## June 10, 2004

### String entropy and black hole correspondence

#### Posted by Urs Schreiber

As with the previous entry, this one here is a reply to a question on sci.physics.strings which seems to have problems to propagate through USENET.

‘Mike2’ wrote in news:Mike2.17k9lf-100000@physicsforums.com

It seems clear that strings represent structure, and various quantities are calculated along the worldsheet. So can one calculate the entropy associated with the information contained in the quantities along the string?

Yes, there are many states of the string which have the same energy and taking the logarithm of this number gives you the entropy of the string at that energy.

There are lots of very deep questions associated to this entropy.

One is the so-called string-black hole correspondence. It is generally said that a black hole carries the highest amount of entropy per volume. But a simple calculation shows that this is true only up to a very small size of the black hole. As the black hole shrinks (due to evaporation by means of Hawking radiation) it will become very tiny and at some point the entropy of a highly excited single string of a given mass will be equal to that of the black hole of that mass. For even lower masses the string’s entropy will even be greater than that of the corresponding black hole. The point at which that happens is called the string/black hole correspondence point.

The interesting thing is that, despite the crudeness of the calculations used in this sort of correspondence, it gives an easy way to calculate the correct order-of-magnitude entropy of all kinds of black holes, Schwarzschild, rotating, various charged ones, etc. It also provides a nice heuristic picture of black hole entropy at the correspondence point. One can sort of imagine the different “bits of string” sitting on the horizon and the entropy comes from the different ways in which these bits are connected inside the whole by the string.

Gary T. Horowitz & Joseph Polchinski: A Correspondence Principle for Black Holes and Strings (1996)

and

Thibault Damour & Gabriele Veneziano: Self-gravitating fundamental strings and black-holes (1999)

I have once written a little more detailed description of this correspondence principle on sci.physics.research:

In its more refined form this ‘principle’ amounts to noting that there is a critical excitation energy where a massive string collapses under its own gravity to the size of the order of the string scale (becoming a ‘string ball’) and that precisely at this point its rms radius coincides with its Schwarzschild radius and furthermore all its thermodynamical properties (temperature, entropy, radiation, decay rate) coincide, up to some unknown factors of order unity, with that of a BH of the same mass (e.g. hep-th/9907030).

(I am not sure how this relates to the D5/D1 brane models, which I don’t know well, but I seem to recall that these brane configurations are describable, and are described, by ‘effective long strings’, too.)

Anyway, the string/BH correspondence principle gives rise to a neat mental picture of the BH degrees of freedom which is actually rather similar to the LQG picture of a ‘pierced horizon’. It is roughly the following:

A highly excited string with the large mass $M\gg 1/{l}_{s}$ (${l}_{s}=\sqrt{{\alpha }^{\prime }}$ is the string scale) is in strikingly good approximation a random walk of $n=\sqrt{N}$ steps of step size ${l}_{s}$, where $N$ is the level number of the string, i.e. $N={M}^{2}{l}_{s}^{2}$. It follows that its entropy is to leading order

(1)${S}_{s}\sim n\sim M{l}_{s}\phantom{\rule{thinmathspace}{0ex}}.$

This is quite unlike the entropy dependence of a black hole, which goes as

(2)${S}_{\mathrm{BH}}\sim {M}^{q}$

with $q>1$. But it so happens that at the above mentioned correspondence point, which is reached when the mass of the string becomes the critical value

(3)${M}_{c}=\frac{1}{{g}^{2}}\frac{1}{{l}_{s}}$

($g$ is the string coupling) and where the string collapses under its self-gravity to a ball of diameter $\sim {l}_{s}$, the entropy of the string and that of a BH of the same size coincide. Still, the entropy of a random-walk-like string, even in the collapsed form, has a simple interpretation, it counts the number of decisions one can make while stepping along the random walk.

Now imagine how that collapsed ‘random walk’ looks like: A chain of n segments, each of length ${l}_{s}$ is restricted to lie within a ball whose diameter is also about ${l}_{s}$. A typical such state looks somewhat star-shaped with all the vertices of the random walk on the outside, forming a sphere. This sphere about coincides with the event horizon of a BH which has the same mass as our string. The edges of the random walk cross the interior of this sphere, pierce the horizon, deposit their vertex there, then return to a point near the corresponding antipode and so on, thereby covering the sphere with all n vertices, all about equally spaced (for a typical state). What is the mean area ${A}_{v}$ of the sphere occupied by one such vertex? It is the number of vertices divided by the area of the horizon, i.e.

(4)${A}_{v}\sim n/{l}_{s}^{\left(d-1\right)}\sim n{g}^{\left(d-1\right)}/{l}_{p}^{\left(}d-1\right)\sim 1/{l}_{p}^{\left(d-1\right)}\phantom{\rule{thinmathspace}{0ex}}.$

Here $d$ is the number of spatial dimensions and ${l}_{p}$ is the Planck length, given in string theory by

(5)${l}_{p}^{\left(d-1\right)}={g}^{2}{l}_{s}^{\left(d-1\right)}\phantom{\rule{thinmathspace}{0ex}}.$

It follows that (for instance) in 1+3 dimensions each of the above vertices occupies an area of about a square Planck length of the event horizon.The entropy of this system is, due to the nature of a random walk and by the above formula, proportional to the number of vertices and hence to the area of the event horizon in Planck units.

This is the picture of black hole microstates at the string/BH correspondence point, i.e. for BH that are about to decay into a string state or for strings that are about to become black holes. (In fact the string ball configuration has been used to predict the signature of decaying black holes that may be detected in accelerators one day).

What happens to this crude picture when the mass is increased further? I am not sure how solid the knowledge obout the answer to this question is, but there is a lot of literature about “strings on the stretched horizon”. The basic idea is that once the BH description takes over the above mentioned vertices are somehow frozen on the event horizon. Since the temperature of the string and the Hawking temperature agree at the correspondence point and hence the rates of change of horizon area with mass do, one can show that further quanta of mass that one throws into a BH at correspondence point correctly translates into further string ‘vertices’ appearing on the horizon. But this only holds in the vicinity of the correspondence point. Farther away one has to take into account the fact that the energy of an object near a BH horizon is different when measured by an asymptotically far away observer. When this red-shifting effect is accounted for one can apparently consistently imagine the BH entropy being due to a (very) long string which is lying on the “stretched” event horizon in form of a random walk.

Of course, all this is nowhere near the technical sophistication of D5/D1 brane-system calculations. It is rather like a Bohr-atom model of quantum black holes.

Posted at June 10, 2004 10:24 PM UTC

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