1993 黑洞蒸发中的裸奇点与霹雳奇点-精品文档资料整理.pdf

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1、arXiv:hep-th/9207105v1 30 Jul 1992Naked And Thunderbolt SingularitiesIn Black Hole EvaporationS. W. Hawking&J. M. StewartDepartment of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeSilver StreetCambridge CB3 9EWUKJuly 1992AbstractIf an evaporating black hole does not settle down

2、to a non radiating remnant, adescription by a semi classical Lorentz metric must contain either a naked singularity orwhat we call a thunderbolt, a singularity that spreads out to infinity on a spacelike or nullpath. We investigate this question in the context of various two dimensional models thath

3、ave been proposed. We find that if the semi classical equations have an extra symmetrythat make them solvable in closed form, they seem to predict naked singularities butnumerical calculations indicate that more general semi classical equations, such as theoriginal CGHS ones give rise to thunderbolt

4、s. We therefore expect that the semi classicalapproximation in four dimensions will lead to thunderbolts. We interpret the prediction ofthunderbolts as indicating that the semi classical approximation breaks down at the endpoint of black hole evaporation, and we would expect that a full quantum trea

5、tment wouldreplace the thunderbolt with a burst of high energy particles. The energy in such a burstwould be too small to account for the observed gamma ray bursts.11 IntroductionIt has been known for some time that classical general relativity predicts singularitiesin gravitational collapse. At the

6、 singularities, the Einstein equations will not be defined.Thus there will be a limit as to how far into the future one can predict spacetime. However,it seems that singularities formed in gravitational collapse always occur in regions that arehidden from infinity by an event horizon, so the breakdo

7、wn of the Einstein equations atthe singularity does not affect our ability to predict the future in the asymptotic region ofspace. This assumption that the singularities are hidden is known as the Cosmic CensorshipHypothesis and is fundamental to all the work that has been done on black holes.Itrema

8、ins unproven but it is almost certainly true for classical general relativity with asuitable definition of a singularity that is so bad it cant be smoothed out or continuedthrough.On the other hand, in the semi classical approximation to quantum gravity a blackhole formed in a gravitational collapse

9、 will emit thermal radiation and evaporate slowly.If the black hole has a charge that is coupled to a long range field and which cant beradiated, such as a magnetic charge, it may be able to settle down to a non radiatingstate such as the extreme Reissner-Nordstrm solution. But for black holes witho

10、ut sucha charge, there are no zero temperature classical solutions they can settle down to. Onemight suppose they settled down to some stable or semi stable remnant that was not aclassical solution but was maintained by quantum effects. However, quite apart from thefact that there is nothing very ob

11、vious to stabilize such remnants, their existence wouldcreate severe problems. If they had a mass of the order of the Planck mass, one might haveexpected that there would be more than the cosmological critical density of the remainsof black holes formed in the very early universe. While if they had

12、zero mass, they wouldlead to infinite degeneracy of the vacuum state.The most natural assumption would seem to be that black holes without a conservedcharge disappear completely.To suppose that black holes could be formed but neverdisappear would violate CPT unless there were also a separate species

13、 of white holeswhich would have existed from the beginning of the universe. On the other hand, if blackholes disappear completely, black and white holes can be different aspects of the sameobjects, which would be an aesthetically satisfying solution to the CPT problem. Holeswould be called black whe

14、n they were large and classical, and not radiating much, butthey would be called white when the quantum emission was the dominant process.If black holes disappear completely, this can not be described by a Lorentzian metricwithout some sort of naked singularity, or what would be even worse, a region

15、 of closedtime like curves. Spreading out from the naked singularity or region of chronology violationwould be a Cauchy horizon. Beyond this horizon the semi classical equations would notuniquely specify the solution, but one would hope that it would determined by a fullquantum treatment, though may

16、be with loss of quantum coherence. Otherwise, we couldbe in for a surprise every time a black hole on our past light cone evaporates.Within the context of the semi classical approximation there is however an alternativeto a naked singularity that has not received much attention. We shall call it a t

17、hunderbolt.It is a singularity that spreads out to infinity on a space like or null path. It is not a2naked singularity because you dont see it coming until it hits you and wipes you out.It would mean that the semi classical equations could not only not be evolved uniquely(as with a naked singularit

18、y), but they could not be evolved at all more than a finitedistance into the future. If the thunderbolt was null, one could regard it as the singularCauchy horizon produced by some would-be naked singularity. This would be like whatis believed to happen to the inner Cauchy horizons of classical blac

19、k holes under genericperturbations.One might therefore expect that although the semi classical equationscould lead to naked singularities in special situations, one would get a thunderbolt if oneperturbed the equations or the initial data slightly.If the semi classical equations were to predict a th

20、underbolt singularity as the endpoint of black hole evaporation, one would have to conclude that the singularity would besoftened and smeared out by quantum effects because surely many black holes must haveevaporated in the past, and yet we survived. Nevertheless, if the semi classical equationspred

21、ict thunderbolts, this might indicate that something fairly dramatic happens in thefull quantum theory.In four dimensions, the one loop corrections are quadratic in the curvature.Thismeans that the semi classical equations including one loop back reaction are fourth orderand have unphysical runaway

22、solutions. It is therefore hard to use them to decide whetherthe evaporation of black holes leads to naked singularities or thunderbolts. On the otherhand, the the one loop corrections in two dimensions are proportional to the curvaturescalar. This means that the semi classical equations are second

23、order even when the backreaction is taken into account. It should therefore be possible to decide what they predictas the outcome of black hole evaporation. Hopefully, this will give an indication of whatmight happen in four dimensions.In two dimensions the Einstein Hilbert Lagrangian R is a diverge

24、nce. This meansthat to get a non trivial interaction with the metric, one has to multiply the EinsteinHilbert term by a function of a dilaton field . An interesting model in which the metric iscoupled to a dilaton field and N minimal scalars has been proposed by Callan, Giddings,Harvey and Strominge

25、r 1, (henceforth referred to as CGHS). In the classical version ofthis theory one can form a black hole by sending in a wave of one of the scalar fields fromthe asymptotic region. Quantum field theory on this classical black hole background thenshows that the black hole will radiate thermally in eac

26、h of the fields. Presumably thismeans that the black holes will evaporate but a full quantum treatment of the problemseems too difficult even in this simple theory. However, Callan et al suggested that inthe large N limit, one could neglect ghosts and quantum fluctuations of the metric anddilaton in

27、 comparison with those of the scalar fields. The effective action arising fromthe scalar quantum loops would be completely determined by the trace anomaly and theconservation equations together with boundary conditions. One could therefore add itto the classical action for the metric and dilaton fie

28、lds and obtain a set of semi classicalhyperbolic differential equations for the metric and dilaton.Even these relatively simple equations have not been solved in closed form. Callanet al hoped that the result of including the action of the scalar loops would be to cause ablack hole to evaporate comp

29、letely without any singularity and tend at late times to thelinear dilaton solution, which is the analogue of Minkowski space, and which is the natural3candidate for a ground state. However, later work showed that there was necessarily asingularity, and that the solution could not settle down to a s

30、tatic state in which thesingularity remained hidden behind an event horizon.These results presumably indicate that the semi classical equations lead either to anaked singularity or a thunderbolt.But which?The original semi classical equationsproposed by CGHS do not seem to admit closed form solution

31、s. Various authors havesuggests modifications to the semi classical equations that introduce an extra symmetryand make the equations solvable in closed form. We shall show the exact solutions havenaked singularities.However they also continue to emit radiation at a finite rate andthe mass becomes ar

32、bitrarily negative. Such behaviour is presumably unphysical, or atleast one hopes so.The conservation of energy would lose its practical significance ifone could have negative mass naked singularities.In one case at least, one could usethe non uniqueness of the solution after the naked singularity h

33、as appeared to cut offthe analytically continued exact solution at the Cauchy horizon produced by the nakedsingularity and glue on a non radiating solution. This procedure however transforms theCauchy horizon into a thunderbolt singularity, although a fairly mild one.In the four dimensional case, th

34、e equations dont have symmetries that allow oneto solve them in closed form. There is thus no reason to expect special properties likeconformal symmetry in two dimensional models of black holes. We shall therefore investi-gate the behaviour of solutions of the original semi classical equations propo

35、sed by CGHSwhich we expect to be more typical of the general case.Since these equations do notadmit solutions in closed form, there seems no alternative but to integrate the equationsnumerically. Fortunately hyperbolic equations in 1+1 dimensions are relatively easy andthere are reliable and numeric

36、ally stable routines available. To test their accuracy, we firstapplied them to the equations without back reaction. We obtained excellent agreementwith the known solution, the Witten two dimensional black hole. Encouraged by this, weincluded the back reaction terms and obtained results that strongl

37、y indicate a thunderbolt.This supports our view that while naked singularities may occur for certain sets of semiclassical equations with special symmetries, more general two dimensional models of blackhole evaporation will exhibit thunderbolts.In section 2 the model and the various sets of semi cla

38、ssical equations are described.Those with special symmetries that allow exact solutions are shown to lead to nakedsingularities in section 3 while in section 4 the numerical results of integrating more generalequations are presented. A test is given to distinguish a thunderbolt from an eternal black

39、hole. The implications for black holes in four dimensions are discussed in section 5. Thenumerical algorithm used is described in an appendix.42. The semi classical modelCGHS assume the spacetime contains a dilaton field and N minimally coupled scalarfields fi, described by a classical LagrangianL =

40、12ge2?R + 4()2+ 42?12NXi=1(fi)2#(2.1)where R is the Ricci scalar and is a coupling constant.Any two dimensional spacetime is of course conformally flat, so one can introduce nullcoordinates xand write the line element asds2= e2dx+dx.(2.2)CGHS suggested that in the limit of a large number N of scalar

41、 fields fione could neglectthe quantum fluctuations of the dilaton and the metric, and treat the back reaction in thescalar fields semi classically by adding to the action a trace anomaly term+.(2.3)CGHS took = N/12. However taking ghosts into account leads to =N 2412,(2.4)in that theory. For consis

42、tency with refs 35 we henceforth define by (2.4). Occasionallywe shall use the earlier value in the form = N/12; obviously = +2. We shall call thetheory defined by equations (2.1), (2.3) and (2.4) the original theory.Strominger 2 has suggested that the ghosts should be coupled to a different metric.

43、This leads to the action of the original theory (with replaced by ), plus an additionalterm2(+ + + + +).(2.5)We shall call this the decoupled ghost theory, though in fact the ghosts are still coupledto the geometry, only differently.de Alwis 3 and Bilal and Callan 4 have suggested that the cosmologi

44、cal constant2term be multiplied by a function D() to make the theory conformally invariant whereD() =14(1 + y)2exp?1 y1 + y?(2.6)and y =1 e2. We shall call this the conformal theory. It can be solved in closedform.Another Lagrangian with a special symmetry that has a conserved current j=( ) has been

45、 proposed by Russo, Susskind and Thorlacius 5. It is the Lagrangianof the original theory plus the additional term+(2.7)5We shall call this the conserved current theory.The general solution of the conformal and conserved current theories with an asymp-totically flat weak coupling region will be give

46、n in section 3. It will be shown they havenaked singularities for positive . Here we give the field equations for the two Lagrangianswithout special symmetries, the original and decoupled ghost theories.The evolutionequations can be written in the form+fi=0,(2.8a)+ =P1(2+ + Y ),(2.8b)+ =Q+,(2.8c)whe

47、re we have introduced the quantitiesP =1 e2Q =1 12e2,(2.9a)in the original theory, andP =1 e2+12 e4Q =1 12 e2,(2.9b)in the decoupled ghost theory. HereY =122e2.(2.10)In addition there are two constraint equations. In the original theory they aree2?22+ 4+? ?2+ (+)2 t+(x+)?=12Xi(+fi)2,(2.11a)e2?22 4?

48、?2 ()2 t(x)?=12Xi(fi)2,(2.11b)where tare arbitrary functions. They are constraints in the following sense. (2.11a,b)need be imposed only on surfaces x= const. and x+= const. respectively. They holdthen throughout the spacetime as a consequence of the evolution equations. The con-straints for the dec

49、oupled ghost theory involve replacing by as well as adding someextra terms which vanish when = . Since we only impose the constraints on the initialsurfaces where we may also set = (see later), we do not need to write down explicitlythe constraints for this theory. One may easily recover the classic

50、al equations, i.e., withoutthe trace anomaly term, by setting = 0 in the equations of the original theory.We consider first solutions of the classical equations.Equations (2.8b,c) have thesolutione2= e2=M 2x+x,(2.12)6where M is a constant, and arbitrary additive constants to xhave been ignored. If M