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P–v criticality in the extended phase space of a noncommutative geometry inspired Reissner–Nordström black hole in AdS space-time

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Abstract

The P–v criticality and phase transition in the extended phase space of a noncommutative geometry inspired Reissner–Nordström (RN) black hole in Anti-de Sitter (AdS) space-time are studied, where the cosmological constant appears as a dynamical pressure and its conjugate quantity is thermodynamic volume of the black hole. It is found that the P–v criticality and the small black hole/large black hole phase transition appear for the noncommutative RN-AdS black hole. Numerical calculations indicate that the noncommutative parameter affects the phase transition as well as the critical temperature, horizon radius, pressure and ratio. The critical ratio is no longer universal, which is different from the result in the van de Waals liquid–gas system. The nature of phase transition at the critical point is also discussed. Especially, for the noncommutative geometry inspired RN-AdS black hole, a new thermodynamic quantity \(\varPsi \) conjugate to the noncommutative parameter \(\theta \) has to be defined further, which is required for consistency of both the first law of thermodynamics and the corresponding Smarr relation.

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Notes

  1. When the variation of \(\varLambda \) is included in the first law, the black hole mass M should be identified with enthalpy rather than with internal energy [21,22,23,24,25].

  2. (16) can be derived from a scaling (dimensional) argument [21]. In particular, it should be emphasized that, since \(\theta \) ia a dimensionful parameter, the corresponding term will inevitably appear in the Smarr relation.

  3. The reverse isoperimetric inequality is [25]

    figure a

    where V is the thermodynamic volume of AdS black hole in d space-time dimensions, A is the horizon area of the black hole and \(\mathcal {A}_{d-2}\) is the volume of the unit \((d-2)\)-sphere, i.e.,

    figure b

    When \(d=4\), (22) reduces to

    figure c

    It is easy to check (24) holds for the noncommutative geometry inspired RN-AdS black hole.

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Acknowledgements

This work was supported by the Natural Science Foundation of Education Department of the Shannxi Provincial Government under Grant No. 15JK1077 and the Doctorial Scientific Research Starting Fund of Shannxi University of Science and Technology under Grant No. BJ12-02.

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Authors and Affiliations

  1. College of Arts and Sciences, Shannxi University of Science and Technology, Xi’an, 710021, People’s Republic of China

    Jun Liang, Zhi-Hua Guan & Bo Liu

  2. Library, Shannxi University of Science and Technology, Xi’an, 710021, People’s Republic of China

    Yan-Chun Liu

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  1. Jun Liang
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  2. Zhi-Hua Guan
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Correspondence to Jun Liang.

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Liang, J., Guan, ZH., Liu, YC. et al. P–v criticality in the extended phase space of a noncommutative geometry inspired Reissner–Nordström black hole in AdS space-time. Gen Relativ Gravit 49, 29 (2017). https://doi.org/10.1007/s10714-017-2189-8

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