Relativistic correction to black hole entropy

Abstract

In this paper, we study the relativistic correction to Bekenstein–Hawking entropy in the canonical ensemble and isothermal–isobaric ensemble and apply it to the cases of non-rotating BTZ and AdS-Schwarzschild black holes. This is realized by generalizing the equations obtained using Boltzmann–Gibbs (BG) statistics with its relativistic generalization, Kaniadakis statistics, or $\kappa$-statistics. The relativistic corrections are found to be logarithmic in nature and it is observed that their effect becomes appreciable in the high-temperature limit suggesting that the entropy corrections must include these relativistically corrected terms while taking the aforementioned limit. The non-relativistic corrections are recovered in the $\kappa \rightarrow 0$ limit.

Appendices

Appendix A: Kaniadakis statistics ($\kappa$-statistics): a brief review

$\kappa$-statistics is a relativistic generalization of the Boltzmann–Gibbs (BG) statistics. The $\kappa$-entropy emerges from the relativistic generalization of the Boltzmann–Gibbs–Shannon (BGS) entropy and generates power law-tailed distribution which in the limit $\kappa \rightarrow 0$ reproduces the ordinary exponential distribution. This $\kappa$-generalized statistics has been applied successfully to a wide range of problems. Formally, it is a one-parameter deformation of the ordinary exponential and logarithmic functions as follows

$$\begin{aligned}{} & {} e_\kappa (x)=(\sqrt{1+\kappa ^2x^2}+\kappa x)^{\frac{1}{\kappa }} \end{aligned}$$

(A.1)

$$\begin{aligned}{} & {} \ln _\kappa (x)=\frac{x^\kappa -x^{-\kappa }}{2\kappa } \end{aligned}$$

(A.2)

The $\kappa$-exponential and $\kappa$-logarithm for the case $0<\kappa <1$ can also be written as

$$\begin{aligned}{} & {} e_\kappa (x)=e\bigg (\frac{1}{\kappa }{{\,\textrm{arcsinh}\,}}(\kappa x)\bigg ) \end{aligned}$$

(A.3)

$$\begin{aligned}{} & {} \ln _\kappa (x)=\frac{1}{\kappa }\sinh \bigg (\kappa (\ln (x))\bigg ) \end{aligned}$$

(A.4)

Some of the basic properties of the $\kappa$-exponential are as follows:

$$\begin{aligned}&e_\kappa (x)\in {\mathbb {C}}^\infty ({\mathbb {R}}) \end{aligned}$$

(A.5)

$$\begin{aligned}&\frac{d}{dx}e_\kappa (x)>0 \end{aligned}$$

(A.6)

$$\begin{aligned}&\frac{d^2}{dx^2}e_\kappa (x)>0 \end{aligned}$$

(A.7)

$$\begin{aligned}&e_\kappa (-\infty )=0^+ \end{aligned}$$

(A.8)

$$\begin{aligned}&e_\kappa (+\infty )=+\infty \end{aligned}$$

(A.9)

$$\begin{aligned}&e_\kappa (0)=1 \end{aligned}$$

(A.10)

$$\begin{aligned}&e_\kappa (-x)e_\kappa (x)=1 \end{aligned}$$

(A.11)

For a real number r, the following property holds

$$\begin{aligned}{}[e_\kappa (x)]^r=e_{\kappa /r}(rx) \end{aligned}$$

(A.12)

Similarly, the $\kappa$-logarithm has following basic properties:

$$\begin{aligned}&\ln _\kappa (x)\in {\mathbb {C}}^\infty (\mathbb {R^+}) \end{aligned}$$

(A.13)

$$\begin{aligned}&\frac{d}{dx}\ln _\kappa (x)>0 \end{aligned}$$

(A.14)

$$\begin{aligned}&\frac{d^2}{dx^2}\ln _\kappa (x)<0 \end{aligned}$$

(A.15)

$$\begin{aligned}&\ln _\kappa (0^+)=\infty \end{aligned}$$

(A.16)

$$\begin{aligned}&\ln _\kappa (1)=0 \end{aligned}$$

(A.17)

$$\begin{aligned}&\ln _\kappa (+\infty )=+\infty \end{aligned}$$

(A.18)

$$\begin{aligned}&\ln _\kappa (1/x)=-\ln _\kappa (x) \end{aligned}$$

(A.19)

For a real number r, the following property holds

$$\begin{aligned} \ln _\kappa (x^r)=r\ln _{r\kappa }(x) \end{aligned}$$

(A.20)

For any x, $y\in {\mathbb {R}}$ and $|\kappa |<1$, the $\kappa$-sum is defined as

$$\begin{aligned} x{{\mathop {\oplus }\limits ^{\kappa }}}y=x\sqrt{1+\kappa ^2y^2}+y\sqrt{1+\kappa ^2x^2} \end{aligned}$$

(A.21)

which is equivalent to

$$\begin{aligned} x{{\mathop {\oplus }\limits ^{\kappa }}}y=\frac{1}{\kappa }\sinh ({{\,\textrm{arcsinh}\,}}(\kappa x)+{{\,\textrm{arcsinh}\,}}(\kappa y)) \end{aligned}$$

(A.22)

The two important relations based on $\kappa$-sum, useful for our discussion are:

$$\begin{aligned} e_\kappa (x{{\mathop {\oplus }\limits ^{\kappa }}}y)&=e_\kappa (x)e_\kappa (y) \end{aligned}$$

(A.23)

$$\begin{aligned} \ln _\kappa (xy)&=\ln _\kappa (x){{\mathop {\oplus }\limits ^{\kappa }}}\ln _\kappa (y) \end{aligned}$$

(A.24)

Finally, we define $\kappa$-Laplace transform and its inverse as

$$\begin{aligned} F_\kappa (s)&={\mathcal {L}}_\kappa \{f(t)\}(s)=\int \limits _0^\infty f(t)[e_\kappa (-t)]^s dt \end{aligned}$$

(A.25)

$$\begin{aligned} f(t)&={\mathcal {L}}_\kappa ^{-1}\{F_\kappa (s)\}(t)=\frac{1}{2\pi i}\int \limits _{c-i\infty }^{c+i\infty } \frac{F_\kappa (s)[e_\kappa (t)]^s}{\sqrt{1+\kappa ^2t^2}}ds \end{aligned}$$

(A.26)

The ordinary Laplace transform and its inverse are recovered in the limit $\kappa \rightarrow 0$.

Appendix B: relation between $\kappa$-deformed entropy and $\kappa$-deformed partition function

We start with the $\kappa$-distribution given as

$$\begin{aligned} \rho ^i_\kappa =\frac{e_\kappa (-\beta E_i)}{{\mathcal {Z}}_\kappa } \end{aligned}$$

(B.1)

$\mathcal {Z_\kappa }$ is the normalization constant called the $\kappa$-deformed partition function. The $\kappa$-deformed entropy gives

$$\begin{aligned} S_\kappa&=-\sum _i\rho ^i_\kappa \ln _\kappa \rho ^i_\kappa =-\sum _i \rho ^i_\kappa [\ln _\kappa (e_\kappa (-\beta E_i)){{\mathop {\oplus }\limits ^{\kappa }}}(-\ln _\kappa {\mathcal {Z}}_\kappa )] \end{aligned}$$

(B.2)

$$\begin{aligned}&=-\sum _i \rho ^i_\kappa [-\beta E_i{{\mathop {\oplus }\limits ^{\kappa }}}(-\ln _\kappa {\mathcal {Z}}_\kappa )] \end{aligned}$$

(B.3)

Here, we have used the properties of $\kappa$-deformed log, $\ln _\kappa (xy)=\ln _\kappa x{{\mathop {\oplus }\limits ^{\kappa }}}\ln _\kappa y$ and $\ln _\kappa (\frac{1}{x})=-\ln _\kappa x$. Using $\kappa$-sum, we obtain the above relation as

$$\begin{aligned} S_\kappa =\sum _i \rho ^i_\kappa [\beta E_i\sqrt{1+(\kappa \ln _\kappa {\mathcal {Z}}_\kappa })^2+\ln _\kappa {\mathcal {Z}}_\kappa \sqrt{1+(\kappa \beta E_i)^2}] \end{aligned}$$

(B.4)

Since $\sum _i\rho ^i_\kappa =1$ and $\sum _i\rho ^i_\kappa E_i=\langle E\rangle =U$, we have

$$\begin{aligned} S_\kappa&=\beta \bigg (\sum _i\rho ^i_\kappa E_i\bigg )\sqrt{1+(\kappa \ln _\kappa {\mathcal {Z}}_\kappa )^2}+\ln _\kappa {\mathcal {Z}}_\kappa \sqrt{\bigg (\sum _i\rho ^i_\kappa \bigg )^2+\bigg (\kappa \beta \sum _i\rho ^i_\kappa E_i\bigg )^2} \end{aligned}$$

(B.5)

$$\begin{aligned}&=\beta U\sqrt{1+(\kappa \ln _\kappa {\mathcal {Z}}_\kappa )^2}+\ln _\kappa {\mathcal {Z}}_\kappa \sqrt{1+(\kappa \beta U)^2} \end{aligned}$$

(B.6)

This gives

$$\begin{aligned} S_\kappa =\ln _\kappa {\mathcal {Z}}_\kappa {{\mathop {\oplus }\limits ^{\kappa }}}\beta U \end{aligned}$$

(B.7)

which is equal to Eq. (2.4) of the main text.