If M-theory is compactified on a d-torus it becomes a D = 11 – d dimensional theory with Newton constant
GD = G11/Ld = l911/Ld —– (1)
A Schwartzschild black hole of mass M has a radius
Rs ~ M(1/(D-3)) GD(1/(D-3)) —– (2)
According to Bekenstein and Hawking the entropy of such a black hole is
S = Area/4GD —– (3)
where Area refers to the D – 2 dimensional hypervolume of the horizon:
Area ~ RsD-2 —– (4)
S ~ 1/GD (MGD)(D-2)/(D-3) ~ M(D-2)/(D-3) GD1/(D-3) —– (5)
From the traditional relativists’ point of view, black holes are extremely mysterious objects. They are described by unique classical solutions of Einstein’s equations. All perturbations quickly die away leaving a featureless “bald” black hole with ”no hair”. On the other hand Bekenstein and Hawking have given persuasive arguments that black holes possess thermodynamic entropy and temperature which point to the existence of a hidden microstructure. In particular, entropy generally represents the counting of hidden microstates which are invisible in a coarse grained description. An ultimate exact treatment of objects in matrix theory requires a passage to the infinite N limit. Unfortunately this limit is extremely difficult. For the study of Schwarzchild black holes, the optimal value of N (the value which is large enough to obtain an adequate description without involving many redundant variables) is of order the entropy, S, of the black hole.
Considering the minimum such value for N, we have
Nmin(S) = MRs = M(MGD)1/D-3 = S —– (6)
We see that the value of Nmin in every dimension is proportional to the entropy of the black hole. The thermodynamic properties of super Yang Mills theory can be estimated by standard arguments only if S ≤ N. Thus we are caught between conflicting requirements. For N >> S we don’t have tools to compute. For N ~ S the black hole will not fit into the compact geometry. Therefore we are forced to study the black hole using N = Nmin = S.
Matrix theory compactified on a d-torus is described by d + 1 super Yang Mills theory with 16 real supercharges. For d = 3 we are dealing with a very well known and special quantum field theory. In the standard 3+1 dimensional terminology it is U(N) Yang Mills theory with 4 supersymmetries and with all fields in the adjoint repersentation. This theory is very special in that, in addition to having electric/magnetic duality, it enjoys another property which makes it especially easy to analyze, namely it is exactly scale invariant.
Let us begin by considering it in the thermodynamic limit. The theory is characterized by a “moduli” space defined by the expectation values of the scalar fields φ. Since the φ also represents the positions of the original DO-branes in the non compact directions, we choose them at the origin. This represents the fact that we are considering a single compact object – the black hole- and not several disconnected pieces.
The equation of state of the system, defined by giving the entropy S as a function of temperature. Since entropy is extensive, it is proportional to the volume ∑3 of the dual torus. Furthermore, the scale invariance insures that S has the form
S = constant T3∑3 —– (7)
The constant in this equation counts the number of degrees of freedom. For vanishing coupling constant, the theory is described by free quanta in the adjoint of U(N). This means that the number of degrees of freedom is ~ N2.
From the standard thermodynamic relation,
dE = TdS —– (8)
and the energy of the system is
E ~ N2T4∑3 —– (9)
In order to relate entropy and mass of the black hole, let us eliminate temperature from (7) and (9).
S = N2∑3((E/N2∑3))3/4 —– (10)
Now the energy of the quantum field theory is identified with the light cone energy of the system of DO-branes forming the black hole. That is
E ≈ M2/N R —– (11)
Plugging (11) into (10)
S = N2∑3(M2R/N2∑3)3/4 —– (12)
This makes sense only when N << S, as when N >> S computing the equation of state is slightly trickier. At N ~ S, this is precisely the correct form for the black hole entropy in terms of the mass.