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Black Hole Evaporation and Semiclassicality at Large D 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 .
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