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1、arXiv:hep-th/9702045v1 5 Feb 1997DAMTP/R-96/56EVOLUTION OF NEAR-EXTREMAL BLACK HOLESS.W. Hawkingand M. M. Taylor-RobinsonDepartment of Applied Mathematics and Theoretical Physics,University of Cambridge, Silver St., Cambridge. CB3 9EW(February 1, 2008)AbstractNear extreme black holes can lose their
2、charge and decay by the emission ofmassive BPS charged particles. We calculate the greybody factors for low en-ergy charged and neutral scalar emission from four and five dimensional nearextremal Reissner-Nordstrom black holes. We use the corresponding emissionrates to obtain ratios of the rates of
3、loss of excess energy by charged andneutral emission, which are moduli independent, depending only on the inte-gral charges and the horizon potentials. We consider scattering experiments,finding that evolution towards a state in which the integral charges are equalis favoured, but neutral emission w
4、ill dominate the decay back to extremalityexcept when one charge is much greater than the others. The implications ofour results for the agreement between black hole and D-brane emission ratesand for the information loss puzzle are then discussed.PACS numbers: 04.70.Dy, 04.65.+eTypeset using REVTEXE
5、-mail: swh1damtp.cam.ac.ukE-mail: mmt14damtp.cam.ac.uk1I. INTRODUCTIONIn the last year, there has been rapid progress in the use of D-branes to describe andexplain the properties of black holes. In a series of papers, starting with 1, the Bekenstein-Hawking entropies for the most general five dimens
6、ional BPS black holes in string theorywere derived by counting the degeneracy of BPS-saturated D-brane bound states. Laterthese calculations were extended to near-extremal states 2, in the particular sector of themoduli space accessible to string techniques described by Maldacena and Strominger as t
7、he“dilute gas” region. There is some evidence, though no rigorous derivation as yet, that theagreement can be extended throughout the moduli space of the near-extremal black holes3.These ideas were then extended to supersymmetric four-dimensional black holes withregular horizons 4, 5. In 6, 7, 8, it
8、 was argued that it is useful to view the four-dimensional black holes as dimensionally reduced configurations of intersecting branes inM-theory.Such configurations again permit the derivation of the entropy of the four-dimensional state in terms of the degeneracy of the brane bound states.More rece
9、ntly, attention has been focused on the calculation of decay rates of five-dimensional black holes and the corresponding D-brane configurations. It was first pointedout in 9 that the decay rate of the D-brane configuration exhibits the same behaviour asthat of the black hole 10, when we assume that
10、the number of right moving oscillationsof the effective string is much smaller than the number of left moving ones. In a surprisingpaper by Das and Mathur 11, the numerical coefficients were found to match and it hasrecently been shown 12 that the string and semiclassical calculations also agree whe
11、n wedrop the assumption on the right moving oscillations. For four dimensional black holes inter-secting brane models of four-dimensional black holes also give agreement between M-theoryand semi-classical calculations of decay rates 13, 14. In the last month, a rationale for theagreement between the
12、 properties of near extremal D-brane and corresponding black holestates in the dilute gas region has been proposed 22.These D-brane and M-theory calculations are restricted to certain limited regions of theblack hole parameter space. In this paper, we calculate the semi-classical emission rates ina
13、sector of the moduli space which is out of the reach of D-brane and M-theory techniques(at present). We then obtain moduli independent quantities describing the ratio of chargedand neutral scalar emission rates and confirm that they are in agreement with the ratescalculated in the dilute gas region
14、of the moduli space. Thus scattering from black holesdisplays a certain universal structure for states not too far from extremality.One can get an idea of when charged emission will be important compared to neutralemission by considering the expression for the entropy. For the five dimensional extre
15、meblack hole this isS = 2n1n5nK,(1)where n1,n5,nKare integers that give the 1 brane, 5 brane and Kaluza-Klein charges respec-tively. The emission a massive charged BPS particle will reduce at least one of the integers(say nK) by at least one. This will cause a reduction of the entropy of2S =sn1n5nK.
16、(2)The emission of Kaluza-Klein charge will be suppressed by a factor of exp(S) and will besmall unlessnK n1n5.(3)Thus it seems that charged emission will occur most readily for the greatest charge and willtend to equalise the charges. However, when the charges are nearly equal, charged emissionof a
17、ny kind will be heavily suppressed. On the other hand, neutral emission can take placeat very low energies and so will not cause much reduction of entropy. One would thereforeexpect it to be limited only by phase space factors and to dominate over charged emissionexcept when one charge is much great
18、er than the others. The situation with four dimensionalblack holes is similar except that there are four charges. Again charged emission will tendto equalise the charges but neutral emission will dominate except when one charge is muchgreater than the others. In what follows we shall consider the fi
19、ve dimensional case andtreat four dimensional black holes in the appendix.In section II we start by calculating the rates of emission of neutral and charged scalarsfrom near extremal five-dimensional Reissner-Nordstrom black holes. We find that the ratioof the rates of energy loss by charged and neu
20、tral emission are moduli independent; theydepend only on the integral charges1and the horizon potentials. Neutral emission alwaysdominates charged emission, unless one of the integral charges is much greater than theproduct of the other two.We then discuss the implications for scattering from the bl
21、ack hole; it was suggestedin 12 that under some circumstances the black hole will decay before we can measure itsstate. We point out an error in their analysis, and show that it should be possible to obtainentropy in the outgoing radiation equal to that of the black hole state without the blackhole
22、decaying.Finally, in section IV, we discuss the implications of our results for the information lossquestion. It has been explicitly shown that the emission rates from near extremal black holesand D-branes agree in the sectors of the moduli space accessible to string calculations. Onewould expect th
23、at this agreement between the D-brane and black hole emission rates wouldcontinue throughout the entire moduli space of near BPS states, although a verification isnot yet possible. Now for the D-brane configuration we can determine the microstate whenthe entanglement entropy in the radiation is equa
24、l to that of the D-brane system. Since itis possible to obtain such an entropy in the outgoing radiation from the black hole before itdecays, it might seem as if we can extract enough information to determine the black holemicrostate without it decaying. That is, there would seem to be no obstructio
25、n to scatteringradiation from the black hole and obtaining information from the outgoing radiation. Onemight then expect any further scattering to be unitary and predictable.1We distinguish here between charges normalised to be integers, which we call integral charges,and the physical charges, which
26、 depend also on moduli.3This however by no means settles the information question. Although scattering offaD-brane regarded as a surface in flat space is unitary, it is not so obvious that informationcannot be lost if one takes account of the geometry of the D-brane. The causal structuremay have pas
27、t and future singular null boundaries like horizons and, as with horizons, thereis no reason that what comes out of the past surface should be related to what goes into thefuture surface. In the case of a static brane of one kind, there will be no information lossand the scattering will be unitary b
28、ecause this corresponds under dimensional reduction toa black hole of zero horizon area. However, in the case of four and five dimensional blackholes with four and three non zero charges respectively, the effects of the charges balance togive a non singular horizon of finite area and one might expec
29、t non unitary scattering withinformation loss.II. FIVE DIMENSIONAL SCATTERINGIn this section, following 9, 11 and 12, we consider scattering from a five dimensionalblack hole carrying three electric charges; such black hole states were first constructed in 3and 15. We will work with a near extremal
30、solution which is a solution of the low energyaction of type IIB string theory compactified on a torus. Then, the five-dimensional metricin the Einstein frame is:ds2= hf2/3dt2+ f1/3(h1dr2+ r2d23),(4)whereh = (1 r20r2), f = (1 +r21r2)(1 +r25r2)(1 +r2Kr2).(5)and the parameters riare related to r0by:r2
31、1= r20sinh21, r25= r20sinh25, r2K= r20sinh2K.(6)We require here only the metric in the Einstein frame; the other fields in the solution maybe found in 12. The extremal limit is r0 0, i with rifixed; we shall be interestedin the sections of the moduli space where the BPS state is the extreme Reissner
32、-Nordstromsolution, where the limiting values of riare equal to re, the Schwarzschild radius.We may regard the black hole as the compactification of a six-dimensional black stringcarrying momentum about the circle direction; we will be using this six-dimensional solutionin the following sections, an
33、d the metric (in the Einstein frame) is given by:ds2= (1 +r21r2)1/2(1 +r25r2)1/2dt2+ dx25+r20r2(coshKdt + sinhKdx5)2+ (1 +r21r2)1/2(1 +r25r2)1/2(1 r20r2)1dr2+ r2d23#.(7)We assume that we are in the very near extremal region where r0 re, and moreover willconsider all three hyperbolic angles to be fin
34、ite. It is here that our analysis differs fromprevious work; with this choice of parameters, we move away from the dilute gas region anda straightforward D-brane analysis of emission rates is not possible.4The entropy is:S =Ah4G5=22r30Qicoshi4G5(8)whilst the Hawking temperature is defined by:TH=12r0
35、Qicoshi.(9)We may define symmetrically normalised charges by:12r20sinh2i= Qi.(10)For simplicity of notation, we assume throughout the paper that all charges are positive;obviously for negative charges we simply insert appropriate moduli signs. Our notation forthe three charges Q1, Q5, QKindicates th
36、eir origin in D-brane models, from 1D-branes,5D-branes, and Kaluza-Klein charges respectively. The energy in the BPS limit is:E =4G5Q1+ Q5+ QK(11)where G5is the five dimensional Newton constant, with the excess energy for a near extremalstate beingE =r204G5Xie2i.(12)It was stated in 3 that the near
37、extremal solution is specified by six independent parameters,which we may take to be the mass, three charges, and two asymptotic values of scalar fields.However, there are in fact only five independent parameters; once we fix the three charges, aswell as r0and one hyperbolic angle, the other two hyp
38、erbolic angles are fixed. So we specifythe state of the black hole by its mass, three charges and only one extremality parameter.If the BPS state is Reissner-Nordstrom, then excitations away from extremality leave thegeometry Reissner-Nordstrom, since the three hyperbolic angles are the same. For sm
39、allexcitations, the relationship between the temperature and the excess energy isTH=2resG5Er2e,(13)which will be useful in the following. With appropriate normalisations, we can define thepotentials associated with the charges as:Ai=Qidt(r2+ r2i),(14)with the potentials on the horizon r = r0being:Ai
40、=Qidt(r20+ r2i).(15)5For perturbations which leave the compactification geometry passive, we obtain the standardReissner-Nordstrom solution by the rescaling r2= (r2+ r2i) which gives the solution in thefamiliar form:ds2= (1 r2+ r2)(1 r2 r2)dt2+1(1 r2+ r2)(1 r2 r2)d r2+ r2d23,TH=12(r2+ r2r3+),(16)Ai=
41、Qdt r2,where in the extremal limit r2are equal to Q.A. Neutral scalar emissionIn this section we compute the absorption probability for neutral scalars by the slightlynon-extremal black hole. Our discussion parallels that in 12, and we hence give only a briefsummary of the calculation. We solve the
42、Klein Gordon equation for a massless scalar onthe fixed background; taking the field to be of the form = eitR(r), we find that:hr3ddr(hr3ddr) + 2fR = 0.(17)where we have taken l = 0 since we will be interested in very low energy scalars. We assumethe low energy condition:re 1,(18)where we treat the
43、ratios ri/reas approximately one.Solutions to the wave equation may be approximated by matching near and far zonesolutions. We divide the space into two regions: the far zone r rfand the near zoner 1, the entropy of the extreme state vanishes, with the formal temperaturediverging; the existence of m
44、ass gaps was then suggested to prevent radiation at the extreme.For extreme states in which the entropy is finite and the temperature is zero - the type ofstates which we are analysing here - there are no such objections to the black hole absorbingor emitting arbitrarily small amounts of energy, and
45、 no such justifications for introducingmass gaps in the classical solutions.In 20, and more recently in 21, the thermal factors in black hole emission rates werederived taking account of self-interaction.This approach gives the appropriate thermalfactors for both the high energy tail of the emission
46、 spectrum of a non-extremal black holeand also for the emission spectrum of a very near-extremal black hole, and it is found thatthey differ significantly from those in the free field limit. There are however no physicalreasons for requiring the excitation spectrum to be quantised in the very near e
47、xtremal limitin the semi-classical theory.One would expect the spectrum of the classical black hole to be continuous with arbi-trarily small amounts of energy being emitted and absorbed. In the parametrisation of theprevious section, the implies that the potentials iare continuous and not discrete.
48、Ouremission rates will only be valid provided that the total excitation energy above the ex-tremal state is greater than the uncertainty in the kinetic energy of the state according tothe uncertainty principle; below this temperature our rates should be modified in the wayssuggested in 20 and 21.It
49、is important to note that individual ican correspond to excitation energies which aresmaller than the uncertainty in the kinetic energy provided that the total excitation energyis much greater. This will occur if one physical charge, let us say the Kaluza-Klein charge,is much smaller than the other
50、two. It may at first appear as though this implies that thekinetic energy of emitted scalars carrying the other two charges must be smaller than theuncertainty in kinetic energy, and much smaller than the temperature. However the divisionof the excitation energy into three sectors is artificial in t