PHYSICAL REVIEW D 102, 026016 (2020)
Black hole evaporation and semiclassicality at large D
† ‡ Frederik Holdt-Sørensen ,1,2,* David A. McGady,1,2,3, and Nico Wintergerst 1, 1Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark 2Niels Bohr International Academy, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark 3Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden
(Received 29 August 2019; accepted 16 June 2020; published 15 July 2020)
Black holes of sufficiently large initial radius are expected to be well described by a semiclassical analysis at least until half of their initial mass has evaporated away. For a small number of spacetime dimensions, this holds as long as the black hole is parametrically larger than the Planck length. In that case, curvatures are small, and backreaction onto geometry is expected to be well described by a time-dependent classical metric. We point out that at large D, small curvature is insufficient to guarantee a valid semiclassical description of black holes. Instead, the strongest bounds come from demanding that the rate of change of the geometry is small and that black holes scramble information faster than they evaporate. This is a consequence of the enormous power of Hawking radiation in D dimensions due to the large available phase space and the resulting minuscule evaporation times. Asymptotically, only black holes with entropies S ≥ DDþ3 log D are semiclassical. We comment on implications for realistic quantum gravity models in D ≤ 26 as well as relations to bounds on theories with a large number of gravitationally interacting light species.
DOI: 10.1103/PhysRevD.102.026016
I. INTRODUCTION treatment in quantum gravity. On the other hand, resolution of the black hole information paradox—usually phrased as Generic (nonextremal) black holes famously have the tendency of semiclassical black holes to turn pure both finite entropies and temperatures, which together lead quantum states into mixed ones [1], a phenomenon that to Hawking radiance/luminosity and, eventually, complete would be in clear tension with basic postulates of quantum evaporation. Absent a full treatment within quantum — gravity, one studies evaporation within the semiclassical mechanics suggests a breakdown at much earlier stages. approximation, where the length scale given by Newton’s There is compelling evidence that this should happen at the so-called Page time [2] by which an initially l ≔ 1=ðD−2Þ ¼ −1 ¼ constant, P GN MP tP, vanishes compared classical black hole has lost roughly half its initial area to the length scales of the geometry. Geometric back- via evaporation. reaction via quantum mechanical fluctuations then can be In this paper, we do not further explore the issue of safely ignored, which lets the radiation be cleanly com- information loss, even though we hope that our findings puted. A posteriori, one then assumes that the flux might provide nontrivial insights. Instead, we focus on the calculated in the semiclassical approximation is accurate much more innocent question about the properties that even for a finite mass black hole. allow for a semiclassical treatment of the early stages of The validity of this scheme has been the subject of active black hole evaporation. In four spacetime dimensions, this debate for several decades. It necessarily breaks down at the is well understood. Whenever the Schwarzschild radius is very late stages of black hole evaporation, when curvatures large compared to the Planck scale, a black hole is semi- become large compared to the Planck scale, requiring a full classical at least up to its Page time [3]. We show that when the spacetime dimension D is sufficiently large, this ceases “ ” *[email protected] to be true for a class of large black holes. † [email protected] Recent studies of the limit of a large number of ‡ [email protected] spacetime dimensions D have led to a better understanding of the aspects of classical general relativity, especially for Published by the American Physical Society under the terms of black holes [4,5]. Yet, semiclassical (and fully quantum) the Creative Commons Attribution 4.0 International license. – Further distribution of this work must maintain attribution to features of black holes [6 9] within these new formulations the author(s) and the published article’s title, journal citation, of this large-D limit seem to be relatively unexplored. This and DOI. Funded by SCOAP3. paper is yet another partial step in this direction.
2470-0010=2020=102(2)=026016(7) 026016-1 Published by the American Physical Society HOLDT-SØRENSEN, MCGADY, and WINTERGERST PHYS. REV. D 102, 026016 (2020)
2 The main thrust of our work stems from the evaporation 2 2 dr 2 ds ¼ −hDðrÞdt þ þ r dΩD−2; ð4Þ and scrambling timescales of Schwarzschild black holes in hDðrÞ D dimensions. The scrambling time is the time that a black D−3 D−3 16π Ω hole needs to process and obscure infalling information ð Þ¼1 − RH RH ¼ MH = D−2 hD r ; l ð − 2Þ ; [10]. It has been conjectured to saturate various bounds, r P MP D bounds which are necessary for an evaporating black hole ð5Þ not to violate basic properties of quantum mechanics
[10,11]. At large D, black holes coupled to ND massless D−1 −1 2 D where ΩD−2 ≡ 2π =Γð 2 Þ is the area of the (D − 2)- modes, with Bekenstein-Hawking entropy SBH and Hawking temperature T , evaporate and scramble on the dimensional unit sphere. The periodicity properties of H ð Þ¼− ð Þ timescales: g00 r hD r in the Euclidean signature directly give the Hawking temperature: t 4π Dþ1=2 S 1 evap ∼ BH × SD−2; ð1Þ 1 ð − 3Þ t D N BH ∼ þ β ¼ ¼ D ð Þ P D tE tE BH;TH β 4π : 6 BH RH t M 1 log S scr ¼ P ∼ D−2 BH ð Þ log SBH S × 1 2 ; 2 D t 2πT BH D = This factor of is responsible for the high luminosities that P H are the focus of this paper. It is straightforward to see that where we have assumed the usual expression [10,11] for the semiclassical Bekenstein-Hawking entropy is the scrambling time to hold in general D, a premise that we Ω D−2 4π will discuss more towards the end of the paper. We have ≔ ABH ¼ D−2 RH ¼ ð Þ SBH 4 4 l − 2 RHMH: 7 parametrized the timescales in terms of the entropy SBH GN P D as it is the only dimensionless quantity in pure gravity in ’ ≔ lD−2 asymptotically flat spacetimes. As such, it allows us to The black hole s area is ABH; again GN P . We frame capture the entire D scaling without having to consider the our main discussion in terms of the entropy SBH. individual D dependence of dimensionful quantities, such as Newton’s constant or the Schwarzschild radius. B. Evaporation times and scrambling times Even from a very superficial view on black hole We now combine properties of D-dimensional black information processing, it seems clear that a black hole holes with those of blackbodies in D dimensions. cannot evaporate faster than it scrambles information. Straightforward computations, for example in [12],give Any black hole that appears to do so cannot possibly the following luminosity of a spherical blackbody with evaporate thermally and thus be described by semiclassical radius R and temperature T in D dimensions: physics. Yet, for fixed SBH, the ratio of the scrambling ∼ and evaporation times indeed increases as tscr=tevap ðTRÞD D − 1 ζðDÞ D P ¼ðN DÞ : ð8Þ D ND log SBH=SBH. Thus, any fixed entropy black hole D D R2 D π can be described by semiclassical physics only up to some ð Þ critical dimension Dcrit SBH . Properly semiclassical black Apart from dimensionless factors of D, this expression is holes have bounded entropy: simple to understand. Since the power of blackbodies is D−2 proportional to their area, one has PD ∼ R . Dimensional ≳ Dþ3 ð Þ SBH D log D: 3 analysis then fixes the scaling with T. Finally, summing over all decay channels gives a factor of ND. Note that RH 3=2 D−1 This implies that semiclassical black holes have l ≳ D . ζð Þ P D D rapidly goes to 1 at large D. We stress that we work with manifestly dimensionless We now use the properties of black holes as approximate l ¼ ¼ 1 ratios and thus may safely ignore how P tP =MP blackbody radiators in D dimensions. The D-dimensional ¼ scale with the spacetime dimension D. Bose-Einstein distribution for temperature T TH peaks near II. SCHWARZSCHILD IN D DIMENSIONS D − 3 D2 In this section, we first give general properties of E ∼ E⋆ ≔ ðD − 1ÞT ¼ðD − 1Þ ≃ : ð9Þ H 4πR 4πR D-dimensional Schwarzschild black holes. We then find H H their semiclassical evaporation and scrambling times. Thus, when D ≫ 1, black hole radiance is dominated by wavelengths that are small compared to RH. This excises A. Metric, entropy, and temperature greybody physics—which we parametrize and denote by γ ð Þ— The metric for the D-dimensional Schwarzschild black the factor D RH from contributing to black hole radi- ≳ 8 hole with ADM mass MH is well known to be ance for even moderate values of D (e.g., D ) [6].
026016-2 BLACK HOLE EVAPORATION AND SEMICLASSICALITY AT … PHYS. REV. D 102, 026016 (2020)
There is a further, slight, modification of the absorption/ To this end, it is useful to discuss black hole families in D ˆ emission area of the black hole. Rather than being a dimensions, indexed by positive numbers k and S0: function of R , it is parametrized by the maximum critical H b kþ1 impact parameter C, below which null rays are captured by R ∼ l D 2 ; −1 −1 ˆ Dk H P bC ¼ðD Þ1=ðD−3ÞðD Þ1=2 S ðk; DÞ ≔ S0D 2 ⇒ ð13Þ a black hole [12]: 2 −3 . BH Dkþ1−k RH D M ∼ M D 2 : Combining these yields the black hole luminosity H P At large D, these families of black holes (i.e., these large-D 8π Dþ2 ¼ − dMH ¼ ND D γ ð Þ ð Þ limits) exactly correspond to those studied by Emparan PBH 2 2 4π × KD × D RH ; 10 dt e RH et al. [4] (and are related to the large-D limits studied by ≔ ˆ l=2 Battacharya et al. [5]), where they fix RH R0D and where both KD [defined in the Appendix in Eq. (A1)] and study families of black holes with l ¼ 1 and l ¼ 2. (Note γ ð Þ D RH (which is a proxy for greybody physics contri- that their Dl=2 is our Dðkþ1Þ=2.) buting to the luminosity) approach unity as D grows [6]. Note that the most important factor in P , ðD=4πÞD, BH A. Where should semiclassical gravity apply? ¼ D−3 fundamentally comes from the fact that RHTH 4π . S We need clear criteria where the above semiclassical Thus, a black hole with entropy BH evaporates after l analysis applies. Clearly, we must have RH > P and rffiffiffi ≫ Oð1Þ 2 −ð þ1Þ SBH . However, there are further conditions. t e 2 D D 2 S 1 L evap ¼ BH D−2 D ð Þ Sub-Planckian curvature: First, we should require 8π 4π × SBH × ; 11 tP e ND KD that the length scale defined by the curvature invariant 2 αβμν R ≔ RαβμνR j is sub-Planckian: αβμν r¼RH where LD, defined in Eq. (A2), also goes to unity for increasing D. This expression for tevap comes from rewrit- 2 2 Rαβμν ðD − 1ÞðD − 2Þ ðD − 3Þ ing M in terms of S . We recover Eq. (1) as K =L → 1 ¼ H BH D D 4 ð l Þ4 ð − 3Þ2 ð ð − 1ÞÞ → 1 MP r¼R RH= P and D = D D . H pffiffiffiffi 4 Finally, the scrambling time is [11] D D ≃ ≪ 1: ð14Þ 1 SBH 4 −2 tscr ≔ MP ¼ 2 log SBH SBH D ð Þ 2π log SBH − 3 × Ω : 12 tP TH D D−2 This bound serves to ensure the subdominance of higher curvature corrections to the Einstein-Hilbert action under As is well known, in D ¼ 4 a large black hole, with the assumption of technical naturalness. ≈ ð l Þ2 ≫ 1 SBH RH= P , will scramble significantly faster Softness of radiation: Second, we fix the energy of the ≪ ⋆ than it evaporates: tscr tevap. However, the factor of most likely quanta E to be lighter than the black hole: S =ð D ÞD t BH 4π in evap makes room for large black holes in ⋆ ≫ 1 ≫ 1 ≫ E D D that have both SBH and tscr tevap. ≃ ≪ 1: ð15Þ MH SBH III. SEMICLASSICAL PHYSICS AND LARGE D ⋆ ∼ 1 The radiation cannot match a blackbody if E MH. Towards the end of the evaporation process, the semi- Quasistatic geometry: Third, we would like the black classical analysis is expected to break down for any black hole geometry to be relatively static during evaporation: hole. For instance, we expect that once the evaporation has proceeded to a sufficiently advanced point, the dynam- dRH ¼ dMH dRH ¼ PBH RH ≪ 1 ð Þ − 3 : 16 ics of the black hole is no longer well described by the dt dt dMH D MH background spacetime evolving solely according to the classical Einstein equations. Nonetheless, for sufficiently Explicitly, as the Schwarzschild solution is static in large initial entropies, the evaporation and scrambling times Einstein gravity, if there is any appreciable departure from j j≪1 are very well approximated by the expressions in Eqs. (11) static geometry, characterized by dRH=dt , then we and (12). In this section, we make this statement more 1 ⋆ precise. In particular, we identify sources of the breakdown We may also require E 026016-3 HOLDT-SØRENSEN, MCGADY, and WINTERGERST PHYS. REV. D 102, 026016 (2020) 2 assume that the system ceases to be semiclassical. In pure gravity ND counts distinct graviton polarizations _ 2 ≪ 1 DðD−3Þ 2 This constraint is closely related to requiring T=T and grows quadratically: ND ¼ 2 ∼ D . Thus, the for an approximately thermal emitter. However, since “k ¼ 2” black holes evolve slowly at large D; they should _ 2 ∼ j _ j T=T RH =D, this gives a considerably weaker con- be well described by semiclassical physics. straint at large D. As a consequence of the above considerations, semi- Similarly, we may demand the black hole’s decay width, classical black holes at large D should have entropies that D−2 given by its inverse lifetime, to be much smaller than its grow at least as quickly as SBH >D .However,itis mass. However, this related constraint is much weaker than straightforward to see that j j ≪ 1 the constraint dRH=dt . Short scrambling times: In the next section, we show 4π D 1 2 tevap SBH e that there is a range of black holes that satisfy all of the ¼ ; ð20Þ t ≫1 D N log S 2 above conditions, even though their scrambling times (12) scr D D BH 2 are longer than their semiclassical half-life, tevap= (11). where, for ease, we have used the simplified scalings in If true, this would imply that information would leak out Eqs. (1) and (2) with properly restored order-one factors. of the black hole essentially unobscured. This is clearly 2 2 Minimally, ND is at least DðD − 3Þ=2 ≃ 8π ðD=4πÞ . This incompatible with semiclassical Hawking radiation and lets us bound t =t from above: forces us to impose the new condition evap scr t 4π Dþ2 S e2 t 026016-4 BLACK HOLE EVAPORATION AND SEMICLASSICALITY AT … PHYS. REV. D 102, 026016 (2020) ≫ 1 ≥ 2 t log S log S For D , this bounds k , which in turn bounds the scr ¼ BH ¼ BH : ð26Þ l ≳ 3=2 4π − 3 radii of semiclassical black holes by RH= P D .An t× RHTH D interesting consequence of this analysis is that semiclass- Now, we focus on the interesting regime where S ¼ ical black hole temperatures are now bounded from above BH pffiffiffiffi ˆ ð 4πÞDþl 1 ¼ð − 3Þ ð4π l Þ−1 ð4π Þ−1 S0 D= . We see that tevap=t× < is still stronger by TH=MP D × RH= P < D . j j ≪ 1 than the two semiclassical constraints dRH=dt and 2 4 ≪ 1 Rαβγδ=MP . This follows straightforwardly from the C. Light crossing and scrambling fact that for these black holes, we have Equation (2) for the scrambling time is rather natural, ∼ ð Þ irrespective of the number of dimensions. A black hole as a tscr=t× log D; 27 strongly coupled quantum system is characterized by its and thus the constraints from Eqs. (20) and (21) translate temperature T and an effective number of qubits ∼S . H BH into the constraint that Fast scrambling is associated with an exponential growth 4π Dþ2 2 of mixing, with a rate naturally set by the temperature. tevap SBH e ≲ : ð28Þ Complete scrambling then sets in when all qubits mix, after t D D ð4πÞ2 ∼ −1 × D≫1 a time t TH log SBH, in agreement with (2). Nonetheless, one may reasonably wonder whether the This more stringent constraint still leaves almost the physics of scrambling at large D has hidden powers of D in entire window open. Explicitly, black holes with entropies j j ≪ 1 ð D ÞD−2 ð D ÞDþ3 it which could close the window where dRH=dt and within the range 4π 026016-5 HOLDT-SØRENSEN, MCGADY, and WINTERGERST PHYS. REV. D 102, 026016 (2020) tied to a loss of perturbative unitarity [14,16]. We leave the question of exploring such a connection in the context of large-D physics [17,18] for the future. V. CONCLUSIONS AND FUTURE WORK Analyzing the first quantum corrections to black holes in a large number of dimensions, D ≫ 1, reveals rather FIG. 1. Comparison of typical evaporation curves for D ¼ 4 curious properties. The presence of this large dimensionless (red) and D ≫ 1 (blue), where for concreteness we picked number has the capacity to significantly alter the naive ¼ 100 ð Þ intuition, gleaned from black holes in four and five D for the latter. In the left panel, we plot RH t=tevap , ð Þ while on the right we display SBH t=tevap . dimensions, for the timescales in unitary evaporation. Indeed, in D ≫ 1, black hole temperatures grow linearly ∼ with D. Explicitly, THRH D, which comes from the Here, we briefly note that the lowest value of D where R D gradient of g00ðrÞ ∼ 1 − ð HÞ at the horizon. Further, t ∼ t for an otherwise “semiclassical” Schwarzschild r scr evap the Hawking luminosity grows factorially with D. This black hole, i.e., one whose curvature scales are sub- Planckian and whose geometry varies slowly in time growth is mainly due to the growth of the available phase D (due to Hawking radiation), is significantly larger than space with . Neither of these scalings is surprising. Yet, D ¼ 26. In other words, the new condition for semi- they inexorably lead to very short evaporation timescales for even relatively massive black holes, with MH ≲ ð D ÞD, classicality in this paper only applies above the upper MP 4π critical dimension for consistent string theories. which may lead to new observations. Within this paper, we have taken the conservative IV. CONNECTION TO PREVIOUS WORK approach of interpreting all possible sources of tension that appear within the semiclassical approximation as Previous work has also noted that black hole informa- pointing to the latter’s demise. In particular, we have tion, entropy, and evaporation can exhibit qualitatively new identified a range of parameters in which otherwise semi- features at large D. Though recent work on the large-D classical black holes appear to evaporate much faster than limit of general relativity has focused on classical features they scramble information. Since this is incompatible with of the theory, they are explicitly aware of the high everything we understand about the microscopic dynamics luminosity (and short timescales) that would be associated of scrambling, we have taken this as a signal that in a large with semiclassical, and fully quantum, gravity in large D. number of dimensions, semiclassical physics breaks down Moreover, semiclassical aspects of black hole physics at scales significantly larger than naively expected. – at large D have been a direct focus of [6 9]. In particular, Interestingly, this realization helps us to clarify nontrivial the Hawking luminosity was explicitly found in [6], and puzzles that have arisen in the literature, such as the – Refs. [7 9] have discussed the relative size of black hole possibility of having hyperentropic matter at sufficiently entropy as compared to the entropies of unbound systems large D [7] that violates the Bekenstein bound [19].3 While of weakly gravitating matter at large D. recent arguments argue against the validity of such con- In particular, Ref. [8] prominently discusses possible clusions on different grounds [9], we think that our picture tension between basic principles of black hole entropy and provides a complementary and conceptually very simple properties of weakly gravitating systems when the entropy explanation for why such apparent tension arises outside ð Þ ≲ ð 4πÞDþl l of a black hole is SBH D D= if lies in the the regime of validity of semiclassical gravity. l ∈ ð1 1Þ interval 2 ; . In our present context, it is very One of the main assumptions underlying this work is amusing to note that these black holes fall squarely within the validity of expression (12) for the scrambling time in the family of black holes for which the scrambling bound is any D. From a certain point of view, this may seem the strongest. In fact, our inequality (22) resolves possible unnatural. After all, it implies that the entire D dependence tension by demonstrating that a semiclassical analysis of of the scrambling time lies only in the D dependence of such black holes should not be trusted. temperature and entropy. One may thus wonder whether Our results are similar in spirit to results derived in the corrections to this expression, relevant only at sufficiently context of a large number N of gravitationally interacting large D, could lead to a decrease of thermalization species [14]. Their presence opens up the phase space in a similar fashion as going to large D. Consequently, evapo- 3Such matter does not violate the covariant entropy bound, ration times and scrambling times can become comparable 2 ≠ 4 which is a factor of D−2 weaker in D [20]. It would be for sufficiently large N, which again bounds the validity of interesting to see whether the fact that the Bekenstein bound is the semiclassical approximation [15]. 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