Feb 20, 1989 - gards notations, weshall refer tothe treatmentgiven chy problem for the null radial outgoing geodesic in refs. [12,131, and use the notations of ...
Volume 135, number 3
PHYSICS LETTERS A
20 February 1989
COSMIC CENSORSHIP AND TOLMAN-BONDI SPACETIMES Vittorio GORINI Dipartimento di Fisica, Gruppo Teorico, Via Amendola 173, 70126 Ban, ha/v and INFN Sezione di Ban, Ban, Italy
Gabriele GRILLO International SchoolforAdvanced Studies, Strada Costiera 11, 34100 Miramare-Grignano, Trieste, Italy
and Mauro PELIZZA Dipartimento di Fisica, Sezione di Fisica Teorica, Via Celonia 16, 20133 Milan, Italy Received 7 July 1988; revised manuscript received 16 November 1988; accepted for publication 19 December 1988 Communicated by J.P. Vigier
We classify a class of naked singularities in Tolman—Bondi spacetimes and prove that some ofthem are “gravitationally strong”. This provides a counterexample to a recently formulated cosmic censorship conjecture.
There have been two different approaches concerning cosmic censorship. Indeed, one can take the attitude of censoring globally visible singularities (i.e. visible from “infinity”). However, weak censorship conditions need a well defined infinity in Penrose compactification and, furthermore, general relalively is a scale-invariant theory. Thus, it seems more reasonable, following Penrose [1,2], to discuss strong censorship conjectures, in which the local visibility of a singularity is related to its stability, From this point of view, even though many examples of violation ofglobal hyperbolicity are known, they are not necessarily counterexamples to strong cosmic censorship cOnjectures; in some particular cases, instability can be tested by the well known “blue shift” criterion, in analogy with perturbation theory for Reissner—Nordstrøm spacetime [3]. Newman [4] has formulated a strong hypothesis of a new kind, with the purpose of relating the “strength” of a naked singularity to its stability, as tested by the blue shift criterion. Roughly speaking, the strength of a singularity is measured by the behaviour ofthe proper volume defined by Jacobi fields 154
along a congruence of non-spacelike geodesics. The idea is that singularities in whose proximity the proper volume vanishes, and which are at least locally visible, are unstable with respect to the blue-shift criterion. This notion can be formalized through the concept of generalized Jacobi field, and of strong limiting focusing condition (strong LFC). With the notations of ref. [51,the resulting conjecture is the following (assuming also one of the usual energy conditions). Conjecture [4]. Let JI be the maximal global hyperbolic development of some initial data (hab, Zab) on a partial Cauchy slice C. Assume that there exists a further extension J~?ofif, which is not globally hyperbolic. Then, if on each incomplete non-spacelike geodesic with at most one endpoint in strong LFC holds, then VpEH~(C),[(p)flC is non-compact. .~‘
The above conjecture has been supported by Newman himself [6], who has investigated a class ofTolman—Bondi spacetimes [7] whose naked singulanties, though locally visible, are “gravitationallyweak”,
Volume 135, number 3
PHYSICS LETTERS A
20 February 1989
in the sense that they do not satisfy strong LFC. In this note, we single out another set ofTolman—Bondi metrics for whichNewman’s conjecture fails (see also ref. [8] for a similar behaviour in a numerical selfsimilar solution to Einstein equations). On the cornparison between Newman’s results and ours we shall return below. It is also worth pointing out that previous analyses [9,10] have shown that at least in two particular cases within the Tolman—Bondi class, the so-called marginally bound and time-symmetric collapses, naked singularities may occur. Ourpurpose is thus to provide an example in which strong LCF holds. To this end, we shall rely on the following sufficient condition for strong LFC given by Clarke and Krdlak [11] in terms ofthe behaviour
where b, c and h are positive constants. We recall that when m (r) = const x [— 2E( r) ] 3/2, the spacetime is isometric to a closed spatially homogeneous (Friedmannian) universe, so that the constants c and h furnish a measure of the local non-homogeneity of the model. Eq. (3) has to be compared with Newman’s assumption that the metric and the energy density be smooth at the origin ofthe spatial coordinates. Now, it can be shown with straightforward but somewhat lengthy calculations that our comoving coordinate r (as defined below) is asymptotically proportional to Newman’s one, obtained by setting Y( t, r) = r, for some fixed, but otherwise arbitrary In (0, 2ith). Furthermore, for example, the energy density p ( I, r) behaves asymptotically as f~ (1) +f 2 (I) r’, where f1 and
of the Ricci tensor. Precisely, if y: (0, 1] ii is an incomplete geodesic (for example null), and if
are smooth functions of tin the interval (0, 27th). Therefore, in contrast to Newman, we do not require smoothness at r= 0. In fact, Newman’s assumption is crucial for his method of proof to work (see ref. [6],section 6, and in particular the fact that the constant h0 defined following formula (29) either vanishes or is infinite if h ,~2). On the other hand, in the case h = 2, Newman’s results coincide with ours (see our following eqs. (10) and (20). In the limiting case 2 in r=0. h =Returning 3 (see below), theanalysis, resultingby manifold is C to our the arbitrariness in the choice of the radial comoving coordinate t we are allowed to set (locally) (— 2E)’ /2 = r. The Cauchy problem for the null radial outgoing geodesic equation ds2 = 0 reads
—+
1
H(s)=j
~aI Rab_a—j
~bl
1
-~d~--~ dv=j’ Ricc(~’J’)dv
(1)
is non-integrable in (0, 1] (v is an affine parameter), then strong LFC holds. In Tolman—Bondi spacetimes, in which C’~4= 0 (i, j= 1, 2) holds in a pseudo-orthonormal basis parallel transported along y, the condition is necessary and sufficient [11]. It is first necessary to classify the initial conditions which give rise to a locally naked singularity. As regards notations, we shall refer to the treatment given in refs. [12,131, and use the notations of ref. [12], with the replacements R—~r, r—~Y. A dot and a prime will denote partial derivatives with respect to t and r respectively. We omit the consideration of shellcrossing and surface layer singularities [14,15]. We also set t 0 ( r) = 0. In the case in which the arbitrary function E(r) is negative in a neighborhood of the
origin and E(0) =0, the corresponding fluid elernents are “absorbed” by a singularity at a finite coordinate time (2) m(r)3”2~ t~(r)=2E [—2E(r)] Then, the requirement Y’ >0 (absence of shellcrossing) is equivalent to the condition that t, (r) is monotonically increasing. Set m
f(r):= (_2E)3/2_+~+~~~
(3)
dt t(0)=t,(0) dr — (1 + 2E)”2’ —
_________
Since E(0)
= 0,
.
(4)
in order to solve (4) asymptotically
we approximate by Y’. From the can parametric formthe ofright-hand the metricside (compare ref. [12]), we have sin ~ (5) Y’=(f’r+J)(l—cos,1)—f’ sin,1 cos and asymptotically, in a neighborhood ofr=0, i=2,t, ~
b (~—2it)2 2chr”2E 2 ,~— 2~t —
(6)
We are looking for solutions t(r) 3, the apparent horizon is strictly decreasing,exists so thatastrong censorship holds.behaving If h
