THE STRING-BLACK HOLE TRANSITION

planck
Aug 17
6 min read

Recently, at a fundamental physics conference, one of the fathers of string theory, the great physicist Leonard Susskind, was asked to provide a simple explanation for a basic audience of a fundamental feature of string theory. In this article, we will briefly describe Susskind's surprising explanation: with just four very simple formulas and a couple of straightforward concepts, he is able to explain one of the deepest and most surprising features of string theory: the so-called "string-black hole correspondence."

The parameters of a string

In string theory, fundamental extended objects are described by a set of physical quantities: the coupling constant (g), which determines how strongly strings interact with each other, the fundamental string length (ls), and the mass (M). In this article, we will simply consider four fundamental relations, common to all types of string theories:

1st) Relationship between Newton's constant (G), the coupling constant (g) and the length of the string (ls):

(1)

2nd) Schwarzschild Radius (Rs):

(2)

3º) Schwarzschild radius/chord length:

(3)

4th) Entropy of the string:

(4)

Suppose we have a set of strings and we have the ability to slowly and continuously vary the value of their total length L. If we start decreasing

the value of L we will reach a point where the total length of the strings is less than the Schwarzschild radius (RS). At that moment the set of strings "collapses" and a black hole is formed. This will happen when the length of the string reaches Rs, more specifically, when the Schwarzschild radius per unit length of string Rs/ls (3) is of the order of unity. Similar to the vibrations of a guitar string, strings have different "vibration frequencies". The higher the "vibration frequency" the higher the mass corresponding to that vibration state. Since the beginning of string theory it has been known that there are massive vibrations of strings whose Schwarzschild radius is less than Rs/ls and therefore these states will produce black holes.

What we want to know is: at what point does this critical transition point between string and black hole occur?

Strings and black holes

In 1993, Susskind was invited by the Rutgers group to give a seminar on strings and black holes. By then, it was already known that black holes radiated energy and therefore had to have temperature and entropy. Specifically, the total entropy of a black hole is given by the famous Bekenstein-Hawking expression:

This entropy must come from an enormous number of "microstates" corresponding to vibrations of some fundamental object or objects. What could these objects be? Susskind was convinced that this entropy could only come from the motions or vibrations of certain fundamental objects that were compressed into the horizon. Of course, Susskind was referring to vibrational states of strings, the fundamental objects contemplated in the promising (super)string theory.

But how could the dynamics of these objects be calculated? Then he had an idea: if we start from an initial black hole with total entropy S, formed from very massive vibrations of strings, and we begin to slowly reduce the coupling constant (g) without changing the entropy (what is known in thermodynamics as an adiabatic curve), then below a certain value of g the horizon will disappear and we will have only a set of free strings with the same entropy as the original black hole.

As we reduce the value of g, there will come a point where the horizon disappears, and we will have only one set of vibrating strings. The question is: can we identify and explain what happens at that very transition point?

Therefore if we can calculate the entropy of the final set of free strings we can know the total entropy S of the original black hole.

The black hole-string transition

The transition point from a black hole to a set of free strings will occur when Rs/ls=1. Taking expression (3) we have that this will happen when: 1=Mg2ls, that is, when M=1/g2ls. Below we draw this line on a graph where we represent the variation of M with respect to g (red line):

The red line represents all possible critical transition points, that is, all points where Rs/ls=1 holds.

Below, we draw the adiabatic curves associated with a black hole. These curves are hyperbolas that represent all the points where entropy remains constant (purple curves). The point we're looking for will be at the intersection of the red curve and the constant-entropy curves.

The question we ask ourselves now is: What happens to entropy just beyond the transition point? From that point on, we will have a free rope, so entropy will be independent of g (free ropes do not interact). This means that from the transition point on, entropy will be practically a constant straight line:

If we start with a black hole of mass Mo and coupling go and start decreasing g keeping the entropy constant following, for example, the upper adiabatic curve, the transition point will be:

The transition point occurs at the green cross in the figure. At that point, Rs/ls = 1 and Mg has not changed relative to Mog, so the following holds:

Solving the two previous equations we obtain that for the common point of both graphs the following is fulfilled:

Taking into account expressions (1) and (4) we arrive at the result:

This is the Bekenstein-Hawking entropy! We have obtained the same result Hawking achieved through rigorous semiclassical analysis simply by using a few fundamental "string" expressions! It should be noted that this calculation is only approximate (it does not take into account the 1/4 factor), however, it serves to easily understand certain concepts of string theory and clearly points to a "string" explanation of the microstates that make up a black hole.

Entropy and area of a black hole

The famous Bekenstein-Hawking equation tells us that the entropy of a black hole is proportional to the area of the horizon. This characteristic is very strange: it's as if the interior of the black hole didn't exist and all the microstates were stored on the surface of the horizon. The question we now ask is: could this strange behavior of black holes be an indication of the "string" origin of a black hole's microstates?

To answer the previous question, we'll perform a very simple calculation. We only need to know two simple general expressions in string theory:

Where L is the total length of the string and Ss is the entropy of the string.

Combining these two expressions we obtain the expression (4) that we saw previously:

Let's consider that the horizon is indeed filled with vibrating strings, and therefore, we can speak of the horizon as a surface with fields and energy. Then, as with any surface, any particle incident on it can be scattered or absorbed. The cross section of a surface is a value used to calculate the probability that a particle incident on it will be scattered. Let's consider a massless scalar particle (the simplest possible particle) incident on the surface of the black hole, and suppose that said surface is filled with free strings. What is the probability that the incident particle will collide with a string and be scattered? A string following a random path on the surface of the black hole can be represented as follows:

The blue lines represent the random path traced by the string and the purple square represents the particle that strikes the surface perpendicularly.

Each of the grids has length ls and the total length of the string is L. Then we have that the total mass of the string is: M=L/ls2. Now imagine a scalar particle of length 1 (i.e. length ls) falling into the black hole. This particle will be represented as a grid of area ls2 passing perpendicularly through the x,y plane. The probability of the incident particle (purple square) colliding with an individual string is proportional to the area of the square ls2 and to the interaction capacity of the string, i.e., g2. Therefore

The individual cross-section is g2ls2. The total cross-section of the surface will be the sum of all the individual sections: