The extended uncertainty principle inspires the R\'{e}nyi entropy

TL;DR

This paper derives the Rényi entropy for black holes using the extended uncertainty principle, linking non-extensivity to quantum states, and explores thermodynamic properties and evaporation times of excited black holes.

Contribution

It introduces a novel approach connecting EUP to Rényi entropy for black holes and analyzes thermodynamic behavior of excited states.

Findings

Rényi entropy depends on black hole quantum states.

Temperature can have a positive minimum even when entropy vanishes.

Evaporation time of excited black holes is characterized.

Abstract

We use the extended uncertainty principle (EUP) in order to obtain the R\'{e}nyi entropy for a black hole (BH). The result implies that the non-extensivity parameter, appeared in the R\'{e}nyi entropy formalism, may be evaluated from the considerations which lead to EUP. It is also shown that, for excited BHs, the R\'{e}nyi entropy is a function of the BH principal quantum number, i.e. the BH quantum excited state. Temperature and heat capacity of the excited BHs are also investigated addressing two phases while only one of them can be stable. At this situation, whereas entropy is vanished, temperature may take a non-zero positive minimum value, depending on the value of the non-extensivity parameter. The evaporation time of excited BH has also been studied.

Equations

$$S_T=\frac{1}{1-q}\sum _{i=1}^W(P_i^q-P_i),$$

$$\mathcal{S}=\frac{1}{\delta }\ln (1+\delta S_T),$$

$$\Delta x\Delta p\ge [1+\beta (\Delta x{)}^2+\eta (\Delta p{)}^2+\gamma ].$$

$$\Delta x\Delta p\ge [1+\frac{\beta \pi }{4}(\Delta x{)}^2],$$

$$\Delta E\ge \frac{1}{\Delta x}[1+\frac{\beta \pi }{4}(\Delta x{)}^2].$$

$$\Delta A_{min}\ge 8\pi [1+\frac{\beta \pi }{4}(\Delta x{)}^2].$$

$$\Delta A_{min}\simeq 8\pi \lambda [1+\frac{\beta }{4}A],$$

$$\frac{dS}{dA}=\frac{\Delta S_{min}}{\Delta A_{min}}=\frac{b}{8\pi \lambda [1+\frac{\beta }{4}A]},$$

$$S=\frac{b}{2\pi \lambda \beta }\ln [1+\frac{\beta }{4}A].$$

$$\lambda =\frac{b}{2\pi },$$

$$S=\mathcal{S},\delta =\beta .$$

$$\frac{A}{4}=4\pi \left(M^2-\frac{n}{2}\right),$$

$$S={\beta }^{-1}\ln \left[1+4\beta \pi \left(M^2-\frac{n}{2}\right)\right].$$

$$T=\frac{1+4\beta \pi \left(M^2-\frac{n}{2}\right)}{8\pi M},$$

$$C=\frac{\partial M}{\partial T}=\frac{8\pi M^2}{2\beta \pi \left(2M^2+n\right)-1},$$

$$T=\frac{T_B}{2}[3-\frac{n}{2M^2}],$$

$$\frac{dM}{dt}=-16\pi M^2\sigma T^4(M),$$

$$t=-\frac{4(4\pi {)}^3}{\sigma }{\int }_M^0\frac{M^{2}dM}{(1+4\beta \pi \left(M^2-\frac{n}{2}\right){)}^4},$$

$$\displaystyle\tilde{t}=\frac{3\tan^{-1}[\frac{\sqrt{8\pi\beta}M}{\sqrt{2-4\pi\beta n}}]}{\sqrt{1-2\pi\beta n}}+\frac{1}{(4\pi\beta(n-2M^{2})-2)^{3}}\bigg{(}$$

$$\displaystyle 2\sqrt{4\pi\beta}M\Big{(}(4\pi\beta)^{2}[3n^{2}+16nM^{2}-12M^{4}]+12$$

$$\displaystyle-16\pi\beta(3n+8M^{2})\Big{)}\bigg{)},$$

$$t=\frac{4(4\pi {)}^3}{3\sigma }{M}^3,$$

Full text

I Introduction

In one hand, uncertainty principle inspires Bekenstein entropy, and indeed, various generalized uncertainty principles (GUP) add different modifications to Bekenstein entropy Bek ; maju1 ; ahmed1 ; ahmed2 ; ahmed3 . On the other hand, Bekenstein entropy is a non-extensive entropy measure, a property which motivates some physicists to consider it as a suitable candidate for the Tsallis entropy in calculating the Rényi entropy kom ; prd ; rhd ; non21 ; prdinf . In fact, due to the long-range nature of gravity pla , the use of generalized entropy formalisms such as those introduced by Tsallis tsa and Rényi ren has recently been taken into consideration kom ; prd ; rhd ; non21 ; prdinf ; Tavayef ; smm .

A long-range interacting system with $W$ discrete states while each state $i$ has probability $P_{i}$ may follow a power probability distribution instead of the ordinary distribution pla ; tsa ; ren . Working with $G=c=k_{B}=\hbar=\frac{1}{4\pi\epsilon_{0}}=1$ (Planck units), where $k_{B}$ denotes the Boltzmann constant, the Tsallis entropy of such system is defined as tsa

[TABLE]

where $q$ is an unknown parameter pla . It is worthwhile mentioning that one can reach $S_{T}=\frac{A}{4}$, where $A$ denotes the horizon of system (boundary), by applying the Tsallis entropy definition (1) to the gravitational systems 5 . This result is in agreement with the cosmological studies in which authors assumed $S_{T}=\frac{A}{4}$ in calculating Rényi entropy kom ; prd ; rhd ; non21 ; prdinf written as pla

[TABLE]

in which $\delta\equiv 1-q$. In addition to the successes of this entropy in describing cosmos kom ; prd ; rhd ; prdinf , it can also be combined with the Verlinde’s hypothesis ver to give us a theoretical basis for the MOND theory and its modifications non21 . In this paper, we are going to show that EUP can also lead to the emergence of Rényi entropy. In addition, relation between $\delta$ (and thus $q$) and the quantum mechanical parameter appeared in EUP is also derived. After getting the mentioned aim in the next section, we study some thermodynamic properties of excited BHs meeting Rényi entropy in the third section. The last section is devoted to summary and concluding remarks.

II From EUP to Rényi entropy

In the framework of the high energy physics, such as quantum gravity, various GUP and EUP are derived prdgup ; eup1 ; eup2 ; eup3 ; Bek , which can generally be written as prdgup

[TABLE]

Here, $\gamma$ is positive and depends on the expectation values of $p$ and $x$ prdgup ; norazi . Bearing the fact that the minimum uncertainty is obtainable for $\gamma=0$ in mind norazi , we consider the $\eta=\gamma=0$ case which leads to EUP written as prdgup ; eup1 ; eup2 ; eup3 ; Bek ; prd0

[TABLE]

where $\beta$ is positive and independent of the values of $\Delta x$ and $\Delta p$ prdgup . The non-zero minimal values of $\Delta x$ and $\Delta p$, called $\Delta x_{0}$ and $\Delta p_{0}$, respectively, are obtainable whenever $\beta>0$ prdgup . The above EUP affects the early universe thermodynamics norazi , and in general, there are deep connections between EUP and GUP and $i$) the dispersion relation dis1 ; dis2 , $ii$) the Chandrasekhar and Jeans limits and the dark energy problem prd0 ; mnras . More studies on the outcomes of employing GUP in various branches of physics can also be found in Bek ; maju1 ; ahmed1 ; ahmed2 ; ahmed3 ; deltag ; deltax1 ; deltax2 ; deltax3 ; deltae2 ; deltae1 ; pedram .

Whenever the EUP (4) is valid, one can write $\Delta E\approx\Delta p$ for the uncertainty of the particle energy ($\Delta E$) ahmed1 ; ahmed2 ; ahmed3 ; deltax3 ; deltae2 ; deltae1 , leading to

[TABLE]

In BH physics, whenever a BH with area $A$ absorbs or emits a quantum particle with energy $E$ and size $R$, then the changes in the BH area follows the $\Delta A\geq 8\pi ER$ relation deltax1 ; deltax2 ; deltax3 . Since the size of a quantum particle cannot be less than the uncertainty in its position deltaamin ; deltax2 ; deltax3 , one reaches $\Delta A_{min}\geq 8\pi E\Delta x$ for a quantum particle deltax2 ; deltax3 ; ahmed3 . Combining this result with Eq. (5), we reach at

[TABLE]

As it has been argued in Refs. deltax1 ; deltax2 ; deltax3 ; ahmed3 , one can write $(\Delta x)^{2}\approx\frac{A}{\pi}$ and insert it in Eq. (6) to obtain

[TABLE]

where $\lambda$ is an unknown coefficient fixed later ahmed3 .

Therefore, $\Delta A_{min}$ is the minimum changes in the boundary $A$ whenever EUP (4) is valid. It is also obvious to assume that the corresponding entropy changes is also minimum and equal to one bit of information $\Delta S_{min}=b=\ln 2$ ver ; deltaamin ; ahmed3 . The above argument motivates us to write

[TABLE]

leading to

[TABLE]

In the limit of $\beta\rightarrow 0$, the Bekenstein entropy ($\frac{A}{4}$) should be recovered Bek ; maju1 ; ahmed1 ; ahmed2 ; ahmed3 ; deltaamin ; serd which yields

[TABLE]

whereby we get

[TABLE]

Thus, one can realize that $i$) EUP may allow us to employ the Rényi entropy, and in this situation, $ii$) the EUP parameter $\beta$ determines the value of the non-extensivity parameter $\delta$. Finally, it is also worthwhile mentioning that the value of $\lambda$ obtained in Eq. (10) is the same as that of the previous work by other authors ahmed3 in which the $\beta=\gamma=0$ case has been studied.

III Applications to the excited BHs

For excited BHs, i.e. the BHs which emitted a large amount of Hawking quanta, the recent Bohr-like approach to BH quantum physics in Bohr ; Bohr2 ; Bohr3 permits to write the Bekenstein entropy in terms of the BH quantum level as Bohr3

[TABLE]

where $M$ is original BH mass and ${n}$ is the BH principal quantum number if the BH is seen as ${gravitational}$ ${atom}$, see e.g., Bohr ; Bohr2 ; Bohr3 . We indeed recall that, the intuitive but general belief Bohr ; Bohr2 ; Bohr3 : the BHs result in highly excited states representing both the ${Hydrogen}$ ${atom}$ and the ${quasi-thermal}$ emission in quantum gravity, has been shown to be correct, because the Schwarzschild BH results in somewhat similar to the historical semi-classical hydrogen atom introduced by Bohr in 1913, see Bohr ; Bohr2 ; Bohr3 for more details. Thus, by using Eqs. (2) and (11), the Rényi entropy becomes function of the BH principal quantum number, i.e., the BH excited state, given as

[TABLE]

Accepting the $E=M$ relation and bearing the $T=\frac{\partial E}{\partial S}$ relation in mind, one reaches

[TABLE]

for temperature of the Hawking radiation in this formalism. As a check, the temperature $T_{B}=\frac{1}{8\pi M}$, obtained by using the Bekenstein entropy, is also recovered at the appropriate limit $\beta\rightarrow 0$. In this manner, for $n\rightarrow n_{max}=2M^{2}$ Bohr3 , we have $S\rightarrow 0$ and $T\rightarrow T_{B}$ independent of the value of $\beta$. Moreover, the heat capacity evaluated as

[TABLE]

includes a singularity at $n=n_{max}$ whenever $\beta=\frac{1}{8\pi M^{2}}\equiv\beta_{0}$. For this critical value of $n$, we have $C>0$ ($C<0$) for $\beta>\beta_{0}$ ($\beta<\beta_{0}$), and in neighboring of this point one can write $C\sim\frac{1}{\beta-\beta_{0}}$. This means that the $\beta>\beta_{0}$ ($\beta<\beta_{0}$) phase can be stable (unstable) nonex ; smbh ; callen . For the critical value $\beta_{0}$, one can write (14) as

[TABLE]

indicating $T>T_{B}$ for $n<n_{max}$ which means that $T_{B}$ is the minimum possible temperature at this situation. More studies on the non-excited BHs ($n=0$) as well as their thermodynamics in the framework of the Rényi entropy can also be found in nonex ; nonex1 ; nonex2 .

Now, let us look at the radiation of excited BH as a black body radiation, and write smbh ; nonex1

[TABLE]

where $\sigma$ denotes the Stefan-Boltzman constant, and additionally, we assumed that the $r=2M$ relation is still valid Bohr3 . Therefore, by using Eq. (14), the time that a BH needs to lose its mass $M$ can be evaluated as

[TABLE]

leading to

[TABLE]

where $\tilde{t}\equiv 3\sigma t(\frac{\beta}{4\pi})^{\frac{3}{2}}$. Whenever $\beta=0$ or even $n=n_{max}$, one obtains

[TABLE]

the evaporation time of a Schwarzschild BH nonex1 . The reason is clear, for both addressed cases, Eq. (14) reduces to the ordinary temperature of Schwarzschild BH Bohr3 . It is also worthwhile mentioning some similarities and differences between the properties of an excited Schwarzschild BH in the Bekenstein and Rényi entropy formalisms. $i$) Both the Bekenstein (12) and Rényi (13) entropies are vanished for $n=n_{max}$, $ii$) in the framework of the Bekenstein entropy, temperature is always independent of $n$, and $iii$) heat capacity is always negative, independent of the value of $n$, in the regime of the Bekenstein entropy (the $\beta=0$ limit of (15)), while $C$ can be positive depending on the values of $n$ and $\beta$. For example, whereas $n=n_{max}$, we have $C>0$ if $\beta>\beta_{0}$.

IV Summary and Concluding Remarks

The Rényi entropy for a BH has been obtained through EUP. By using the recent Bohr-like approach to BH quantum physics, we have also shown that, for excited BHs, the Rényi entropy is a function of the BH principal quantum number, i.e. the BH quantum excited state. Some thermodynamic properties of excited BH meeting Rényi entropy have also been addressed. The results show that although there are two phases at $n\rightarrow n_{max}$, only the phase with $\beta>\beta_{0}$ can be stable. Moreover, whenever $n\rightarrow n_{max}$, then entropy is vanished, and temperature takes its non-zero possible minimum value ($T_{B}$) if $\beta=\beta_{0}$.

Evaporation time of the excited BHs has also been studied. Although we addressed the $n=n_{max}$ case in our study, in reality, the maximum possible value of $n$, namely $N_{max}$, may be limited as $N_{max}=n_{max}-1$ whenever the Planck mass and the Planck distance are approached 33 ; Bohr3 . In this manner, non of the Bekenstein and Rényi entropies are vanished when the maximum value of $n$ is taken by the excited BH. It leads to the interesting results in the Rényi entropy framework, in agreement with the third law of thermodynamics, $i$) based on Eq. (13), entropy takes its minimum value which is non-zero, and $ii$) from Eq. (14), the minimum of temperature can be positive or even greater than $T_{B}$ depending on the value of $\beta$.

Acknowledgment

The work of H. Moradpour has been supported financially by Research Institute for Astronomy & Astrophysics of Maragha (RIAAM).