A Statistical Derivation of Bekenstein-Hawking Entropy for Schwarzschild Black Holes
Naman Kumar




School of Basic Sciences, Indian Institute of Technology, Bhubaneswar, 752050
namankumar5954@gmail.com


§ ABSTRACT 
    A microscopic derivation of the Bekenstein-Hawking entropy for the Schwarzschild black hole was presented earlier by using a non-trivial phase space. It was argued that the Schwarzschild black hole behaves like a 1D quantum mechanical system. In this paper, we show that if we assume the phase space to obey the holographic principle and take the microscopic particles inside the quantum gravitational system to be ideal bosonic gas, we can derive the Bekenstein-Hawking entropy. The assumption of the phase space to follow the holographic principle such that the Schwarzschild black hole behaves as a 2D system is very much in the spirit of our understanding of black holes than their behavior as a 1D system. However, the argument suggests that the black hole be treated as a system with the equation of state P=ρ.
    

  
*Keywords:
Bekenstein-Hawking Entropy, Statistics of Black Holes, Holographic principle, 2D phase space.


A Statistical Derivation of Bekenstein-Hawking Entropy for Schwarzschild Black Holes
    Naman Kumar
     
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§ INTRODUCTION

It is well known that black holes have thermodynamic behaviour. Hawking<cit.> first showed that the area of black holes is a non-decreasing function of time

    dA/dt≥0

Bekenstein<cit.> argued that this similarity can be attached to the thermodynamic concept of entropy and proposed the equation

    S=η k_BA/l^2_p

where η was an undetermined dimensionless constant. When Hawking<cit.> derived the temperature of the black hole, the speculations turned robust, and the entropy was obtained as

    S=k_BA/4l^2_p

thereby fixing the dimensional constant η. This is the famous Bekenstein-Hawking entropy. This result is very interesting as it suggests that the entropy scales as the area rather than the conventional volume. This equation further led to the idea of the holographic principle<cit.>, which suggests that the information about the volume of space is stored on its boundary. The best realization of the holographic principle is the AdS/CFT correspondence<cit.>. In<cit.>, a microscopic derivation of the Bekenstein-Hawking entropy for Schwarzschild black hole was presented by assuming a non-trivial phase space. The independent quantized modes were calculated by gc^3Vdp/2π Għ^2 rather than the conventional Vd^3p⃗/(2πħ)^3, thereby effectively treating the 3+1 Schwarzschild black hole as a 1D quantum mechanical system. For example, in the conventional case, the logarithm of the partition function of a photon gas system is given by<cit.>

    lnΞ=2V/(2πħ)^3∫ln(1-e^-β cp)d^3p=π^2/45c^3ħ^3V/β^3

It was further argued that the black hole be viewed as a system with the equation of state P=ρ. This equation of state has also been attached with black hole thermodynamics earlier (see <cit.> and references therein). Each point in the phase space represents a state of the system. In the spirit of the holographic principle, we take the phase space to be 2D such that the state of the system is essentially contained completely on the 2D phase space. The paper is organized as follows. We first review the earlier derivation presented in<cit.>. Then we derive the Bekenstein-Hawking entropy formula using a statistical approach. Finally, we discuss the motivation behind statistical derivation based on the emergent nature of gravity and information limit due to the holographic principle. 


§ A BRIEF REVIEW OF EARLIER DERIVATION

The following assumptions were made: First, the particles inside the quantum gravitational system are bosonic and massless. Second, they obey the energy-momentum relation ϵ=pc with p=|p⃗| and third, the calculation of quantized modes is done by gc^3Vdp/2π Għ^2. Next, the Schwarzschild black hole of radius R in 3+1 dimensions is modelled as a system consisting of these particles. The logarithm of the partition function, in this case, is given by

    ln Z=-gc^3V/π Għ^2∫_0^∞ln(1-e^-β cp)dp=gπ c^2V/6Għ^2β

Then we obtain the expressions for entropy and energy as

    E=-∂/∂βZ=gπ k_B^2c^2/6Għ^2VT^2



    S=k_B(ln Z+β E)=gπ k_B^2c^2/3Għ^2VT

The pressure is given by

    P=k_BT∂ln Z/∂ V=gπ k_B^2c^2/6Għ^2T^2

Comparing with ρ=E/V gives the equation of state as P=ρ. The Komar mass as the gravitational source corresponds to (ρ+3P)V, which gives

    M=4E=2gπ k_B^2c^2/3Għ^2VT^2=2gπ k_B^2c^2/3Għ^24/3π R^3T^2

Taking M to be the mass of the black hole and choosing g=9 along with R=2GM/c^2 for a Schwarzschild black hole gives the Hawking temperature, T=ħ c^3/8π Gk_BM. Substituting into Eq.(<ref>), we get the Bekenstein-Hawking entropy.



§ A STATISTICAL DERIVATION BASED ON 
HOLOGRAPHIC PRINCIPLE

We make the following assumptions about the microscopic particles inside a quantum gravitational system. First, the particles are ideal bosonic gas with mass m such that their partition function follows Bose-Einstein statistics and they obey the energy-momentum relation of an ideal gas which is given by ϵ=p^2/2m where p=|p⃗|. Second, the calculation of independent quantized modes follows the holographic principle such that the states of this system are completely specified by a 2D phase space and is evaluated by gV/l_pd^2p⃗/(2π)^2ħ^2=gV√(c^3/Għ)d^2p/(2πħ)^2, where g a dimensionless constant is introduced to include any other degrees of freedom and the gravitational system is effectively 2D with area A=V/l_p. We model the Schwarzschild black hole of radius R in 3+1 dimensions as a system consisting of these particles.
The logarithm of the partition function(Z) is

    ln Z=-√(c^3/Għ)2gV/(2πħ)^2∫_0^∞ln(1-e^-p^2/2mk_BT)d^2p⃗

It is important to remark that if we consider the black hole to consist of a massive particle (ideal bosonic gas in this case), then 2D phase space follows inevitably, as can be verified from Eq.(<ref>) otherwise, we can not recover the Bekenstein-Hawking entropy. Eq.(<ref>) can be re-written as

    ln Z =-√(c^3/Għ)2gV/(2πħ)^2∫_0^∞ 2πln(1-e^-p^2/2mk_BT)pdp=√(c^3/Għ)2gV/(2πħ)^2π^3m/3β

where β=1/k_BT and V=4/3π R^3. We, therefore, get the expressions for the energy and entropy as

    E=-∂/∂βln Z=√(c^3/Għ)2gπ^3k_B^2/3(2πħ)^2mVT^2


    S=k_B(ln Z+β E)=√(c^3/Għ)4gπ^3k_B^2/3(2πħ)^2mVT

The pressure is given by

    P=k_BT∂ln Z/∂ V=√(c^3/Għ)2gπ^3m/3(2πħ)^2k_B^2T^2

Comparing with ρ=E/V gives the equation of state as P=ρ.
Komar mass as the source then gives M=4E. Thus, we get

    Mc^2=4E=√(c^3/Għ)8gπ^3k_B^2/3(2πħ)^2mVT^2

Taking gm=9ħ/cl_p=9m_p where m_p is Planck mass, and V=4/3π R^3 such that R=2GM/c^2 for a Schwarzschild black hole, we finally obtain using Eq.(<ref>)

    T=ħ c^3/8π Gk_BM

which is the Hawking temperature. Putting this value in Eq.(<ref>) we get

    S=k_Bπ R^2c^3/Għ=k_BA/4l_p^2

where l_p^2=Għ/c^3. Thus, we obtained the Bekenstein-Hawking entropy by assuming that the phase space follows the holographic principle. On comparing Eq.(<ref>) and Eq.(<ref>) we get the relation

    2TS=M

This is the Smarr formula for the 3+1 dimension Schwarzschild black hole. It is important to emphasize here the importance of Eq.(<ref>). Deducing this equation in this microscopic derivation is necessary to derive the correct form of the Bekenstein-Hawking Entropy; otherwise, we will end up with the wrong coefficient even if we obtain the correct Hawking temperature. Another thing to note is that the equation of state P=ρ is critical to get the correct Smarr formula, and as discussed earlier, the failure to obtain the correct Smarr formula leads to the wrong coefficient. This suggests that the black hole is to be viewed as a system with the equation of state w≡P/ρ=1, as already argued in <cit.> and references therein. The mass m of each microscopic particle is then interpreted as m=m_p and the dimensionless constant g=9.


§ CONCLUSION AND DISCUSSION

In this paper, we successfully derived the Bekenstein-Hawking Entropy of a Schwarzschild black hole by considering it to consist of ideal bosonic gas at the microscopic level with a phase space that obeys the holographic principle. It is remarkable that the correct Bekenstein-Hawking temperature can be exactly derived from such a simplistic picture. Moreover, in<cit.>, the authors reproduced correct Bekenstein-Hawking entropy by taking w≡P/ρ=1 although the reasons were not fully understood. In<cit.>, the authors managed to exactly "derive" this result. Our derivation in a different setting strongly supports this idea. The phase space is taken to be two-dimensional such that all the information about the bulk system is contained in it. This essentially means treating the black hole as a two-dimensional system at the microscopic scale. The earlier derivation presented in<cit.> treated black holes as a 1D system where the quantum particles were taken to be equivalent to a photon gas. Both these pictures do not seem to be very realistic. On the other hand, treating the black hole as a 2D system is very much in the spirit of our understanding of black holes. The assumption of quantum particles to be ideal boson gas is much more realistic too. It is also widely believed that gravity and spacetime have emergent origins. Verlinde<cit.> argued that gravity is an entropic force arising from the tendency of a microscopic theory to maximize its entropy. This shows that there is a maximum limit on the amount of information in a volume of space. This is also the result of the holographic principle. In this view, it becomes important to microscopically derive the Bekenstein-Hawking entropy. This strongly motivated us to study a statistical derivation based on the holographic principle. This derivation also suggests that the black hole be treated as a system with the equation of state w≡P/ρ=1. A series of works called holographic cosmology has been studied based on this equation of state<cit.>. We are hopeful that these results will inspire further investigation.


§ CONFLICT OF INTEREST

The author declares no conflict of interest.

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