EFTA01164244.pdf

Hawking Evaporation is Inconsistent with an Event Horizon at r = 2M Boron D. Chowdhury' and Lawrence M. Krauss" 'Department of Physics, Arizona State University, Tempe, AZ 85287 and 'School of Earth and Space Exploration, Arizona Stale University, Tempe, AZ 85287, USA, and Mount Stromlo Observatory, Research School of Astronomy and Astrophysics, Australian National University, Weston, ACT, Australia, 2611 (Dated: August 28, 2014) A simple classical consideration of black hole formation and evaporation times as measured by an observer at infinity demonstrates that an infall cutoff outside the event horizon of a black hole must be imposed in order for the formation time of a black hole event horizon to not exceed its evaporation time. We explore this paradox quantitatively and examine possible cutoff scales and their relation to the Planck scale. Our analysis suggests two different possibilities, neither of which can be resolved classically and both of which require new physics associated with even large black holes and macroscopic event horizons: either an event horizon never forms, for example due to radiation during collapse (resolving the information loss problem), or quantum effects may affect space-time near an event horizon in ways which alter infall as well as black hole evaporation itself. I. INTRODUCTION II. INFALL AND OBSERVATION TIMES FOR A TEST PARTICLE NEAR THE EVENT HORIZON Evaporating black holes present a number of paradoxes Consider a massive particle starting from rest at the that have motivated a great deal of work in classical and location 2. = R 7 O(M). For inward radial motion the quantum gravity over the past 30 years. Most notable, four velocity is given by as pointed out by Hawking, black hole radiance appears to be in conflict with unitarity, as pure states appear to dt dr 1— R /2M 2M evolve into mixed states, implying an information loss tri ( dr' dr ) ( ) (1) 1 _ 2111 ' V r paradox that has yet to be fully resolved. Several ideas have been proposed. from the possibility that all the in- where T is the proper time. The coordinate velocity formation stored in a black hole is accessible at its hori- (ti- n =— 14) dr can be integrated to yield zon Il, 2), to the possibility that black hole event horizons 1? Jr' vri do not form [3-5). tinfoil = (2) 2m -1 ft dr — There is, however, a classical black hole paradox which While the result can be expressed analytically in closed is less often recognized. Because of the infinite redshift form, it is not particularly illuminating. We shall later factor at r = 2M, infalling objects appear to take an plot specific results for a variety of cases. infinite time to cross the event horizon as observed by a For now, we observe that near the horizon distant observer, whereas the same observer will measure dt 2M the lifetime of the black hole against evaporation given (3) by the standard relation r ...v. M3 for a black hole of mass dr r — 2M M. This implies that a black hole would be observed by giving a distant observer to evaporate before it forms. tinfau = —2M log(r — 2M) + canal. (4) Here we explore this paradox in more detail and deter- This illustrates the fact that infall times as seen by a mine conditions on infall that might allow causality to be distant observer diverge as the horizon is approached. We preserved in the process of black hole evaporation. Our can cut off these divergences by considering infall times argument relies purely on classical general relativity con- to regions arbitrarily close to the classical horizon. siderations and hence is not subject to the many vagaries Consider for example, cutting off infall at a of interpretation often associated with considerations of Schwarzchild coordinate distance of r = 2M + 1 (recall quantum effects and gravity. that we are using units here where Alp = 1). In order to determine the time taken for a distant ob- We conclude with a brief discussion of possible impli- server to observe this infall we need to add to the infall cations of our analysis, all which would appear to require time estimated above, the time it takes for a signal to quantum effects to be significant even for the horizons come out to r = R from r = 2M -F 1 around large black holes, where one would think that R dr -2cr classical CH should be sufficient to describe space-time toutgoing = — r -F 2Al log(r — 2M.)1:41+1 and associated phenomena in the vicinity of the event 12 at -F1 I — horizon. R— (2411 +1) + 2AI log(R — 2M) . (5) EFTA01164244  2 lb determine specific numbers we consider R = 20M. If we consider initial configurations of infall such that For large Al one finds R> M. Then the two roots become tinfrin/M 2 log M + 112 , (6) m M,2R. (11) to„ta„ing/M 2 log Al + 18 . (7) The first root is then the appropriate one to choose in order to describe negative velocities, i.e. infall, and to We plot in figure 1 the infall, outgoing and total time in get the correct ADM rest mass at infinity. We can plug units of mass for various masses assuming R = 20M. The this back into (8), and for the purposes of comparison asymptotic behaviour described above is clearly visible. take r = 20M so that again we can derive an expression for tr' which is not particularly illuminating, so we do not present it here. We can again integrate this expres- sion with respect to radial position to get the infall time, which we do numerically and plot in figure 2. FIG. I: The solid curve is the time to get to r = 2M + 1, starting from It = 20M, the dashed curve is the time for a light signal to get back front that location and the dotted curve is the total time. The x-axis is logarithmic in Al and the times are measured in units of M. For reference, one solar mass is was. FIG. 2: The collapse time for a self-gravitating shell to reach r = 2M + 1. The convention for the curves are as given in figure 1. III. COLLAPSE TIME FOR A Comparing the two figures we see that infall times vary SELF-GRAVITATING SPHERICAL SHELL by at mast 20-30% between the two cases, so that the analytically derived times for single test particle provide For completeness, we can compare this analysis of a reasonable estimates to determine causality. particle falling inside the event horizon to the more rele- vant case of a spherical shell of material collapsing under IV. TEMPERATURE AND DISTANCE its own gravity, using equations of motion worked out by ESTIMATES AT VARIOUS CUTOFF RADII Israel [15]. The speed in outside coordinates (there is a disconti- nuity in coordinates across the shell) is In order to consider various cutoff distances which maintain causality, we consider both local temperatures Al m and proper distances from the horizon. The local temper- dt m 2r — = V — _ 2A/ m (8) ature measured by a static observer at coordinate radius dr 2AI r, outside of the event horizon, is 1/(Fa —1 + where m is an integration parameter which can be taken Tr — , 1141 (12) to be the rest mass of the shell, and M is the mass pa- 1/1 — aL—u rameter for the external geometry. The shell comes to a rest at For p = r — 2M < 2fif this becomes m 2 1 R= (9) T,. (13) 2(rn - Al) 2NfisrVirp which allows us to calculate m in terms of It In particular at the location r — 2M = 1 one finds 2M 1 m = R(1 - - ) (10) T2A1-1-1 (14) 2v/firOtt EFTA01164245  3 so that the local temperature a coordinate distance of formation timescale. It is interesting to note that the the Planck length away from the horizon is well below latter factor in the distance estimate is reminiscent of a the Planck temperature. tunneling scale but determining whether or not this is a We can understand this quite simply by considering coincidence would require a full quantum treatment. instead the proper distance from the horizon, i, V. CONCLUSIONS d, „ dr (15) `12M r Our calculations explicitly demonstrate the quantita- For p — 2M < 2M we get tive scale of the problem associated with timescales for evaporating black holes, but of course they do not deter- P dp mine how to resolve this problem. Several possibilities d„f-•-• v/Wf f 2 . (16) do suggest themselves, however. o v'P Perhaps the fact that the existence of a horizon at so a coordinate distance of 1 from the horizon is actually r = 2M implies that the evaporation time of a black hole d2,w+i = 2 2A9. is longer than the formation time as seen by a distant If instead we consider distance p = M-" we get observer suggests a literal solution-namely that a hori- zon does not have time to form, aS would be the case if d2m+m ^ = 2 \,5M(1-n)12 (17) radiation by infalling material was sufficient to cause full evaporation before horizon formation. As we have noted so for n = 1 we get the proper distance from the horizon this would also resolve the information loss paradox as- to be Planckian. sociated with black hole evaporation which was the chief The local temperature at this distance (i.e. for n = 1) motivation of earlier proposaLs of this possibility [3-5). is Alternatively, some exotic quantum effects could ei- „ 1 1 1 ther cause space-time fluctuations in the horizon radius, W,1 — O(1) . (18) 87rM 2V2if causing particles to be absorbed inside of the horizon in a finite time as observed by a distant observer. This how- So a proper Planck distance from the horizon, corre- ever would likely alter Hawking's radiation calculation, sponding to a coordinate distance AI-1, also corresponds since emitted radiation at late times comes from very to a local Planck temperature. near the horizon, and thus would also be subject to the Using (5) and we can see that the time it takes a light effects of a fluctuating horizon. signal to reach a distance n- R from a distance r — 2M n- Finally, perhaps some other catastrophic quantum M- " is, for large M gravity effects manifest themselves near the event horizon which would affect infall just outside of the horizon. This to„/going =n- 2(n + 1)M log M . (19) possibility is reminiscent of the suggestion of fuzzballs Our earlier estimates imply the infall time for massive [3, 6-9), or firewalls [10-14). shell will also have a similar logarithmic dependence on All of these possibilities imply a dramatic shift in our M. understanding of black hole physics and in particular the Since MlooM c M3 it is clear that if we cut off quantum processes that lead to Hawking radiation and infall at distances from the horizon comparable to regions evaporation. While they might resolve the semiclassical where the local temperature is of order the Planck mass, temporal paradox we have focused on here, all them beg objects will be observed by a distant observer to take an equally perplexing question: Why should quantum significantly less time to infall than the evaporation time gravity processes be relevant to understanding physics of the black hole. near the event horizon even for arbitrarily large black We can ask at what distance from the horizon the out- holes, where the event horizon occurs at a macroscopi- going time for a light ray to reach a distant observer will cally large distance scale where quantum effects should be of the order of evaporation timescale. That will oc- naively be negligible? cur when r - 2M n- Me-m2 = Me-suck. The physical Whatever the resolution, it remains remarkable that distance corresponding to this coordinate distance is relatively simple classical or semiclassical considerations associated with black holes such as we have presented dvd+aft _„2 = (20) here point so directly to exotic requirements for quantum gravity that may filter into even macroscopic phenomena, As long as we cut off infall before this distance the affect possibly cherished classical or quantum mechanical black hole evaporation timescale will be longer than the principles. EFTA01164246  4 (1] G. 't Hooft, Nuel. Phys. B 335. 138 (1990). [9] 1. Bena and N. P. Warner. arXiv:1311.4538 (hep-th]. (2] L. Susskind, L. Thorlacius and J. Uglum, Phys. Rev. D [10] A. AIntheiri, D. Marolf, J. Polchinski and J. Sully, JHEP 48, 3743 (1993) [hep-th/9306069]. 1302, 062 (2013) [arXiv:1207.3123 [hep-th]]. (3] S. D. Madan, Fortseh. Phys. 53, 793 (2005) [hap- [11] S. G. Avery, B. D. Chowdhury and A. Puhm, JHEP th/0502050]. 1309, 012 (2013) [arXiv:1210.6996 [hep-th]]. [4] T. Vachaspati, D. Stojkovic and L. M. Krauss, Phys. Rev. [12] R. Bousso, Phys. Rev. D 88, 084035 (2013) D 76, 024005 (2007) [gr-gc/0609024]. [arXiv:1308.2665 [hep-th]]. [5] S. W. Hawking, arXiv:1401.5761 [hep-th]. [13] B. D. Chowdhury, JHEP 1310, 034 (2013) [6] 1. Bena and N. P. Warner, Adv. Timor. Math. Phys. 9, [arXiv:1307.5915 [hep-th]]. 667 (2005) [hap-di/04081061. [14] R. Bousso, Phys. Rev. Lett. 112, 041102 (2014) [7] K. Skenderis and M. Taylor, Phys. Rept. 467, 117 (2008) (arXiv:1308.3697 [hep-th]]. [tirXiv:0804.0552 [hep-th]]. [15] W. Israel, Nuovo Cim. B 44510, 1 (1966) [Erratum-ibid. [8] B. D. Chowdhury and A. Virmani, arXiv:1001.1444 [hea- B 48, 463 (1967 NUC1A,B44,1.1966)]. th]. EFTA01164247