(Thermo)Dynamics of D-Brane Probes

Trieste Meeting of the TMR Network on Physics beyond the SM1

PROCEEDINGS

(Thermo)dynamics of D-brane probes

E. Kiritsis∗,T.Taylor# ∗ Physics Department, University of Crete, 71003, Heraklion, GREECE # Physics Department, Northeastern University, Boston, MA 02115, USA E-mail: [email protected], [email protected]

Abstract: We discuss the dynamics and thermodynamics of particle and D-brane probes moving in non-extremal black hole/brane backgrounds. When a probe falls from asymptotic infinity to the horizon, it transforms its potential energy into heat, TdS, which is absorbed by the black hole in a way consistent with the first law of thermodynamics. We show that the same remains true in the near-horizon limit, for BPS probes only, with the BPS probe moving from AdS infinity to the horizon. This is a quantitative indication that the brane-probe reaching the horizon corresponds to thermalization in gauge theory. It is shown that this relation provides a way to reliably compute the entropy away from the extremal limit (towards the Schwarzschild limit).

2 1. Introduction and conclusions Hooft coupling gYMN. The SYM/AdS correspondence and its ther- The black D-brane solutions of type II super- mal black hole generalizations emerge in a par- gravity [1] and their near-horizon geometry [2] ticular limit of N D-branes coinciding at one are the central elements of CFT/Anti-deSitter point in the transverse space; this corresponds (AdS) correspondence [3, 4, 5]. In particular, to a conformal point in the SYM moduli space, the (3+1)-dimensional world-volume of N coin- with zero vacuum expectation values (vevs) of N ciding, extremal D3-branes is the arena of =4 all scalar fields. Before taking the near-horizon supersymmetric SU(N) Yang-Mills (SYM) the- limit one could also consider some other configu- ory which in the large N limit, according to the rations, obtained for instance by placing a num- Maldacena conjecture [3], is linked to type IIB ber of D-branes in the bulk of AdS; this corre- superstrings propagating on (the near-horizon) sponds to switching on some scalar vevs on the × 5 AdS5 S background geometry. Recently, there Higgs branch of the gauge theory. In this work, has been an interesting proposal [6] linking the we consider the case of a single D-brane probe N thermodynamics of large N, =4 supersymmet- in the background of a near-extremal black hole ric SU(N) Yang-Mills theory with the thermo- with a large number of coinciding D-branes. We dynamics of Schwarzschild black holes embed- consider a probe moving from asymptotic infin- ded in the AdS space [7]. The classical geom- ity towards the black hole horizon. As the probe etry of black holes with Hawking temperature moves through the horizon, the black hole re- T encodes the magnetic confinement, mass gap ceives the quantity of heat that is determined and other qualitative features of large N gauge by the first law of thermodynamics. The cor- theory heated up to the same temperature. At responding change in black hole entropy is con- the computational level, the quantity that has sistent with thermodynamical identities. Hence been discussed to the largest extent [8, 6, 9] is from the thermodynamical point of view, the D- the Bekenstein-Hawking entropy which, in the brane gas is physically located at the black hole near-horizon limit, should be related to the en- horizon. tropy of Yang-Mills gas at N →∞and large ’t We can then consider a similar process in ∗Talk presented by the second author. the near-horizon limit description. That is, a Trieste Meeting of the TMR Network on Physics beyond the SM1 E. Kiritsis∗,T.Taylor# probe brane is falling from the boundary of the junction with entropy calculations, done near the appropriate space (AdS5 for D3-branes) rather limit where supersymmetry is restored (and thus than from spatial infinity, to the horizon. We the calculation is reliable) and then extrapolated find again, for BPS probes, that the potential to the Schwarzschild region where supersymme- energy released is equal to the heat, TdS,ab- try is completely broken and calculations are dif- sorbed by the black hole. Thus, this process can ficult to control. be used to calculate the entropy of gauge theory This talk is based on results obtained in col- in an alternative way by integrating back the po- laboration with Constantin Bachas [12]. We will tential energy. Moreover, this process has now first describe in detail the case of Reissner-Nordström an interpretation in terms of the spontaneously black holes in four dimensions and their near- broken SU(N +1)→SU(N) × U(1) gauge the- horizon limit. Then we will describe arbitrary ory. At T = 0, any Higgs expectation value is black Dp-branes and finally focus on the near- stable, since it is a modulus protected by su- horizon limit of D3-branes. persymmetry. Once T>0, supersymmetry is broken, and the Higgs acquires a potential with 2. Reissner-Nordström black holes an absolute minimum at zero expectation value. Starting with the theory at the top of the po- In order to see how probes can be used to study tential (very large expectation value) the Higgs black hole thermodynamics, it is instructive to will start rolling down. The Higgs field reaching consider first the case of a point particle propa- an expectation value of the order of the electric gating in the background geometry of a charged mass corresponds in supergravity to the probe black hole. The Reissner-Nordström (RN) met- reaching the horizon. At this point, the energy ric for a charged black hole with ADM mass M and electric charge Q reads stored in the Higgs field is thermalized and equal ! 2 2 2 2 to the overall heat received by the thermal Yang- 2lpM lpQ 2lpM ds2 = − 1 − + dt2+ 1 − + Mills system. This makes more precise a similar r r2 r picture described in [10]. From the supergrav- !−1 2 2 ity point of view the brane continues to move, lpQ + dr2 + r2dΩ2 (2.1) crossing the horizon. It is not obvious what this r2 2 motion corresponds to in gauge theory (but see 2 where GN = l is Newton’s constant. There is [10] for a proposal). p also a static electromagnetic potential, which can We consider also the effects of higher order Q be obtained from Gauss’ law, A0 = .Theouter α0 corrections to the probe action. Such correc- r and inner horizons are located at r+ and r− with tions have been partially computed for the appro- q 2 2 − 2 2 priate backgrounds directly in the near-horizon r = lp M M Q /lp . The standard limit [11]. We argue that that although there are expression for the Bekenstein-Hawking entropy, 0 2 α corrections to the D-brane world-volume ac- Area πr+ S(M,Q)= = 2 , (2.2) tion, they do not influence the calculation of the 4GN lp heat in our argument. can be thought of as the ‘equation of state’ in the In all of the examples analyzed here, the microcanonical ensemble. From it we can obtain probe method can be viewed as an independent the temperature and ‘chemical potential’ of the method of “measuring the entropy” by integrat- black hole ing the heat, TdS. In particular, it provides a 2 1 ∂S 4πr+ ∂S Q first order partial differential equation for the en- ≡ = ,µ≡−T = . 0 T ∂M r+ − r− ∂Q r+ tropy in terms of mass M and charge Q. α - Q M corrections do not modify the leading equation. (2.3) The only source of stringy corrections to the en- The free energy and grand canonical potential tropy equation comes from bulk α0-corrections can also be obtained by the standard thermo- to the RR gauge potential at the horizon. This dynamic expressions, F ≡ M − TS and A ≡ equation is important since it can be used in con- M − TS−µQ.

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Consider next the process in which a probe This partial diﬀerential equation can be integrated particle with mass m and electric charge q falls for the equation of state S(M,Q), provided we inside the black hole. The black hole plus par- know already the answer on some (initial) curve ticle form an isolated system, with total mass in the (M,Q) plane, such as for Schwarzschild M + m and total charge Q + q, which will even- black holes Q = 0 or extremal black holes, M = tually reach thermal equilibrium. The entropy Q/lp. Notice that even though the thermody- will therefore change by an amount namic properties of a Schwarzschild black hole

are in an essential way quantum (¯h enters in the ∂S ∂S δS = m + q. (2.4) expressions for both temperature and entropy), ∂M ∂Q Q M the extension to charged black holes follows from the simple classical argument outlined here. Such Using the explicit form of the chemical potential, a differential equation may turn out to be prac- derived from the ‘equation of state’, we find tically important. Several thermodynamic prop- Qq TδS = m− . (2.5) erties seem to be more easily computable on or r+ close to the extremal limit. There, supersymme- This equation admits a simple interpretation as try is of help as evidenced by recent D-brane/black the heat released by the probe particle while falling hole calculations [13]. Being able to calculate at inside the black hole. The action for such a par- the extremal boundary of the (M,Q) plane, one ticle indeed reads can use (2.9) to extrapolate the calculation to Z Z p the whole plane, most importantly in the region µ ν µ Γ=m dτ −Gµν x˙ x˙ + q dτAµx˙ . (2.6) Q = 0 where supersymmetry is completely broken. It should be noted though that imposing In the static gauge, t = τ, it takes the form a boundary condition at the extremal boundary Z M = Q maybe problematic since in most cases Γ= dt V (r) + velocity terms (2.7) the partial derivatives diverge there (T =0). This is the case here as well as for the Dp-branes with V the static potential, with p<5. We could however impose bound- s 2 2 2 ary conditions on a line just outside the extremal 2lpM lpQ qQ V (r)=m 1− + + . (2.8) boundary (at near-extremality) where computa- r r2 r tions are still reliable. This point is conceptually Notice that the potential includes the self-energy important and needs further investigation. m of the probe, and is constant in the extremal The reader may of course object that these limit for both source (M = Q/lp) and probe considerations depend on our choice of a minimal (m = q/lp), consistently with the absence of a probe action. Quantum gravity effects or stringy static force in this case. Now as the particle effects (controlled respectively by lp and ls)can moves from spatial infinity to the outer horizon, give rise to curvature terms and/or non-minimal the difference in potential energy, δV = V (∞) − electromagnetic couplings, which would modify V (r+), is converted to kinetic energy and then the potential (2.8). Nevertheless, assuming ther- eventually dissipated as heat. This is precisely modynamic equilibrium, the relation the content of eq. (2.5). The argument can also be run backwards. TδS = V(∞)−V(r+)=m−qA0(r+) (2.10) Starting from the static potential eq. (2.8), and assuming that the potential energy of the probe continues, as we will now argue, to be valid. This is converted to heat at the outer horizon, leads to is of course consistent with the fact that for a eq. (2.5). Comparing with eq. (2.4) then gives neutral particle TδS = m is simply the first law

of thermodynamics. Q ∂S ∂S µ = A0(r+)= ⇐⇒ − = A0(r+) . To see why the potential diﬀerence is always r+ ∂Q ∂M M Q given by eq.(2.10), consider possible corrections (2.9) to the action Γ of the particle. These must be of

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µ the form and so on. Notice that Ωn are tensors both of the Z target space diffeomorphisms as well as world- δΓ= ds f(R, F, x˙) , (2.11) line reparametrizations. The piece of the acceleration that contributes to the potential is p ≡ µ with ds dτ x˙ x˙ µ the invariant proper time µ 1 G00,0 µ µ Ω2 pot = − δ0 +Γ00 (2.14) element, and f a scalar made out of the electro- 2 G00 magnetic field strength, the Riemann tensor, the This in principle can give singular contributions probe velocity, and covariant derivativesR thereof. µ µ on the horizons via invariants of the form Gµν Ωm Note that corrections of the form dx Aµf˜ are Ων . We can show however that in the usual on- not allowed, because gauge invariance forces the n shell perturbative derivation of the higher α0 cor- scalar function f˜ to be a constant.1 Now in the rections such invariants cannot appear. The rea- quasi-static limit xµ =(τ,0···0), the invariant son is that the first non-trivial correction is com- element ds tends to dt at spatial infinity, and puted by matching on-shell some scattering am- vanishes at the horizon, when expressed in terms plitude involving the particle with a combination of the asymptotic time. The scalar f on the of higher velocity or acceleration terms. How- other hand must vanish at spatial infinity where µ µ ν ever, on-shell Ω ∼ F ν x˙ .Thus,theΩ2is re- both F and R go to zero. Thus, provided f stays 2 dundant and does not appear on-shell in the first smooth at the event horizon, these higher-order correction terms. Consequently the corrected ac- terms will not contribute to the static potential tion involves only velocities, and the corrected either at r = r+ or at r = ∞, as advertised. equations will equate Ω2 to velocities again. Thus, To show that f cannot indeed diverge at the to any finite order of perturbation theory, we do horizon, note first that this is automatic for a not have the dangerous acceleration terms. This scalar function of the background fields, since the is supported by a recent calculation of (part of) horizon singularity is a coordinate artifact. Sup- the O(α02) terms for D0-branes [11]. pose next that f is the pull back on the world-line The upshot of the previous argument is that of a space-time tensor the right-hand side of eq. (2.10) is universal, and n dxµ1 dxµn dt hence so is the differential equation (2.9) which f = Tµ ···µ ··· = T0···0 + ··· 1 n ds ds ds one can integrate for the equation of state. One (2.12) immediate corollary, assuming the entropy stays with T a function of F , R and their covariant smooth in the extremal limit where T =0,is derivatives and the dots stand for kinetic terms. that A0(r+) =1/lp always. dt 2 00 extremal Near the horizon ( ds ) = G diverges. Never- There is, to be sure, still a lot of room for theless f must remain regular, or else the scalar string or quantum-gravity corrections to the ther- µ1 ···µn invariant Tµ1 ···µn T could not possibly be modynamic functions. Both the chemical po- finite. tential, A0(r+), away from extremality, and the The only remaining possibility is that f de- equation of state for, say, neutral holes S(M,0), pends on extrinsic invariants, or on other higher- are expected in general to receive such correc- derivative terms of the coordinate functions. They tions. It is also conceivable that, like finite-size are constructed by using the covariant acceler- effects, string and/or quantum gravity correc- µ ations Ωn.Forn= 1, this is the usual four- tions invalidate our thermodynamic treatment of µ µ velocity, Ω1 =˙x.Forn= 2, this is the acceler- the problem. ation

µ µ 1 ν ρ µ µ ν ρ 3. Near-extremal near-horizon limit Ω =¨x − ∂τlog(Gνρx˙ x˙ )˙x +Γνρx˙ x˙ 2 2 (2.13) To discard quantum gravity eﬀects we will now assume that lp is vanishingly small compared to 1This does not imply that the electromagnetic coupling must be minimal, since (2.11) may depend non- all other length scales in the problem. We will trivially on the ﬁeld strength. furthermore take the near-extremal limit, and

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3 concentrate on the near-horizon geometry of the Q Q 2 4 3 2 3 U = + (2πT) lp +2Q (2πT) lp + O(lp) black hole, lp 2 (3.9) p ∼ + − − + 1 2 2 2 2 l δr (r r ) r . (3.1) Φ= −2πTQ − 6π Q T lp + O(lp) (3.10) lp In order to analyze this limit, it is convenient to Note here that the leading contributions to U define the new coordinates and Φ are singular. This will also be the case for Dp-branes. ≡ 2 ≡2 ˜ r lpM (1 + u)andt lpMt, (3.2) The static potential of a point probe reads in this limit [16] in terms of which the metric reads q 2 q − q 2− 2 2 −2 2 f(u) 2 1 V (u)= u+m u u0 . (3.11) (lpM) ds = − dt˜ + 1+ × lp lp 1 2 u 1+ u For a generic probe m>q/lp, so that the poten- du2 tial grows linearly at the spatial infinity of AdS2 × + u2dΩ2 , (3.3) f(u) 2 space. For an extremal probe, on the other hand, the potential goes to a constant at infinity, and where the potential difference δV = V (∞) − V (u0)= q 0 u 2 2 l u0 is well defined. We can therefore use our f(u)=1−( ) and r ≡ lpM (1u0) . (3.4) p u thermodynamic argument to derive the expres- The outer and inner horizons are located in the sion for the chemical potential new coordinates at u = u0, while the electric 1 − u0 µ = , (3.12) potential takes the form lp p 2 2 −1 1 − u0 in agreement with the near-extremal expansion A0 ≡ (lpM) A˜0 = . (3.5) lp(1 + u) of the electrostatic potential at the horizon µ = A0(u0). Note that, as with most other thermody- Finally the potential of a point probe can be namic quantities, one has to keep the first sub- worked out easily with the result leading correction in the near-extremal expan- p p sion of µ, in order to find the leading temperature u2−u2 q 1−u2 V (u)=m 0 + 0 . (3.6) dependence. 1+u lp 1+u A heuristic rephrasing of the main message of Consider now the limit (3.1) which can be this section is as follows: a near-extremal black −1 hole exerts no net force on an extremal probe written equivalently as (lpM) u, u0 1. In this limit the metric simplifies to at long distance. The potential energy, is thus converted to kinetic energy and eventually re- du2 leased as heat while the probe falls in the near- (l2M)−2 ds2 = −f(u)u2 dt˜2 + + dΩ2 . p f(u)u2 2 horizon geometry. To leading order in the ex- (3.7) tremality parameter one can therefore compute 2 The extremal case ( f =1)givestheAdS2 × S the chemical potential by ignoring the physics in space, also known as the Bertotti-Robertson uni- the asymptotically-flat region. verse. For finite u0, on the other hand, one has a two-dimensional black hole embedded in this 4. Black Dp-branes asymptotic geometry.2 For the thermodynamic 3 quantities, we obtain We consider now the background geometry (in the string frame) of a near-extremal black hole 2 2 3 O 2 S = πQ +4π Q Tlp + (lp) (3.8) describing a number of coinciding Dp-branes [1]: 2This solution is different from the one discussed in q − 2 · 2 [14]. 2 f(r)pdt + d~x d~x dr 3 ds10 = + Hp(r) + A similar result was obtained in [15]. Hp(r) f(r)

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2 2 (7−p)/2 +r dΩ8−p (4.1) L Φ=VpTp q (4.8) 7−p 7−p 7−p 7−p L r0 r0 + L Hp(r)=1+ ,f(r)=1− (4.2) q r7−p r7−p 4πΩ8−pVp (9−p)/2 7−p 7−p The parameters L and r0 determine the AdS S = 2 r0 r0 + L , (4.9) 2κ10 throat size and the position of horizon, respec- It is easy to check that these quantities sat- tively. They are related to the ADM mass M isfy the thermodynamic identitydU = TdS + and the (integer) Ramond-Ramond charge N in ΦdN with U = M. the following way: We consider now a Dp-brane probing the above h i Ω8−pVp 7−p 7−p solution, with zero background values for all other M = 2 (8 − p)r0 +(7−p)L 2κ10 fields. In this case, the D-brane probe action is q Z Z − p (7 p)Ω8−p (7−p)/2 7−p 7−p −φ ˆ N = 2 L r0 + L (4.3) Γp = Tp e detg ˆ + Tp C (4.10) 2κ10Tp where Ωn is the volume of a unit n-dimensional wherewehavealsosettheworld-volumeFαβ = sphere, and Vp is the common p-dimensional D- 0. Using the solution above we obtain the static brane (flat) volume. The relations (4.3) involve potential [17] the D-brane tension Tp and the 10-dimensional "p # "p gravitational constant κ10. The RR charge N f(r) f(r) V (r)=VpTp +C(r) =VpTp + is quantized, with each D-brane carrying a unit Hp(r) Hp(r) charge so that N is equal to the number of D- s  7−p − branes. Finally, r0 Hp(r) 1 s + 1+ 7−p (4.11) L Hp(r) 2 2 7−p 2κ10TpN 1 2(7−p) − 1 7−p L = + r0 r0 ≡ (7 − p)Ω8−p 4 2 where C(r) C012···p(r). Note that, like in the (4.4) RN case, the horizon value C(r0) is equal to the The RR charge is the source of the p-form chemical potential Φ. The values of the potential field at infinity and at the horizon are, respectively, 2 V (∞)=VpTp and V (r0)=Φ. 2κ10TpN C012···p(r)= 7−p 7−p = ADp-brane probe is a BPS state with the Ω8−p(7 − p)(r + L ) s mass ∆M = VpTp and charge ∆N =1.Asit 7−p − moves from infinity to the horizon, its potential r0 Hp(r) 1 − ∞ = 1+ 7−p . (4.5) energy changes by ∆E = V (r0) V ( ), and L Hp(r) the quantity of heat received by the black hole is All other components vanish, except in the case again dE = −∆E. On the other hand, the black of p = 3, when the self-duality condition hole gains mass dM =∆M=VpTp and charge

1 ν1···ν5 dN =∆N= 1. This process is described by the Fµ ···µ = √ µ ···µ ν ···ν F (4.6) 1 5 5! det g 1 5 1 5 equation requires non-zero p-form components in the trans- dE = V (∞) − V (r0)=dM − ΦdN = verse directions. There is also a dilaton back- φ (3−p)/4 − ground (constant for p =3): e =Hp (r). = dU ΦdN = TdS (4.12) By using standard methods of black hole ther- which does indeed hold. As expected, the probe modynamics, it is straightforward to determine motion, governed by the background ﬁeld ac- the Hawking temperature, entropy and chemical tion of eq.(4.10), is consistent with black hole potential corresponding to the solution (4.1,4.5). thermodynamics and provides a similar partial They are respectively: equation of state as in the RN case. The ar- 0 (5−p)/2 gument concerning the absence of α corrections 7 − p r T = q 0 (4.7) here is more involved. Unlike the case of one- 4π 7−p 7−p r0 + L dimensional world-volumes, here the equations

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of motion do not set the accelerations (second 2 2 4 +R dΩ5 + O(ls ) , (5.1) fundamental forms) to be a function of velocities [11]. At this point we can argue that to order where O 02 4 (α ) there are no corrections using the explicit u0 4 f(u)=1− ,R≡4πgsN = λ (5.2) results of [11]. It turns out that the second fun- u4 damental form enters in such a way that there are 2 u0 = πTR , (5.3) no extra corrections again on the horizon. More- over the appearance of a non-zero five form is where λ is the t’Hooft coupling. The limiting value of the four-form (4.5) is not expected to change the previous statement. 4 4 Here we must also investigate the higher anoma- 4 (πTR) u 8 C0123 =1+l − + O(l ), (5.4) lous CP-odd couplings. The relevant case is that s 2 R4 s of D3 branes and the coupling As in the near-horizon limit of the RN black-hole, Z (4πα0)2 here also the gauge field diverges at the boundary Sθ = −T3 a [p1(T ) − p1(N )] (4.13) →∞ 48 of AdS5, u . A D3-brane probe in the bulk of the AdS where p1 denotes the first Pontriagin class and space corresponding to N background D3-branes T , N stand for the tangent and normal bundles can be thought of as a realization of SU(N +1) to the brane respectively It can be shown that gauge theory in the SU(N) × U(1) symmetric this vanishes for diagonal metrics with the re- Higgs phase. In the following, we examine some quired Poincaré symmetry. Thus, there is no aspects of the probe dynamics and thermody- 02 correction to the heat up to order O(α )and namics in the near-horizon limit, in order to show we suspect that this is true to all orders. This that it is well-defined and it leads to sensible re- would imply again that all corrections come from sults also in the Higgs phase. To that end, we will the corrected background fields. use the following expansions in the string length scale ls: 5. D3-branes in the near-horizon limit 4 4 4 1 4 4 4 4 12 L = R ls 1 − π R T ls + O(ls ) (5.5) 2 The case of D3-branes is particularly interesting 2 2 1 4 4 4 4 8 because the world-volume action of N coincid- r0 = πTR ls 1+ π T R ls +O(ls) (5.6) 4 ing D-branes involves a four-dimensional N =4 which follow from relations written in the previ- supersymmetric SU(N) Yang-Mills theory. Ac- ous section. Similarly, cording to the Maldacena conjecture [3], the large N limit of this gauge theory is related to the near- 3 2 2 4 4 M = NV3T3 + π V3N T +O(ls) (5.7) horizon AdS geometry of the extremal (r0 =0) 8 black D3-brane solution (4.1). Witten [6] has ex- 1 2 2 3 4 S = π V3N T + O(ls) (5.8) ploited the AdS/SYM correspondence in order to 2 study the large N dynamics of non-supersymmetric 1 2 4 4 Φ=V3T3− π V3NT +O(ls) (5.9) SYM, with N =4 supersymmetries broken by non- 4 ∼ 4 →∞ zero temperature effects. According to this pro- Note that T3 1/ls . As pointed out be- posal, the non-extremal solution (4.1) may be fore in ref.[9], the limiting entropy (5.8) is 3/4 of used to study SYM at T identified with the Hawk- the corresponding quantity in the weakly coupled ing temperature (4.8) as long as T 1/L,so SU(N)SYM. that the metric remains near-extremal (r0 L). Taking the limit in the static potential (4.11), 0 2 0 we obtain4 In the Maldacena limit, α ≡ ls → 0atu≡r/α   s and T fixed, the solution (4.1) describes an AdS-  4 2 4 4 u  πTR Schwarzschild black hole [7]: V (u)=V3T3 1+ls 1− − 1+  R4 u 2 2 4 2 2 u 2 2 du We disagree with the potential obtained in the near- ds = ls (−f(u)dt + d~x · d~x)+R + R2 u2f(u) horizon limit in [18].

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# ) 2 4 1 πTR 8 References + + O(ls) (5.10) 2 u [1] G.T. Horowitz and A. Strominger, Nucl. Phys. so that the interaction energy is B 360 (1991) 197.

4 [2] G.W. Gibbons and P.K. Townsend, Phys. Rev. int − V3 u × Lett. 71 V (u)=V(u) V3T3 = 3 4 (1993) 3754. (2π) gs R   [3] J. Maldacena, hep-th/9711200. s 2 4 2 4 [4] S.S. Gubser, I.R. Klebanov and A.M. Polyakov,  πTR 1 πTR  4 × 1 − − 1+ +O(ls) hep-th/9802109. u 2 u [5] E. Witten, hep-th/9802150. (5.11) [6] E. Witten, hep-th/9803131. and has a smooth limit as ls → 0. Since the probe is BPS, V int(∞) = 0 as in the RN case. [7] S.W. Hawking and D. Page, Comm. Math. The thermodynamic argument is still valid. Phys. 87 (1983) 577. We let the probe fall until it reaches the horizon. [8] S.S. Gubser, I.R. Klebanov and A.W. Peet, The heat supplied by the black hole to the probe Phys. Rev. D54(1996) 3915. is [9] S.S. Gubser, I.R. Klebanov and A.A. Tseytlin, hep-th/9805156. dE = V (∞) − V (u0)=V3T3−V(u0)= [10] T. Banks, M. Douglas, G. Horowitz and E. Mar-

1 2 4 tinec, hep-th/9808016 =∆M−Φ∆N = π V3NT (5.12) 4 V. Balasubramanian, P. Kraus, A. Lawrence, S. Trivedi, hep-th/9808017. where ∆N = 1. On the other hand, taking the limit of the equations of the previous sec- [11] C. P. Bachas, P. Bain, M. B. Green, hep- tion we obtain that the black hole cools down by th/9903210. dT = −T/(2N) which is sufficient to prove di- [12] C. P. Bachas, E. Kiritsis and T.R. Taylor, in rectly that dE = TdS, with the entropy given preparation. by eq.(5.8). We thus find as before consistent [13] A. Strominger and C. Vafa, hep-th/9601029, thermodynamics, when we allow the probe brane Phys. Lett. B 379 (1996) 99; to move from the boundary of AdS space to the C. G. Callan and J. Maldacena, hep- B472 horizon. This (gravitational) equality corresponds th/9602043, Nucl. Phys. (1996) 591; to the qualitative expectation that an ultraviolet A. Dhar, G. Mandal and S. R. Wadia, hep- th/9605234, Phys. Lett. B388 (1996) 51; fluctuation in N=4 SYM spreads until it reaches S. R. Das, and S. D. Mathur, hep-th/9606185, the size of the thermal wavelength and thus ther- Nucl. Phys. B478 (1996) 561; hep-th/9607149, malizes [10]. Nucl. Phys. B482 (1996) 153; J. Maldacena and A. Strominger, hep- Acknowledgments th/9609026, Phys. Rev. D55 (1997) 861; hep-th/9702015, Phys. Rev. D56 (1997) 4975; We would like to thank C. Bachas for his col- S. F. Hassan and S. R. Wadia, hep-th/9712213, laboration and fruitful discussions. E. Kiritsis Nucl. Phys. B526 (1988) 311. would like to thank the Ecole Polytechnique for [14] A. Strominger, hep-th/9809027. hospitality and support while parts of this work [15] J. Maldacena, J. Michelson and A. Strominger, was done. He is also grateful to the SISSA node JHEP 9902 (1999) 011; hep-th/9812073. for financially supporting his participation to the [16] P. Claus, M. Derix, R. Kallosh, J. Kumar, P. TMR midterm meeting. The work of T.R. Tay- Townsend and A. Van Proyen, hep-th/9804177. lor was done while he was visiting Laboratoire de [17] J. Maldacena, Phys. Rev. D57 (1998) 3736, Physique Théorique et Hautes Energies at Uni- hep-th/9705053. versité de Paris Sud, Orsay. He is grateful to [18] A.A. Tseytlin and S. Yankielowicz, hep- Pierre Binétruy and all members of LPTHE for th/9809032. their kind hospitality and support.