On the black hole interior in string theoy
aa r X i v : . [ h e p - t h ] M a y ON THE BLACK HOLE INTERIORIN STRING THEORY
Roy Ben-Israel , Amit Giveon , Nissan Itzhaki and Lior Liram Physics Department, Tel-Aviv University, IsraelRamat-Aviv, 69978, Israel Racah Institute of Physics, The Hebrew UniversityJerusalem, 91904, Israel
Abstract
The potential behind the horizon of an eternal black hole in classical theories isdescribed in terms of data that is available to an external observer – the reflectioncoefficient of a wave that scatters on the black hole. In GR and perturbative stringtheory (in α ′ ), the potential is regular at the horizon and it blows up at the singu-larity. The exact reflection coefficient, that is known for the SL (2 , R ) k /U (1) blackhole and includes non-perturbative α ′ effects, seems however to imply that there isa highly non-trivial structure just behind the horizon. Introduction
Describing the interior of a black hole (BH) and determining the fate of an infallingobserver is a difficult task in string theory. The observables in string theory are on-shellquantities – S-matrix elements in flat space-time and correlation functions at the boundaryof an AdS background. As such they live at infinity and are natural from the point ofview of an external observer that by causality has no access to the BH interior.The standard way to address this challenge is to find an off-shell effective action thatyields the same S-matrix elements as string theory and to look for BH solutions in thiseffective action. At low energies (relative to the string mass) the relevant effective actionis supergravity (SUGRA) which respects the equivalence principle. Consequently, thehorizon of a BH in SUGRA is smooth. At higher energies corrections to SUGRA becomerelevant. Perturbative corrections (both in α ′ = l s and in the string coupling, g s ) can bedescribed in terms of higher orders terms (roughly of the type R ) [1] that also respectthe equivalence principle and are small at the horizon of a large BH. Therefore, they canhave an important effect only at the singularity (see e.g. [2]). In particular, the horizonremains smooth when perturbative corrections are taken into account.What happens in the full string theory, including all corrections, perturbative or other-wise? In particular, can non-perturbative stringy effects violate the equivalence principleand render the horizon singular? One might view this as a philosophical question since wecannot solve string theory exactly. We do not have an effective action that takes accountof all non-perturbative corrections that can be used to test the equivalence principle.There is, however, a setup in which this question becomes quite precise. Some cor-relation functions were calculated exactly in the coset conformal field theory (CFT) SL (2 , R ) k /U (1) [3]. These include the exact reflection coefficient associated with k near-extremal NS5-branes in the classical limit, g s = 0. The goal of this note is to takeadvantage of this exact stringy data in order to determine whether the horizon is regularor not in this setup. More precisely, we wish to calculate the potential outside and behindthe horizon in terms of the reflection coefficient.It is natural to expect that for large k the answer must be that the horizon is smoothand that large deviations from the SUGRA potential take place only near the singularity.This expectation is based on the fact that we consider a classical setup while argumentsso far for non-trivial structure at the BH horizon are quantum mechanical [4–9]. However,2s we shall see, things are more interesting.The paper is organized as follows: in the next section we describe how to calculatethe potential behind the horizon of an Eternal Black Hole (EBH) in terms of data thatis available to an external observer – the reflection coefficient. In section 3 we showthat in SUGRA (including perturbative corrections) the potential blows up only at thesingularity while in the exact SL (2 , R ) k /U (1) BH model the potential blows up justbehind the horizon. In section 4 we discuss some aspects associated with coarse-grainingthe potential behind the horizon. We summarize in section 5. In this section we show how to determine the potential behind the horizon of an EBH interms of data that is available to an external observer – the reflection coefficient associatedwith a scattering of a wave, with some momentum p , on a classical black hole, R ( p ).Let us suppose that we know R ( p ). At least in classical cases, which we focus on here,we expect R ( p ) to drop exponentially fast with p since most of the wave gets absorbed bythe black hole. Still the reflected wave contains information about the surrounding of theblack hole. In particular, one can ask what kind of a potential, V ( x ), in a Schr¨odinger-likesetup, − ∂ x ψ ( x ) + V ( x ) ψ ( x ) = p ψ ( x ) , (2.1)yields R ( p ). In quantum mechanics this is known as the inverse scattering problem.In GR (2.1) can be found by transforming the Klein-Gordon equation in the EBHbackground into a Schr¨odinger equation. In this case x is the tortoise coordinate, where x = ∞ is the asymptotic region and x = −∞ is the horizon. From the point of view ofGR, the input is the BH background and the output is the reflection coefficient.In string theory the logic is reversed: the basic quantities that we know how to calculateare on-shell scattering amplitudes, from which we can read R ( p ). Once known, one can askwhat kind of a background yields R ( p ). In particular, we wish to know if this backgroundadmits a smooth horizon. To do so we solve the inverse scattering problem and find thepotential V ( x ). The background is singular when V ( x ) blows up. In other words we canview V ( x ) as a singularity detector. A refinement of this statement appears at the end ofthis section.The range of x indicates that the only information about the EBH we can extract3 irectly this way is about the region outside the black hole. This is as it should be sincecausality implies that the reflection should take place outside the horizon for the reflectedwave to make it back to infinity. To find the potential behind the horizon, which wedenote by V int ( x ), we can analytically continue V ( x ). From V int ( x ) we can tell if thereis a singularity behind the eternal BH horizon and where it is located. In GR V int ( x )can be found by writing the Klein-Gordon equation behind the horizon in a Schr¨odingerequation form. As will be discussed momentarily this equation can also be found fromthe analytic continuation of the Schr¨odinger equation outside the horizon.Our goal here is to find the relation between R ( p ) and V int ( x ). Such a relation can beused in order to determine in the exact SL (2 , R ) k /U (1) BH model the potential behindthe horizon. In particular, it can be used to tell if something unusual happens to V whenwe cross the EBH horizon. To do so we first recall how the potential outside the BH andthe reflection coefficient are related.In 1D, when there are no bound states, and when the potential decays sufficientlyfast as x → ±∞ , the potential is related to the scattering data in the following way (seee.g. [10]), V ( x ) = 2 iπ Z ∞−∞ p R ( p ) T ∗ ( p ) T ( p ) ( ψ ∗ ( p, x )) dp. (2.2) T ( p ) is the transmission coefficient and ψ ( p, x ) is the Jost solution to the Schr¨odingerequation that asymptotically behaves as ψ ( p, x ) ∼ ( e − ipx , x → −∞ , T ( p ) e − ipx + R ( p ) T ( p ) e ipx , x → ∞ . (2.3)In the case of EBHs V ( x ) is smooth and its maximal value is of the order of the curvature.Consequently, for wavelengths much smaller than the curvature scale, T ( p ) approaches 1and R ( p ) decays exponentially fast. In fact, both for Schwarzchild EBH (see appendix A)and for SL (2 , R ) k /U (1) EBH (see next section) the asymptotic behaviour of the reflectioncoefficient is R ( p ) ∼ e − β | p | , (2.4)where β is the inverse temperature.The potential at the horizon (from the outside) is smooth. Combining (2.2) and (2.3)we see that near the horizon it takes the form V Hor − out = lim x →−∞ iπ Z ∞−∞ p R ( p ) T ∗ ( p ) T ( p ) e ipx dp. (2.5)4e already know that V Hor − out is smooth. Still it is worthwhile to see how this is consistentwith (2.5). A divergence in (2.5) can come only from the UV. However, since at the UV T ( p ) →
1, we get from (2.4) that V UVHor − out ∼ lim x →−∞ Z p e − β | p | e ipx dp, (2.6)which converges rapidly.What happens when we cross the horizon? The relation between the tortoise andKruskal coordinates x = β π log( uv ) , (2.7)implies that crossing the horizon amounts to taking x → x ± iβ . (2.8)The ‘ ± ’ is due to the two simplest possible branch choices in (2.7).In the interior the horizon is at Re ( x ) = −∞ and the singularity at Re ( x ) = 0. Eq.(2.8) implies that the potential behind the horizon, V int ( x ), is related to V ( x ) via V int ( x ) = V ( x ± iβ/ . (2.9)Hence the potential just behind the horizon reads V Hor − In = lim x →−∞ iπ Z ∞−∞ p R ( p ) T ∗ ( p ) T ( p ) e ipx cosh( βp/ dp. (2.10)We emphasise that T ( p ) and R ( p ) in this equation are the transmission and reflectioncoefficients associated with scattering outside the EBH. The cosh( βp/
2) is the only dif-ference between the potential inside and the potential outside. This follows from (2.9).Note that we keep the two branches of (2.9) which together with R ∗ ( p ) = R ( − p ) , T ∗ ( p ) = T ( − p ) (2.11)make the reality of V Hor − In manifest.It is clear that V Hor − out is finite. Again a divergence can appear at the UV where T ( p ) →
1. If (2.4) holds we get that V UVHor − In ∼ lim x →−∞ Z p e − β | p | e ipx cosh( βp/ dp, (2.12) The asymptotic region behind the singularity, that plays no role here, is at Re ( x ) = ∞ . • Both V Hor − In and V Hor − Out vanish when x → −∞ . Hence there is no discontinuity aswe cross the horizon. In the next section we show this explicitly for the SL (2 , R ) k /U (1)EBH. This is a general result that follows from the equivalence principle held in GR andperturbative string theory. • If we relax the condition x → −∞ in (2.12) we see that its RHS blows up at x = 0exactly where the singularity is located. This is not a coincidence. At the UV the wavefunction ψ ( x, p ) is well approximated by e ipx which implies that the contribution of theUV modes to the potential is given by the RHS of (2.12) for any x . As is evident fromthe RHS of (2.12) the UV modes are sufficient to detect the singularity.In the next section we shall see that in the exact SL (2 , R ) k /U (1) EBH (2.4) does nothold and that consequently there is a non-trivial structure just behind the horizon. Beforewe do so we make a slight refinement. A Refinement
Below we refine the definition of the singularity detector. This refinement does notaffect any of the conclusions that follow.To motivate the refinement we consider a free massive mode in Rindler space. Rindlerspace is merely a reparametrization of Minkowski space, and so an ideal singularity de-tector should vanish everywhere in Rindler space. However, as we now show, V ( x ) doesnot vanish in Rindler space-time. In fact, it blows up at infinity. To see this we considerthe massive Klein-Gordon equation of a mode with energy ω in the Rindler metric ds = − (cid:18) πβ (cid:19) ρ dt + dρ , (2.13)where the Unruh (inverse) temperature, β , is written explicitly. It takes the form1 ρ (cid:20) β π ρ ω φ + ∂ ρ ( ρ∂ ρ φ ) (cid:21) = m φ. (2.14)To put it in a Schr¨odinger form, we switch to the tortoise coordinate x = β π log (cid:18) πρβ (cid:19) , (2.15)to find − ∂ x φ ( x ) + e πβ x m φ ( x ) = ω φ ( x ) . (2.16)6herefore V ( x ) = e πβ x m , which blows up at infinity. Clearly, the fact that the potentialblows up at infinity does not indicate there is a singularity there. To fix this we define D ( x ) = ∂ x ( e − πβ x V ( x )) . (2.17)The factor e − πβ x is basically the redshift factor between the Rindler and Minkowski ob-servers.Indeed in Rindler space D ( x ) vanishes and in the BH background, much like V ( x ),it blows up only at the singularity. In the following section we show that D ( x ) (andalso V ( x )) blows up at the horizon of the SL (2 , R ) k /U (1) BH. One may wonder if somealternative definition of D ( x ) would not be divergent at the horizon. While this is true,the definition (2.17) could be related to tidal forces acting on infalling observers which isa measurable physical quantity. As such, it is a valid singularity detector. SL (2 , R ) k /U (1) BH In this section we consider classical ( g s = 0) EBH associated with the SL (2 , R ) k /U (1)model. First, we ignore non-perturbative stringy corrections to the reflection coefficientand obtain the expected result that the horizon is smooth. Then we show that the fullstringy reflection coefficient, that includes non-perturbative effects, alters this conclusionquite dramatically. At the semi-classical level the SL (2 , R ) k /U (1) EBH can be described by the followingbackground [11–13] ds = − tanh (cid:18) ρ √ k (cid:19) dt + dρ , exp(2Φ) = g (cid:16) ρ √ k (cid:17) , (3.1)where Φ is the dilaton. We are interested in the classical limit of the theory, so we take g → t, ρ ) this background is defined only forthe exterior region with the horizon at ρ = 0, but it can be extended to the region behind7he horizon, as usual, by switching to Kruskal coordinates, u = sinh (cid:18) ρ √ k (cid:19) e t/ √ k , v = sinh (cid:18) ρ √ k (cid:19) e − t/ √ k , (3.2)in terms of which the background takes the form ds = 2 k uv dudv, Φ − Φ = −
12 log(1 + uv ) . (3.3)We see that the background is smooth at the horizon ( uv = 0) and that it is singular at uv = −
1. This, of course, is well known. Here we wish to show that V ( x ) (or D ( x )) leadto the same conclusion.The starting point is the reflection coefficient from which we can deduce the potential.At the perturbative level, the reflection coefficient can be determined from the underlying SL (2 , R ) structure of the L and ¯ L generators [14]. In the supersymmetric case it reads R SUGRA ( p ) = Γ( i √ kp )Γ (cid:16) (1 − i √ kp − i √ kω ) (cid:17) Γ( − i √ kp )Γ (cid:16) (1 + i √ kp − i √ kω ) (cid:17) , (3.4)where p is the momentum in the ρ direction as measured at infinity and ω is the energy(associated with t ) of the wave. They are related by the on-shell condition ω = ± r p + 12 k . (3.5)The branch choice for the square root is such that for ω, p ∈ R , sign( ω ) = sign( p ). Thisassures that the reflection coefficient, as required, has the property (2.11) that is usedbelow.At large p (or ω ) we get (2.4) (with β = 2 π √ k ). Therefore the discussion in theprevious section guarantees that the potential blows up only at the singularity and that, inparticular, the horizon is regular. Still it is worthwhile to do this exercise here since, as weshall see, non-perturbative corrections will alter this conclusion quite severely. Note thatthe exact same expression for the reflection coefficient is obtained by solving the Klein-Gordon equation in the background (3.1). This implies that there are no perturbative α ′ corrections in the supersymmetric case [15, 16].The potential that gives this reflection coefficient is V ( x ) = 12 k − (cid:16) e √ k x (cid:17) . (3.6)8uch like in Schwarzschild BH, the potential goes to zero exponentially fast at the hori-zon x → −∞ . However, unlike in Schwarzschild BH it does not go to zero at infinity, butrather to a positive constant. This is a feature that is related to the linear dilaton at infin-ity and is not expected to affect the physics at the horizon. To find its behaviour behindthe horizon we write down the tortoise coordinate in terms of the Kruskal coordinates(3.2), x = r k uv ) , (3.7)which gives V ( u, v ) = 12 k (cid:18) − uv ) (cid:19) . (3.8)We see that except at the singularity, where it diverges, the potential is smooth every-where. In particular, it is continuous across the horizon where it vanishes. D ( x ) (2.17) associated with this potential is, D ( x ) = tanh (cid:16) x √ k (cid:17) − √ k / cosh (cid:16) x √ k (cid:17) . (3.9)It is clear that everywhere outside the BH D ( x ) is small. As for the potential, we writeit in terms the Kruskal coordinates to see how it behaves behind the horizon, D ( u, v ) = − uv (3 + uv ) √ k / (1 + uv ) . (3.10)Hence for large k we see that D ( x ) is small everywhere but at the singularity. α ′ corrections The SL (2 , R ) k /U (1) model also illustrates neatly that perturbative α ′ corrections arenegligible at the horizon of a large BH. In particular, they do not render the horizonsingular.In the bosonic case the reflection coefficient obtained via the underlying SL (2) struc-ture of the L and ¯ L generators differs a bit from (3.4) [14]. This difference, that is dueto the shift k → k −
2, implies that the classical background (3.1) receives perturbative α ′ corrections. To determine the modified background, one seeks a background in whichthe Klein-Gordon equation leads to the corrected reflection coefficient [14]. We do not9resent the details of this calculation (that can be found in [14]) since, not too surpris-ingly, this background is regular at the horizon. In the next subsection we shall see thatnon-perturbative α ′ corrections are more interesting. A nice feature of the SL (2 , R ) k /U (1) EBH is that, much like in the Liouville model [17],the reflection coefficient can be calculated exactly on the sphere [3]. It reads R exact ( p ) = R SUGRA ( p ) R non − per ( p ) , (3.11)with R non − per ( p ) = − Γ (cid:16) i q k p (cid:17) Γ (cid:16) − i q k p (cid:17) , (3.12)where R SUGRA is given in (3.4).The exact reflection coefficient in the SL (2 , R ) k /U (1) EBH is determined by the exactreflection coefficient in the Euclidean H +3 [3]. There is no transmission coefficient in H +3 .Therefore we do not know the exact transmission coefficient in the SL (2 , R ) k /U (1) EBH.It is natural to assume that for p ≫ /kT ( p ) → , (3.13)since this is a general result that follows from having a smooth potential outside the EBHthat is bounded by the curvature that scales like 1 /k . It is possible that in the exactstring theory (3.13) does not hold, but that would mean that non-perturbative effects inclassical string theory affect drastically the region outside the EBH. We find this hard tobelieve and so we assume (3.13).We would like to argue now that (3.11) and (3.13) imply that there is a singularityjust behind the horizon. This too is a bit hard to believe, but it is less dramatic than thealternative that follows from a violation of (3.13).At first sight it is hard to see how such a startling result can come about. The non-perturbative correction, R non − per , is merely a phase. Instead of (2.4), the leading UVbehaviour is R ( p ) ∼ e − β | p | + iθ ( p ) . (3.14)10t is clear from (2.6) that V Hor − Out still converges rapidly (which is part of our assumptionin reaching (3.13)). On the other side of the horizon the UV contribution to V Hor − In reads V UVHor − In ( x ) = lim x →−∞ iπ Z p e i (2 px + θ ( p )) dp. (3.15)If θ ( p ) was a function that approaches a constant at large p then the RHS of (3.15) stillwould have blown up only at the location of the GR singularity ( x = 0) and the effect of θ ( p ) was to smear it a bit. Similarly if at the UV we had θ ( p ) ∼ c p then the singularityin the RHS of (3.15) would have been shifted by c . A combination of these two options,shifting a bit and smearing the singularity, is something that is natural to expect from α ′ corrections.This, however, is not what happens in the exact string theory. Using Stirling’s ap-proximation, one finds that for p ≫ k we have − Γ (cid:16) i q k p (cid:17) Γ (cid:16) − i q k p (cid:17) ∼ i sign( p ) e iθ ( p ) , with θ ( p ) ∼ r k p log r k | p | e ! . (3.16)This means that the rate by which the phase is growing, keeps on increasing indefinitely.This fact leads to some interesting effects already in the corresponding cigar geometry[18, 19] and consequently on the relevant Hartle-Hawking wave function [20].Here this implies that the singularity gets expelled all the way to the horizon. This isbecause the shift of the singularity is dependent on p and it keeps on growing indefinitelyat the UV, c ( p ) ∼ p /k log( | p | ). A more precise way to see this is to consider thepotential behind the horizon, (2.12), that now reads V UVHor − in ( x ) = lim x →−∞ iπ Z p >k p e i (cid:16) √ k p log (cid:16) √ k | p | e (cid:17) +2 px (cid:17) dp, (3.17)where we made use of (3.16). The integral is controlled by saddle points at p = ± r k e − √ k x (3.18)that give V UVHor − in ∼ k e − √ k x cos (cid:16) e − √ k x (cid:17) . (3.19)We see that while in perturbative string theory the potential behind the horizon is small(of order 1 /k ) until close to the singularity, in the full classical string theory it is large (of11rder k ) and it blows up exponentially fast while oscillating with an exponentially largefrequency as we get closer to the horizon.The same holds for D ( x ) that reads D Hor − in ( x ) ∼ k e − √ k x sin (cid:16) e − √ k x (cid:17) . (3.20)We conclude that while the non-perturbative stringy corrections in α ′ have a tiny effectoutside the horizon, they render the region just behind the horizon singular. In this section we discuss two aspects associated with coarse-graining the results of theprevious section. One from the point of view of an external observer and the other fromthe point of view of an infalling observer.
Let us suppose that an external observer attempts to check experimentally that there is,as we claimed, a non-trivial structure just behind the horizon of a classical BH. Assumingthere is no flaw in our reasoning, such an observer should measure the reflection coefficientand see if (3.11) is correct. The non trivial effects we discussed come from the deep UVand so a natural question to ask is: what could an external observer, that has access toenergies below some cutoff Λ, concludes experimentally about the structure behind thehorizon (without extrapolating her findings to arbitrarily high energies)?From (3.18) we see that if we cut p (or ω ) off at Λ, then the singularity is not pushedall the way to the horizon, but to x = − r k log Λ p k/ ! . (4.1)To understand the physical meaning of this it is useful to switch to ρ , the invariantdistance from the horizon. Using (2.7) and the fact that beyond the horizon ρ → iρ wehave x = √ k log (cid:18) sin (cid:18) ρ √ k (cid:19)(cid:19) . (4.2)The classical singularity is at ρ sin = π p k/ (cid:18) sin (cid:18) ρ √ k (cid:19)(cid:19) = − k log Λ p k/ ! . (4.3)In the large k limit and for Λ that is not exponentially large the singularity is pushedonly slightly away from x = 0: writing ρ = ρ sin − δρ and expanding in δρ √ k we find δρ = 4 log Λ p k/ ! . (4.4)As long as Λ is not exponentially large in some power of k , the singularity appears toinflate by a stringy distance with a mild logarithmic dependence on the cutoff.This is reminiscent of the root mean square variation of the transverse directions instring theory [24] h (∆ X ) i = α ′ X n> n (4.5)where the sum is over the stringy modes. This sum diverges, but a finite resolution of themeasuring device gives h (∆ X ) i = 2 log( n max ) . (4.6) n max plays the role of the cutoff and the factor of 2 is due to the fact that we work with α ′ = 2. Since the mass of the string at level n is M = n α ′ it is natural to relate n max toΛ and so there is an agreement between (4.6) and (4.4). It would be nice if this technicalagreement could shed light on the origin of our results; perhaps in relation with [25–27]in which it was speculated that (4.6) might play an important role in BH physics.To conclude experimentally that the potential is large at macroscopic distances awayfrom the classical singularity, the external observer should probe the BH with exponen-tially large energies. Since we work with g s = 0, these energies are still negligible comparedto the mass of the BH. This suggests that small but finite g s could completely modify thepicture. A finite g s does not induce a cutoff, but it does modify dramatically the physicsat energies of the order of 1 /g s [28]. We are, unfortunately, in no position to comment onthat. We have just concluded that an external observer will have to reach exponentially largeenergies in order to conclude that there is structure just behind the horizon. Does this13ean that an infalling observer with a smooth wave function is not sensitive to thisstructure and can fall freely through it?The fact that V ( x ) oscillates wildly seems to support this possibility. If, for example,we average V ( x ) with a Gaussian wave function with some width ∆ V ( x, ∆) = 1∆ p π/ Z dx ′ V ( x ′ ) e − ( x − x ′ )2∆2 , (4.7)then it is easy to see that despite the fact that the amplitude of V grows faster than itsfrequency, V ( x, ∆) does not blow up at the horizon.However, x is not the coordinate associated with an infalling observer. Near thehorizon, the infalling observer coordinates are U = √ ku, V = √ kv, (4.8)so that at the horizon we have ds = dU dV . Moreover, instead of V ( x ) we should consider D ( x ) that is more closely related to the tidal forces an infalling observer would experience.In terms of these coordinates we have D ( U, V ) ∼ k / (cid:18) U V k (cid:19) − k sin (cid:18) U V k (cid:19) − k ! . (4.9)Since D ( U, V ) blows up at finite values of U and V (at V, U = 0) smearing it with awave function that is natural for an infalling observer does not wash away the singularityat the horizon. We conclude that the fact there is a non-trivial structure just behind thehorizon is something an infalling observer experiences.
In this note we showed how data that is available to an external observer – the reflectioncoefficient – can be used to calculate the potential behind the horizon of an EBH. Asexpected in perturbative string theory, the potential is small and smooth at the horizonand it blows up only at the singularity. However, the exact reflection coefficient that isknown for the SL (2 , R ) k /U (1) EBH appears to suggest that the region just behind thehorizon is singular.At first sight this conclusion appears to be too dramatic since we considered classicalstring theory ( g s = 0) while previous arguments for structure at the horizon are quantum14echanical [4–9]. However, [4–9] merely state what should happen for the informationto be emitted in the radiation: the Hawking particles must be on-shell very close tothe horizon [4, 9] and not just at infinity. Or alternatively, the Hawking particle and itspartner cannot form a pure state [5–8]. None of these papers, however, explain how thiscomes about. In other words, they do not find a mistake in Hawking’s derivation [32]that the information is lost. Rather they argue which of his conclusions must be wrongfor the information to be recovered. It is possible, we believe, that the results presentedhere could fill in this gap. Namely, the classical non-perturbative stringy effects are theseeds for the quantum effects discussed in [4–9].For this to happen we should be able to answer the following question: what is theorigin, in classical string theory, of the structure just behind the horizon? Simply put,how come that in classical string theory the EBH horizon is not smooth? Currently we donot have an answer to this question. However, we would like to point out that a relatedeffect occurs in the Euclidean version of the BH – the cigar geometry [18, 19]. There it isbelieved that the source is the condensation of the winding tachyon [29–31]. It is not clearto us what is the Lorentzian analogue of the winding tachyon. This, we think, is likely tobe a key ingredient for improving the understanding of the results presented here.We wish to end by spelling out the various assumptions made in reaching the conclusionthat the horizon is singular: • We assumed that the exact reflection coefficient is given by (3.11). The originalcalculation of Teschner was done for H +3 [3] and simple manipulations (gauging and Wickrotation) give (3.11). The results of [3] (before and after gauging) were rederived usingother methods (see e.g. [21–23]). • We assumed that T ( p ) → SL (2 , R ) k /U (1) EBH non-perturbative effects in classical stringtheory have dramatic effects. • We assumed a Schr¨odinger-like setup or equivalently a Klein-Gordon equation insome background and studied if this background is regular at the horizon. The surprisingresults we encountered come from the UV and it is natural to wonder if the Klein-Gordonequation is the right equation of motion to use. In particular, the effects are due to scalessuch that p ≫ k (see (3.16)) and at such high scales other terms might be important. Thefact that in perturbative string theory a Klein-Gordon equation in a regular backgroundgives the correct reflection coefficient [14] does not appear to support this possibility.15either does the Euclidean setup [19]. Nevertheless, it is clearly worthwhile to explorethe possibility that corrections to the Klein-Gordon equation at the UV can render thehorizon smooth. • A related assumption is that the relation between the tortoise and Kruskal coordi-nates is not modified at the deep UV. This, we believe, is the same as assuming that theequivalence principle holds.
Acknowledgments
We thank D. Kutasov for discussions. This work is supported in part by the I-COREProgram of the Planning and Budgeting Committee and the Israel Science Foundation(Center No. 1937/12), and by a center of excellence supported by the Israel ScienceFoundation (grant number 1989/14). LL is thankful for the support from the AlexanderZaks fellowship.
A Schwarzschild BH Reflection Coefficient
In this appendix we compute the high energy behavior of the reflection coefficient in thecase of a Schwarzschild BH and show that it obeys (2.4). As mentioned above, whenput in terms of the tortoise coordinate, the Klein-Gordon equation can be reduced to aSchr¨odinger-like equation. For the Schwarzschild metric, the tortoise coordinate is r ∗ = r + 2 M log (cid:16) r M − (cid:17) , (A.1)where r is the radial coordinate and M is the BH mass. The potential reads (see e.g. [33]) V ( r ) = (cid:18) − Mr (cid:19) (cid:18) Mr + ℓ ( ℓ + 1) r (cid:19) , (A.2)where ℓ is the angular momentum and r is understood to be an implicit function of r ∗ .Since we are only interested in high energy behavior, we exploit the complete anal-ogy with quantum mechanics and use the Born approximation in which the reflectioncoefficient reads R ( p ) = 12 i p Z ∞−∞ V ( r ) e − i p r ∗ dr ∗ . (A.3)16or the potential (A.2), this has a closed form expression, R ( p ) = − Γ( − iM p ) (cid:20) iM p Γ(4 iM p, iM p ) + (4 iM p ) iMp e − iMp (cid:18)
12 + ℓ ( ℓ + 1) (cid:19)(cid:21) , (A.4)where Γ( a, z ) is the incomplete gamma function. At high energies this behaves as R ( p ) ∼ − iπ e − β p , (A.5)when written in terms of the temperature, β = 8 πM . We see that, indeed, this obeys(2.4).We note that an expression for the high frequency behavior of the reflection coefficientwas already given in [34]. However, there, the solutions to the Klein-Gordon equationare given in terms of r (and not r ∗ ). Additionally, the asymptotic functions according towhich the reflection coefficient is defined, were chosen to be exp [ ± ip ( r + 2 M log (2 pr ))].This choice gives a reflection coefficient that differs from (A.5). However, upon makingthe switch to the tortoise coordinate (A.1) and the consequent redefinition of the reflectioncoefficient, one gets an expression which is in agreement with (A.5). References [1] D. J. Gross and E. Witten, “Superstring Modifications of Einstein’s Equations,”Nucl. Phys. 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