Thermal Entropy of Black Hole Spacetime

In classical general relativity, a black hole is depicted as a "black hole" opened in spacetime, from which an object that has been sucked in can never escape. However, the situation changes when quantum field theory is incorporated. As Dr. Hawking pointed out, black holes are not completely black; they are thought to emit thermal radiation consisting of light of various wavelengths. This phenomenon is now known as "Hawking radiation."

Just as measuring the temperature of light emitted from heated iron reveals the temperature of the iron itself, the temperature of Hawking radiation is considered to be equal to the temperature of the black hole. (Note, however, that this is the temperature of a part of the black hole that emits radiation, and not necessarily the temperature of the whole. I will write about this possibility in a future note.)

In this way, a black hole is no longer understood as a spacetime without temperature, but as an entity with thermodynamic properties such as temperature and entropy. The idea that spacetime, which was understood as something that contains matter, can have temperature and entropy like matter itself, is what Dr. Hawking's theory suggested. This brought a completely new understanding of time and space to theoretical physics.

Next, we write the equation stating that the temperature of Hawking radiation is equal to the temperature of the black hole, and then multiply the black hole's mass M by the square of the speed of light to define the black hole's energy. As a result, it is shown that a black hole has the following thermodynamic entropy, which is proportional to the area of its event horizon. (In the following, I will write in a natural unit system where the Planck constant ℏ, the speed of light c, and the Boltzmann constant are set to 1.)

Here, G is the gravitational constant. This entropy is now called "Bekenstein-Hawking entropy" after the two physicists. One is, of course, Dr. Hawking, and the other is Jacob Bekenstein.

When he proposed the thermodynamic entropy of black holes, Bekenstein was a graduate student of Dr. John Wheeler, who coined the name "black hole." At the time, Bekenstein proposed a famous thought experiment: if you throw a cup of hot coffee into a black hole, does the thermodynamic entropy that the cup possessed disappear from the universe? In an era when it was not yet thought that spacetime had thermal properties, Bekenstein proposed a revolutionary hypothesis that a black hole should have thermodynamic entropy proportional to its area, respecting the second law of thermodynamics that entropy does not decrease.

It was Dr. Hawking himself who strongly criticized this idea from Bekenstein, who was a graduate student at the time. Bekenstein argued that black holes have thermodynamic properties based on Dr. Hawking's research results that the area of a black hole never decreases, but the doctor thought that Bekenstein was misinterpreting his results. At the time, Dr. Hawking did not think that spacetime itself had thermal properties.

So Dr. Hawking began calculations to prove that Bekenstein's hypothesis was wrong. However, an irony of fate occurred. Through his own calculations, he discovered that black holes emit thermal radiation. This discovery made Bekenstein's hypothesis seem correct. Furthermore, Dr. Hawking confirmed that there is a proportionality constant of 1/(4G) between entropy and the event horizon area. Today, that entropy is widely known as "Bekenstein-Hawking entropy (BH entropy)."

In recent years, research into quantum gravity theory has progressed, and using string theory and loop quantum gravity, it has become possible to partially derive BH entropy by summing up the number of microstates, just like entropy in statistical mechanics. In particular, in string theory, it has been shown that the statistical mechanical entropy of special black holes matches the BH entropy. This result is attracting attention as an indication of the success of string theory.

However, just because string theory derived BH entropy from the micro level does not mean it is the correct theory. It is certain that a correct quantum gravity theory must derive BH entropy, but the reverse is not necessarily true.

This point can be understood in terms of the renormalization group. Leonard Susskind, a researcher in string theory, developed the following argument. Suppose there is a quantum gravity theory that can accurately explain regions of extremely high energy and momentum, such as the Planck scale. We do not know if it is string theory, loop quantum gravity, or an unknown theory. In any case, assume that some quantum gravity theory exists.

On the other hand, observations and experiments to date have shown that Einstein's general relativity provides a correct description in regions of energy and momentum far lower than the Planck scale. Therefore, a quantum gravity theory must necessarily derive Einstein's theory in the low-energy region. According to the renormalization group explanation, the low-energy effective theory obtained as a result of integrating out the degrees of freedom in the high-energy region must be the Einstein-Hilbert action of general relativity. Furthermore, in the low-energy region, the gravitational constant G must appear in the action. Complex quantum effects in the high-energy region are absorbed into the renormalization effects of G in the low-energy region.

Let's think about it a little more concretely. Let's assume that an accurate quantum gravity theory defines the partition function of the inverse temperature parameter β using some micro degrees of freedom u by the following path integral.

Assuming that the integration parameters of this path integral can be decomposed into momentum p mode degrees of freedom u(p),

we write it as follows. This equation is written as if there is an upper limit to the momentum, but there does not need to be such an upper limit. We perform the integration of high momentum modes and leave the path integral for low momentum modes as follows.

In the effective action that appears at this time, all quantum effects of high momentum modes are included as renormalization in the coupling constants of the low-energy interactions. And the crux of Susskind's argument is that this low-energy path integral should, as an effective theory, become the path integral of general relativity as follows from observations and experiments.

Since the gravitational constant G is measured in the low-energy region, it is theoretically defined using the coupling constant within this effective action. The first term of the action is the famous Einstein-Hilbert action, but a surface term is attached as the second term. This surface term appears because gravitational interaction produces a long-range force. While there is some ambiguity in the specific form of this surface term, its role is common to all. When taking the variation of the Einstein-Hilbert action, which is the volume term, to derive the Einstein equations, this second surface term acts to adjust the action so that no extra conditions that unnecessarily restrict the solutions to the equations of motion emerge from the surface region.

If we believe Susskind's natural assumption that such an effective action appears, the path integral in equation (2) is evaluated as follows. (Note that the calculation itself using Euclidean path integrals in this context had already been performed by Dr. Hawking and his collaborators before Susskind. Susskind's argument provided a renormalization group significance to that calculation.)

Here, it is also shown from the argument of the saddle-point method for path integrals that the following relationship holds between β and the mass of the black hole.

From the black hole mass M, the radius of its event horizon is obtained as follows.

And the area of the event horizon is also given by the following.

Therefore, it is possible to provide the area of the event horizon as a function of β. Here, when calculating the thermodynamic entropy from the partition function Z(β), the BH entropy is properly derived from equation (3) as follows.

To obtain this result, Susskind did not use the details of quantum gravity theory. He derived this without specifying superstring theory or loop quantum gravity theory. Therefore, deriving BH entropy is merely a necessary condition for a correct quantum gravity theory, and not a sufficient condition to guarantee the correctness of a specific quantum gravity theory.