The Mudcat Café TM
Thread #59418 Message #1230822
Posted By: Rapparee
21-Jul-04 - 03:09 PM
Thread Name: BS: The Mother of all BS threads
Subject: RE: BS: The Mother of all BS threads
Amos, here's your answer:
The thermodynamic partition function, Z, for a system at temperature, beta to the minus one, is the expectation value of e to the minus beta, times the Hamiltonian, summed over all states. As is now well known, this can be represented by a Euclidean path integral, over all fields that are periodic in Euclidean time, with period beta at infinity. Similarly, the partition function for a system with angular velocity, omega, will be given by a path integral over all fields, that are periodic under the combination of a Euclidean time translation, beta, and a rotation, omega beta. One can also specify the gauge potential at infinity. This gives the partition function for a thermodynamic ensemble, with electric and magnetic type charges. The mass, angular momentum, and electric charges of the configurations in the path integral, are not determined by the boundary conditions at infinity. They can be different for different configurations. Each configuration, will therefore be weighted in the partition function, by an e to the minus the charge, times the corresponding potential. On the other hand, the magnetic type charges, are uniquely determined by the boundary conditions at infinity, and are the same for all field configurations in the path integral. The path integral therefore gives the partition function, for a given magnetic charge sector.
The lowest order contribution to the partition function, will be e to the minus I, where I, is the action of the Euclidean black hole solution. The action can be related to the Hamiltonian, as integral H, minus pq dot. In a stationary black hole metric, all the q dots will be zero. Thus the action, I, will be the time period beta, time the value of the Hamiltonian. As I said earlier, the Hamiltonian surface term at infinity, is mass, plus omega J, plus Phi Q, and the Hamiltonian surface term on the horizon, is zero. If one uses the contribution from this action to the partition function, and uses the standard formula, one finds the entropy is zero.
However, because the surfaces of constant Euclidean time, all intersected at the horizon, one had to introduce an inner boundary there. The action, I, = beta times Hamiltonian, is the action for region between the boundary at infinity infinity, and a small tubular neighbourhood of the horizon. But the partition function, is given by a path integral over all metrics with the required behavior at infinity, and no internal boundaries or infinities. One therefore has to add the action of the tubular neighbourhood of the horizon. What ever supergravity theory one is using, and what ever dimension one is in, one can make a conformal transformation of the metric to the Einstein frame, in which the coefficient of the Einstein Hilbert action, capital R, is on over 16 pi G, where G is Newton's constant in the dimension of the theory. The surface term associate with the Einstein Hilbert action, is one over 8 pi G, times the trace of the second fundamental form. This gives the tubular neighbourhood of the horizon, an action of minus one over 4 G, times the codimension two area of the horizon. If one adds this action to the beta times Hamiltonian, one gets a contribution to the entropy, of area over 4 G, independent of dimension, or of the particular supergravity theory. Higher order curvature terms in the action, would give the tubular neighbourhood an action, that was small compared to area over 4 G, for large black holes. Thus the quarter area law, is universal for black holes. It can be traced to the non trivial topology of Euclidean black holes, which provides an obstruction to foliating them by a family of time surfaces, and using the Hamiltonian to generate a unitary evolution of quantum states. Because the entropy is given by the horizon area in Planck units, one might think that it corresponded to microstates, that are localized near the horizon. However, gravitational entropy, like gravitational energy, can not be localized, but can only be defined globally. This can be seen most clearly in the case of the three dimensional BTZ black hole, to which all four or five dimensional black holes, can be related by a series of U dualities, which preserve the horizon area. The BTZ black hole, is a solution of the 2+1 Einstein equations, with a negative cosmological constant.
Locally, the only solution of these equations, is anti de Sitter space, but the global structure can be different. To see this, one can picture anti de Sitter space, as conformal to the interior of a cylinder in 2+1 Minkowski space. The surface of the cylinder, represents the time like infinity of anti de Sitter space. Similarly, Euclidean anti de Sitter space, is conformal to the interior of a cylinder in three dimensional Euclidean anti de Sitter space. If one now identify this cylinder periodically along its axis, one occasions the background geometry for quantum fields in anti de Sitter space, at a finite temperature.