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Steven Hawking ..interesting stuff.
I was reading Steven Hawkings lectures and felt that I had to lift this for those who may be interested... its from his web site full of interesting stuff.
alex
Gravitational Entropy (June '98)
Extract.......
It was Paul Dirac, my predecessor at Cambridge, who first realized that time evolution in quantum theory, could be formulated as a unitary transformation, generated by the Hamiltonian. This worked well in non relativistic quantum theory, in which the Hamiltonian was just the total energy. It also worked in special relativity, where the Hamiltonian, could be taken to be the time component of the four momentum. But there were problems in general relativity, where neither energy, nor linear momentum, are local quantities. Energy and momentum can only be defined globally, and only for suitable asymptotic behavior.
Dirac himself, developed the Hamiltonian treatment for general relativity. In d dimensions, one can write the metric in the ADM form. That is, one introduces a time coordinate, tau, which I take to be Euclidean, and shift and lapse functions. The Hamiltonian, can then be expressed as an integral, over a surface of constant time. However, the difference from special relativity, was that all the terms in this volume integral, vanished for configurations that satisfied the field equations. I must admit that when I came across the Hamiltonian formulation as a student, I thought, why should one bother working out the volume terms, since they are zero. The answer is, of course, that although the volume terms are zero on solutions, they have non zero Dirac brackets, which become commutators in the quantum theory. Because the volume contributions to the Hamiltonian are zero, its numerical value has to come from surface integrals, at the boundaries of the volume. Such surface terms arise in any gauge theory, including Maxwell theory, when one integrates the constraint equations by parts. In the gravitational case, the Hamiltonian also gets a contribution to the surface term, from the trace K surface term in the action, that is required to cancel the variation in the Einstein Hilbert action, capital R. The surface term in general, makes both the action, and the Hamiltonian, infinite. It is therefore sensible to consider only the difference between the action or Hamiltonian, and those of some reference background solution, that the solutions approach at infinity. This reference background acts as the vacuum, for that sector of the quantum theory. It is normally taken to be flat space, or anti de Sitter space, but I will consider other possibilities.
In asymptotically flat space, if the surfaces of constant tau at infinity, are related by just a time translation, the shift is zero, and the lapse is one. The Hamiltonian surface term at infinity, is then just the mass, plus the electric charge, Q, times the electro static potential, Phi. In a topologically trivial spacetime, one could make the electro static potential zero, by a gauge transformation. However, this will not be possible in non trivial topologies. As I will explain later, magnetic charges do not contribute to the surface term in the Hamiltonian. If the surface of constant tau at infinity, are related by a time translation, plus a rotation through phi, of omega, the surface term at infinity picks up an extra omega J term.
Normally, one considers solutions, which can be foliated by surfaces of constant time, that have boundaries only at infinity, in asymptotically flat space. In such situations, the total Hamiltonian, that is the volume integral, plus the surface terms, will generate unitary transformations, that map the Hilbert space of initial states, into the final ones.
All the quantum states concerned, can be taken to be pure states. There are no mixed states, or gravitational entropy.
However, solutions like black holes, have a Euclidean geometry with non trivial topology. This means that they can't be foliated by a family of time surfaces, that agree with the usual notion of time. If you try, the family of surfaces will necessarily have intersections, or other singularities, on surfaces of codimension two or more. In fact Euclidean black holes, are the simplest examples. So I will show how the break down of unitary Hamiltonian evolution, gives rise to black hole entropy. I will then go on to more exotic possibilities, like Taub Nut, and Taub bolt. In these cases, the entropy is not necessarily a quarter the area, of a codimension two surface.
As Jim Hartle and I first discovered in 1975, black holes, have a regular Euclidean analytical continuation, if and only if the Euclidean time, tau, is treated like an angular coordinate. It has to be identified with a period, beta, =2 pi over kappa, where kappa is the surface gravity of the horizon. This means that the surfaces of constant tau, all intersect on the horizon, and the concept of a unitary Hamiltonian evolution, will break down there. The surfaces of constant tau, will therefore have an inner boundary at the horizon, and the Hamiltonian will also have contributions from surface terms, at this boundary. If one takes the Hamiltonian vector, to be the combination of the tau and phi Killing vectors that vanishes on the horizon, then the lapse and shift vanish on the horizon. This means that the gravitational part of the surface term, is zero. If the vector potential is also regular on the horizon, the gauge field surface term is also zero.
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.
alex
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