Abstract
There are several well-established methods for computing thermodynamics in single-horizon spacetimes. However, understanding thermodynamics becomes particularly important when dealing with spacetimes with multiple horizons. Multiple horizons raise questions about the existence of a global temperature for such spacetimes. Recent studies highlight the significant role played by the contribution of all the horizons in determining Hawking’s temperature. Here we explore the Hawking temperature of a rotating and charged black hole in four spacetime dimensions and a rotating BTZ black hole. We also find that each horizon of those black holes contributes to the Hawking temperature. The effective Hawking temperature for a four-dimensional rotating and charged black hole depends only on its mass. This temperature is the same as the Hawking temperature of a Schwarzschild’s black hole. In contrast, the effective Hawking temperature depends on the black hole’s mass and angular momentum for a rotating BTZ hole.
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CS thanks the Saha Institute of Nuclear Physics (SINP) Kolkata for financial support. We thank the reviewer for all the valuable comments and suggestions that helped us to improve the manuscript’s quality.
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Appendices
A Derivation of Eq. (25)
Equation (17) is the exact Dirac equation in curved space-time. Now to solve the equation we apply the Hamilton–Jacobi method for that we take the limit \(\hbar \rightarrow 0\) and consider the equation upto \(O(\hbar )\). Here, also we consider a mass-less charged particle, so in our case, \(m=0\). Now upon substituting the ansatz (24) into the Eq. (17), it becomes evident that within an approximation up to \(O(\hbar )\), we can neglect the spin coefficient \(\omega _\mu ^{\alpha \beta }\). So, we start with an approximated Dirac equation by neglecting the spin coefficient,
If we consider only nonzero tetrad, then the above equation reduces to the following
Four Gamma matrices are
By inserting the values of gamma matrices, we can write the equation (48) as,
where A, B, and C are
Now, using the expression of A, B, C, and D in equation (50), we get
Thus, we arrive at the following four equations,
B Derivation of Eq. (41)
Here, we apply a similar procedure for writing an approximated Dirac equation for a rotating BTZ black hole. The approximated Dirac equation for the rotating BTZ black hole spacetime is then given by,
There are three gamma matrices in three dimensions \(\gamma ^i =(i\sigma ^2,\sigma ^1,\sigma ^3)\). Where \((\sigma ^1,\sigma ^2,\sigma ^3)\) are the three spin Pauli matrices. Now, considering the nonzero tetrads for BTZ black hole, we can write Eq. (54) as,
So we get a set of two equations as follows,
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Singha, C., Nanda, P. & Tripathy, P. Hawking temperature of black holes with multiple horizons. Gen Relativ Gravit 55, 106 (2023). https://doi.org/10.1007/s10714-023-03154-z
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DOI: https://doi.org/10.1007/s10714-023-03154-z