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.
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Notes
See Appendix B for a derivation.
We set \(k_B=1\) in this paper.
Analogous to the single Laplace transform, this is the \(\kappa \)-deformed generalization of the double Laplace transform.
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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
The \(\kappa \)-exponential and \(\kappa \)-logarithm for the case \(0<\kappa <1\) can also be written as
Some of the basic properties of the \(\kappa \)-exponential are as follows:
For a real number r, the following property holds
Similarly, the \(\kappa \)-logarithm has following basic properties:
For a real number r, the following property holds
For any x, \(y\in {\mathbb {R}}\) and \(|\kappa |<1\), the \(\kappa \)-sum is defined as
which is equivalent to
The two important relations based on \(\kappa \)-sum, useful for our discussion are:
Finally, we define \(\kappa \)-Laplace transform and its inverse as
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
\(\mathcal {Z_\kappa }\) is the normalization constant called the \(\kappa \)-deformed partition function. The \(\kappa \)-deformed entropy gives
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
Since \(\sum _i\rho ^i_\kappa =1\) and \(\sum _i\rho ^i_\kappa E_i=\langle E\rangle =U\), we have
This gives
which is equal to Eq. (2.4) of the main text.
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Kumar, N. Relativistic correction to black hole entropy. Gen Relativ Gravit 56, 47 (2024). https://doi.org/10.1007/s10714-024-03228-6
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DOI: https://doi.org/10.1007/s10714-024-03228-6