Friday, February 16, 2007 ... /////

Black Saturn

Henriette Elvang (MIT) spoke about the Black Saturns, a work with Pau Figueras and very recently also with Roberto Emparan. Because I teach during the duality seminars (today: angular dependence of the photoelectric effect in advanced QM), she was nice to repeat the talk for me, in a more interactive form, and it was even more interesting than I expected.

A black Saturn is a black hole surrounded by a black ring. The ring's angular momentum creates a force that repels it from the black hole in the middle. The solution will have negative modes signaling instabilities but it is a classical solution anyway. In their case, they construct solutions to five-dimensional pure gravity i.e. Ricci-flat geometries with an unusual structure of horizons.

The basic method to calculate these solutions goes back to a 1917 paper by Hermann Weyl - a paper whose content is described in Wald's book on GR. I didn't know about it but Weyl discovered an early version of the LLM construction 90 years ago.

I find it quite impressive that a mathematician could find such an LLM-like construction in 1917 - the same year when crackpots in less advanced countries, such as Vladimir Lenin, were able to impress whole nations with their dumb leftist ideologies - just a year after general relativity was completed. Recall that in the AdS5 x S5 version of LLM, you want to fill a two-dimensional plane with two colors (black and white, for example). For each such picture, you can construct a solution.

However, as we noticed, there exists a similar construction due to Weyl that one may use to construct various solutions in pure four-dimensional gravity. We have two colors "t" and "phi" that can be used to draw a picture on line. If you fill most of the line by the "phi" color except for a line interval whose color is "t", you obtain the Schwarzschild black hole. The length of the line interval is correlated with the size of the black hole. Evidently, the limit where the length goes to infinity corresponds to an infinitely large black hole - i.e. flat space or Rindler space in different coordinates. A similar limit - a plane divided to black and white - gives you the Penrose pp-wave in the AdS5 x S5 case.

Analogous constructions for five-dimensional gravity were found more recently. They use them to construct solutions of five-dimensional pure gravity.

In five spacetime dimensions, the massive little group is SO(4) whose rank is two which is why you can have two independent angular momenta. Let's look at stationary solutions with one angular momentum only, e.g. J_{12}. The metric will have three Killing vectors - two angles generated by the J_{12} and J_{34} rotations and time translations (that would be true even if you had both angular momenta). That's why the metric won't depend on five coordinates but only two.

It turns out that the Einstein equations for this Ansatz are non-linear but integrable differential equations in two variables. You can use a trick that I recently heard from Zack Guralnik - if I remember well - how to transform non-linear partial differential equations in two variables to linear differential equations for some generating functions of a higher number of variables. The procedure is somewhat analogous to replacing a system of non-linear differential equations for a classical system by the linear Schrödinger equation for its quantization because the generating function is analogous to a wavefunction: such a procedure can surprisingly simplify the calculations in some cases. These integrable systems lead to similar equations as the planar limit of the N=4 theory although this is most likely a mathematical coincidence only.

The relevant five-dimensional version of the LLM-Weyl procedure involves three colors on a line. For a suitable configuration of colors, you may obtain a black Saturn solution.

It is interesting to draw the phase diagram of these solutions. Create a two-dimensional graph and believe me that they can construct a two-parameter family of solutions. The x-axis is the total angular momentum "J" while the y-axis is the total area "A" i.e. the entropy. All points of this graph below a certain line can be identified as black Saturns with different parameters describing the ring and its distance from the black hole.

Black rings themselves are special examples of black Saturns for which the size of the spherical black hole at the center vanishes. In the phase diagram, the family of the pure rings looks like a union of two semi-infinite lines connected with a cusp. The point at the cusp maximizes the entropy and minimizes the angular momentum - but there are two lines along which you can go if you want to lower the entropy or increase the angular momentum. The angle at the cusp is zero.

There exists another, very similar pair of lines in the phase diagram (whose detailed position is however different) corresponding to black Saturns at equilibrium. Note that the horizon of a black Saturn is disconnected: it is made out of a three-sphere (hole) and a Cartesian product of a circle and a two-sphere (ring). These two components can therefore have different chemical potentials for energy (also known as the temperature - in the context of black objects, it is proportional to the surface gravity at the horizon) and different chemical potentials for the angular momentum (also known as the angular velocity). However, some black Saturns happen to have the same angular velocity for both components and the same surface gravity for both components. They are described by the one-dimensional pair of semi-infinite lines connected at a cusp.

We discussed how to get black bi-Saturn, among other possible solutions. You can also imagine a "black bi-Saturn" as a black hole near the Southern pole connected with another black hole near the Northern pole by a negative-tension (repulsion-inducing) cosmic string (that creates an excess angle, unlike the usual deficit angles for positive-tension strings) - and both of them are surrounded by a black ring wrapped near the equator of the Earth. Such a configuration has the same symmetric as the black Saturn and it is conceivable that you can write an exact metric for it, too.

Because the Ricci-flatness for the black Saturn Ansatz defines an integrable system, you may also believe that various other physical questions about this system - such as the perturbations, geodesics, classical worldsheets and worldvolumes of various branes in this geometry etc. - could be exactly solvable, too.

We took the speaker for a dinner. It was the first time when Henriette visited Henrietta's table in the Charles Hotel. ;-)