Maxwell-Boltzmann type Hawking radiation

TL;DR

This paper proposes that black hole Hawking radiation follows a Maxwell-Boltzmann distribution rather than the traditional Planck spectrum, based on the distinguishability of horizon area quanta.

Contribution

It introduces a novel derivation showing Hawking radiation spectrum as Maxwell-Boltzmann, challenging the standard Bose-Einstein assumption.

Findings

Hawking radiation spectrum is Maxwell-Boltzmann, not Bose-Einstein.

Black hole horizon area quanta are distinguishable, affecting entropy calculations.

Revises understanding of black hole radiation spectrum based on quantum area quantization.

Abstract

Twenty years ago, Rovelli proposed that the degeneracy of black hole (i.e. the exponential of the Bekenstein-Hawking entropy) is given by the number of ways the black hole horizon area can be expressed as a sum of unit areas. However, when counting the sum, one should treat the area quanta on the black hole horizon as distinguishable. This distinguishability of area quanta is noted in Rovelli's paper. Building on this idea, we derive that the Hawking radiation spectrum is not given by Planck radiation spectrum (i.e., Bose-Einstein distribution) but given by Maxwell-Boltzmann distribution.

Equations

$$A_i=A_1,A_2,A_3,A_4,A_5,A_6....$$

$$A=\sum _i{N}_i{A}_i,$$

$$\Delta A=-A_i$$

$$\Delta Q=-kT\frac{A_i}{4}$$

$$f_i=\frac{kT}{h}\frac{A_i}{4}$$

$$Q=N!\prod _{i=1}^{\infty }\frac{d_i^{N_i}}{N_i!}$$

$$A=\sum _{n=1}^{\infty }{N}_i{A}_i$$

$$G\equiv \ln Q+\alpha \left[A-\sum _{n=1}^{\infty }{N}_i{A}_i\right],$$

$$\ln Q(\cdots ,N_{i-1},N_i,N_{i+1},\cdots )=N_i\ln d_i+\ln N!$$

$$-(\cdots +\ln N_{i-1}!+\ln N_i!+\ln N_{i+1}!+\cdots )$$

$$\ln Q(\cdots ,N_{i-1},N_i-1,N_{i+1},\cdots )=(N_i-1)\ln d_i+\ln (N-1)!-(\cdots +\ln N_{i-1}!$$

$$+\ln (N_i-1)!+\ln N_{i+1}!+\cdots )$$

$$0=\frac{\partial G}{\partial N_i}=\ln Q(\cdots ,N_{i-1},N_i,N_{i+1},\cdots )-\ln Q(\cdots ,N_{i-1},N_i-1,N_{i+1},\cdots )-\alpha A_i$$

$$=\ln d_i+(\ln N!-\ln (N-1)!)-(\ln N_i!-\ln (N_i-1)!)-\alpha A_i$$

$$\frac{N_i}{N}=\frac{d_i}{e^{\alpha A_i}}$$

$$N_i=\frac{d_i}{e^{h{f}_i/(kT)}-1}$$

$$\frac{N_i}{N}=\frac{d_i}{e^{h{f}_i/(kT)}}=\frac{d_i}{e^{A_i/4}}$$

$$\sum _i\frac{N_i}{N}=\sum _i{d}_i{e}^{-A_i/4}$$

Full text

1 Background

According to loop quantum gravity [1, 2, 3], the eigenvalues of the area operator are quantized, and the black hole area, as much as any area, is the sum of these eigenvalues. For example, let us say that we have the following area eigenvalues (i.e., the unit areas):

[TABLE]

Then, the black hole area $A$ must be given by the following formula:

[TABLE]

where the $N_{i}$s are non-negative integers. Here, we can regard the black hole as having $\sum N_{i}$ partitions, each of which has one of the $A_{i}$ as its area.

Using the discreteness of the area, in 1996, Rovelli [4] proposed that the degeneracy of black hole (i.e., the exponential of the Bekenstein-Hawking entropy) is given by the number of ways the black hole horizon area can be expressed as a sum of unit areas. However, when counting the sum, one should treat the area quanta on the black hole horizon as distinguishable. This distinguishability of area quanta is noted in Rovelli’s paper: one should treat quanta as distinguishable if they have fixed locations. Indeed, area quanta have fixed locations on the black hole horizon.

In [5] we derived a selection rule for the Hawking radiation using elementary derivation of Bose-Einstein distribution. The selection rule was that upon an emission of a photon by a black hole, the horizon area $A$ decreases by a unit area. In other words,

[TABLE]

As the Bekenstein-Hawking entropy is given by $S=kA/4$, and we know $\Delta Q=T\Delta S$, the energy decrease is given by

[TABLE]

Since this energy must be equal to the energy of photon emitted (i.e., $\Delta Q=-hf$) the frequency of the photon emitted during the Hawking radiation is given by

[TABLE]

In the next section, we explain why we should use the Maxwell-Boltzmann distribution instead of the Bose-Einstein distribution for the Hawking radiation.

2 Maxwell-Boltzmann distribution

This section closely uses the method presented in the famous quantum mechanics textbook by Griffiths [6]:

Let us say that the unit areas $A_{1},A_{2},A_{3},\cdots$ have degeneracies $d_{1},d_{2},d_{3},\cdots$. Suppose we have a black hole with area $A$ which satisfies $A=\sum_{i}N_{i}A_{i}$ as explained before. For a given configuration ($N_{i}=N_{1},N_{2},N_{3},\cdots$), how many different ways can this be achieved?

Now, recall that the area quanta is distinguishable. Then, the answer is given by

[TABLE]

where $N=\sum_{i}N_{i}$. Recall that we also have the following condition

[TABLE]

To find the most probable configuration $(N_{1},N_{2},N_{3},\cdots)$, we need to maximize $\ln Q$ as follows:

[TABLE]

where $G$ is to be maximized and $\alpha$ is a Lagrange multiplier. Let us maximize it by differentiating with respect to $N_{i}$. First, note

[TABLE]

Then, using $N-1=N_{1}+\cdots+N_{i-1}+N_{i}-1+N_{i+1}+\cdots$, we have

[TABLE]

Using these, we have

[TABLE]

The conclusion is

[TABLE]

On the other hand, if Hawking radiation were given by Bose-Einstein distribution, we know that the Hawking radiation for large photon frequency is given by

[TABLE]

This expression must reduce to (12) for large $f_{i}$. Using (5) we conclude $\alpha=1/4$. Therefore, (12) becomes

[TABLE]

We can check that our calculation is indeed correct. Summing the both sides, we get

[TABLE]

The left-hand side is 1 by the definition of $N$. The right-hand side is also 1 by Domagala-Lewandowski-Meissner formula [7, 8].

3 Discussions and Conclusions

Even though our result that the Hawking radiation follows the Maxwell-Boltzmann distribution is different from the currently accepted Bose-Einstein distribution, it may not be easy to experimentally confirm this even if Hawking radiation is observed, as $e^{A_{i}/4}$ is much bigger than 1. For example, for $A_{1}$, the smallest unit area, $e^{A_{1}/4}$ is about 85 [9].