THE INTERIOR STRUCTURE OF A BLACK HOLE

The title of this post is seemingly puzzling: How can we even attempt to describe what happens inside a black hole? Although it may seem incredible, we can use certain general aspects of string theory to try to glimpse the fundamental internal structure of these strange objects.

In this article, we'll delve into the interior of black holes and uncover some aspects of their "fundamental interior components." These components could hold the key to the deepest problem in fundamental physics: establishing the primordial nature of space-time itself.

Compacted dimensions and Kaluza-Klein states

The Standard Model of particle physics explains all the fundamental forces except gravity and is based on the so-called gauge fields. As we saw in this article , physicists Teodor Kaluza and Felix Klein (KK) discovered that the electromagnetic force field arises if we compact an extra dimension into a very small radius. At first glance, the possible existence of new, very small dimensions may seem like an overly speculative idea; however, as we saw in this other article , the concept of dimension is not absolute but depends on the energy at which we analyze a specific physical system. A straight line appears one-dimensional at low energy, but if we analyze it closely at high energy, we will observe that it actually has a thickness and is therefore two-dimensional like a cylinder. In fact, the existence in nature of point-like objects with zero diameter is inconsistent. KK theory implies the existence of a fifth dimension compacted into a very small circle. When examining the dynamics of this five-dimensional space-time, we find something very interesting: for an observer located in the four macroscopic dimensions , the motion of a particle in the fifth dimension is interpreted as the existence of an electric charge.

This implies something fascinating: compact dimensions have a very specific physical effect on the usual dimensions. The electromagnetic force arises from the coupling of the U(1) gauge field to a charged particle, and the latter can be interpreted as the momentum of a particle in a compact dimension. The Standard Model is based on the SU(3)xSU(2)xU(1) gauge symmetry. The first symmetry explains the strong nuclear force, the second the weak nuclear force, and the third, as we have seen, the electromagnetic force. The obvious question then is: Can we describe the rest of the fundamental forces by adding new compact dimensions? After an enormous theoretical effort to answer this question, all attempts lead to the same theory: superstring theory (ST). Since this theory is based on 10 dimensions and our macroscopic Universe has 4 dimensions, there must be 6 compactified dimensions. The way in which these 6 extra dimensions are compactified produces different charge and matter contents and gives rise to the different classes of string theories.

The microstates of extreme black holes

Next, we consider a black hole with total electric charge Q, specifically an extreme black hole with charge Q=M. Charged black holes are described by the Reissner-Nordström metric:

Where:

The event horizon will be positioned at the value of r where the metric diverges. To calculate its position, simply set delta=0:

This black hole has two event horizons: R+ and R- and we can see that for a given value of the mass M the position of the horizons only depends on the charge Q. We are interested in the specific case in which the black hole has the maximum allowed charge: Q=M, this case is called an extremal black hole. In this case R+=R-=M, that is, the black hole has only one horizon. Furthermore, it can be easily shown that in extremal black holes the Hawking radiation is zero and therefore the charges Q are stable.

The Bekenstein-Hawking entropy formula is a universal expression that tells us that the entropy of any black hole with any number of dimensions is: S=A/4G . Where A is the area of the black hole (AN) and G is Newton's gravitational constant. This entropy has to be generated by an enormous number of microstates that must conform the fundamental quantum structure of these AN. The key question we ask in this article is: What is the nature of this enormous number of microstates? Following the work initiated by KK and from the point of view of the use of new compact dimensions, can we try to understand the fundamental quantum structure of black holes? As we will see below, the answer is affirmative and the path to the answer is fascinating.

Charges, solitons and BPS states in string theory

Before continuing, we need to briefly explain three fundamental "ingredients" of string theory that we will need to study black hole microstates: the so-called BPS states, the two types of charges, and the so-called string theory solitons.

The BPS states

The coupling g measures the "strength" with which the strings interact. The main problem we face is that increasing this parameter above a certain value causes the metric to diverge, and we obtain a black hole. However, in the specific case of extreme ANs, we have a certain number of very specific states called BPS states. The main advantage of using these states is that they remain invariant when changing g, so their number is the same at low and high couplings.

The charges of string theory

The electromagnetic force arises when a charged point particle couples to the electromagnetic force field. In CT, point charges do not exist; the charge is concentrated at the two ends of a string. The electromagnetic force in CT arises when a charged string couples to the equivalent of the electromagnetic field in CT: the so-called Kalb-Ramond field . This field can be of two types, giving rise to the two types of charges in CT: NS (Neveu-Schwarz) charges and RR (Ramond) charges . These charges can be either electric or magnetic. Although all this may seem very exotic, it is nothing more than a generalization of the electromagnetic field to an extended, non-point object. A charged string coiled in a compactified dimension is viewed from the macroscopic dimensions as a charged point particle.

Solitons of string theory

Finally, in addition to the usual states corresponding to string vibrations, there are other types of states called solitons. These states are solutions to the low-energy equations of motion and constitute stable configurations that carry a fundamental unit of "magnetic" charge of topological origin.

The five-dimensional extreme black hole in string theory

Next, we'll make the leap from general relativity to quantum gravity using our best tool available: superstring theory. In this article, our primary object of study will be a five-dimensional extreme black hole. This choice is only for simplicity, it should be clear that the study of a four-dimensional black hole can also be carried out following an equivalent process .

Following Kaluza and Klein's original idea but with much more advanced theoretical tools, let's see what happens when compactifying several dimensions. Since superstring theory requires 10 dimensions, we'll use five macroscopic dimensions and five microscopic dimensions. The process we'll follow is as follows:

1) We will compactify 4 dimensions, for example 6, 7, 8 and 9, into a "4D sphere", more specifically, into a 4D torus. In a similar way to how in KK theory we obtain a new field by compactifying an extra dimension, by compactifying a 3-sphere we obtain a new field that we will call H. The action of string theory in 6 dimensions (obtained by compactifying four) contains the following term:

2) We compact dimension 5 into a circle with a radius R much larger than that of the other 4 compacted dimensions. The previous term then becomes:

Now we have three gauge fields with U(1) symmetry: H+, H- and G. H+ is obtained by compactifying dimension 5 into a circle, H- is obtained by compactifying dimensions 6, 7, 8 and 9 and G is the usual KK field associated to the momentum of a particle (string) in dimension 5. Following the purest KK style we now ask: what charges will an observer located in the five macroscopic dimensions observe emanating from the compacted dimensions? Our macroscopic observer will measure three different charges: one associated to the H+ field in the circular dimension 5, another associated to the H- field in the 4 dimensions compacted into a 4D torus and another associated to the momentum of a string in the circular dimension 5, we will call these charges Q1, Q5 and n respectively. The value of the charges Q1 and Q5 will be proportional to the compacted surface from which the H force field emanates. Therefore we have:

The value of the last charge will be proportional to the G field which in turn comes from the movement in dimension 5:

Now we ask ourselves the following question: What is the entropy of this black hole?

To answer this question we must first solve a "small" problem:

The black hole horizon is singular, and therefore we cannot evaluate its surface area to calculate the entropy. However, the value of the parameter alpha, called the "dilaton," can be conveniently adjusted. There is a certain value of alpha for which the sources of Q1 and Q5 cancel each other out and therefore "stabilize." This value is:

By adjusting the dilaton to this value we manage to stabilize the metric on the horizon and obtain the following metric:

This expression seems very complex and intimidating but seen from the 5 macroscopic dimensions where x5 is negligible and taking the case of an extreme NA where the two horizons coincide, that is: r+=r-=ro we simply obtain:

This is the classical metric obtained by general relativity for a five-dimensional extreme NA! The classical five-dimensional extreme NA is a solution to the low-energy equations of superstring theory!

The area of the black hole is:

And therefore the entropy of our black hole is:

Since the momentum P in the compactified fifth dimension is quantized and equals P=n/R we finally have:

Can we find the fundamental microstates that generate this entropy?

Counting the ground states of the black hole

Now we face the key question: What kinds of objects can generate this entropy? Can we identify them using string theory?

As we saw in this article, a black hole is produced when we increase the coupling g above a certain value. Above this value, the metric diverges, and we cannot make concrete calculations. However, because the BPS states do not vary when we increase the coupling, we can count them at low coupling and be sure that as g increases, the AN we obtain has the same number of BPS states. The fundamental question is: Is there any type of object that produces BPS states, contributes a unit of charge, and exists at low coupling? In the case of RR charges, the answer is yes: this object is called a D-brane. In fact, a D-brane is a soliton similar to a magnetic monopole and therefore carries exactly one unit of R charge at low coupling. Therefore, the process we must follow is the following:

1) We start with a loose coupling with several BPS states containing R-charges. The total value of these charges will be Qi. These states can be considered a bound state of several D-branes.

2) We increase the coupling g until we obtain a five-dimensional extreme black hole. Thanks to the invariance with respect to g, the BPS states do not change, and therefore we will have the same value of the total charge Qi.

At this point, the question that remains to be answered is: at low coupling, how many BPS states did our initial charge Qi have? This number of initial states will be those that produce the entropy of our final AN!

In the previous section we saw that our black hole carries 3 types of charges: Q1, Q5 and n. What kind of objects can these charges generate? How many BPS states do they contain at low energy? A 5-Dbrane wound once over the five compacted dimensions carries a charge Q5=1 . A 1-Dbrane (a string) wound once over dimension 5 carries a charge Q1=1 . 1-Dbranes are stuck to the 5-Dbrane but are free to move around the four compacted dimensions. This produces 4Q1 massless bosons, therefore, this configuration carries a total number of BPS states of 4Q1Q5. Finally, a string with momentum in dimension 5 carries a momentum P=n/R. In field theory, for n much larger than R the entropy on a two-dimensional surface is given by the formula:

Where c is a quantity known in field theory as the central charge. The value of the central charge depends on the number of fermion bosons. For 4Q1Q5 bosons, the value of c is 6Q1Q5. Therefore, the final entropy we obtain is:

This is exactly the Bekenstein-Hawking entropy of our black hole!

Left: Recreation of D-branes rolling up several compactified dimensions and producing the charges Q5 and Q1. The vibrations of these extended objects produce the black hole microstates in five dimensions and determine the position of the event horizon. Right: If instead of compactifying the hidden dimensions into a circle or a 4D torus, we compact them into a geometry called a Calabi-Yau manifold, we can obtain the fields and particles of the Standard Model of particle physics.

The fundamental components of space-time

In our five-dimensional black hole, entropy is generated by the vibration of extended five-dimensional objects coiled around dimensions 5, 6, 7, 8, and 9, and one-dimensional objects coiled around dimension 5 (along with momentum strings traveling in dimension 5).

Similarly, it can be calculated that for a more realistic four-dimensional black hole, the entropy is generated by 6D-branes curled up in the six compact dimensions, 5D-branes curled up in five compact dimensions, and 2D-branes curled up in two of the compact dimensions (along with momentum strings traveling in one compact dimension). This shows us something truly fascinating: the configuration of extended objects on the geometry of the compact dimensions determines the microstates and horizon of the black hole. Therefore, a black hole, seen from our macroscopic dimensions, is made up of branes and strings vibrating through the hidden dimensions.

Have we finally found the fundamental components of space-time?

Although it may seem incredible, new theoretical studies seem to indicate that one of the compact hidden dimensions of string theory must be on the micrometer scale and could be detected in upcoming experiments underway.

As we will see in the next article , fascinating times are coming for fundamental physics!

Sources:

Microscopic Origin of the Bekenstein-Hawking Entropy Counting States of Near-Extremal Black Holes The origin of black hole entropy in string theory