^{1}

^{2}

We provide a simple way for calculating the entropy of a Schwarzschild black hole from the entropy of its Hawking radiation. To this end, we show that if a thermodynamic system loses its energy only through the black body radiation, its loss of entropy is always 3/4 of the entropy of the emitted radiation. This proposition enables us to relate the entropy of an evaporating black hole to the entropy of its Hawking radiation. Explicitly, by calculating the entropy of the Hawking radiation emitted in the full period of evaporation of the black hole, we find the Bekenstein-Hawking entropy of the initial black hole.

In the early 70’s, based on some thermodynamic analyses, it was shown that black holes (BHs) possess nonzero entropy [

In spite of these seminal achievements, and after more than four decades since the realization of BHs as thermodynamic systems, the question about microscopic origin of the BH entropy has not given a well-established answer. Nonetheless, there have been interesting progresses towards answering this question, which have employed somehow different approaches. The first derivation of BH entropy has been in the context of Euclidean canonical [

In this paper, we calculate the entropy of a Schwarzschild BH, based on the same method which Hawking used to calculate the temperature of this BH. To be more specific, we will focus on the physics of the emitted radiation, in order to find the entropy of the BH. For convenience, we use the

In order to provide intuitions for the reader, we first consider an unknown thermodynamic system, which our knowledge about it is only through its black body radiation. We want to find the temperature and the entropy of such a system. However, finding the temperature is an easy job by studying the spectrum of its thermal radiation. On the other hand, finding the entropy is composed of two steps: 1) relating the entropy loss of the system and the entropy carried outside by the radiation 2) finding the entropy of the black body radiation emitted from the beginning of the process of radiation till the end of it. The latter would be the state at which all possible internal energy that the system can radiate is finished. This can be assumed to be the state with zero entropy. Therefore, by the second step, we find the entropy of the overall radiation; and then by the first step, the initial entropy of the system can be found.

The first step of the two above can be taken by considering the proposition below, which is a general and interesting one^{1}, enabling us to relate the entropy loss of a black body to the entropy of its black body radiation:

Proposition 1. If a thermodynamic system at equilibrium loses its internal energy only through its black body radiation into the vacuum, then its loss of entropy is

(approximately) ^{2}.

Proof. Let us denote the internal energy, temperature, and entropy of our thermodynamic system by U, T, and S respectively. Consider that the system loses an infinitesimal amount of internal energy

On the other hand, we consider the black body radiation emitted in the vicinity of the surface of the thermodynamic system, (approximately) as a canonical ensemble of photon^{3} gas at thermal equilibrium with the system. The internal energy, temperature and entropy of this photon gas can be denoted by

In the equilibrium,

amount of radiation to be emitted, the equilibrium temperature gradually varies; but this relation for the entropy variations remains always valid. So, denoting the entropy loss of the system during the radiation process by

W

In Proposition 1, the factor 4/3 is an approximated factor. The approximation can originate from different issues: 1) The black body radiation around the surface of an object has some preferred directions of propagation. Hence, considering it as a perfect canonical ensemble is an approximation. 2) Interactions between these particles are ignored. 3) It has been explicitly assumed that the radiation propagates into the vacuum, while one can consider an ambient matter. 4) The particles in the radiation are considered to be massless. Among the four approximations above, the first and second ones would be cancelled out from the analysis together with the factor 4/3, as will be described later. Hence, they are irrelevant to the analysis. Nonetheless, details of these approximations can be found in Refs. [

Another important, but relevant hidden assumption in Proposition 1 is the flatness of the background metric; in the derivation of the relation (2) for canonical photon gases, mode expansions in flat spacetime has been used. Hence, in order to make the proposition applicable for the BH thermodynamic system, we assume that in the process of evaporation of a Schwarzschild BH, propagation of Hawking radiation is an adiabatic process. In other words, entropy of the Hawking radiation does not change by its propagation towards the infinity. Notice that this assumption is not in contradiction with the increase in the entropy through the emission process at the moment of creation of the radiation. Therefore, we can perform the calculations at spatial infinity, and then equate the resulted entropy with the BH entropy. So, the first step in calculation of the entropy for our BH is taken.

The second step is to calculate the entropy of the Hawking radiation emitted during the whole process of evaporation of the BH. Consider a Schwarzschild BH with the initial mass

temperature which is related to the remaining mass M of the black hole by

[

In order to find the entropy in the thin shell mentioned above, let us study the Hawking radiation within it. At the time when the Hawking temperature of the radiation within the shell is some

in which

The

in which

At the end, the following remarks are necessary to be mentioned.

・ In Ref. [

of BH to be

namical properties of photon gases, he finds the entropy of Hawking radiation to be approximately 4/3 of the entropy of initial BH. In spite of his splendid work, he does not well appreciate that this increasing of the entropy via black body radiation by a factor of

infinity to be

・ As it was advertised before, the factor appearing in the Equation (4) is the same factor in Proposition 1, with exactly the same approximations about directional motion of interacting particles. Hence, this factor cancels out of the analysis, justifying irrelevance of studying these approximations in details. This cancellation clarifies the difference of our analysis with Zurek’s analysis: if one wants to calculate the entropy of the black body radiation from the assumed entropy loss of the object, then this factor and approximations in its derivation would play a crucial role. It is simply because the entropy of the radiation would be eventually the entropy loss of the object multiplied by this factor. In contrast, if one calculates the entropy of the radiation by studying its statistical mechanics, and wants to find the entropy loss of the object using her results, then this factor cancels out. Explaining in another way, the goal of the former work has been elaborating the entropy of the Hawking radiation, which would not result to a clear-cut outcome. In contrast, the aim of the analysis here is the entropy of the BH, which yields the area of it as an accurate result. If we pay enough attention to this subtle but very important issue, then the difference between the two works would become clear, and the irrelevance of accurate determination of the factor 4/3 would be justified. We strongly encourage the reader to contemplate this issue, in order to find the logic behind this document. However, the independence of the analysis from the specific choice of this factor is a positive aspect of the analysis; it makes the analysis to be applicable when one goes beyond the approximation of canonical non-interacting photon emission.

・ One might be skeptical about triviality of the analysis, because by invoking the Hawking temperature

black hole, one can integrate the black hole entropy to find the final result. Nonetheless, the proposed derivation of Bekenstein-Hawking entropy in this paper is nontrivial. It is because of nontriviality of physical steps for calculating the entropy of the whole radiation, relating it to the entropy of BH, and finally using the peculiar temperature of Hawking radiation accompanied by geometrical nontrivial relation between energy and surface area of the BH. Moreover, the analysis sheds light on similar approaches such as the brick-wall model [

・ The analysis shows that invoking the statistical mechanics of the quantum fields around a BH to describe its microstates, eventually would lead to the analysis presented here.

・ It can be easily checked that considering the gray body factor for the BHs would not alter our analysis, because Equation (4) remains unchanged.

・ At the end, we mention that the analysis presented here has been motivated by studying number of photons vs. the entropy of Hawking radiation elaborated in Ref. [

To conclude, we emphasized that one can find the entropy of a black body object solely from studying its radiation, if its temperature is known during the time interval of the complete evaporation. Based on this issue, we promoted a method to derive the Bekenstein-Hawking entropy of a Schwarzschild black hole by analyzing the statistical mechanics of its Hawking radiation at infinity. The main inputs of the calculation, in addition to the Hawking temperature, were adiabatic propagation, conservation of the energy, and the specific constraint between the mass and radius of the black hole. Not surprisingly, the final result matches the area law of the black hole entropy in General Relativity.

SA thanks Prof. Golshani for his supports. KH thanks high energy group at University of Barcelona for their hospitality. We would also like to thank members of quantum gravity group at IPM, Roberto Emparan, and Hrvoje Nikolic, for their inspiring and helpful discussions. Besides, we thank Ms. Pileroudi for her kind helps in the period of developing the paper. This work has been supported by the Allameh Tabatabaii Prize Grant of National Elites Foundation of Iran and the Saramadan grant of the Iranian vice presidency in science and technology.

Aghapour, S. and Hajian, K. (2016) Black Hole Entropy from Entropy of Hawking Radiation. Journal of Modern Physics, 7, 2063-2070. http://dx.doi.org/10.4236/jmp.2016.715181