Large D gravity and low D strings
LLarge D gravity and low D strings Roberto Emparan a,b , Daniel Grumiller c , Kentaro Tanabe ba Instituci´o Catalana de Recerca i Estudis Avan¸cats (ICREA)Passeig Llu´ıs Companys 23, E-08010 Barcelona, Spain b Departament de F´ısica Fonamental, Institut de Ci`encies del Cosmos,Universitat de Barcelona, Mart´ı i Franqu`es 1, E-08028 Barcelona, Spain c Institute for Theoretical Physics, Vienna University of TechnologyWiedner Hauptstrasse 8-10/136, A-1040 Vienna, Austria [email protected], [email protected], [email protected]
Abstract
We point out that in the limit of large number of dimensions a wide class of non-extremal neutral black holes has a universal near horizon limit. The limiting geometry isthe two-dimensional black hole of string theory with a two-dimensional target space. Itsconformal symmetry explains properties of massless scalars found recently in the large D limit. In analogy to the situation for NS fivebranes, the dynamics near the horizon doesnot decouple from the asymptotically flat region. We generalize the discussion to chargedblack p -branes. For black branes with string charges, the near horizon geometry is thatof the three-dimensional black strings of Horne and Horowitz. The analogies betweenthe α (cid:48) expansion in string theory and the large D expansion in gravity suggest a possibleeffective string description of the large D limit of black holes. We comment on applicationsto several subjects, in particular to the problem of critical collapse. a r X i v : . [ h e p - t h ] M a r . The study of General Relativity and its black holes simplifies drastically in the limitthat the number D of spacetime dimensions diverges [1] (see also [2]). The reason is thatwhen D is very large, the gravitational field is strongly localized within a region very closeto the black hole horizon. Near a horizon of radius r the potential develops a very largegradient, ∼ D/r , with the result that the geometry further than a distance ∼ r /D fromthe horizon is essentially flat spacetime.The appearance of two separate scales r /D (cid:28) r can be used to identify two differentregions in the black hole geometry: a ‘far region’, defined by r − r (cid:29) r /D , and a ‘near-horizon region’, where r − r (cid:28) r . The dynamics in each of them is quite different: inthe far region there are waves that propagate in flat spacetime; the near region containsthe dynamics intrinsic to the black hole. These two sets of degrees of freedom interactin an ‘overlap region’, r /D (cid:28) r − r (cid:28) r , common to both. Note that this two-regionstructure is a property of the geometry, and not the result of taking a long-wavelengthapproximation to field propagation in the black hole background.In ref. [1] it has been argued that the existence of a far region in which the metricbecomes exactly flat in the limit D → ∞ is generic for very wide classes of black holes— essentially, all black holes whose horizon length scales, as well as their asymptoticgravitational field, remain finite as D → ∞ . In this article we investigate the propertiesof the near-horizon region. We find that for many neutral black holes — including allknown vacuum and AdS non-extremal black holes whose size, again, remains finite in thelimit D → ∞ — this region is universally described by a well-known geometry: the two-dimensional (2D) string-theory black hole of [3, 4, 5]. The conformal symmetry of thissolution is the explanation for the properties of the amplitudes for massless scalar fields inthese backgrounds found in [1]. Our results imply that the same symmetry is also presentin the limit D → ∞ near the horizon of many other neutral black holes, including rotatingblack holes and Anti-de Sitter (AdS) black holes.We also explore the large D limit of a general class of dilatonic black p -branes, chargedunder a ( q + 2)-form field strength ( q ≤ p ). We find that for non-dilatonic solutions withelectric string charge the near-horizon region is the three-dimensional black string of Horneand Horowitz [6]. II.
Let us begin with the Schwarzschild–Tangherlini solution in D = 3 + n dimensions ds = − f ( r ) dt + dr f ( r ) + r d Ω n +1 , f ( r ) = 1 − (cid:16) r r (cid:17) n , (1)where d Ω n +1 is the line-element of the round ( n + 1)-sphere, and introduce the coordinate R = ( r/r ) n , in terms of which ds = − R − R dt + r n R /n d R R ( R −
1) + r R /n d Ω n +1 . (2)1he near-horizon region at large n is defined by ln R (cid:28) n , so we find ds = − R − R dt + r n d R R ( R −
1) + r d Ω n +1 . (3)We see that the size along the radial direction is very small when n is large. As observedin [1], this region would be traversed very quickly, on a time ∼ r /n , by freely fallingobservers. It is then convenient to rescale the time to ˆ t = nt/ (2 r ). Furthermore, if wechange R = cosh ρ the near-horizon metric becomes ds = 4 r n (cid:0) − tanh ρ d ˆ t + dρ (cid:1) + r d Ω n +1 . (4)The part of the metric in parenthesis is the 2D string black hole of [3, 4, 5], which isthe coset manifold SL (2 , R ) /U (1). The appearance of this geometry in this context isactually expected from an observation made in [7, 8]. It is well known that the dimensionalreduction on a sphere, ds = g µν dx µ dx ν + r e − / ( n +1) d Ω n +1 (5)where g µν ( x λ ) is a 2D metric and Φ( x λ ) a scalar field, of the Einstein-Hilbert action yieldsa specific 2D dilaton gravity action, I = Ω n +1 r n +10 πG (cid:90) d x √− g e − (cid:18) R + 4 nn + 1 ( ∇ Φ) + n ( n + 1) r e / ( n +1) (cid:19) . (6)Maybe less well known [7, 8] is that in the limit n → ∞ the action (6) is equivalent to the2D string action I = 116 πG (cid:90) d x √− g e − (cid:0) R + 4( ∇ Φ) + 4 λ (cid:1) (7)with G = lim n →∞ G Ω n +1 r n +10 (8)after we identify the cosmological constant parameter λ = n/ (2 r ). Notice that keeping r /n finite amounts to keeping finite the Hawking–temperature, T H = λ π (9)of both the large D Schwarzschild–Tangherlini and the 2D string black hole. The stretchingof the time coordinate performed above has the effect of rescaling the Euclidean time circleto finite size. The large difference in the sizes of the 2D metric and the transverse S n +1 in(4) makes the 2D gravitational constant very small if we keep fixed the 2D size r /n andalso fix G . This is particularly important in the quantum theory, as we discuss later.The presence of the 2D string-theory black hole geometry in (4) implies that the am-plitudes for propagation of waves in this background will realize the conformal symmetry SL (2 , R ), thus providing a rationale for the results of [1].2he Minkowski vacuum in the limit of large n corresponds, after rescaling t by a factorof n , to the linear dilaton vacuum of the 2D theory.Other features of fields in the background (1) also have a nice interpretation upondimensional reduction on the sphere. The action for a minimal scalar Ψ = ψ ( t, r ) Y ( l ) n +1 (Ω),where Y ( l ) n +1 (Ω) are spherical harmonics on S n +1 , in the geometry (5) is I [Ψ] = 12 (cid:90) d D x (cid:112) − g ( D ) (cid:0) ∇ ( D ) Ψ (cid:1) = Ω n +1 r n +10 (cid:90) d x (cid:112) − ˆ g e − (cid:20) ( ˆ ∇ ψ ) + 4 e / ( n +1) ln (cid:18) ln + 1 (cid:19) ψ (cid:21) → Ω n +1 r n +10 (cid:90) d x (cid:112) − ˆ g e − (cid:20) ( ˆ ∇ ψ ) + 4 ln (cid:18) ln + 1 (cid:19) ψ (cid:21) (10)where we have rescaled the 2D metric g µν = λ − ˆ g µν . Consider first fields with l ∼ O ( n ).These propagate in the 2D black hole geometry as massless, non-minimally coupled scalars.If their frequency in ‘far region time’ t is ωr ∼ O ( n ), then in ‘near-horizon time’ ˆ t it isˆ ω = 2 ωr /n ∼ O ( n − ). These excitations encounter a dilaton barrier much higher thantheir energy and have vanishingly small amplitude for tunelling between the near and farregions, so that they can be said to decouple. Instead, waves of frequency ˆ ω ∼ O ( n ) havenon-zero probability to penetrate or to pass above the barrier [1]. Thus the large D limitis not a decoupling limit. In this sense, these near-horizon geometries are similar to thenear-horizon region of near-extremal NS fivebranes [9].Waves with large angular momentum ˆ l = 2 l/n = O ( n ) have an effective mass ∼ ˆ l/r .They must have frequency ˆ ω > ˆ l + 1 in order to escape to the asymptotic region [1]. Theseexcitations probe scales much smaller than the radius of the n + 1-sphere and effectivelysee the geometry ds = 4 r n (cid:0) − tanh ρ d ˆ t + dρ + d x n +1 (cid:1) . (11)We can verify the presence of the same geometry in other neutral black holes. Considerthe Myers-Perry solutions with (for simplicity) a single rotation [10]. In terms of thecoordinate R introduced above, they take the form ds = − dt + 1 σ R (cid:0) dt + αr sin θ dφ (cid:1) + R /n r (cid:32) σn δ d R R + σ dθ + (cid:16) α R − /n (cid:17) sin θ dφ + cos θ d Ω n − (cid:33) (12)where σ = 1 + α cos θ R /n , δ = 1 + α R /n − R . (13)The usual rotation parameter is a = αr . We see again that the metric in radial directionsis small. So we consider waves of frequency ω = O ( n ), which are the ones that can probethe full geometry. The metric that a partial wave sees will be different depending on thevalue of the component l φ of angular momentum along the rotation direction φ . When3he wave has small impact parameter l φ /ω in the plane of rotation, as is the case when l φ = O ( n ), it probes the region near the pole at θ = 0. At larger impact parameters, with l φ = O ( n ), it probes the region around a finite angle θ . Denoting v = α sin θ / √ α ,the appropriate rescalings areˆ t = n r t √ − v , ˆ y = n (1 + α )2 α v √ − v φ , ˆ θ = n √ α θ − θ ) . (14)If we set R = (1 + α ) − cosh ρ we find ds = 4 r n (1 − v ) (cid:18) − d ˆ t + d ˆ y + (cid:0) d ˆ t + v d ˆ y (cid:1) (1 − v ) cosh ρ + dρ + d ˆ θ + n − v ) cos θ d Ω n − (cid:19) . (15)The (ˆ t, ˆ y, ρ ) part of the metric is the result of adding a line ˆ y to the 2D string black holeand performing a boost of velocity v along ˆ y . So locally this geometry is equivalent tothe 2D string black hole. For angular momenta that are O ( n ) the wave probes the poleregion where v = 0 and the geometry is the product of the static 2D string black holetimes R × S n − . When the impact parameter is maximum, so that θ = π/
2, the waveis strongly localized in the angular directions transverse to the rotation direction and the(ˆ θ, Ω n − ) part of the metric is actually R n .These conclusions generalize to the case in which the black hole has several non-zerospins: for O ( n ) frequencies and angular momenta along a direction in which the blackhole rotates, we find the 2D string black hole with a boost along the rotation direction.For small, O ( n − ), impact parameters we recover the static 2D black hole.When the number of non-zero spins grows like n/ a i = αr . The radialcoordinate of the 2D black hole, in terms of the Boyer-Lindquist radius r [10], iscosh ρ = (cid:18) rr H (cid:19) n (1 − α ) . (16)Here the horizon radius r H is r H = r (cid:112) − α (cid:18) n ln(1 − α )1 − α + O ( n − ) (cid:19) (17)and we assume that α < n → ∞ . Then, rescaling t and the anglesappropriately as before, we recover the boosted 2D string black hole. In the cases whereall the spins of the black hole are turned on, the solution admits an extremal limit. Thiswould correspond in this example to α →
1, for which the above expansion breaks down.Extremal rotating black holes have zero temperature and we expect their near-horizongeometry to be locally inequivalent to the 2D string black hole. We shall not discuss inthis article the detailed analysis required to study their near-horizon geometry.4et us now consider non-rotating AdS black holes. Their metric is of the same formas (1) but with f ( r ) = 1 − (cid:16) r r (cid:17) n + r L . (18)Thus, when we take the large n limit keeping the coordinate R = ( r/r ) n finite we find ds = − R / R − R dt + r n d R R ( R / R −
1) + r d Ω n +1 , (19)where R = L r + L . (20)The only change relative to (4) is a shift in the location of the horizon to R = R < L → ∞ ) ofAdS holography could be particularly simple in the large D limit. It would be interestingto verify this and to consider also the large D limit on the field theory side.We conjecture that the appearance of the 2D string black hole is a universal featureof the large D limit of non-extremal neutral black holes, at least when the limit is takenin such a manner that the length scales of the horizon remain finite in the limit. Eq. (11)is the geometry perceived by waves of frequencies and angular momenta ω, l ∼ O ( D ).We now examine black branes and include the effects of charge. We consider a largeclass of them, corresponding to dilatonic black p -branes with electric q -brane charge whichare solutions of the theories I = 116 πG (cid:90) d D x √− g (cid:18) R − ∇ ϕ ) − q + 2)! e − aϕ H q +2] (cid:19) . (21)We refer to appendix A.2 of [11] for details of the solutions. For a p -brane we denote D = n + p + 3. The brane carries q -brane charge, with q ≤ p , and the dilaton coupling is a .The charge is parametrized in terms of a charge-boost velocity u , so the more conventionalcharge rapidity β is tanh β = u . The parameter u varies between 0 for neutral branes and u = 1 for extremal ones, and it is kept finite as D → ∞ , and so are p and q , too. In thislimit, the near-horizon geometries become (we set r = 1 for simplicity) ds = (cid:18) − u R (cid:19) N (cid:18) − R − R − u dt + d y q (cid:19) + 1 n d R ( R − u )( R −
1) + d Ω n +1 + d z p − q , (22)where N = 42( q + 1) + a . (23)All the dependence of metric functions on p , q and a has been reduced to a single parameter N . This may be an indication of universality classes of near-horizon solutions labeled bythe values of N and u . 5he presence of charge deforms the near horizon geometry away from the 2D stringblack hole to other black holes of dimensionally-reduced (multi-)dilaton theories. Let usconsider the case q = 1, a = 0, i.e., N = 1. After rescaling the t and y coordinates by afactor of n we obtain n ds = − (cid:18) − R (cid:19) d ˆ t + (cid:18) − u R (cid:19) d ˆ y + d R ( R − u )( R −
1) + n d Ω n +1 + n d z p − . (24)The sector (ˆ t, ˆ y, R ) in this metric is the geometry of the three-dimensional black strings of[6], which are a solution of three-dimensional string theory with a known exact conformalfield theory. This geometry arises whenever q = 1 and a = 0, independently of p . Itsappearance should not be a surprise when we consider that it can be obtained by addinga line y to the 2D string black hole followed by a boost and T-duality along y [12].These geometries arise in the near-horizon region of non-extremal black holes, withno requirement that the solutions be any close to extremality. For extremal solutions thevariety of near-horizon geometries can be expected to be much larger. We can easily findsome examples. For instance, the large D limit of the Reissner-Nordstrom black holes isobtained setting a = 0 = p = q in the solutions above, and then the extremal limit isreached when u →
1. We find n ds = − (cid:0) − R − (cid:1) d ˆ t + d R ( R − + n d Ω n +1 = − e − ρ ) d ˆ t + dρ + n d Ω n +1 , (25)where in the last line we have changed to ρ = ln( R − t, ρ ) part of the geometryapproaches 2D flat spacetime in the asymptotic region ρ → ∞ , while near the horizonat ρ → −∞ it becomes the infinite throat of the AdS spacetime. So the large D limitresults in the replacement of the non-extremal horizon of the 2D string black hole withan extremal throat characteristic of the solution at finite D . We expect this type ofreplacement to be a general feature of large D extremal charged solutions. III.
The emergence of a 2D conformal symmetry offers, at the very least, the prospect ofa significant degree of control over the classical theory of very wide classes of neutral, non-extremal large D black holes. Even more tantalizing is the appearance of low-dimensionalstring geometries near the horizon of large D black holes. Regarding D as a parameterthat can be made large is essential for the stringy interpretation. The connection to the2D string black hole instructs us to identify √ α (cid:48) ∼ r D , (26)so the large D expansion corresponds to the α (cid:48) expansion in string theory. The near-horizon geometries are ‘stringy geometries’, of size ∼ r /D . Note also that the Bekenstein–Hawking entropy S BH of Schwarzschild–Tangherlini black holes, as a function of the mass6 , is S BH ∝ M D − , (27)and therefore in the large D limit asymptotes to the Hagedorn behavior expected fromstring theory, S BH ∝ M [1]. Moreover, the black hole temperature (9) corresponds tothe string scale T H ∼ / √ α (cid:48) . This may be an indication of a string-like nature of theexcitations near the horizon.In view of these observations, the two-region picture of large D black hole spacetimeswould have the following interpretation: in the far region we keep r fixed, so when D → ∞ we have α (cid:48) → r /D finite,and we obtain a string-scale geometry with string excitations. We note, however, that α (cid:48) corrections to the 2D black hole are known [13], and they do not coincide with the 1 /D corrections to the near-horizon geometry. Probably we should not be surprised by thesediscrepancies: since full quantum string theory cannot be formulated consistently in theselarge D spacetimes, presumably any strings in the near-horizon region should be effectivestrings and not fundamental ones.Nevertheless, it would be remarkable to find, similarly to what happens in the large N limit of Yang-Mills theories, that an effective string theory emerges in the large D limit ofgravity — in this case, one has to look for the strings near a black hole horizon. If indeedthis occurs, the symmetry SO ( D −
2) of the angular sphere will likely play a role. Morework is needed to put these ideas on a firmer ground and, if they are correct, identify thekind of effective string theories that can arise in this context.The quantum theory may also be constrained by the near-horizon conformal symmetry.The strength of quantum gravitational effects can be controlled by suitably choosing howthe Planck length scales with D as the number of dimensions increases. Notice that the‘effective string length’ r /D is a purely classical length scale (as it is in string theory)which can be arbitrarily separated from the quantum Planck scale, e.g., it is possible tohave large α (cid:48) effects but small quantum gravity effects.It should be interesting to study the evaporation of these black holes through quantumHawking radiation. If the Planck length is chosen to not scale with D , then there is (atleast) one significant difference with the situation for near-extremal NS five branes, whichdecay by the slow leakage of radiation of energy ∼ T H that reaches the asymptotic regionof the 2D black hole. For large D black holes (and with a D -independent Planck scale)the typical energy of Hawking quanta is much larger, ω ∼ DT H ∼ D /r [14, 1]. Thewavelength of these quanta is so small that they do not distinguish between the near andfar regions, and they accelerate enormously the decay rate of the black holes. However,if the Planck length ( G (cid:126) ) / ( D − is made to shrink like 1 /D then typical Hawking quantawill have string-like energies ∼ D/r (analogous to the ’t Hooft large- N limit in whichthe gauge coupling g Y M is made to shrink like 1 /N ). This may give a better chance ofcontrolling the evaporation process by using the near-horizon 2D conformal symmetry7nd, possibly, an effective string description.Finally, from a purely pragmatic viewpoint, the large D limit can lead to considerablesimplifications [1], and the effective 2D formulation discussed in the present work can bea useful tool. As an example let us consider critical collapse `a la Choptuik [15]. So far,analytic derivations of the critical exponent γ are generally not possible. One relevantexception is the derivation in [16] of the value γ = in the RST model. This theorydiffers from the 2D model (7), (10) in three aspects: it takes into account semi-classicalcorrections, it has a large number of scalar fields instead of just one, and the scalar fieldsdo not couple to the dilaton field in 2D. Nevertheless, it seems plausible to us that theirresult for the critical exponent is the correct one in the large D limit, since the geometricpart of the RST model coincides classically with the action (7). There is also numericalevidence at D ∼ O (10) that backs up our conjecture [17, 18]. It would be gratifying toderive our conjectured result for the critical exponent, γ = , analytically and to probenumerically the large D regime. We believe that both avenues are accessible. Acknowledgments
RE was partially supported by MEC FPA2010-20807-C02-02, AGAUR 2009-SGR-168 andCPAN CSD2007-00042 Consolider-Ingenio 2010. DG was supported by the START projectY 435-N16 of the Austrian Science Fund (FWF). KT was supported by a grant for researchabroad by JSPS.
References [1] R. Emparan, R. Suzuki and K. Tanabe, “The large D limit of General Relativity,”arXiv:1302.6382 [hep-th].[2] V. Asnin, D. Gorbonos, S. Hadar, B. Kol, M. Levi and U. Miyamoto, “High and LowDimensions in The Black Hole Negative Mode,” Class. Quant. Grav. (2007) 5527[arXiv:0706.1555 [hep-th]].[3] G. Mandal, A. M. Sengupta and S. R. Wadia, “Classical solutions of two-dimensionalstring theory,” Mod. Phys. Lett. A (1991) 1685.[4] S. Elitzur, A. Forge and E. Rabinovici, “Some global aspects of string compactifica-tions,” Nucl. Phys. B (1991) 581.[5] E. Witten, “On string theory and black holes,” Phys. Rev. D (1991) 314.[6] J. H. Horne and G. T. Horowitz, “Exact black string solutions in three-dimensions,”Nucl. Phys. B (1992) 444 [hep-th/9108001].87] J. Soda, “Hierarchical dimensional reduction and gluing geometries,” Prog. Theor.Phys. (1993) 1303.[8] D. Grumiller, W. Kummer and D. V. Vassilevich, “Dilaton gravity in two-dimensions,” Phys. Rept. (2002) 327 [hep-th/0204253].[9] J. M. Maldacena and A. Strominger, “Semiclassical decay of near extremal five-branes,” JHEP (1997) 008 [hep-th/9710014].[10] R. C. Myers and M. J. Perry, “Black Holes in Higher Dimensional Space-Times,”Annals Phys. (1986) 304.[11] M. M. Caldarelli, R. Emparan and B. Van Pol, “Higher-dimensional Rotating ChargedBlack Holes,” JHEP (2011) 013 [arXiv:1012.4517 [hep-th]].[12] J. H. Horne, G. T. Horowitz and A. R. Steif, “An Equivalence between momentumand charge in string theory,” Phys. Rev. Lett. (1992) 568 [hep-th/9110065].[13] R. Dijkgraaf, H. L. Verlinde and E. P. Verlinde, “String propagation in a black holegeometry,” Nucl. Phys. B (1992) 269.[14] S. Hod, “Bulk emission by higher-dimensional black holes: Almost perfect blackbodyradiation,” Class. Quant. Grav. (2011) 105016 [arXiv:1107.0797 [gr-qc]].[15] M. W. Choptuik, “Universality and scaling in gravitational collapse of a masslessscalar field,” Phys. Rev. Lett. (1993) 9.[16] A. Strominger and L. Thorlacius, “Universality and scaling at the onset of quantumblack hole formation,” Phys. Rev. Lett. (1994) 1584 [hep-th/9312017].[17] E. Sorkin and Y. Oren, “On Choptuik’s scaling in higher dimensions,” Phys. Rev. D (2005) 124005 [hep-th/0502034].[18] J. Bland, B. Preston, M. Becker, G. Kunstatter and V. Husain, “Dimension depen-dence of the critical exponent in spherically symmetric gravitational collapse,” Class.Quant. Grav.22