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Inflation: An Open and Shut Case (April '98)
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Gravitational Entropy (June '98)
Quantum Cosmology, M-theory and the Anthropic Principle (January '99).
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Physics Colloquiums - Rotation, Nut charge and Anti de Sitter space (February '99).

Please note that (·) marks indicate progression of slide show. The slide show is a PowerPoint97 file. It may be viewed on Windows PCs without Powerpoint97 (to do this download and extract the zip file, and run PNGSETUP.EXE).

This lecture is the intellectual property of Professor S.W. Hawking. You may not reproduce, edit or distribute this document in anyway for monetary advantage.
 

The work I'm going to talk about has been carried out with Chris Hunter and Marika Taylor Robinson at Cambridge, and Don Page at Alberta. It is described in the first three papers shown on the screen. Related work has been carried out by Chamblin, Emparan, Johnson, and Myers. However, they seemed a bit uncertain what reference background to use. I have also shown a reference to Dowker which is relevant.

It has been known for quite a time, that black holes behave like they have entropy. The entropy is the area of the horizon, divided by 4 G, where G is Newton's constant.

The idea is that the Euclidean sections of black hole metrics are periodic in the imaginary time coordinate. Thus they represent black holes in equilibrium with thermal radiation. However there are problems with this interpretation. First, one can not have thermal radiation in asymptotically flat space, all the way to infinity, because the energy density would curve the space, and make it an expanding or collapsing Friedmann universe. Thus, if you want a static situation, you have to resort to the dubious Gedanken experiment, of putting the black hole in a box. But you don't find black hole proof boxes, advertised on the Internet. The second difficulty with black holes in equilibrium with thermal radiation is that black holes have negative specific heat. In many cases, when they absorb energy, they get larger and colder. This reduces the radiation they give off, and so they absorb faster than they radiate, and the equilibrium is unstable. This is closely related to the fact that the Euclidean metric has a negative mode. Thus it seems that asymptotically flat Euclidean black holes, describe the decay of hot flat space, rather than a black hole in equilibrium with thermal radiation. The third difficulty with the idea of equilibrium is that if the black hole is rotating, the thermal radiation should be co-rotating with it. But far away from the black hole, the radiation would be co-rotating faster than light, which is impossible. Thus, again one has to use the artificial expedient, of a box of finite size.

Way back in pre-history, Don page and I, realized one could avoid the first two difficulties, if one considered black holes in anti de Sitter space, rather than asymptotically-flat space. In anti de Sitter space, the gravitational potential increases as one goes to infinity. This red shifts the thermal radiation, and means that it has finite energy. Thus anti de Sitter space can exist at finite temperature, without collapsing. In a sense, the gravitational potential in anti de Sitter space, acts like a confining box.

Anti de Sitter space can also help with the second problem, that the equilibrium between black holes and thermal radiation, will be unstable. Small black holes in anti de Sitter space, have negative specific heat, like in asymptotically flat space, and are unstable. But black holes larger than the curvature radius of anti de Sitter space, have positive specific heat, and are presumably stable.

At the time, Don page and I, did not think about rotating black holes. But I recently came back to the problem, along with Chris Hunter, and Marika Taylor Robinson. We realized that thermal radiation in anti de Sitter space could co-rotate with up to some limiting angular velocity, without having to travel faster than light. Thus anti de Sitter boundary conditions, can solve all three problems, in the interpretation of Euclidean black holes, as equilibria of black holes, with thermal radiation.

Anti de Sitter black holes may not seem of much interest, because we can be fairly sure, that the universe is not asymptotically anti de Sitter. However, they seem worth studying, both for the reasons I have just given, and because of the Maldacena conjecture, relating asymptotically anti de Sitter spaces, to conformal field theories on their boundary. I shall report on two pieces of work in relation to this conjecture. One is a study of rotating black holes in anti de Sitter space. We have found Kerr anti de Sitter metrics in four and five dimensions. As they approach the critical angular velocity in anti de Sitter space, their entropy, as measured by the horizon area, diverges. We compare this entropy, with that of a conformal field theory on the boundary of anti de Sitter space. This also diverges at the critical angular velocity, when the rotational velocity, approaches the speed of light. We show that the two divergences are similar.

The other piece of work, is a study of gravitational entropy, in a more general setting. The quarter area law, holds for black holes or black branes in any dimension, d, that have a horizon, which is a d minus 2 dimensional fixed point set, of a U1 isometry group. However Chris Hunter and I, have recently shown that entropy can be associated with a more general class of space-times. In these, the U1 isometry group can have fixed points on surfaces of any even co-dimension, and the space-time need not be asymptotically flat, or asymptotically anti de Sitter. In these more general class, the entropy is not just a quarter the area, of the d minus two fixed point set.

Among the more general class of space-times for which entropy can be defined, an interesting case is those with Nut charge. Nut charge can be defined in four dimensions, and can be regarded as a magnetic type of mass.

Solutions with nut charge are not asymptotically flat in the usual sense. Instead, they are said to be asymptotically locally flat, or ALF. In the Euclidean regime, in which I shall be working, the difference can be described as follows. An asymptotically flat metric, like Euclidean Schwarzschild, has a boundary at infinity, that is a two-sphere of radius r, cross a circle, whose radius is asymptotically constant.

To get finite values for the action and Hamiltonian, one subtracts the values for flat space, periodically identified. In asymptotically locally flat metrics, on the other hand, the boundary at infinity, is an S1 bundle over S2. These bundles are labeled by their first Chern number, which is proportional to the Nut charge. If the first Chern number is zero, the boundary is the product, S2 cross S1, and the metric is asymptotically flat. However, if the first Chern number is k, the boundary is a squashed three sphere, with mod k points identified around the S1 fibers.

Such asymptotically locally flat metrics, can not be matched to flat space at infinity, to give a finite action and Hamiltonian, despite a number of papers that claim it can be done. The best that one can do, is match to the self-dual multi Taub nut solutions. These can be regarded as defining the vacuums for ALF metrics.

In the self-dual Taub Nut solution, the U1 isometry group, has a zero dimensional fixed point set at the center, called a nut. However, the same ALF boundary conditions, admit another Euclidean solution, called the Taub bolt metric, in which the nut is replaced by a two dimensional bolt. The interesting feature, is that according to the new definition of entropy, the entropy of Taub bolt, is not equal to a quarter the area of the bolt, in Planck units. The reason is that there is a contribution to the entropy from the Misner string, the gravitational counterpart to a Dirac string for a gauge field.

The fact that black hole entropy is proportional to the area of the horizon has led people to try and identify the microstates, with states on the horizon. After years of failure, success seemed to come in 1996, with the paper of Strominger and Vafa, which connected the entropy of certain black holes, with a system of D-branes. With hindsight, this can now be seen as an example of a duality, between a gravitational theory in asymptotically anti de Sitter space, and a conformal field theory on its boundary.

It would be interesting if similar dualities could be found for solutions with Nut charge, so that one could verify that the contribution of the Misner string was reflected in the entropy of a conformal field theory. This would be particularly significant for solutions like Taub bolt, which don't have a spin structure. It would show whether the duality between anti de Sitter space, and conformal field theories on its boundary, depends on super symmetry. In fact, I will present evidence, that the duality requires super symmetry.

To investigate the effect of Nut charge, we have found a family of Taub bolt anti de Sitter solutions. These Euclidean metrics are characterized by an integer, k, and a positive parameter, s. The boundary at large distances is an S1 bundle over S2, with first Chern number, k. If k=0, the boundary is a product, S1 cross S2, and the space is asymptotically anti de Sitter, in the usual sense. But if k is not zero, they are what may be called, asymptotically locally anti de Sitter, or ALADS.

The boundary is a squashed three sphere, with k points identified around the U1 direction. This is just like asymptotically locally flat, or ALF metrics. But unlike the ALF case, the squashing of the three-sphere, tends to a finite limit, as one approaches infinity. This means that the boundary has a well-defined conformal structure. One can then ask whether the partition function and entropy, of a conformal field theory on the boundary, is related to the action and entropy, of these asymptotically locally anti de Sitter solutions.

To make the ADS, CFT correspondence well posed, we have to specify the reference backgrounds, with respect to which the actions and Hamiltonians are defined. For Kerr anti de Sitter, the reference background is just identified anti de Sitter space. However, as in the asymptotically locally flat case, a squashed three sphere, can not be imbedded in Euclidean anti de Sitter. One therefore can not use it as a reference background, to make the action and Hamiltonian finite.

Instead, one has to use Taub Nut anti de Sitter, which is a limiting case of our family. If mod k is greater than one, there is an orbifold singularity in the reference backgrounds, but not in the Taub bolt anti de Sitter solutions. These orbifold singularities in the backgrounds could be resolved, by replacing a small neighbourhood of the nut, by an ALE metric. We shall take it, that the orbifold singularities are harmless.

Another issue that has to be resolved, is what conformal field theory to use, on the boundary of the anti de Sitter space. For five dimensional Kerr anti de Sitter space, there are good reasons to believe the boundary theory is large N Yang Mills. But for four-dimensional Kerr anti de Sitter, or Taub bolt anti de Sitter, we are on shakier ground. On the three dimensional boundaries of four dimensional anti de Sitter spaces, Yang Mills theory is not conformally invariant.

The folklore is that one takes the infrared fixed point, of three-dimensional Yang Mills, but no one knows what this is. The best we can do, is calculate the determinants of free fields on the squashed three sphere, and see if they have the same dependence on the squashing, as the action. Note that as the boundary is odd dimensional, there is no conformal anomaly. The determinant of a conformally invariant operator, will just be a function of the squashing. We can then interpret the squashing, as the inverse temperature, and get the number of degrees of freedom, from a comparison with the entropy of ordinary black holes, in four dimensional anti de Sitter.

I now turn to the question, of how one can define the entropy, of a space-time. A thermodynamic ensemble, is a collection of systems, whose charges are constrained by La-grange multipliers.

One such charge, is the energy or mass, with the Lagrange multiplier being the inverse temperature, beta. But one can also constrain the angular momentum, and gauge charges. The partition function for the ensemble, is the sum over all states, of e to the minus, La-grange multipliers, times associated charges.

Thus it can be written as, trace of e to the minus Q. Here Q is the operator that generates a Euclidean time translation, beta, a rotation, delta phi, and a gauge transformation, alpha. In other words, Q is the Hamiltonian operator, for a lapse that is beta at infinity, and a shift that is a rotation through delta phi. This means that the partition function can be represented by a Euclidean path integral.

The path integral is over all metrics which at infinity, are periodic under the combination of a Euclidean time translation, beta, a rotation through delta phi, and a gauge rotation, alpha. The lowest order contributions to the path integral for the partition function will come from Euclidean solutions with a U1 isometry, that agree with the periodic boundary conditions at infinity.

The Hamiltonian in general relativity or supergravity, can be written as a volume integral over a surface of constant tau, plus surface integrals over its boundaries.

The volume integral vanishes by the constraint equations. Thus the numerical value of the Hamiltonian, comes entirely from the surface terms. The action can be related to the Hamiltonian in the usual way. Because the metric has a time translation isometry, all dotted quantities vanish. Thus the action is just beta times the Hamiltonian.

If the solution can be foliated by a family of surfaces, that agree with Euclidean time at infinity, the only surface terms will be at infinity.

In this case, a solution can be identified under any time translation, rotation, or gauge transformation at infinity. This means that the action will be linear in beta, delta phi, and alpha. If one takes such a linear action, for the partition function, and applies the standard thermodynamic relations, one finds the entropy is zero.

The situation is very different however, if the solution can't be foliated by surfaces of constant tau, where tau is the parameter of the U1 isometry group, which agrees with the periodic identification at infinity.

The break down of foliation can occur in two ways. The first is at fixed points of the U1 isometry group. These occur on surfaces of even co-dimension. Fixed-point sets of co-dimension-two play a special role. I shall refer to them as bolts. Examples include the horizons of non-extreme black holes and p-branes, but there can be more complicated cases, like Taub bolt.

The other way the foliation by surfaces of constant tau, can break down, is if there are what are called, Misner strings.

To explain what they are, write the metric in the Kaluza Klein form, with respect to the U1 isometry group. The one form, omega, the scalar, V, and the metric, gamma, can be regarded as fields on B, the space of orbits of the isometry group.

If B has homology in dimension two, the Kaluza Klein field strength, F, can have non-zero integrals over two cycles. This means that the one form, omega, will have Dirac strings in B. In turn, this will mean that the foliation of the spacetime, M, by surfaces of constant tau, will break down on surfaces of co-dimension two, called Misner strings.

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 the Misner strings. This modifies the treatment, in two ways. First, the surfaces of constant tau now have boundaries at the fixed-point sets, and Misner strings, as well as the boundary at infinity. This means there can be additional surface terms in the Hamiltonian.

In fact, the surface terms at the fixed-point sets are zero, because the shift and lapse vanish there. On the other hand, at a Misner string, the lapse vanishes, but the shift is non zero. The Hamiltonian can therefore have a surface term on the Misner string, which is the shift, times a component of the second fundamental form, of the constant tau surfaces. The total Hamiltonian, will be the sum of this Misner string Hamiltonian, and the Hamiltonian surface term at infinity.

As before, the action will be beta times the Hamiltonian. However, this will be the action of the space-time, with the neighborhoods of the fixed-point sets and Misner strings removed. To get the action of the full space-time, one has to put back the neighborhoods. When one does so, the surface term associated with the Einstein Hilbert action, will give a contribution to the action, of minus area over 4G, for the bolts and Misner strings. Here G is Newton's constant in the dimension one is considering. The surface terms around lower dimensional fixed-point sets make no contribution to the action. The action of the space-time, will be the lowest order contribution to minus log Z, where Z is the partition function. But log Z is equal to the entropy, minus beta times the Hamiltonian at infinity. So the entropy is a quarter the area of the bolts and Misner strings, minus beta times the Hamiltonian on the Misner strings. In other words, the entropy is the amount by which the action is less than the value, beta times the Hamiltonian at infinity, that it would have if the surfaces of constant tau, foliated the space-time.

This formula for the entropy applies in any dimension and for any class of boundary condition at infinity. Thus one can use it for rotating black holes, in anti de Sitter space. In this case, the reference background is just Euclidean anti de Sitter space, identified with imaginary time period, beta, and appropriate rotation.

The four-dimensional Kerr anti de Sitter solution, was found by Carter, and is shown on the slide. The parameter, a, determines the rate of rotation. When a-l approaches 1, the co-rotation velocity approaches the speed of light at infinity. It is therefore interesting to examine the behavior of the black hole action, and the conformal field theory partition function, in this limit.

To calculate the action of the black hole is quite delicate, because one has to match it to rotating anti de Sitter space, and subtract one infinite quantity, from another.

Nevertheless, this can be done in a well-defined way, and the result is shown on the slide. As you might expect, it diverges at the critical angular velocity, at which the co-rotating velocity, approaches the speed of light.

The boundary of rotating anti de Sitter, is a rotating Einstein universe, of one dimension lower. Thus it is straightforward in principle, to calculate the partition function for a free conformal field on the boundary. Someone like Dowker might have calculated the result exactly. However, as we are only human, we looked only at the divergence in the partition function, as one approaches the critical angular velocity.

This divergence arises because in the mode sum for the partition function, one has Bose-Einstein factors with a correction because of the rotation. As one approaches the critical angular velocity, this causes a Bose-Einstein condensation in modes with the maximum axial quantum number, m.

The divergence in the conformal field theory partition function has the same divergence as the black hole action, at the critical angular velocity. I haven't compared the residues. This is difficult, because it is not clear what three-dimensional conformal field theory one should use on the boundary of four dimensional anti de Sitter. The case of rotating black holes in anti de Sitter five, is broadly similar, but with some differences. One of these is that, because the spatial rotation group, O4, is of rank 2, there are two rotation parameters, a & b. Each of these must have absolute value less than l to the minus one, for the co-rotation velocity to be less than the speed of light, all the way out to infinity. If just one of a & b, approaches the limiting value, the action of the black hole, and the partition function of the conformal field theory, both diverge in a manner similar to the four dimensional case. But if a = b, and they approach the limit together, the action and the partition function, both have the same stronger divergence. Again, I haven't compared residues, but this might be worth doing. It may be that in the critical angular velocity limit, the interactions between the particles of super Yang Mills theory, become unimportant. If this is the case, one would expect the action and partition function to agree, rather than differ by a factor of four thirds, as in the non rotating case. I now turn the case of Nut charge. For asymptotically locally flat metrics in four dimensions, the reference background is the self-dual Taub Nut solution. The Taub bolt solution, has the same asymptotic behavior, but with the zero-dimensional nut fixed point, replaced by a two-dimensional bolt. The area of the bolt is 12 pi N squared, where N is the Nut charge. The area of the Misner string is minus 6 pi N squared.

That is to say, the area of the Misner string in Taub bolt, is infinite, but it is less than the area of the Misner string in Taub nut, in a well-defined sense. The Hamiltonian on the Misner string, is N over 8. Again the Misner string Hamiltonian is infinite, but the difference from Taub nut, is finite. And the period, beta, is 8pi N. Thus the entropy, is pi N squared. Note that this is less than a quarter the area of the bolt, which would give 3 pi N squared. It is the effect of the Misner string that reduces the entropy.

We would like to confirm the effect of Misner strings on entropy, by seeing what effect they have on conformal field theories, on the boundary of anti de Sitter space. For this purpose we constructed versions of Taub nut and Taub bolt, with a negative cosmological constant. The Taub nut anti de Sitter metric is shown on the transparancy. The parameter E, is the squashing of the three-sphere at infinity. If E=1, the three spheres are round, and the metric is Euclidean anti de Sitter space. However, if E is not equal to one, the metric can not be matched to anti de Sitter space at large distance. Each value of E, therefore, defines a different sector of ALADS metrics. This is an important point, which did not seem to have been realized by Chamblin et al. One can also find a family of Taub bolt anti de Sitter metrics, with the same asymptotic behavior. These are characterized by an integer, k, and a positive quantity, s. These determine the asymptotic squashing parameter, E, and the area of the bolt, A-. K is the self-intersection number of the bolt. Thus the spaces do not have spin structure if k is odd. At infinity, the squashed three sphere has k points identified around the U1 fiber. This means that the reference background, is Taub nut anti de Sitter, with k points identified. If k is greater than one, the reference background will have an orbifold singularity at the nut. However, as I said earlier, I shall take it that such singularities are harmless. To calculate the action, one matches the Taub bolt solution on a squashed three sphere, to a Taub nut solution. To do this, one has to re-scale the squashing parameter, E, as a function of radius. The surface term in the action, is the same asymptotically for Taub nut and Taub bolt. Thus the action comes entirely from the difference in volumes. For k greater than one, the action is always negative, while for k=1, it is positive for small areas of the bolt, and negative for large areas. This behavior is similar to that for Schwarzschild anti de Sitter space, and might indicate a phase transition in the corresponding conformal field theory. However, as I will argue later, there are problems with the ADS, CFT duality, if k=1. On the other hand, our results seem to indicate that there will be no phase transition, if more than one point is identified around the fiber. It will be interesting to see if this is indeed the case, for a conformal field theory on an identified squashed three-sphere.

In these Taub bolt anti de Sitter metrics, one can calculate the area of the Misner string, and the Hamiltonian surface term. Both will be infinite, but if one matches to Taub nut anti de Sitter on a large squashed three-sphere, the differences will tend to finite limits. As in the asymptotically locally flat case, the entropy is less than a quarter the area of the bolt, because of the effect of the Misner string.

One can also calculate the entropy from the partition function, by the usual thermodynamic relations. The mass will be given by taking the derivative of the action with respect to beta. This is equal to the Hamiltonian surface term at infinity. The mass or energy, is the only charge that is constrained in the ensemble. The nut charge is fixed by the boundary conditions, and so doesn't need a La-grange multiplier. Thus the entropy is beta M, minus I. This agrees with the entropy calculated from the bolts and Misner strings, showing our definition, is consistent.

Formally at least, one can regard Euclidean conformal field theory on the squashed three sphere, as a twisted 2+1 theory, at a temperature, beta to the minus one. Thus one would expect the entropy to be proportional to beta to the minus two, at least for small beta. This has been confirmed by calculations by Dowker, of the determinants of scalar and fermion operators on the squashed three sphere, for k=1.

Dowker has not so far calculated the higher k cases, but one would expect that these would have similar leading terms, but with beta replaced by beta over k. The next leading order terms in the determinant, are beta to the minus one, log beta. No terms like this appear in the bulk theory, so if there really is an ADS, CFT duality in this situation, the log beta terms have to cancel between the different spins.

In fact, the scalar and fermion log beta terms will cancel each other, if there are twice as many scalars as fermions. This would be implied by super symmetry, suggesting that super symmetry is indeed necessary for the ADS, CFT duality.

The Misner string contributions to the entropy are of order beta squared. Thus Dowker's calculations will have to be extended to this order, to k greater than one, to fermion fields with anti periodic boundary conditions, and to spin one fields. All this is quite possible, but it will probably require Dowker to do it.

One might ask, how can a conformal field theory on the Euclidean squashed three sphere, correspond to a theory in a spacetime of Lorentzian signature. The answer is that, unlike the Schwarzschild anti de Sitter case, one has to continue the period, beta, to imaginary values. This makes the spacetime periodic in real time, rather than imaginary time. One gets a 2+1 rotating spacetime, rather like the Goedel universe, with closed time like curves. Although field theory in such a spacetime, may seem pathological, it can be obtained by analytical continuation, and is well defined despite the lack of causality. It is interesting that the analytically continued entropy, is negative, suggesting that causality violating spacetimes, are quantum suppressed. However, it is probably a mistake, to attach physical significance, to the Lorentzian conformal field theory.

To sum up, I discussed the ADS, CFT duality in two new contexts. That of rotating black holes and that of solutions with nut charge. I showed how gravitational entropy can be defined in general. The partition function for a thermodynamic ensemble can be defined by a path integral over periodic metrics. The lowest order contributions to the partition function will come from metrics with a U1 isometry, and given behavior at infinity. The entropy of such metrics will receive contributions from horizons or bolts, and from Misner strings, which are the Dirac strings of the U1 isometry, under Kaluza Klein reduction. One would like to relate this gravitational entropy, to the entropy of a conformal field theory on the boundary. For this reason, we considered a new class of asymptotically locally anti de Sitter spaces. Other people have investigated the Maldacena conjecture, by deforming the compact part of the metric, but this is the first time deformed anti de Sitter boundary conditions, have been considered.

We studied Taub bolt anti de Sitter solutions, with Taub nut anti de Sitter, as the reference background. The entropy we obtained obeyed the right thermodynamic relations, and had the right temperature dependence, to be the entropy of a conformal field theory, on the squashed three sphere. Because the Taub bolt solutions for odd k, do not have spin structures, this may indicate that the anti de Sitter, conformal field theory correspondence, does not depend on super symmetry.

I will end by saying that gravitational entropy, is alive and well, 34 years on. But there's more to entropy, than just horizon area. We need to look at the nuts and bolts.

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