``On the collapse of a spherically symmetric star'' : Emphasizes That ``Singularity Theorems'' Essentially Assume What They Intend To Prove!

On the collapse of a spherically symmetric starMarcus Kriele Citation: Journal of Mathematical Physics 36, 3676 (1995); doi: 10.1063/1.530990 View online: http://dx.doi.org/10.1063/1.530990 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/36/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Conformal diagrams for the gravitational collapse of a spherically symmetric dust cloud AIP Conf. Proc. 1256, 349 (2010); 10.1063/1.3473876 On spherically symmetric gravitational collapse in the Einstein‐Gauss‐Bonnet theory AIP Conf. Proc. 1122, 356 (2009); 10.1063/1.3141325 Non‐linear perturbations of a spherically collapsing star AIP Conf. Proc. 1122, 205 (2009); 10.1063/1.3141259 Spherically symmetric collapse of an anisotropic fluid body into an exotic black hole J. Math. Phys. 38, 4202 (1997); 10.1063/1.532002 Analytical solutions of spherically symmetric collapse of an anisotropic fluid body into a black hole J. Math. Phys. 36, 340 (1995); 10.1063/1.531309

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On the collapse of a spherically symmetric star Marcus Krielea) The Fields Institute for Research in Mathematical Sciences, 185 Columbia Street West, Waterloo, Ontario N2L 5Z5, Canada

(Received 18 July 1994; accepted for publication 28 February 1995)

The initial value problem for a compact, spherically symmetric star with a rather general equation of state is discussed. For any given time it is possible to calculate the area of the spheres of symmetry which intersect the boundary, if space-time is Cs and the initial energy density is smooth (i.e., C”) and satisfies some minor technical conditions. The physical relevance of this result is also discussed. 0 1995 American Institute of Physics.

I. INTRODUCTION

It is generally believed that concentration of matter leads to its collapse into a black hole or, on a cosmological scale, into the final big crunch. The main support for this idea is the singularity theorems of Hawking, Penrose, and others (e.g., Refs. 1, 2) and the derivation of mass bounds for spherically symmetric stars pioneered by Chandrasekha? (cf. Ref. 4 for an extensive review).

The assumptions of the general singularity theorems are (even in principle) very difficult to justify experimentally, because some of them are of a global nature. For instance, the absence of causality violations is crucial to most singularity theorems (cf. Refs. 5 and 6). The presence of a closed trapped surface which is assumed in those singularity theorems that (hopefully) predict the collapse of a star is not entirely global. But under normal circumstances such a trapped surface can only occur inside the black hole region (Ref. 1, Proposition 9.2.1). Thus the singularity theorems seem to be inapplicable for predicting the formation of a singularity before a black hole forms.

The usual derivation of a mass bound is to calculate the condition under which a spherically symmetric configuration that is embedded into the Schwarzschild solution is static. But the failure to be static is not a very good indication for the actual collapse of a star. For instance, the radius of its boundary could oscillate or approach a value r*>O(t --+ m) (cf., Theorem III.1 below). Moreover, if a nonstatic star really collapses, we would like to know how this process takes place.

A particularly interesting problem in the theory of space-time singularities is to find conditions on the matter distribution outside the black hole region which ensure the collapse of the configuration. It seems that in order to find answers one has to study the initial value problem. Up to now such conditions have only been given in very special situations: Oppenheimer and Snyder7 studied spherically symmetric dust solutions. The case of a spherically symmetric, massless scalar field has been studied by Christodoulou.8 A spherically symmetric perfect fluid configuration with linear equation of state and a global condition to the effect that the energy density is decreasing in radial direction has been studied in Ref. 9.

The purpose of the present article is to study the initial value problem for smooth stars, i.e., for (1) smooth, spherically symmetric perfect fluid solutions which have fairly general equation of state and are (2) embedded into vacuum such that the energy density has compact support. Until recently it was even unknown whether (apart from the dust case) such solutions exist. However, Rendall” has proven an existence theorem for the polytropic equation of state, p(E)=KE(“+l)‘n(K,n ERf\{O}), under the assumption that initially Ed’ is smooth. (For sim- plicity, he also assumes that the initial surface is time symmetric; he does not assume spherical sy~etry.)

a@resent address: Technische Universitl Berlin, Fachbereich Mathematik, Sekr. MA 8-3, Strasse des 17. Juni 136, 10623 Berlin, Germany.

3676 J. Math. Phys. 36 (7), July 1995 0022-2488/95/36(7)/3676/18/$6.00

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Marcus Kriele: On the collapse of a spherically symmetric star 3677

Our approach is complementary to Rendall’s. We will not solve the initial value problem but under the assumption that a smooth solution exists (and a few more technical ones) we will calculate the motion of the boundary of the star.

The main theorem of this article states that there are only two extreme cases left: The matter at the boundary is free falling as in the dust case or the comoving velocity field has vanishing expansion at the boundary. In both cases it is possible to calculate the (area) radius of the boundary for any given time (we will refer to this as “calculation of the motion of the boundary,” because the area radius describes the boundary completely), independently of the precise matter distribution inside the star and of the precise equation of state (Theorem 111.1). It also follows that the boundary in the solutions guaranteed by Rendall” is free falling (cf. Proposition IV.1).

Our result may appear rather strange. So let us briefly discuss which property of the initial value problem is responsible for this result. Writing down the initial value problem in characteristic form (cf. Appendix) one sees that at the boundary the system is not hyperbolic in the sense of Courant and Hilbert.” Moreover, all characteristic directions coincide with the velocity direction of the comoving observers. The reason for this nonregularity is our assumption that the star is embedded into vacuum. This mathematical circumstance suggests to try to solve the initial value problem only along the boundary of the star. This is the key idea of the present article.

We will now state the conditions we are imposing on our solutions (these assumptions will be restated more carefully in the third paragraph of Sec. III):

(i) The matter is assumed to be a perfect fluid with energy density .YZ and pressure p. (ii) There exists an initial surface X0 orthogonal to the flow lines of matter such that

supp(e)flC, is compact and e/g, is smooth, i.e., C”. (iii) There exists an equation of state, p = f ( E), and an q>O such that

(1) f~c0.q E c3, (2) f”((O,qJ)CR+, if f ‘(0) exists.

(iv) Some monotonicity assumptions concerning the first two radial derivatives of E near the boundary of the star.

(v) E does not approach 0 too fast (this condition is not needed for Proposition IV.l). (vi) Space-time is C3 and E is also C3.

Condition (ii) states that initially the star is a smooth, compact object. By “smooth” we mean C”. However, sounding quite innocent, condition (ii) is our key assumption and smoothness of ~1”~ cannot be relaxed significantly. In particular, our results would not hold if one just assumed traditional Cl-matching conditions at the boundary. Condition (iii) roughly states that the equation of state and its derivative do not oscillate too much near E=O and that pressure is positive for small energy densities. It is satisfied by natural candidates for an equation of state, for instance, by a polytropic fluid [however, in this particular case, our main theorem can be strengthened (cf. Proposition IV.l)]. Condition (iv) is also technical and probably does not have much bearing on the result. Condition (v) is only a condition on the initial data and can therefore be satisfied, if the initial value problem makes sense. It should not raise much physical concern because it applies only to an arbitrarily small neighborhood of the boundary in the initial slice. In particular, the initial condition of Rendall can still be satisfied.

The article is organized as follows: Section II contains preparational material on spherically symmetric perfect fluid space-times. In this section only the equation of state, the existence of f’(0) [the existence of f’(0) need not be assumed for the main Theorem 111.11, and the field equations are needed. The main result of section II, Proposition 11.3, may be known, but I could not find a proof in the literature. In Sec. III the motion of the boundary of the star is calculated. In Sec. IV the physical relevance of the main theorem and the conditions we impose are discussed, In the Appendix we put the initial value problem into characteristic form. Its only purpose is to illustrate the failure of hyperbolicity of the Einstein equation stated in the Introduction.

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3678 Marcus Kriele: On the collapse of a spherically symmetric star

II. SPHERICAL SYMMETRY

Let (M,g) be a spherically symmetric perfect fluid space-time. We choose comoving coordinates (t,q, e,(p) in which the metric has the form

g=-e 2v(t,q) dt2+e2h(t*q) dq2+ r2(t,q)(d02+ sin2( B)dq2). (2.1)

(It should be stressed that the argument presented in this article is independent of the global existence of such coordinate systems. The argument is essentially local. Parts of the argument which are global are given in manifestly invariant form.) We will assume that there is a center of symmetry given by r=O. The velocity vector field of the flow lines of the fluid is then given by u=e-“a, and the energy momentum tensor by T=(~+p)u b@u b +pg, where E denotes the energy density and p the pressure of the fluid. We also assume an equation of state

f:R -+ w,

E H f(e)=p.

In this article we will express all quantities with respect to the (partial) orthonormal frame {u,L}, where L: = e-’ d, is the unique unit vector that is orthogonal to u and to the orbits of SO(3) and satisfies dr(L)sO near the center of symmetry. Thus L is geometrically defined. The quantities F: = L. r and y: = u. r are therefore also of geometric significance [in the following we will for any function F and any vector u denote the directional derivative dF( v) by u. F]. It is easy to check that

[u, L]=(L*v)u-(u.X)L, (2.2)

which gives an invariant meaning to L. v and u. X. Misner and Sharp12 have shown that

r= Jy2+ 1-(2&r), (2.3)

where

I r(t,q)

m(t,q): =4rr F2e di (t fixed) 0

is the total mass enclosed by a centered sphere with surface area 4 w2. For m the following equations hold: l2

u-m= -4rrr2yp, (2.5)

L=m=47i-r2TE. (2.6)

Lemma Ill: In our notation, the field equation, tic-s/2g= 8aT, is given by (i) u.y=l?(L.u)-mlr2-4rrrp, (ii) u.r=y(L. v), (iii) u~u+X+(u~X)2=2mlr3fL~L~v+(L~v)2-47r(fz+p), (iv) hy=ryu.x), (v) L.r=mlr2+y(u.X)-47rrE. Proof: Formulas (i), (iii), (iv), (v) follow from Ref. 13, 14.17, the definitions of y, F, L, u, and

Eq. (2.3). (ii) follows from (iv) and Eq. (2.2). cl Lemma 11.2: The equation of motion, div( T) = 0, is given by (i) L.p=-(E+p)L.v, (ii) U-E=-(~+~)div(u)=-(e+p)(u.X+2ylr).


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Proof Reference 13, 14.18. q Let x0 be a spacelike hypersurface orthogonal to u. We consider & as initial value surface. With the exception of two unphysical cases (cf. Remark II.4 below) the exterior of a star may

be characterized by the mass m being constant and is locally isometric to the Schwarzschild solution. This is the main content of the following proposition.

Proposition 11.3: Suppose int(f- ’ ((0))) =0 and g does not satisfy r2 = l/( 8 TE) = - l/( 8 TP) =const in any open set.

(i) Let ?KM be open and connected. Then ml!/= cons@ g1 tL is Schwarzschild @elfA = 0.

(ii) Zf supp(e)fl& is compact, then there exists a qtl==o such that 4e=min{qIE(I,<)=OVi>q} for all PER.

(iii) ml{(t,q)lq>qB}=const. Proof: (i): Assume ml //= const. Equations (2.5) and (2.6) imply that at each point x E ‘ZG one

of the following holds:

(A) g(x)=0 and p(x)=0 or

(B) e(x)=0 and y(x)=0 or

(C) I’(x)=0 and y(x)=0 or

(D) r(x)=0 and p(X)=o.

Since all these sets are closed and because of int(f- ’ ({ 0))) =8, we find for each neighborhood %‘C ?6 of x an open set ,?J?I?%’ such that on V one of the following holds:

(4 ~(~-0 and p17,=0 or

(b) EI~.=O and y1~,=0 or

(c) lY17,=0 and ylr.=O or

(d) lTlr.=O and plr=O and ~1~’ is constant.

We show first that m =const implies that g17’ is Schwarzschild. In case (a) we have TIT’ = 0 and the claim follows from Birkhoff’s Theorem. Because of the equation of state we have in case (b) ply ,=f(O) = const. In view of Lemmas

II. 1 (i) and II.2 (i) we obtain 0 = - mlr2-4vrp, if p f: 0. Hence r is also constant, whence T=L.r=O. Lemma II.1 (v) implies now m=O. From Eq. (2.3) we obtain rlj. = 2m = 0 which is impossible since g is a Lorentzian metric.

In case (c) we have r=const. From Lemma II.1 (v) we infer m =4rr3e. Hence e=const. From Lemma II.1 (i) we obtain E= -p. Now I?=0 implies r2=ml(4rrer)= 1/(87re) = - l/( 8 7rp) =const.

In case (d) we can assume y # 0. Then Lemma II.1 (i) and (ii) give u .y = -m/r2 and L. Y= 0. From Lemma II.2 (ii) we get u. X = - 2ylr. We apply these equations to Lemma II. 1 (iii) to obtain

( 1 2 2m -2 u.; +4;=7-4lrE.

r

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Hence we have u.y= -mlr2f2gEr-3y2/r= -mlr2, and therefore y/r=const. Thus u. A =const and we can again use Lemma II.1 (iii) which now gives mlr3=const. But this means y=u.r=O.

The union of all T is dense in i%. Thus gi,, must also be Schwarzschild. gl,, Schwarzschild * cl,, = 0 is clear. So because of the connectiveness of % we must only

show that EI,, = 0 j rnlJl is locally constant. p =const because of our equation of state. We can assume that p # 0. From Lemma II.2 we get u. A = - 2ylr and L * v= 0. Lemma II. 1 (i) gives

u-y= -mlr2-4nrp (2.7)

and Lemma II.1 (iii) gives -2(~.y)lr+6(ylr)~=2m/r~-4~p. Both equations together imply y/r= k (k E W) and therefore u. X =const. A further application of Lemma II.1 (iii) gives 2mlr’=const. Thus O=(u.m)lr3-3mylr4 and therefore by Eq. (2.5) 0= -4~(ylr)p-3mylr4 which gives 4rrp= -3mlr3 or y=O. If y # 0, then u+y=k2r and Eq. (2.7) imply k2=2mlr3. Inserting this equation and y= kr into Eq. (2.3) we obtain r= 1. Thus Lemma II.1 (v) gives O=mlr2+y( -2ylr) and therefore k2=ml(2r3). We conclude k=y=O. Now it follows from Eqs. (2.5) and (2.6) that m is locally constant.

(ii): Let t,ER and q>min{~:~(t~,~)=O,p(t~,q)=OVq”>~}. First we show that e(t,q) =O, p(t,q) = 0 for all t ER. From Lemma II.2 (ii) and the equation of state we obtain

Since c(t.+.,q)=p(t* ,q)=O, this differential equation implies e(t,q)+p(t,q)=O for all tER. Lemma II.2 (ii) implies then E( t,q) = 0 [and p( t,q) = 0] for all t E R and our assertion is proven.

Since supp(c)nxo is compact and because of (i) there exists a point at which c=p = 0. Thus the equation of state implies min{G:E(t, ,<)=O,p(t, ,q)=OV@>;}=min{<:e(t, ,+)=OVq”>G} for all t, EW.

(iii): This is an immediate consequence of (i), (ii), and the fact that {(;,i) E t(M)Xq(M):G>qB} is connected.

Remark 11.4: If intv-‘((0))) # 0, then there exists an energy density interval in which the equation of state resembles dust. In Theorem 13.7 of Ref. 13 the existence of a (essentially unique) dust solution [i.e., f - ‘((0)) =W] with constant mass m but nonzero energy density E is established. But this solution is rather unphysical, because the area of the spheres of symmetry depends only on time.

g=--(Aq+a)’ dt2+(Bt+b)2 dq2+r2(d82+sin2(8)dp2) is an example of a spherically symmetric metric satisfying r2 = l/( 8 ne) = - l/( 8 7~p) =const.

III. THE BOUNDARY OF A STAR

Proposition II.3 allows us to define the boundary of a star to be the submanifold B = d(supp(~))={x E M\q(x) = qe}. We denote the flow of a vector field by exp. Let T be the proper time with respect to u. Then Proposition II.3 implies B={exp(~u(x))]x E BflZ,}. For each ? we can define a unique hypersurface 2; by demanding that it is orthogonal to u and intersects B in {exp(?u(x))]x EB~ZZ~}. Notice that rip; is not constant, but tlZ+ is constant (it is of course possible to reparameterise t so that it coincides with 7 at the boundary of the star). For x E M we will also use the notation Z” to designate the hypersurface x:, that contains x.

On each 2; we can define an invariant, radial coordinate

Q:Z; --+ !R+,

x H distlx;(x,center of symmetry). (3.1)




We denote the value of Q at B with Q,. Notice that the value of Q, depends on the considered hypersurface c ; . It is easy to see that L. Q = 1, L = dQ , and dQlde= l/(L. E). We will use these equations without further mentioning.

General Assumptions: Let us state conditions (ii)- of the introduction more precisely. We demand that for each point x E B there exists a coordinate neighborhood such that the metric is given by Eq. (2.1) with C3 functions v,X,r. On the initial surface Z0 we assume that

supp(e))n& is compact, (3.2)

Epo20. (3.3)

We will also assume that E is C3 and that ~12~ is smooth. We demand that there exists an e+O

such *at &,6,,) CR+ and thatfl(o,eoj is C3. Note that f( 0) = 0 because of Eq. (3.2) and Lemma II.2 (i). Equation (3.3) and the argument in the proof of Proposition II.3 (ii) imply that ~20 everywhere. Moreover, we demand that (L.E)Iz,, (L.L.E)Iz,, (~/L.E)~x,, (eL.L.e/(L . ~)~)lx are monotone functions in 4 near the boundary. These monotonicity conditions ensure

that certkn limits exist. They are satisfied if 61~ (4) = Ae-~‘(e~-q)y for q<qs, where A,P,y > 0. These properties are assumed throughout thl article. (Proposition IV. 1 and the lemmas used in its proof are an exception. For these results relaxed differentiability conditions suffice.) But we may refer to them explicitly when we find it instructive.

The boundary of the star is obviously completely described by its radius r(t,q& at each time t. Because of Lemma II. 1 one needs to know L + v in order to calculate r along B. L. Y which is given in physical terms by Lemma II.2 (i) which reduces to 0=0 at the boundary. Thus the value of (L . V) la depends on the rate at which E and p approach 0. The smoothness assumption for the initial energy density E and the compactness of supp( E) fl Z; are therefore crucial for our discussion. The other assumptions are of a technical nature.

The main theorem is the following: Theorem 111.1: Suppose, our general assumptions are satisjied, in particular E is cl’ and L. u,

L.L. Y, L+L=L. u, u.L+L.v, u.X, L.u.h, L.(div(u)), L+L.(div(u)) are locally bounded near B.

Assume that there exist a, c>O, b E (0, l), a neighborhood JV’ of Bf& in co such that ~1, , is smooth and

E(X)>ae-~I(Q~-Q(~Db for all x EJK

Then the comoving observers at the boundary are either free falling (V,u = 0) or their velocity field has vanishing expansion (div(u) = 0) at the boundary.

If proper time with respect to u/s is denoted by r, then the motion of the boundary is given by

-I/2

dr

for any interval in which the integrand is bounded, or

The radius of a collapsing or expanding star approaches a JCinal radius r* > 0, if suitable initial data are given.

The rest of this section is devoted to the proof of Theorem III.1 and is fairly technical but the idea of the proof is simple. At the boundary one would like to obtain a system of ordinary differential equations with derivative operator u. Inspection of Lemma II.1 shows that in order to

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achieve this one has only to express u. L. v in terms of r,T, y, L. Y. In Lemma III. 10 we show that u. L 1 V= - (u . X) (La V) = - (div( u) - 2 ylr)( L. V) at the boundary, if f’ (0) exists [we give a different argument for the (not very physical) case that f ‘(0) does not exist]. The key argument is to show that if L. v # 0, then div(u) = 0 (cf., Theorem III.12). This is achieved by calculating u. L. L. Y and observing that it contains a summand of the form wdiv( u) and that all other summands of u. L. L. v are bounded. Here we are using that the system of partial differential equations (PDEs) fails to be hyperbolic at the boundary of the star. One may think of this approach as just writing down the Taylor series and noting that because of the special form of the (nonhy- perbolic) Einstein equation there is an ordinary differential equation which decouples.

The reader not interested in the technical details may wish to skip the rest of this section. During the argument the initial smoothness of E is used several times by using that

EL. La eI(L . E)~ approaches 1 at the boundary. This fact is proven in the following three lemmas. Lemma 111.2: Let a>0 and h: [O,a) -+ W+ be a continuousfunction with h-‘({O})=(O). (i) If h is differentiable and h’ is monotone in (0,a) then

lim(h(z)lh’(z))=O. z-0

(ii) If h is two times differentiable and h’, hh”/h” are monotonefinctions in (0,a) then

h(z)h”(z) ;+; (h’(z))2 E[“,ll’

(iii} If h is n times difSerentiable, h’, hh”lh’2 are monotone functions in (O,a), and h’(O)=...= h’“‘( 0) = 0, then

h(z)h”(z) 3 n- 1 f% (h’(z))2 n *

Proof (i): Since h is a positive and h: a monotone function near z =O, h/h’ is bounded away from zero, unless lim,,o h(z)lh’(z)=O. Suppose, there exists b>O such that h’(z)lh(z)<b. Then for all z, zoE(O,a), z>zo, we have ln(h(z))-ln(h(zo))<b(z-zo). This is impossible because -ln(h(zo)) -+ ~0 (z. --t 0).

(ii): Since h(0) =O, h’ # 0, and h>O (z E (O,a)), the function h’ is positive and increasing. Hence h” is positive. If our claim does not hold, then we have

hh” 7’1 h

near z=O. This implies (ln(h’))‘>(ln(h))’ and therefore h’(z)/h(z)>h’(zo)/h(zo) for all z, zo E (O,a), z>zo. This is in contradiction to (i).

(iii): Suppose, there exists a i>O such that

h(z)h”(z) <n - 1 @‘(z))~ n

for all z E (O,z*). This implies

h’(z) ,(;)b-lb

h’(z)< h(z)(“-I)‘”

and therefore there exists a constant k>O with


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h’(z) h(z)(n-l)ln’k

for all z E (O,.?). Integrating once again we obtain h(z)>(kln)“f for all z E (O,?). Hence applying the rule of 1’Hospital n times to the quotient h(z)@ we arrive at a contradiction. cl

Remark 111.3: Clearly, Lemma III.2 holds also for functions h defined on (- a,O] instead of [O,a). In particular, (EL . L . l (L . E)-~)Iz,(x) + 1 (x -+ B).

Lemma 111.4: Assume, f (0) exists and let x” E Co. Choose a compact neighborhood U of x” and let 7*<0< r.+* such that exp( ru (x)) is de$ned for all x E U and all r E [ r.+ , T* .+I. For any x E M denote the hypersu$ace 2; that contains x by c”.

There exist constants K- , K, , A> 0 such that (i) K- distZO(x,B) 6 dist ~exp(7u(x))(exp( ru(x)),B) G K, distsO(x,B), (ii) K-e(x)~~(exp(ru(x)))~K+~(x)

for all rE(r* ,r.+* ) and all x E (2 E X,,fl Ujdist$i,B) < A}. Proof Let xs E B fl ZZo. Since E and f are continuous, lim6+o f '( E) exists, Q= 0, and there

exist a c>O and a compact neighborhood 557 of exp([r, ,r**]u(xJ) with f 0 ~1~ s CEI,~. There exists a A>0 such that {exp([ r* , r.+*]u(x))lx E X0, distzO(x,B) < A}CZ.

(i): .Z can be covered with finitely many comoving coordinate charts. In each such neighborhood we have

u.dist~~(x,B)=e-” I 9ww)

ewixu-h dq. q(x)

Since e-“, e’u. A are bounded, we obtain constants, C,>O depending only on .% such that - C- distZ*(x,B) 6 u . distx,(x,B) s C, dist&,B). Dividing by distnX(x,B) and integrating gives (i).

(ii): From Eq. (3.3) and Lemma II.2 (ii) we obtain

-( 1 +c)e($ldiv(u)ll;G(u. e)l(;)s(l +c)e(i)ldiv(u)ll;.

Since ]div(u)Il.; is bounded in 55, we find a constant K such that

(1 +c)sup{]div(u)ll;]~E~GK.

Integration gives -Kr<ln e(exp(ru(x)))-ln e(x)SKr for all x E {i E Zo]dist~O(x,B) < A} and all rE[r+.,r**]. Our assertion follows by setting K- : = e-kc’**-“*), K, : =e K(7**-r*) Cl

Lemma 111.5: Assume, f' (0) exists and ~12, is smooth. Then

lim qz;b&L- qq(x)

x+B CL* qq(xH2 = l

for all i. Proof We obtain from Lemma III.2 (ii) that

lim qz;(xWL* qqw

x-+E (L.e,&(x))2 S1.

Assume, there exists an n E N such that this limit is smaller than (n - 1 )ln. As in the proof of Lemma III. 2 (iii) we infer




Lemma III.4 implies a similar inequality on &. But this contradicts the smoothness of ~12~. 0 Lemmas 111.6-111.11 will be local. So we use comoving coordinates to make our statements

notationally more transparent. Lemma 111.6: Suppose, EE C’, fis differentiable, f(0) =0, f’(0) and (L . v)(~,~~) exist. Then (9 f’(O)=O, (ii> CL . “)I(~,~~)= -limq+qB,4<4B (f’(dt,q))(L+ 41(f,,)ldt,q)). Proof: (i): By Lemma II.2 (i) we have

CL- q(r’qB)= - ,yyy<,,

f’(4t74)W. 4l(r.q)

dt,q) ff(4oz)) .

Since f(O)=0 and lim,,e f’(e) exists, there exists a constant k>O such that Ie+f(~)l<( 1 +k)lel for 14 ffi su ciently small. Hence near the boundary we get

I I f’(6)L.E 1 , E L-85 E+f(E) ‘l+k If ( )I~-1 E .

Since by Lemma III.2 (i) (and Remark 111.3) (L. E)/E diverges, we obtain (i). (ii): Since for fixed t lim4+4B,4<48 f(e)/6 = f’(0) = 0, (ii) follows from

L.V+f’(e)L.e = -f’(E)L.E+f’(E)L.E E I I ES-f(E) E l=lL.vl~f $1.

Lemma 111.7: Suppose, f ‘(O),(L . Y)I(~,~,) exist, (L . L . E) (t,.J and ( e/L. E) I(~, ) are monotone, and e( t, . ) is a non-negative function in some neighborhood of qB which satisfies the conclusion of Lemma 111.5. For each DO there exists a qr,+qB such that

(9 - (e/L 1 4((L + 41(t,qB) - 4 s f’(e) c -(e/L . e)((L . ~)l(~,~,) + S), (ii) - (e2/L . e)((L . v) Ih7g) - 6) s f(e) s -(2/L * e)((L * 41(r,4,) + 4,

whenever q E (qt,s,q&. Proof (i): A trivial consequence of Lemma III.6 (ii). (ii): Let Is>0 and lq-qs/ be sufficiently small. Then (i) implies

where 25 is given by E(e(t,q))=(L. E)I(,,~). Let Q(q) be defined as in Rq. (3.1). We obtain

q H (e/(L+ E))I(~,~) is a monotone function by assumption. Since q H E(t,q) is also monotone, e ---f - e/%(e) must be a monotone function and an integration of the last equation results in

for



Marcus Kriele: On the collapse of a spherically symmetric star

ll-;~ff#& where Isis defined by 2%Z’(e(t,q))=(L.L.e)l~r,q~. Thus

s

e E2 2de3-(l-b) G

0 -Re)

holds for 14 - qe( small enough and the first inequality follows since b and 8 can be chosen arbitrarily small. For the other inequality remember that e He/Z(e) is a monotone function which implies

f(4 = I,‘f’te)de~(tL~ ~)l(~,~~)+ alI,’ $ -2

des((L. ~)l(~,~~)+ 8) G.

Lemma 111.8: rf(L . v)I(~,~,) > 0, f’(0) exists, and E satis$es the conclusion of Lemma 111.5, then

q!yrB

f'(4t4))dm)

f(4t$?)) = l9 (9

lim f”(E(t,q))L*L.f$t,q)=m. 4-48

(ii)

Proofi (i): Let s>O. For lq-qsl sufficiently small we find

f’(4E -(eIL.e)((L.V)l(t,qg)+~)‘)E (L+)l(f,qg)+S

f(e) -?(L. 41(&q,)- m-(e2/L4)= (L41(r,qg)-S’

Smce S was arbitrary, we get llmq-qB f'(cz)E/f(E) G 1. Suppose. lim,,qB f'(e) e/f(~) < 1. Then for lq - qel sufficiently small we have

(lnCf( e)))‘<(ln(e))‘. Since E is monotonically decreasing we obtain e(q)/f (e(q))< e(G)/f(e($)) for all q E (G,qR), /<-qBj sufficiently small. But

4t4) L. 4t,q) qyyB f(4t,q)) =q!$B f ‘(dt$T))L. 4t4) =O”

because f’(e(t,q)) --t 0 (q -t qB). This is a contradiction. (ii): From Lemma III.7 (i) we obtain

>(tL’ Y)l(t.qB)- *)

Because of the conclusion of Lemma III.5 and (L . ~)l(~,~~) > 0 the first two factors are > (1/2)(L * ~)l(~,~,) for lq-qs/ sufficiently small. Thus it is sufficient to prove that the last factor diverges. Suppose, there exists a b>O such that

f"(E) b f'(c)< -L.E’


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Since de/de= l/(L.c) we conclude ln(f’(E(t,G)))-ln(f’(E(t,q)))s-b(Q(G)-Q(q)) for all 4 E (442s) with IG-- qej sufficiently small. This gives a contradiction sin; -ln(f’( E)) + co(1540).

Lemma 111.9: Zf E satis$es the conclusion of Lemma 111.5, f ‘(0) exists, (L . Y)I(~,~~) > 0 but

CL * L . @l(t,qB) is bounded, then 1imq34B f”(e(t,q))(L . E)I~~,~) = 0. Proof:

(L.L.v)~~~,~~)= lim L. ‘?-“?B

f”(4(e+f(4)-(1 +f’(c))f’(c)

Since limq,qB( 1 - eL . L . e(L . E)-~) = 0 and lim f’( e)=O, our claim follows. Lemma III. 10: (i) u~L~v=Cf”(~)L~~+(L~v)f’(~))div(u)-(L~~)(u~X)+f’(~)L~(div(u)), (ii) (u * L . 41(f,qB) = -CL . 41ct,qB)(U . A)lcr,qB) v provided CL . V)l(r,qB) #

exists, and div( u)lCr,qBj , L. (div(u)), (L . L . v)I(,,~~) are bounded. Proof (i): An application of Eq. (2.2) gives

cl

0, f’(O)

Thus we calculate

-f”(4(e+f(4)+(1 +f’t4lf’t4 f’(E) u.L.v=

(E+f(4)2 (L.e)(u-•)--

e+f( 4 z4.L.E

(ii): Follows immediately from (i) and Lemma 111.9. Lemma zzz.11: If f’(0) exists, w*41(r,qB)>o? (L*L. dl(f,qB)? w*:

.zJ)I(~,~~) are bounded and there exist a, c>O, bE(O,l), a neighborhood .A. of (t,qB) in I2 Cr,qBj such that

e(x)>ae -dQ~-Qtd)~ for all XE.I.~~,




then

lim (f”‘(E(t,q))(L.e)i,,,)+f”(e)L.L.e)=m. q-+48

Proof: Set E(q):=-(f’(~)L~d~)~~r,q~. By Lemma III.6 (ii) we have limq,qB / =(L . ~)l~,,~~~. From the definition of / we obtain

f’(E)L*E=L*f’(E)=-L. & &L./. ( 1

Differentiating, once again gives

f”‘(E)(L*E)2+y(E)L.L.E=L.Cf)l(E)L.E)

=-L.L.(&-2L.(&)(L.&&L.L./. (3.4)

Ld=L*(6-L. v)+L.L. v=(L.L. V)(l +f(E)lE)+(L* V)L*(f(E)IE). Since L-tf(4I4 =&(f(e)I(ef’(e))-1) we obtain

L./=(L*L. v) If ( ~)+(W(-$$-1). (3.5)

Using Eq. (3.5) we calculate

L.L./=(L.L.L.v) i 1

1+f9 f(L.L.y)L. f(E) ( 1

+(L.L.4/(~-1)

+(L..,,..(-&)+(L.,,,&))

f(E) =(L*L*L*v) 1+-

( 1 E +(2(L.L.v)~+(L.v)(L.4)(~-l)

+(L.v)Py l--- (

f(E) f(#Yd Ef’(E) 1 fW2 .

The first summand is bounded by assumption. The second summand converges to 0 because of Lemma III.8 (i) and Eq. (3.5). The last summand, multiplied with E/L. E, is bounded, because f(e)f”(E)lCf’(e)2)C1 by Lemma III.2 (ii). Because of this, Eq. (3.5), and L.(E/L.E)= 1 -EL.L.E/(L.E)~, the only summand in Eq. (3.4) that may not be bounded is the first one.

Suppose, there exists a C>O such that - L. L. (e/L. E) < C. Then an integration gives

-La( &ilQ+L.( &),I=o)

for all QE(&,Qs). Letting Q --+ QB we obtain L . (E/L . e)li) G C(QB - Q) for all Q<Q,. Applying the same argument again results in




A further integration gives - ln(e)lQ + ln(E)l4 > -(2/C)(ll(Q - Q,) - l/(Q - Q,)). We fix n Q=:Q(<) and set k: = e(t,q)e2’(c(Q~-Q)). With this notation we get

e(t,q)Cke -~/[C(QB-Q(~))I

for all q E (G- qB) in contradiction to E 3 ae-c’(qg-q)b, because QB - Q(q) s sup{< E (4,qdleh(r,q)(q~ - q)}. Thus such a C cannot exist and we have - limq,qB L . L * (e/L . E) = m. Our assertion follows since (L . v)loqBj > 0 by assumption. 0

TheoremIII.lZ:Suppose, eissmoothandL~v,L~L~v,L~L~L~u,u~L~L~v,u~X,L~u~X, L. (div( u)), L. L. (div( u)) are locally bounded near B.

Zf there exist a,c>O,b E (O,l), a neighborhood &‘-of Bnzo in x0 such that

e(x)?=ae-C’(Q~-QCX))b for all XEJ~*

then (div(u))ln=O for all t or (L. v)lu=O for all t. Proof: Assume first, f ‘( 0) does not exist and div( u)Iu # 0. By our monotonicity assumptions

on f’ and ImCf)CR+ we have lime-O,e,O f’(e>=m. Lemma 111.10 (i) implies u.L= v=(L.(lnf’oe+ u+ln(div(u))))f’ div(u). The first summand in the inner parentheses is unbounded whereas the other two summands are (without loss of generality) bounded. Thus the sum diverges and therefore also its derivative. This implies f’ div( u) = 0 since u. L. v is bounded by assumption. Hence div( u) = 0. This is a contradiction. Now assume that f ‘(0) exists. Using Eq. (2.2) and Lemma 111.10 (i) we calculate

Let x E B and suppose, (L . V)I, # 0. The last two summands are bounded by our assumptions. The second, third, and fourth summand vanish at x. The assumptions on the falloff of EIZ, hold also for ~12, (with different constants a,c) because of Lemma 111.4. Lemma III.1 1 implies that the first factor of the first term is unbounded, whence (div(u))l, must vanish.

So we have only to show that (L. v)lx=O at one point (t,qg) implies (L+ v)~~=O. But this follows since u~(L~v)~~=-(L~v)~,(u~X)~~ for (L.v)lX # 0. 0

Proof of Theorem III. 1: Since V,u = (L . V) L and the expansion of u is given by div( u) the first claim is only a restatement of Theorem 111.12.

Let x E B. Assume that (L . V) lx # 0. With the help of Lemma III. 10 (ii) and (div( u))l, = 0 we obtain for the motion of the boundary the dynamical system

u.r=y, (3.6)

u+y=TL. v-mlr2, (3.7)

u.T=(L.v)y, (3.8)

u.(L.v)=2(L.v)ylr, (3.9)

2 rp,n~= YI,Z,“E+ 1 -2mlrpof7i3- (3. IO)

Equation (3.10) is necessary to guarantee the constraint (2.3). Observe that L. v=O is included as a special case. We mark quantities evaluated at ~0 with a subscript 0. If y,=O and (L




. ~)a = ml(G&%&), then the solution of the system is given by r=const. Let us now assume that y is a function of r. Since y = U. r, this assumption is satisfied unless r is constant. Equation (3.9) implies L. v=((L. ~),lr$r~ and Eq. (3.8) gives IT=(L. v),l(3r~)(r3--r$+rs. From Eq. (2.3) we obtain

+ ( (L. v,o(r”-r;,

i 2 2m

(34) +l-O +r+

and from Eq. (3.7)

CL- v)o CL. v>o u-y=7 --g- (r3-ri)+r0 r2- $.

7.0 i 0 i

(3.11)

The expression for t follows immediately from Eqs. (3.6) and (3.11). Suppose r is approaching an asymptote r,=const. Then lim,+ y(r)=lim,,,*(u.y)lr

=O. On the other hand, if there exists a point r* with lirn,,,: y(r)=lim,+,*(u~~)l, = 0, then the boundary cannot pass through r= r* , because the solution of the dynamical system passing through r* would have to be r*=const. At r. we can specify (L. ~)~a0 and yo freely, because (L. v)~ depends only on the rate with which E/E:, and plx, approach 0 [Lemma II.2 (i)]. lim,,, * f(r) = lb,, *(u.y)l,=O is equivalent to

i (L. y)o=,:J&~ (3.13)

ro= J1-zmlr,- m(r3,-r;) 3&/W’

(3.14)

We insert Eqs. (3.13) and (3.14) into Eq. (3.11) and obtain

i

m

y2= 3&l-- (r3-r3*)+ t/m

i 2+2mlr- 1

and therefore

f$=,,( 3(r*:2m) ($- $) +2/+ lh). Thus for r.+ > 2m we have sign(dy2/dr) =sign( r- r*). That y2 must approach 0 from above, is satisfied in both cases, ro<r+ and r.+<ro. Moreover, y2 cannot have a zero between r* and ro. Thus the radius of the boundary must approach r* , if Eqs. (3.13) and (3.14) are satisfied and 2m<r*<ro or 2m<ro<r+. If r,<2m or ro<2m, then r* cannot be approached asymptoti- cally, because the star would be contained in the trapped region and must therefore collapse completely. l4 0

IV. DISCUSSION

It appears that the motion of the boundary of the star is determined by its initial velocity Ylz,ns 9 its initial radius rln,nn, the total mass m, and the rate at which energy density, E, and pressure, p, approach 0 at the boundary. If the problem was well posed, then it would even be independent of our falloff condition on the energy density near the boundary (E(X)


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2 ae-dQ~-Q(d’), b ecause any given initial energy distribution can be smoothly approximated by functions which differ only in a compact set and satisfy this condition. However, this assumption was essential for the proof of Lemma III. 11. This lemma does not hold for b Z= 1.

In principle the pressure in radial direction could be different from the pressure tangential to the orbits of isometry. But then we would have to deal with two equations of state which would complicate the problem considerably. Thus we stuck to convention.

Since we are dealing only with the boundary of a star, the existence of an equation of state and positive pressure should be physically justified. A relation between energy density and pressure has to be assumed, because otherwise the problem would be underdetermined. [Notice that if we had assumed the existence of a differentiable function f with e=f(p), then we would have gotten a contradiction immediately. For under the contiition that for large r we have p = E= 0 Lemma II.2 (i) would imply

L.G(p)+p)= -0 +?tP))ti(P)+P)(L. v)

and therefore E= -p. A further application of Lemma II.2 (i) would give p = 0 and the entire space-time would be empty.]

The monotonicity assumptions for E only express that (restricted to any slice Xt) L. L. E and e/L. E should not oscillate in some neighborhood of the boundary. Physically this condition seems very weak. A similar statement is appropriate for the assumption f”(e) L 0 near E=O.

It seems contradictory to physical intuition that the initial energy distribution and the equation of state should have so little an influence. However, at least the free falling case may be reasonably explained by the fact that smoothness of E implies that the pressure near the boundary is so low that the matter resembles dust (this is exactly what happens in the case of Rendall):”

To make this more transparent, we write f in the form

where K E R+. This may be motivated by the example of polytropic gas spheres which satisfy y=const (e.g., Ref. 15). But notice that this does not impose restrictions on f.

Proposition IVl: Assume, f’ (0) exists, f”(e) > 0 near E= 0, and there exists a sugace z,-, orthogonal to u such that qsO is smooth and (L. E)J~~, (L.L. E)J~,, (e/L. E)J+ (EL . L. eI(L . E)~)(x,, are monotone functions near the boundary.

If space-time is C’ then (L. v)pOnB=O, or

lim Y(E)E[O,~] and lim Y’(E)=~. e-+0 C-+0

Proof We can work entirely in the hypersurface X0. Because of f( 0) = 0 and f’ (0) = 0 we get f( E)/E= KEY”) -+ 0. Thus there exists an “E>O such that ylco,;l > 0. Suppose, lim,o ti~)=O. If h-+0 ~‘(4 + 03, then there exists a k>O with y(~)sk~ for E[O,Z]. But then we have lim,, G%=lim,,o &= 1. Contradiction.

Lemma III.7 (ii) implies lim,of( E)/( 2) = 00. Hence we must have fi~)G 1. Suppose now, lim,, Y(E)>O. Lemma III.8 (i) implies lim,+o(~‘(~)Eln(~)+ 1+ tie))= 1. Thus

lim,o(~‘(~)eln(~))<O, whence y’(e) + ~0 (e-0). Notice that we did not use that (M,g) is C3 but only that (L . v)lzO exists. 0

If f’(0) is unbounded, then r(e) + --cx) (6-O). As a corollary to Proposition IV.1 we obtain that the polytropic equation of state is incom-

patible with smoothness of the solution, unless the collapse is dustlike ((L . v)l,= 0). In particular, the boundary in Rendall’s solutions must evolve dustlike.


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Here the importance of our smoothness assumption is apparent. There do exist static solutions with a polytropic equation of state which are embedded into vacuum (Ref. 16, Theorem 2). However, for their solutions the energy density is not C” at the boundary of the star. Inside the star there is no obstruction to E being C”.

Theorem 111.1 implies that the boundary evolves differently if L e v it: 0 at the boundary. I do not know of any solutions which satisfy this condition. Thus it is possible that in this case Cs differentiability is not maintained when space-time evolves, not even near the initial surface [This is different from the “gradient catastrophe” familiar from gas dynamics, because in this case differentiability is only lost after some time has elapsed (Ref. 17, Chap. 1 Sec. ll).] Since this is the only other possibility for the boundary to evolve it would be very interesting to study the existence of solutions of the initial value problem if L. v # 0 is imposed at the boundary.

It may we11 be that our smoothness conditions are too strong for physically realistic solutions. However, it could also be that it is simply not correct to model an isolated object by matching solutions. In order to decide this question (for a given equation of state) one would have to study sequences of perfect fluid space-times whose energy densities (restricted to a given hypersurface) converge to a function with compact support.

This would require an analysis much deeper than ours.

ACKNOWLEDGMENTS

I would like to thank A. D. Rendall and B. Schmidt for their interest and for discussions about the relevance of this work. I would like to thank an anonymous referee for pointing out a mistake in an earlier version and for pointing out Ref. 16.

M. K. was supported by the Ministry of Colleges and Universities of Ontario and the Natural Sciences and Engineering Research Council of Canada.

APPENDIX: DEGENERACITY OF THE EINSTEIN EQUATION AT THE BOUNDARY

In this appendix the initial value problem is formulated in characteristic form. If the collapse is adiabatic, the baryon number density is given by

n(E) = ccl; llG+f(a)d: ,

where c is a constant. The equation of motion (Lemma 11.2) implies that

e -‘=H(t) F, e-‘=h(q)n(e)r2,

where h, H are unspecified constants of integration. These equations are of course only defined in int(supp(e)), but it is plausible that it is possible to continue them to the flat region, because it is known that one can obtain at least a C’ matching of exterior and interior solutions.‘2 We may set H= 1, but for continuing e-x into the flat region h must diverge in order to compensate for n + 0. We obtain from Lemma II. 1


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Marcus Kriele: On the collapse of a spherically symmetric star

0 0 0 r*rhn* df

(E+f)G

0 0 0 r*yhn* df

(E+f)G r*hn* o-

r O 0

r’

Y’ =

ny Eff

-n(l+y*-r*+rm*f)

M~+f)

0

-2ny r

where ri denotes J,a and a’ denotes a,a. Thus we have a quasilinear system of the form

ir+AU’=V,

where A is a matrix. If I is a left eigenvector with eigenvalue cu~R, we obtain

z(a,+aaq)q.q=zV, (A.11

which contains only derivatives in the “characteristic direction” R( d,+ cudJ. The system is in characteristic form if it consists of equations of the form (A. 1). This can be achieved if there exists a basis of left eigenvectors {Zi} of A with real eigenvalues. Then the system is called hyperbolic (Ref. 11, Chap. V Sec. 6).

The left eigenvectors of our matrix are

z’=(1,0,0,0), Z2=(O,E,-D,O), Z3=(0,0,~,~), Z4=(0,0,1/z,- a,,

where

r*hn* c:=-

r*rhn* df r*yhn* df

r 7 D:= (c+f)T2 E:= (Eff)G.

The eigenvalues are given by O,O, rlr 6. The characteristic directions are therefore parallel to d, , d, , d, k &%a, and coincide at the limit E=O. Observe that D, E diverge but C h 0 as we approach the boundary. Hence the system cannot be extended across the boundary as a hyperbolic system, because the eigenvectors Z3,14 would become parallel.

‘S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time (Cambridge University, Cambridge, England, 1973).

‘R. Wald, General Relativity (Chicago University, Chicago, 1984). ‘S. Chandrasekhar, “The maximum mass of ideal white dwarfs,” Astrophys. J. 74, 81-82 (1931). 4J. B. Hartle, “Bounds on the mass and moment of inertia of nonrotating neutron stars,” Phys. Rep. 46, 201-247 (1978). ‘K. Newman, “Black holes without singularities,” Gen. Relativ. Gravit. 18, 981-995 (1989). 6M. Kriele, “A generalization of the singularity theorem of Hawking and Penrose to space-times with causality viola-

tions,” Proc. R. Sot. London, Ser. A 431, 451-464 (1990). ‘I. R. Oppenheimer and H. Snyder, “On continued gravitational collapse,” Phys. Rev. 55, 455-459 (1939). *D. Christodoulou, ‘The formation of black holes and singularities in spherically symmetric gravitational collapse,”

Commun. Pure Appl. Math. XLIV, 339-373 (1991). ‘M. Kriele, “The collapse of a spherically symmetric star with linear equation of state,” Class. Quantum Gravit. 10,

1341-1352 (1993). “A. D. Rendall, “The initial value problem for a class of general relativistic fluid bodies,” J. Math. Phys. 33, 1047- 1053

(1992). ” R Courant and D. Hilbett, Methods of Mathematical Physics (Wiley, New York, 1966), Vol. II. “C’ W Misner and D. H. Sharp, “Relativistic equations for adiabatic, spherically symmetric gravitational collapse,”

Rev.B 136, 571-576 (1964). Phys.

“D. Kramer, H. Stephani, M. Mac Callum, and E. Herlt, Eract Solutions of Einstein’s Field Equations (Cambridge University, Cambridge, England, 1980).




I4 W. C. Hemandez and C. W. Misner, “Observer time as a coordinate in relativistic spherical hydrodynamics,” Astrophys. J. 143, 452-464 (1966).

‘sY. B. Zeldovich and I. D. Novikov, Relativistic Astrophysics: Stars and Relativity (University of Chicago, Chicago, 1971). Vol. 1.

‘*A D Rendall and B. G. Schmidt, “Existence and properties of spherically symmetric static fluid bodies with a given eiuation of state,” Class. Quantum Gravit. 8, 985-1000 (1991).

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``On the collapse of a spherically symmetric star'' : Emphasizes That ``Singularity Theorems'' Essentially Assume What They Intend To Prove!

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