Black Hole Formation at the Correspondence Point
YYITP-13-21
Black Hole Formation at theCorrespondence Point
Norihiro Iizuka ∗ , Daniel Kabat † , Shubho Roy , ‡ and Debajyoti Sarkar , § Yukawa Institute for Theoretical PhysicsKyoto University, Kyoto 606-8502, JAPAN Department of Physics and AstronomyLehman College, City University of New York, Bronx NY 10468, USA Physics DepartmentCity College, City University of New York, New York NY 10031, USA Center for High Energy PhysicsIndian Institute of Science, Bangalore 560012, INDIA Graduate School and University CenterCity University of New York, New York NY 10036, USA
We study the process of bound state formation in a D-brane collision. Weconsider two mechanisms for bound state formation. The first, operativeat weak coupling in the worldvolume gauge theory, is pair creation of W-bosons. The second, operative at strong coupling, corresponds to formationof a large black hole in the dual supergravity. These two processes agreequalitatively at intermediate coupling, in accord with the correspondenceprinciple of Horowitz and Polchinski. We show that the size of the boundstate and timescale for formation of a bound state agree at the correspon-dence point. The timescale involves matching a parametric resonance in thegauge theory to a quasinormal mode in supergravity. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] a r X i v : . [ h e p - t h ] J un Introduction and summary
Understanding black hole microstates from a D-brane or fundamental stringperspective is a long-standing theme in string theory. The original obser-vation that vibrating strings qualitatively resemble a black hole [1, 2] wasfollowed by a quantitative worldvolume derivation of black hole entropy forcertain BPS states [3]. This relationship eventually became a fundamen-tal aspect of the holographic duality between gauge and gravity degrees offreedom [4]. According to this duality, microstates of a black hole are in one-to-one correspondence with microstates of a strongly-coupled gauge theory.This duality also applies to time-dependent processes such as black hole for-mation and evaporation, leading to the viewpoint that these processes shouldbe unitary, contrary to [5].To gain insight into black hole formation, and a better understanding ofthe microstructure of the resulting black hole, in this paper we study theprocess of bound state formation from two perspectives: perturbative gaugetheory and supergravity. In perturbative gauge theory a D-brane bound statecan be formed through a process of open string creation. In supergravity wewill see that open string creation is not possible, and one instead forms abound state through the gravitational or closed-string process of black holeformation.The perturbative gauge theory and supergravity calculations of bound stateformation do not have an overlapping range of validity. But we will show thatthey agree qualitatively at an intermediate value of the coupling, in accordwith the correspondence principle introduced by Horowitz and Polchinski [6].This suggests that there is a smooth transition between the process of openstring creation at weak coupling and black hole formation at strong coupling.As a first test of these ideas, in § § p -branes.Next we consider the time development of the bound states after they haveformed. In § § § § § O ( N ) degrees of freedom are excited [7, 8].There have been many studies of 0-brane black hole microstates from ma-trix quantum mechanics, along with their associated thermalization process.Some previous studies of 0-brane black holes from matrix quantum mechan-ics include [9, 10, 11, 12, 13, 14, 15]. Also see [16, 17] for studies of black holeformation from the gravity perspective, and [18, 19, 20, 21, 22, 23] for stud-ies from the gauge theory perspective. In particular parametric resonancehas been discussed in relation to thermalization in the closely related work[21]. Open string production has been studied as a mechanism for trappingmoduli at enhanced symmetry points in [24], while open string productionin relativistic D-brane collisions has been studied in [25]. Consider colliding two clusters of 0-branes as shown in Fig. 1. We’d liketo understand whether a bound state is formed during the collision. Twomechanisms for bound state formation have been discussed in the literature.1. In a perturbative description of D-brane dynamics, open strings canbe produced and lead to formation of a bound state. This occurs forimpact parameters b (cid:46) √ vα (cid:48) [26]. This can be understood as the2 N v v (cid:0)(cid:1) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6) (cid:6)(cid:7) (cid:8)(cid:8)(cid:9) (cid:10)(cid:11)(cid:12) (cid:12)(cid:13)(cid:14)(cid:14)(cid:15)(cid:16) (cid:16)(cid:17) N Figure 1: Colliding stacks of 0-branes with relative velocity v and impactparameter b .condition for violating the adiabatic approximation. For a review ofthe calculation see appendix A.2. At large N and strong coupling the D-brane system has a dual grav-itational description [27]. In this description, according to the hoopconjecture of Thorne [28, 29, 30], a black hole should form if the twoD-brane clusters are contained within their own Schwarzschild radius.Our goal is to understand in what regimes these two mechanisms for boundstate formation are operative, and whether they are connected in any way.It will be convenient to work in terms of a radial coordinate U with units ofenergy, U = r/α (cid:48) . Here r is the distance between the clusters, r = √ b + v t .The ’t Hooft coupling of the M(atrix) quantum mechanics is λ = g N ,which in string and M-theory units can be expressed as λ = g s N/(cid:96) s = R N/(cid:96) . (1)Here g s is the string coupling, (cid:96) s is the string length, R is the radius of theM-theory circle, and (cid:96) is the M-theory Planck length. The mass of a singleD0-brane is m = 1 g s (cid:96) s = 1 R . (2)
We work in the center of mass frame, with momenta p = N R v p = N R v p + p = 0 (3)3e consider a fixed total energy E , which determines the asymptotic relativevelocity v . 12 N R v + 12 N R v = E , ⇒ v = v − v ∼ (cid:18) N ERN N (cid:19) / = (cid:18) λEl s N N (cid:19) / . (4)In terms of the U coordinate, the asymptotic relative velocity is˙ U = (cid:18) λEN N (cid:19) / . (5)As reviewed in appendix A, open string production sets in when U ∼ (cid:112) ˙ U = (cid:18) λEN N (cid:19) / . (6)Note that the radius at which open strings are produced depends on how wesplit the total D-brane charge. The radius is minimized when N = N = N/
2, which gives the minimum radius for open string production as U ∼ (cid:18) λEN (cid:19) / . (7)This is the case which is interesting for matching to supergravity.There are some checks we should perform to make sure this perturbativeresult is valid. As discussed in [31], the effective action has a double expansionin λ/U and ˙ U /U . The expansion in powers of λ/U is the Yang-Mills loopexpansion, which is valid provided U > λ / . From (7) this requires E > N λ / (8)At the critical point where the loop expansion breaks down, U ∼ λ / , theinequality (8) is saturated.The expansion in powers of ˙ U /U is the derivative expansion, which isvalid when ˙ U < U . Note that the derivative expansion breaks down at thepoint where open strings are produced. Up to this point, i.e. for U > (cid:112) ˙ U ,4ne can trust the two-derivative terms in the effective action, which meansthe asymptotic velocity (4) is a good approximation to the actual velocity. So the only condition for the validity of the perturbative description of openstring production is (8).
At large N the M(atrix) quantum mechanics has a dual gravitational descrip-tion at strong coupling, meaning for U < λ / . So let’s imagine the 0-braneclusters approach to within this distance, and study whether a bound statecan form.At first, one might think a bound state could form via open string pro-duction. As noted in [27], the metric factors cancel out of the Nambu-Gotoaction, and even in the supergravity regime the mass of an open string con-necting the two clusters of D-branes is m W ∼ U . The adiabatic approxima-tion breaks down, and these open strings should be produced, if ˙ U /U > S = 1 g (cid:90) dt U λ − (cid:115) − λ ˙ U U (9)Thus causality bounds the velocity of the probe, λ ˙ U U < . (10)Rather remarkably, the probe has to slow down significantly as U →
0. Inany case, in the supergravity regime we have ˙ U U < U λ , and since U λ < As we will see, this is not the case in the supergravity regime. E is contained within its own Schwarzschildradius. For a 10-dimensional black hole with N units of 0-brane charge, theSchwarzschild radius is U = (cid:18) λ EN (cid:19) / (11)This 10-D supergravity description is only valid if the curvature and stringcoupling are small at the horizon, which requires λ / N − / < U < λ / (12)For smaller U one must lift to M-theory; for larger U the M(atrix) quantummechanics is weakly coupled. At the outer radius where the supergravityapproximation breaks down, U ∼ λ / , eq. (11) tells us that E ∼ N λ / . We’ve found that open string production is only possible at weak coupling,while black hole formation can only occur within the bubble where super-gravity is valid. One could ask if the two phenomena are smoothly connected.Is there a correspondence point where both descriptions are valid?From the perturbative point of view, the transition happens when the con-dition (8) is saturated, E = N λ / . In this case open strings are produced,but at a radius U ∼ λ / where the system is just becoming strongly coupled.From the supergravity point of view, the transition happens when theenergy of the black hole is E = N λ / , corresponding to a Schwarzschildradius U ∼ λ / . In this case the black hole fills the entire region wheresupergravity is valid.This suggests that open string production and black hole formation areindeed continuously connected. Since the transition between the two de-scriptions happens when the curvature at the horizon is of order string scale, α (cid:48) R ∼ ( λ/U ) − / ∼ , (13)this is an example of the correspondence principle of Horowitz and Polchinski[6]. Note that for a given black hole energy, one can view the condition of6eing at the correspondence point, E = N λ / , as fixing the total 0-branecharge, N = (cid:18) E (cid:96) s g s (cid:19) / . (14) p -brane collisions In this section we generalize our 0-brane results and consider D p -braneswrapped on a p -torus of volume V p . We first record some general formu-las then analyze particular cases.The Yang-Mills coupling is g = g s /(cid:96) − ps and the ’t Hooft coupling is λ = g N . In terms of U = r/α (cid:48) , the effective dimensionless ’t Hooftcoupling is λ eff = λU − p . (15)The Yang-Mills theory is weakly coupled when λ eff <
1. It has a dual gravi-tational description at large N when λ eff > p -branes at weak coupling, witha fixed energy density (cid:15) as measured in the Yang-Mills theory. The mass ofa wrapped p -brane is V p /g s (cid:96) p +1 s , so in the center of mass frame the relativevelocity is ˙ U = (cid:18) λ(cid:15)N N (cid:19) / . (16)Open string production sets in when U ∼ (cid:112) ˙ U ∼ (cid:18) λ(cid:15)N N (cid:19) / . (17)The radius at which open strings are produced depends on how we divide thetotal D-brane charge. The radius is minimized by setting N = N = N/ U ∼ (cid:18) λ(cid:15)N (cid:19) / . (18)7his is the case which is interesting for comparison to supergravity.Just as for 0-branes, open string production is not possible in the super-gravity regime. The DBI action for a probe brane is S = 1 g (cid:90) d p +1 x U − p λ − (cid:115) − λ ˙ U U − p (19)Thus the causality bound is ˙ U /U < U − p /λ = 1 /λ eff [32], which rulesout open string production (at least in the probe approximation). Insteadwe have the process of black hole formation, with a horizon radius U =( g (cid:15) ) / (7 − p ) [27].Further analysis depends on the dimension of the branes. p = 0 , , p < U > λ / (3 − p ) and has a dual gravitational description when U < λ / (3 − p ) . Thus openstring production is possible at large distances, while black hole formationis possible at small distances. The correspondence point, where the twodescriptions match on to each other, occurs when (cid:15) = N λ p − p U = λ / (3 − p ) At this energy density open string production occurs just as the Yang-Millstheory is becoming strongly coupled. From the supergravity perspective, theresulting black brane fills the entire region in which supergravity is valid. p = 3In this case the Yang-Mills theory is conformal and dual to AdS × S [4].The ’t Hooft coupling is dimensionless. For λ (cid:46) λ (cid:38) λ = 1. Note that, unlike othervalues of p , the correspondence point is independent of the energy density (cid:15) .As a test of this idea, note that the radius at which open strings form is U = ( λ(cid:15)/N ) / (20)8hile for p = 3 the horizon radius is U = ( g (cid:15) ) / (21)These two expressions for U agree when λ = 1. This suggests that theprocess of open string production for λ (cid:46) λ (cid:38) p = 4 , , p > U > λ / (3 − p ) , while open stringproduction is possible for U < λ / (3 − p ) . The correspondence point where thetwo descriptions match is at (cid:15) = N λ p − p (22) U = λ / (3 − p ) (23) In this section we study the evolution of a bound state formed at weakcoupling by open string creation. We show that the number of open stringsincreases exponentially with time due to a parametric resonance in the gaugetheory. For simplicity we consider 0-brane collisions; the generalization toD p -branes is straightforward and will be mentioned in § N incoming 0-branes collides with a stack of N co-incident 0-branes at rest. We assume weak coupling but do not require large N . In the collision suppose n open strings are produced. These open stringsproduce a linear confining potential, so the system will begin to oscillate.The conserved total energy is E = 12 mv + nτ x (24)Here we’re adopting a non-relativistic description, appropriate to the formof the D0-brane quantum mechanics, while m is the mass of the incoming 0-branes, v is their velocity, n is the number of open strings created, τ = 1 / πα (cid:48)
9s the fundamental string tension, and x is the length of the open strings.The period of oscillation is∆ t = 4 (cid:16) m (cid:17) / (cid:90) E/nτ dx √ E − nτ x ∼ √ mEnτ (25)So up to numerical factors, the frequency of oscillation isΩ = nτ √ mE (26)while the amplitude of oscillation (the maximum value of x ) is L = Enτ (27)We introduce this as a classical M(atrix) background by setting X i = X i cl + x i where X = (cid:18) L sin Ω t N
00 0 (cid:19) X = · · · = X = 0 , (28)We have decomposed the N × N matrix into blocks; N is the N × N unit matrix. Expanding to quadratic order in the fluctuations, the M(atrix)Lagrangian L YM = 12 g Tr (cid:18) ˙ X i ˙ X i + 12 [ X i , X j ][ X i , X j ] (cid:19) (29)reduces to L YM = 12 g Tr (cid:0) ˙ x ˙ x (cid:1) + 12 g (cid:88) i =2 Tr (cid:0) ˙ x i ˙ x i + [ x i , X ][ x i , X ] (cid:1) (30)Note that the potential for x vanishes. We also have the Gauss constraintassociated with setting A = 0, namely (cid:88) i [ X i , ˙ X i ] = 0 (31) We are setting 2 πα (cid:48) = 1 and A = 0.
10o quadratic order this reduces to [ X , ˙ x ] = [ ˙ X , x ] which only constrains x . The simplest solution is to set x = 0.To study the remaining degrees of freedom we decompose x i = (cid:18) a i b i † b i c i (cid:19) (32)where a i is an N × N matrix, b i is an N × N rectangular matrix and c i is an N × N matrix. We will often suppress the index i = 2 , . . . ,
9. Toquadratic order the a and c entries have trivial dynamics, since [ x i , X ] doesnot involve a and c . On the other hand, the equation of motion for b is¨ b + L sin (Ω t ) b = 0 (33)Defining s = Ω t this reduces to Mathieu’s equation, d bds + ( a − q cos 2 s ) b = 0 (34)with the particular values a = 2 q = L / . Mathieu’s equation admitsFloquet solutions b ( t ) = e iγ Ω t P (Ω t ) (35)where P ( · ) is a periodic function with period π . As a function of a and q there are intervals where γ has a negative imaginary part and the solutiongrows exponentially. These intervals correspond to band gaps in the Blochinterpretation of Mathieu’s equation. The imaginary part of γ is plotted asa function of a = 2 q in Fig. 2. There are clearly many intervals where thesolution is unstable, with a typical exponent | Im γ | ∼ . a = 2 q ∼ mE /n (36)After the initial collision the energy E in the oscillating background willdecrease as the system begins to thermalize, while the number n of openstrings gets larger. So we expect the value of a to decrease with time. This Restoring units, we would have L → L τ in (33) and a = 2 q ∼ mE /n τ in (36). a (cid:135) q (cid:160) Im (cid:72) Γ (cid:76)(cid:164) Figure 2: The imaginary part of the Mathieu characteristic exponent as afunction of a = 2 q .means the system will scan across the different instability bands available toit.To summarize, we have found that the oscillating background resultingfrom a 0-brane collision is unstable. The 16 N N real degrees of freedomcontained in b i for i = 2 , . . . , t YM ∼ / Ω ∼ √ mE/nτ (37)Here m is the mass of the N incoming 0-branes, E is the total energy of thesystem, n is the number of open strings present in the off-diagonal block b and τ is the fundamental string tension. We compare the timescale associated with parametric resonance to the quasi-normal modes of a black hole. We consider parametric resonance for D0-branes in § p -branes in § § .1 0-brane parametric resonance As we saw in §
4, the timescale for parametric resonance is determined by theperiod of oscillation. In a 0-brane collision this is given by t YM ∼ / Ω ∼ √ mE/nτ (38)For N incoming D0-branes the mass is m = N /R , where R = g s l s is theradius of the M-theory circle. Also E is the total energy of the system, n is the number of open strings and τ ∼ /l s is string tension. We considerthe case N ∼ N ∼ N , with N large to compare to supergravity. Then theoff-diagonal block b contains O ( N ) elements, so as shown in appendix A O ( N ) open strings are created by parametric resonance.Using R = g s (cid:96) s , τ ∼ /(cid:96) s , n ∼ N and g s ∼ g Y M (cid:96) s we obtain t YM ∼ (cid:114) N ER nτ ∼ √ Eλ / N . (39)At the correspondence point E ∼ N λ / (40)which means t YM ∼ λ − / . (41)At the correspondence point the timescale for parametric resonance is in-dependent of N and is set by the ’t Hooft scale. As we will see in § p -brane parametric resonance It’s straightforward to extend this result to D p -branes. First, the mass of asingle D0-brane in the previous section is replaced by the mass of D p -branewrapped on a volume V p . So we should replace1 /R → V p /g s l p +1 s . (42)13he energy of the incoming D p -branes is related to the energy density (cid:15) by E = (cid:15)V p . (43)The tension of the strings is the same, τ ∼ /(cid:96) s . So for D p -branes, in placeof (38), the oscillation timescale is t YM ∼ √ mEnτ → V p (cid:115) N (cid:15)g s l p +1 s nτ . (44)The number of open strings n is modified. As shown in appendix A, for N ∼ N and p (cid:54) = 3, the number density of open strings at the correspondencepoint is set by the ’t Hooft scale. Thus n ∼ N V p λ p − p . (45)Using this together with g s N = g Y M
N (cid:96) − ps = λ(cid:96) − ps we obtain t YM ∼ V p (cid:115) N (cid:15)g s (cid:96) p +1 s nτ ∼ λ − p − p √ (cid:15)λ / N . (46)From (22) the energy density at the correspondence point is (cid:15) ∼ N λ p − p (47)so the timescale is t YM ∼ λ − − p . (48)Just as for 0-branes, the timescale for parametric resonance is independentof N and set by the ’t Hooft scale.3-branes are a special case since the ’t Hooft coupling is dimensionless.The correspondence point is defined by λ ∼
1. As shown in appendix A, for N ∼ N the number of open strings at the correspondence point is n ∼ N V U (49)where U is the horizon radius of the black brane. The energy density at thecorrespondence point is (cid:15) ∼ N U , so the parametric resonance timescale is t YM ∼ V p (cid:115) N (cid:15)g s (cid:96) p +1 s nτ ∼ U (50)Thus for D3-branes the parametric resonance timescale is 1 /U , which alsohappens to be the inverse temperature of the black brane.14 .3 Comparison to quasinormal modes Quasinormal modes for non-extremal D p -branes were studied in [34, 35] fol-lowing earlier work on AdS-Schwarzschild black holes [36]. The basic idea isto solve the scalar wave equation in the near-horizon geometry of N coinci-dent non-extremal D p -branes, with a Dirichlet boundary condition at infinityand purely ingoing waves at the future horizon. This gives rise to a discreteset of complex quasinormal frequencies, whose imaginary parts govern thedecay of scalar perturbations of the black hole. It was found that the quasi-normal frequencies are proportional to the temperature, with a coefficient ofproportionality that was found numerically in [34].Recall that the temperature, energy density and entropy density of theseblack branes are related to their horizon radius U by [27, 34] T ∼ √ λ U (5 − p ) / (cid:15) ∼ N λ U − p s ∼ N λ / U (9 − p ) / Assuming p (cid:54) = 3, at the correspondence point we have U ∼ λ / (3 − p ) so that T ∼ λ − p (cid:15) ∼ N λ p +13 − p s ∼ N λ p − p These quantities all obey the expected large- N counting, and since the ’tHooft coupling λ has units of (energy) − p , these results could have beenguessed on dimensional grounds. In the special case p = 3 the ’t Hooftcoupling is dimensionless and the correspondence point is defined by λ = 1.At the correspondence point the horizon radius U remains arbitrary, with T = U (cid:15) = N U s = N U Again these results could have been guessed on dimensional grounds.15s we saw in § t YM ∼ (cid:26) λ − / (3 − p ) for p (cid:54) = 31 /U for p = 3 (51)For all p this matches the inverse temperature of the black brane, t YM ∼ /T .Thus at the correspondence point the timescale for parametric resonancematches the timescale for the decay of quasinormal excitations of the blackbrane. It’s interesting to compare the properties of the bound state as initiallyformed to the equilibrium properties of the black hole. This will show usthat, at the correspondence point, very little additional evolution is requiredto reach equilibrium – perhaps just a few e -foldings of parametric resonancewill suffice.First, in a 0-brane collision, note that the total number of open stringsproduced is ∼ N N . With equal charges N = N = N/ O ( N ). At the correspondence point these strings have amass ∼ λ / , so the total energy and entropy in open strings is E ∼ N λ / S ∼ N This matches the equilibrium energy and entropy of the black hole, sug-gesting that black hole formation at the correspondence point is a simpleone-step procedure, in which the open strings that are formed in the initialcollision essentially account for the equilibrium properties of the black hole.The analogous result for p -branes is that the number of open strings at thecorrespondence point is, for p (cid:54) = 3, n ∼ N V p λ p − p (52)where we have used (66) and the fact that U ∼ λ − p . Since the open stringshave a mass ∼ U , this corresponds to a total energy and entropy in open16trings E ∼ N V p λ p +13 − p S ∼ N V p λ p − p which again matches the equilibrium energy and entropy of the black brane.This again suggests that the black hole is essentially fully formed in the initialcollision, with very little additional evolution required to reach equilibrium. Another quantity we can compare at the correspondence point is the sizeof the bound state. At weak coupling, after n open strings have been formed,the amplitude of oscillation of the resulting bound state is, from (27), L = Enτ (53)At the correspondence point for general p we have E ∼ N V p U p +10 (54)while the initial number of open strings created is n ∼ N V p U p (55)Thus the initial amplitude of oscillation as measured in the U coordinate is L/(cid:96) s = E/n ∼ U (56)In other words, the initial oscillation amplitude matches the equilibrium hori-zon radius of the black brane. Again this suggests that after the initialcollision, only a small amount of additional evolution is required to reachequilibrium. So far we have studied bound state formation in a collision between two clus-ters of D-branes, in the geometry shown in Fig. 1. Here we study a differentinitial configuration, in which N D0-branes are uniformly distributed over a17igure 3: A collapsing shell of 0-branes. Initially the 0-branes are spreaduniformly over an S with velocities toward the center.collapsing spherical shell as in Fig. 3. We will see that the correspondenceprinciple applies and a similar outcome is obtained in this case.We consider an initial configuration in which the 0-branes are uniformlyspread over an S of radius U in 9 spatial dimensions. The 0-branes arelocalized but uniformly distributed over the sphere, with velocities directedtoward the center. Intuitively we argue as follows. Since the total volume ofthe sphere scales as U , each 0-brane occupies a volume ∼ U /N , and thedistance between nearest-neighbor 0-branes scales as U/N / . This meansvirtual open strings connecting nearest-neighbor 0-branes are quite light,with a mass ∼ U/N / that goes to zero at large N . However the typicalopen string is much heavier, with a mass ∼ U that is independent of N .We expect these typical open strings to dominate the bound-state formationprocess, and therefore expect to have a well-defined correspondence point atlarge N .To argue this in more detail, it is useful to consider a 0-brane located at When p = 3 the matching is n ∼ N V U , E ∼ N V U , S ∼ N V U . S U θ W Sboson
Figure 4: The 0-branes are spread over an S of radius U . The green S hasradius U sin θ and the red W boson has length 2 U sin θ/ θ to the other 0-brane. See Fig. 4. The number of distinct openstrings dn in the interval ( θ, θ + dθ ) is dn = N π U × π U sin θ ) × U dθ (57)The first factor N/ ( π U ) is the number density of 0-branes on the S , thesecond factor π ( U sin θ ) is the volume of an S located at an angle θ fromthe south pole. Thus the number density of open strings is dndθ = 3532 N sin θ (58)We can also find the mass density of open strings dmdθ . Since an open stringsubtending an angle θ has a mass 2 U sin θ/
2, this is given by dmdθ = dndθ · U sin θ N U sin θ sin θ N dndθ and mass density NU dmdθ are plotted inFig. 5. 19 Π Π Π Π Π Π Θ N (cid:215) (cid:100)(cid:110) (cid:72) Θ (cid:76) (cid:100) Θ Π Π Π Π Π Π Θ
N U (cid:215) (cid:100)(cid:109) (cid:72) Θ (cid:76) d Θ Figure 5: On the left, the W-boson number density N dndθ . On the right, theW-boson mass density NU dmdθ .As can be seen in the figure, there are light open strings at large N . How-ever the number of these strings is tiny, since dndθ ∼ θ at small angles. Mostof the W-bosons are concentrated around θ = π/
2. Therefore a sphericalshell is basically the same as having W-bosons distributed in the interval θ < θ < π − θ , where θ is determined by the fraction of 0-branes pairs weneglect. For example, if we neglect dndθ ≤ − N , then θ ∼ .
1. Since themasses of the W-bosons near θ = π/ O ( U ), we can simply approxi-mate the entire W-boson spectrum by taking m W ∼ U .We now consider what happens when we give the shell of 0-branes somevelocity toward the origin. The analysis is almost identical to the collid-ing clusters considered in §
2. Given N D0-branes with total energy E , theasymptotic relative velocity is E ∼ mass × v ∼ NR v ⇒ v ∼ (cid:18) ERN (cid:19) / = (cid:18) Eλl s N (cid:19) / (60)In terms of the U coordinate, this becomes˙ U = (cid:18) EλN (cid:19) / (61) This is due to the fact that the 0-branes are spread on an S . The distribution wouldbe less sharply peaked in lower dimensions, with dndθ ∼ θ d − on an S d . § N = N ∼ N . Since the W-boson massesare concentrated around m W ∼ U , open string production again sets in when U ∼ (cid:112) ˙ U ∼ (cid:18) EλN (cid:19) / (62)At the correspondence point, where the effective gauge coupling becomesorder one, we have U ∼ λ / (63)and therefore E ∼ N λ / . (64)Just as in §
2, this matches the radius and energy energy of a black hole atthe correspondence point.
In this paper we studied D-brane collisions. We argued that the process ofopen string creation, which leads to formation of a D-brane bound state atweak coupling, smoothly matches on to a process at strong coupling, namelyblack hole formation in the dual supergravity. The transition happens atan intermediate value of the coupling, given by the correspondence principleof Horowitz and Polchinski. The size of the bound state, the timescale forapproaching equilibrium, and the thermodynamic properties of the boundstate all agree between the two descriptions. The latter agreement happensquickly, which suggests that the bound state is formed by the initial collisionin a near-equilibrium configuration.We considered two types of initial configurations, namely colliding clustersof wrapped D p -branes and a collapsing shell of D0-branes. The main dif-ference between the two configurations was that the shell had a tail of lightopen strings which we argued could be neglected. In fact, this distinctionbetween the two configurations is somewhat artificial, since with somewhatmore generic initial conditions the 0-branes which make up the clusters couldhave some small random relative velocities. One would then expect a bit ofopen string production within the clusters, which would put the two exampleson much the same footing. 21n the examples we studied the powers of N were fixed by large- N counting,so at the correspondence point there was essentially only a single lengthscale in the problem, namely the ’t Hooft scale (for p (cid:54) = 3) or the horizonradius (when p = 3). In a sense this guaranteed the matching betweenperturbative gauge theory and gravity results, just on dimensional grounds.To explore this further it would be interesting to study multi-charged blackholes, or to deform the background in a way which introduces another lengthscale, and ask whether there is still a simple transition between perturbativeworldvolume dynamics and black hole formation.A step in this direction would be to consider 0-brane collisions but with N (cid:54) = N . In this case, as we saw in §
6, the matching between perturbativegauge and gravity results must be more complicated, because the energy andentropy in open strings that are created in the initial collision do not matchthe equilibrium energy and entropy of the black hole. This means furtherdynamical evolution is required before the bound state reaches equilibrium.It would be interesting to study this, perhaps by going beyond the linearizedapproximation made when studying parametric resonance in §
4. There areseveral related interesting examples to consider, for example a situation inwhich several concentric layers of shells are collapsing.Another direction would be to use the present results to better understandthe microstructure of black holes. The picture that emerges, that a black holeis a thermal bound state of D-branes and open strings, is reminiscent of thefuzzball proposal [37]. However the real question, relevant for understandingfirewalls [38] or the energetic curtains of [39], is whether this thermal statecould be a dual description of the interior geometry of the black hole.
Acknowledgements
DK is grateful to David Berenstein for numerous discussions on this topic.S.R. thanks Justin R. David for discussions. This work was supported in partby U.S. National Science Foundation grants PHY-0855582 and PHY-1125915and by grants from PSC-CUNY. The research of SR is supported in part byGovt. of India Department of Science and Technology’s research grant underscheme DSTO/1100 (ACAQFT). 22
String production in a D-brane collision
We review the process of open string production in a D-brane collision, fol-lowing [26, 40].Consider colliding two 0-branes with relative velocity v and impact param-eter b . Setting 2 πα (cid:48) = 1, the virtual open strings connecting the two 0-braneshave an energy or frequency ω = √ v t + b . As long as this frequency ischanging adiabatically open strings will not be produced. The adiabatic ap-proximation breaks down when ˙ ω/ω (cid:38)
1. The peak value of this quantityis ˙ ω/ω ∼ v/b when vt ∼ b , so (restoring units) open strings are producedfor b (cid:46) √ vα (cid:48) . In terms of the radial coordinate U = r/α (cid:48) , where r is the dis-tance between 0-branes, the energy of an open string is m W = U/ π . So theadiabatic approximation breaks down and open strings are produced when˙ U /U ∼ Now consider colliding two p -branes wrapped on a torus of volume V p ,with relative velocity v and impact parameter b in the transverse dimen-sions. Consider a virtual open string that connects the two p -branes and hasmomentum k along the p -brane worldvolumes. Setting 2 πα (cid:48) = 1, this virtualopen string has an energy or frequency ω = √ k + v t + b If k = 0 then the condition for open string production is just what it was for0-branes, b (cid:46) √ v . Having non-zero k increases ω and suppresses open stringproduction. Effectively there is a cutoff, that open strings are produced upto a maximum momentum k ∼ b ∼ √ v . Restoring units, the maximummomentum is k ∼ (cid:112) v/α (cid:48) = ˙ U / . This cutoff corresponds to a numberdensity of open strings on the p -brane worldvolume ∼ ˙ U p/ Again these open strings are produced when ˙
U /U ∼ In principle we should distinguish between the asymptotic relative velocity ˙ U = v/α (cid:48) and the actual time-dependent value ˙ U = vα (cid:48) vt √ b + v t . But at vt ∼ b this distinction canbe ignored.
23f we collide two stacks of D p -branes with charges N and N respectively,it’s easy to estimate the total number of open strings that are produced. Atweak coupling the individual brane collisions are independent events. So for0-branes the total number of open strings produced is n ∼ N N while for p -branes the total number of open strings produced is n ∼ N N V p ˙ U p/ (65)or equivalently, in terms of the radius at which open string production takesplace n ∼ N N V p U p (66)There is, however, an important consistency check on this result: we needto make sure the incoming D-branes have enough kinetic energy to producethis number of open strings. Equivalently, we need to make sure that theback-reaction of open string production on the velocities of the D-branes isunder control. Given the number of open strings (66), the energy in openstrings is E string = nU = N N V p (cid:18) λ(cid:15)N N (cid:19) p +14 where we have used (17). On the other hand the kinetic energy of the in-coming branes is E = (cid:15)V p Thus the ratio E string E = λ (cid:18) λ(cid:15)N N (cid:19) p − (67)and the consistency condition E string /E < λ U p − < λ eff <
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