The physical meaning of Sharma-Mittal entropy and its relevance to Giddings' hypothesis on black hole evaporation

Recent research has shown that generalized entropies are suitable for describing the thermodynamic properties of gravitational systems. In 2016, Giddings proposed that the spectrum of Hawking radiation, which characterizes evaporating semiclassical black holes, originates from a quantum "atmosphere" that lies beyond a black hole's horizon. Subsequent studies confirmed that this proposal is possible if the Bekenstein-Hawking entropy of the Schwarzschild black hole is taken to be the Tsallis entropy, and that this result is also compatible with the generalization of the Tsallis entropy to the R & eacute;nyi entropy. In our paper, we show that the Sharma-Mittal entropy, which is another generalization of the Tsallis entropy, rejects the possibility of Hawking's proposal but can be compatible with Giddings' suggestion. So, in this paper, we consider the Sharma-Mittal entropy and study its behavior related to Giddings' suggestion. So we specify a range for the upper limit of the (delta-R) parameter, which enables Giddings' suggestion in the presence of the Sharma-Mittal entropy. We also understood that the Sharma-Mittal entropy, similar to the Tsallis and R & eacute;nyi entropies, exhibits the same behavior under some approximation of its parameters, in accordance with Giddings' suggestion.