284. Kerr/CFT: A paradigm to understand the entropy of real black holes?

This is a guest blog post by Geoffrey Compere from the UCSB.

Recently, a correspondence has been proposed between extremal black holes and a chiral two-dimensional CFT. This correspondence might be of interest for real astrophysical black holes. For example, the massive black hole in the Milky Way whose nickname is GRS 1915+105 has a rotation to mass square ratio

and therefore is only 2% away from the theoretical extremality bound. Since the innermost stable circular orbit is close to the horizon for these black holes, there is a possibility than the details of the physics close to the horizon might be encoded in astrophysical observations.

The cause of the outbursts of X-rays of the supermassive black hole at the center of the galaxy here depicted is not understood but the rapidity with which they rise and fall indicates that they are occurring near the event horizon. The Kerr/CFT has the aim of explaining universal features of such phenomena. (Image copyright – Smithonian Institution)

Right now, the focus is on the theory side. The correspondence is still in his early stages and many open questions remain. I will first review the context in which the new idea fits, the Kerr/CFT itself and I will then comment on some future directions.

Some background : the story of black hole entropy in short

In 1973, Bardeen, Carter and Hawking found that black holes obey laws similar to the ones of thermodynamics. The celebrated computation of the temperature of black holes by Hawking in 1975 showed that there was more than an analogy: black holes are thermodynamical objects with a temperature and therefore an entropy. This entropy was predicted some years before by Bekenstein in order to ensure that the second law of thermodynamics be not violated.

The idea that thermodynamical laws are not fundamental, but can be derived from an underlying microscopic theory goes back to the nineteen century. Indeed, Bolztmann used the idea that gases are composed of atoms and molecules which obey statistical mechanics to derive the laws of thermodynamics. One can then understand that a main result that all theories of quantum gravity should obtain is being able to reproduce the Bekenstein-Hawking entropy.

This test was passed with success both for string theory and loop quantum gravity. On the string theory side, Strominger and Vafa derived in 1995 the entropy of a class of BPS black holes using configurations of D-branes. However, it was understood 3 years later that the success of this test was in fact not surprising. It was realized by Strominger in 1998 that any unitary and UV complete theory of gravity which contains these black holes should predict the correct Bekenstein-Hawking entropy. More precisely, all black holes whose near-horizon geometry contains an factor are described by a conformal field theory (CFT) as found by Brown and Henneaux in 1986, and the entropy of black holes is then a function of only the energy of the corresponding state in the CFT and the central charge of the CFT. One lesson of this result is that the point of string theory is that it is unitary and UV complete; the Bekenstein-Hawking entropy then follows.

These universality results where generalized to the Schwarzschild black hole by Solodukhin and Carlip in 1998/99 but somehow pieces of the puzzle were missing since for example the central charge was depending on an arbitrary parameter while you would expect it to be fixed by the theory.

The Kerr/CFT

The near-horizon geometry of extremal Kerr black holes (NHEK geometry) was studied some time ago by Bardeen and Horowitz. This geometry does not contain a factor but in fact contains a “warped” factor where the warping means that the circle has a non-trivial Hopf fibration on the base.

A first ingredient in the Kerr-CFT program consists in generalizing the Strominger-Brown-Henneaux argument for the existence of a CFT in to the warped case relevant for the extremal Kerr geometries. The techniques to handle that problem in covariant language were created by Brandt and Barnich and I developed them further in my thesis with Glenn Barnich. One has to write down boundary conditions for the fields and at the same time ensure that they obey are invariant under some asymptotic symmetry algebra of interest, here one copy of the Virasoro algebra – this is why the CFT is chiral: there is only one copy. The central charge can then be computed, it is given by

The second ingredient to the correspondence is the derivation of a non-zero temperature at the horizon even though the black hole is extremal and his Hawking temperature vanishes. The argument uses quantum field theory in the curved NHEK geometry and consists in finding the equilibrium vacuum state. It turns out that it has some temperature: !

The entropy of a CFT at fixed temperature is then given by the Cardy formula, and this matches with the Bekenstein-Hawking entropy

Beyond the Kerr/CFT

Many groups have generalized the result to other extremal black holes in various theories. It turns out that the result is very stable: for all cases studied so far, there is an extremal black hole/chiral CFT correspondence.

Many open questions remain on the theory side: what happens for near-extremal black holes? What happens with higher order curvature corrections? What is the vacuum state of the CFT? Is the NHEK stable? Why warped have only one Virasoro, it there a second one? … and so on …

And on the experimental side: can one use Kerr-CFT to explain the results of Gutowski and Chandrashekar of scattering of particles in the Kerr geometry? Does universal characteristics exist for jet emission explained by the dual CFT? What experimental data could be explained by universal features like a CFT ? ….

There is still work to be done! Great news for physicists!