Abstract
In this paper, we have extended the previous study of the thermodynamics and phase transition of the Schwarzschild black hole in the rainbow gravity to the Schwarzschild-AdS black hole where metric depends on the energy of a probe. Making use of the Heisenberg uncertainty principle and the modified dispersion relation, we have obtained the modified local Hawking temperature and thermodynamic quantities in an isothermal cavity. Moreover, we carry out the analysis of constant temperature slices of a black hole. As a result, we have shown that there also exists another Hawking-Page-like phase transition in which case a locally stable small black hole tunnels into a globally stable large black hole as well as the standard Hawking-Page phase transition from a hot flat space to a black hole.
Highlights
The possibility that standard energy-momentum dispersion relations are modified near the Planck scale is one of the scenarios in quantum gravity phenomenology [1,2]
Let us briefly look into the limits of the modified Hawking temperature (2.13) and local temperature (3.1) of the Schwarzschild–AdS black hole in the rainbow gravity enclosed in a cavity
The on-shell free energy of the Schwarzschild–AdS black hole in the rainbow gravity enclosed in a cavity is obtained by the use of the local temperature Tloc in Eq (3.1) and the thermodynamic energy Etot in Eq (3.10), explicitly as
Summary
The possibility that standard energy-momentum dispersion relations are modified near the Planck scale is one of the scenarios in quantum gravity phenomenology [1,2]. Li et al [44] have obtained the Schwarzschild–AdS black hole solution in the framework of rainbow gravity with different rainbow functions from Eq (1.2), and investigated thermodynamic stability without the analysis of phase transition. Gim and Kim (GK) [53] have shown that the Schwarzschild black hole in the rainbow gravity in an isothermal cavity has an additional Hawking–Page phase transition near the event horizon apart from the standard one, which is of relevance to the existence of a locally small black hole. We have shown that a local temperature seen by a freely falling observer depends only on g(ω/ωp) [56] so that the choice of f (ω/ωp) = 1 makes the timelike Killing vector in the rainbow Schwarzschild black hole as usual, and it makes the local thermodynamic energy independent of the test particle’s energy. From the surface gravity κH at the event horizon as follows:
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