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Thermodynamic geodesics of a Reissner Nordström black hole

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Abstract

Starting from a Geometrothermodynamics metric for the space of thermodynamic equilibrium states in the mass representation, we use numerical techniques to analyse the thermodynamic geodesics of a supermassive Reissner Nordström black hole in isolation. Appropriate constraints are obtained by taking into account the processes of Hawking radiation and Schwinger pair-production. We model the black hole in line with the work of Hiscock and Weems (Phys Rev D 41:1142–1151, 1990). It can be deduced that the relation which the geodesics establish between the entropy S and electric charge Q of the black hole extremises changes in the black hole’s mass. Indeed, the expression for the entropy of an extremal black hole is an exact solution to the geodesic equation. We also find that in certain cases, the geodesics describe the evolution brought about by the constant emission of Hawking radiation and charged-particle pairs.

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Notes

  1. To be precise, Weinhold worked in the tangent space defined at a general point of the equilibrium manifold, although it is possible to use his metric as a measure of distance in the manifold itself [14].

  2. \(\wedge \) stands for the exterior product, ‘\(\text {d}\)’ the exterior derivative, and \((\text {d}\varTheta )^n\) is equal to \(\text {d}\varTheta \wedge \cdots \wedge \text {d}\varTheta \), where \(\text {d}\varTheta \) appears n times.

  3. The full value and its uncertainty, in SI units, are given in [40].

  4. Note that it is also possible to choose Q as the dependent variable. This will be treated in greater detail in Sect. 4.

  5. Two values are given in [1]—one for the emission of three massless neutrino species, and another (0.26792) valid when no massless neutrinos are produced, where by massless is meant a mass less than about \(10^{-10}\,\hbox {eV}\) (the black hole would be too cold to emit anything heavier) [1]. Since the upperbounds available nowadays for the mass of neutrino flavours [48] are much greater than this value, we use \(\alpha =0.26792\) when comparing our work with [1].

  6. The full values and uncertainties, in SI units, are given in [40].

  7. The rest energy of the positron (the particle with the same-sign charge as the black hole) is not taken into account; the large charge-to-mass ratio of this particle implies that the energy \(eQ/r_+\) it gains when repelled to infinity is much greater [1].

  8. Here, \(\lambda _\alpha \) stands for the intensive thermodynamic variables of the system. At equilibrium, \(\lambda _\alpha =\frac{\partial U}{\partial E^\alpha }\Bigr |_{\begin{array}{c} E_0^\alpha \end{array}}\) [52].

  9. Several authors are of the opinion that a black hole which is exactly extremal has zero entropy; see, for instance, [54,55,56,57,58].

  10. For instance, in the case of the geodesic with constraints given by Eq. (29), we set \(Q(t=0) = 8\times 10^{8}\,{\hbox {m}}\) and \(M(t=0) = 3\times 10^{9}\,{\hbox {m}}\).

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Acknowledgements

C. F. would like to thank Prof. H. Quevedo of the Universidad Nacional Autónoma de México, Prof. J. Muscat (Department of Mathematics, University of Malta) and Prof. K. Zarb Adami (Institute of Space Sciences and Astronomy, University of Malta).

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  1. Department of Mathematics, Faculty of Science, University of Malta, Msida, MSD 2080, Malta

    Christine Farrugia & Joseph Sultana

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Farrugia, C., Sultana, J. Thermodynamic geodesics of a Reissner Nordström black hole. Gen Relativ Gravit 49, 4 (2017). https://doi.org/10.1007/s10714-016-2169-4

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