301 (1982).

[5] Y. Gursel, V. D. Sandb erg, I. D Novikov and A. A.

Starobinsky,Phys. Rev. D19, p413 (1979); Y. Gursel,

I.DNovikov, V. D. Sandb erg and A. A. Starobinsky,

Phys. Rev. D20, p1260 (1979). 5