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(Thermo)Dynamics of D-Brane Probes

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(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-Nordstr¨om 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-Nordstr¨om 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-Nordstr¨om (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. 2 Trieste Meeting of the TMR Network on Physics beyond the SM1 E. Kiritsis∗,T.Taylor# 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 bro- ken.
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