Gravitational Entropy (June '98)

The first indication of a connection between black holes and entropy, came in 1970, with my discovery that the area of the horizon of a black hole, always increased. There was an obvious analogy with the Second Law of Thermodynamics, which states that entropy always increases. But it was Jacob Bekenstein, who took the bold step, of suggesting the area actually was the physical entropy, and that it counted the internal states of the black hole. I was very much against this idea at first, because I felt it was a misuse of my horizon area result. If a black hole had a physical entropy, it would also have a physical temperature. If a black hole was in contact with thermal radiation, it would absorb some of the radiation, but it would not give off any radiation, since by definition, a black hole was a region from which nothing could escape. If the thermal radiation was at a lower temperature than the black hole, the loss of entropy down the black hole, would be greater than the increase of horizon area.

This would be a violation of the generalized Second Law, that Bekenstein proposed. With hind sight, this should have suggested that black holes radiate. But no one, including Bekenstein and myself, thought anything could get out of a non rotating black hole. On the other hand, Penrose had shown that energy could be extracted from a rotating black hole, by a classical process. This indicated that there should be a spontaneous emission in the super radiant modes, that would be the quantum counter part of the Penrose process. In trying to understand this emission in the super radiant modes, in terms of quantum field theory in curved spacetime, I stumbled across the fact that even non rotating black holes, would radiate. Moreover, the radiation would be exactly what was required, to prevent a violation of the generalized second law. Bekenstein was right after all, but in a way he hadn't anticipated. In this talk, I want to discuss the deep reason, for the existence of such gravitational entropy. In my opinion, it is that general relativity, and extensions like supergravity, allow spacetime to have more than one topology. By topology, I mean topology in the Euclidean regime. The topology of a Lorentzian spacetime can change with time, only if there is some pathology, such as a singularity, or closed time like curves. In either of these cases, one would expect the theory to break down.

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. 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. One could identify under the combination of a rotation along the axis and a rotation, but I shall consider only a translation, for simplicity.

The surface of the identified cylinder, representing the boundary at infinity, will be a two torus, with periodically identified tau and phi, as the two coordinates.

However, given a two torus at infinity, there are two topologically distinct ways, that one can fill it in with a two disk, cross a circle. These two ways, are shown as the two anchor rings, or solid tori, on the screen. The circle can be in either the tau or phi directions. The first corresponds to anti de Sitter at a finite temperature, and the second to the BTZ black hole. In the BTZ black, the orbits of the phi Killing vector, do not shrink to zero, because the black hole has no center in the Euclidean region. On the other hand, the orbits of the tau killing vector, shrink to zero at the horizon, which is the center of the disk, cross the circle. One gets a a non zero action and Hamiltonian for the BTZ black hole, by taking the reference background, to be anti de Sitter space at a finite temperature. As before, this leads to an entropy of a quarter the area of the horizon, which in this case, is the length of the phi orbit at the center of the disk. But the BTZ black hole, is completely homogeneous. Thus one can not localize the entropy on the horizon, which is just like the axis in ordinary three dimensional space. It arises from the global mismatch of Euclidean BTZ, with the reference background, which is Euclidean anti de Sitter, periodically identified in the tau direction.

This analysis of the mismatch between the topology of a reference background, and other Euclidean solutions with the same asymptotic behavior, can be extended to other situations. The intersection of surfaces of constant time on the horizon, is only simplest way in which unitary Hamiltonian evolution can break down. Chris Hunter and I, have been investigating more complicated topologies, in which there are other possible singularities in the foliation of spacetime. One might expect that these singularities, would also have entropy associated with them. Since thermal ensembles are periodic in the Euclidean time direction, we considered reference backgrounds, and other Euclidean solutions, which have a U1 isometry group, with Killing vector, K. The isometry group, will have fixed points where K vanishes. Gary Gibbons and I, classified the possible fixed point sets in four dimensions, into two dimensional surfaces we called bolts, and isolated points that we called nuts. However, one can extend this classification scheme, to Euclidean metrics of any dimension. The fixed point sets will then lie on totally geodesic sub manifolds, of even codimension.

Let tau be the parameter of the U1 isometry group. Then the metric can be written in the Kaluza Klein form, with tau as the coordinate on the internal U1. Here V, omega i, and gamma i j, are fields on the d minus one dimensional space, B, of orbits of the isometry group. B would be singular at the fixed points, so one has to leave them out of B, and introduce d minus two dimensional boundaries to B. The coordinate tau can be changed by a Kaluza Klein gauge transformation, that is, by the addition of a function, lambda on B. This changes the one form, omega, by d lambda, but leaves the field strength, F = d omega, unchanged. If the orbit space, B has non trivial homology in dimension two, the two form, F, can have non zero integrals over two cycles in B. In this case the potential one form, omega, will have Dirac like string singularities, on surfaces of dimension d minus three in B. The foliattion of the spacetime by surfaces of constant tau, will break down both at the fixed points of the isometry, and on the Kaluza Klein string singularities of omega, which I will call Misner strings, after Charles Misner who first realized their nature in the Taub nut solution. Misner strings are surfaces of dimension d minus two in the spacetime.

In order to do a Hamiltonian treatment using surfaces of constant tau, one has to cut out small neighbourhoods of the fixed point sets, and of any Misner strings. The action given by beta times the value of the Hamiltonian, will then be the action of the spacetime, with the neighbourhoods removed. Putting back the neighbourhoods, the Einstein Hilbert term will give a contribution of minus a quarter area, for the Misner strings and the d minus two dimensional fixed point sets. But the contribution to the action from lower dimensional fixed points, will be zero. As before, the Hamiltonian surface terms at the fixed points, will be zero, because the lapse and shift vanish there. But the shift won't vanish on the Misner string, so there will be a Hamiltonian surface term on a Misner string, given by the shift, times a component of the second fundamental form, of the constant tau surfaces. Thus the action will be made up of several contributions. First, there will be beta times the Hamiltonian surface terms at infinity, and on the Misner strings. Then one has to subtract one over 4 G, times the sum of the areas of the bolts, plus the Misner strings. Finally, one has to subtract the same quantities for the reference background. Some or all of these quantities may diverge, but the differences from the reference background will have finite limits, as the boundary is taken to infinity. From now on, I shall mean these finite differences, when I refer to any of these contributions to the action. The partition function, Z, can be related by thermodynamics, to the entropy, and the conserved quantities like energy, angular momentum, and electric charge, whose values are not fixed by the boundary conditions. One has log Z, = the entropy, minus the sum of the conserved quantities, each weighted by its thermodynamic potential. But the Hamiltonian surface term at infinity, multiplied by beta, is by definition, the sum of the conserved quantities, weighted by their thermodynamic potentials. Thus taking the action to be minus log Z, one gets that the entropy is a quarter the area of the bolts, and Misner strings, minus beta times the Hamiltonian surface term on the Misner strings. One can make a Kaluza Klein gauge transformation, by changing tau by a function, lambda, on the orbit space. This will change the position and area of the Misner strings, but the combination, a quarter string area, minus beta the Hamiltonian surface term on the string, will be gauge invariant. Again, this shows that entropy is a global property. It can not be localized in microstates on the Misner string.

The discussion I have given, applies to any gravitational theory in any dimension, which has the Einstein Hilbert action, as the leading term. I shall illustrate it, however, with some examples in four dimensions, that have been worked out by Chris Hunter. Four dimensional metrics, can have several different asymptotic behaviors. In this talk, I shall concentrate on the asymptotically locally flat, and asymptotically locally Euclidean cases, because entropy has not been defined for these spaces previously.

Asymptotically local flat solutions, have a Nut charge, or magnetic type mass, N, as well as the ordinary electric type mass, M. The Nut charge is beta over 8 pi, times the first Chern number of the U1 bundle, over the sphere at infinity, in the orbit space, B. The natural reference backgrounds for solutions with Nut charge, are the self dual multi Taub Nut solutions, which have M=N. When written in Kaluza Klein form, the multi Taub Nut solutions what no bolts, or fixed point sets of dimension two. They do however, have a number of Nuts, or fixed point sets of dimension zero. From each Nut, there is a Misner string, leading to either another Nut, or infinity. The positions and areas of the Misner strings, are gauge dependent. However, a quarter the area of the Misner strings, minus beta times the Hamiltonian surface term on the strings, is gauge invariant, and is independent of the position of the Nuts in the three dimensional flat orbit space. Thus the entropy of the multi Taub Nuts, is zero, as one would expect, since they define the vacuum for that sector of the theory.

There are, however, asymptotically locally flat solutions, that are not multi Taub Nut. The prime example, is the so called Taub bolt solution, discovered by Don page. As its name suggests, this solution has a bolt, with area 12 pi N squared. It also has a Misner string, stretching from the bolt to infinity. The Chern number at infinity, is one. Thus the appropriate reference background, is the single self dual Taub Nut. When one calculates the entropy according to the prescription I have given, the result is, pi N squared. Note that this is not a quarter the area of the bolt, which would have given 3 pi N squared. The difference comes from the different Misner string area and Hamiltonian contributions, in Taub Nut and Taub bolt.

In the asymptotically locally Euclidean case, the appropriate reference backgrounds, are the orbifolds obtained by identifying Euclidean flat space, under a discrete sub group, Gamma, of SU2, that acts freely on the three sphere. The ALE self dual instantons, have these boundary conditions. They can be written in Kaluza Klein form, with bolts, nuts, and Misner strings. The reference backgrounds, can also be written in Kaluza Klein form, with a nut at the orbifold point, and a Misner string from the orbifold point, to infinity. However, the entropy, calculated according to the prescription I have given, turns out to be zero. This is what one would expect, because the ALE instantons, have the same super symmetry as the reference backgrounds. It is only when one has solutions with less super symmetry than the background, that one gets entropy. Examples are non extreme black holes, which have no super symmetry, or extreme black holes with central charges, which have reduced super symmetry.

One can show that there are no vacuum ALE metrics, with less super symmetry than the reference background. The Israel Wilson family of Einstein Maxwell solutions, however, contains non self dual ALE, and even asymptotically Euclidean solutions, with an asymptotically constant self dual Maxwell field at infinity. Since these solutions are not super symmetric, and have different topology to the reference background, one would expect them to have entropy, and this is confirmed by calculation of examples.

One might think instantons with a self dual Maxwell field at infinity, were not physically relevant. However, one can promote them to being Einstein Yang Mills solutions, with a constant self dual Yang Mills field at infinity. One could then match them to Yang Mills instantons in flat space, with large winding numbers, which can have regions where the Yang Mills field, is almost constant.

Finally, to show that the expression we propose for the entropy, can be applied in more than four dimensions, consider the five sphere, of radius, R. This can be regarded as a solution of a five dimensional theory, with cosmological constant. One can take the U1 isometry group, to have a fixed point set on a three sphere of radius R. In this case there are no Misner strings. So the formula gives an entropy of, pi squared, R cubed, over 2G. However, one can choose a different U1 isometry, whose orbits are the Hopf fibration of the five sphere. This isometry group, has no fixed points. So the usual connection between entropy, and the fixed points, does not apply. But the orbit space of the Hopf fibration, is CP2. The Kaluza Klein two form, F, is the harmonic two form on CP2. The one form potential, omega, for this has a Dirac string on a two surface in the orbit space. When promoted to the full spacetime, this becomes a three dimensional Misner string. The area of the Misner string divided by 4G, minus beta times the Hamiltonian surface term on the Misner string, is again the entropy, pi squared R cubed, over 2G. This example is fairly trivial, but it shows that the method can be extended to higher dimensions.

I think there are three morals can be drawn from this work.

The first is that gravitational entropy just depends on the Einstein Hilbert action. It doesn't require super symmetry, string theory, or p-branes.

The second is that entropy is a global quantity, like energy or angular momentum, and shouldn't be localized on the horizon. The various attempts to identify the microstates responsible for black hole entropy, are in fact constructions of dual theories, that live in separate spacetimes.

The third moral, is that entropy arises from a failure to foliate the Euclidean regime, with a family of time surfaces. This would suggest that there would not be a unitary S matrix, for particle scattering described by a Euclidean section, with non trivial topology. No particle scattering situation, with non trivial Euclidean topology, has definitely been shown to exist, but the asymptotically Euclidean solutions with a constant Maxwell field at infinity, are very suggestive. They would seem to point to loss of quantum coherence and information, in black holes. This is the major unresolved question, in the quantum theory of black holes. Let's hope this meeting, gives us new insights into the problems.

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