On the Hagedorn Transition and Collective Dynamics of D0-branes

Banks, Fischler, Klebanov and Susskind have proposed a model for black hole thermodynamics based on the principle that the entropy is of order the number of particles at the phase transition point in a Boltzmann gas of D0-branes. We show that the deviations from Boltzmann scaling found in d<6 noncompact spatial dimensions have a simple explanation in the analysis of self-gravitating random walks due to Horowitz and Polchinski. In the special case of d=4 we find evidence for the onset of a phase transition in the Boltzmann gas analogous to the well-known Hagedorn transition in a gas of free strings. Our result relies on an estimate of the asymptotic density of states in a dilute gas of D0-branes. Matrix theory [1] has motivated a qualitative description of the physics of Schwarzschild black holes in d>4 noncompact spatial dimensions in terms of the thermodynamic properties of a Boltzmann gas of D0branes, interacting via long range gravitational forces [2-9]. According to this proposal neutral black holes may be understood as bound states of the partons of Matrix theory. The analysis is entirely within the mean field approximation to the collective dynamics of D0branes, where the D0-branes are distinguishable Boltzmann particles [3,11,12,13]. All basic qualitative features of the thermodynamics of Schwarzschild black holes are derived using simple scaling arguments. Let us summarize these scaling relations. In [2,3] the Bekenstein-Hawking relation for a black hole in D=d+ 1 spacetime dimensions was deduced from a mean field analysis of the collective dynamics of N Matrix theory partons in the limit of low velocities v and large relative separations r. The mean field Lagrangian of N D0branes, with longitudinal momentum P = N/R, is L = N v 2R +N GD R3 v rD−4 . (1) Application of the Virial Theorem, and the Heisenberg uncertainty principle, v ∼ R RS , to a purported bound state of N D0branes of transverse size, r = RS , gives the scaling N ∼ R S /GD. If we assume the simple rule of thumb S ∼ N , we get the Bekenstein Hawking scaling relation, S ∼ R S /GD , (2) which describes the thermodynamics of a d+1-dimensional black hole with Schwarzschild radius RS and entropy S. The Schwarzschild radius can then be shown to scale with the black hole mass, M , according to the relation RS ∼ (GDM) 1 D−3 , (3)