Quintessence background for 5D Einstein–Gauss–Bonnet black holes
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As we know that the Lovelock theory is an extension of the general relativity to the higher-dimensions, in this theory the first- and the second-order terms correspond to general relativity and the Einstein–Gauss–Bonnet gravity, respectively. We obtain a 5D black hole solution in Einstein–Gauss–Bonnet gravity surrounded by the quintessence matter, and we also analyze their thermodynamical properties. Owing to the quintessence corrected black hole, the thermodynamic quantities have also been corrected except for the black hole entropy, and a phase transition is achievable. The phase transition for the thermodynamic stability is characterized by a discontinuity in the specific heat at \(r=r_C\), with the stable (unstable) branch for \(r < (>) r_C\).
The gravity theory with higher-curvature term, the so-called Lovelock gravity, is one of the natural generalizations of Einstein’s general relativity, introduced originally by Lanczos , and rediscovered by Lovelock . The action of it contains higher-order curvature terms and that reduces to the Einstein–Hilbert action in four-dimensions, and its second-order term is the Gauss–Bonnet invariant. The Lovelock theories have some special characteristics, among the larger class of general higher-curvature theories, in having field equations involving not more than second derivatives of the metric. As higher-dimensional members of Einstein’s general relativity family, the Lovelock theories allow us to explore several conceptual issues of gravity in a broader setup. Hence, these theories have received significant attention, especially when finding black hole solutions. Besides, the theory is well known to be free of ghosts about other exact backgrounds [3, 4, 5]. The theory represents a very interesting scenario to study how higher-order curvature corrections to the black hole physics substantially change the qualitative features, as we know from our experience with black holes in general relativity. Since its inception, steadily attention has been given to black hole solutions, including their formation, stability, and thermodynamics. The spherically symmetric static black hole solution for the Einstein–Gauss–Bonnet theory was first obtained by Boulware and Deser [3, 4, 5], and later several authors explored exact black hole solutions and their thermodynamical properties [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. The generalization of the Boulware-Desser solution has been obtained with a source as a cloud of strings, in Einstein–Gauss–Bonnet gravity [24, 25], and also in Lovelock gravity [26, 27, 28].
The intense activity of studying black hole solutions in Einstein–Gauss–Bonnet theory of gravity is due to the fact that we have, besides theoretical results, cosmological evidence, e.g., dark matter and dark energy. Quintessence is a hypothetical form of dark energy postulated as an explanation of the observation for an acceleration of the Universe, rather than due to a true cosmological constant. If quintessence exists all over in the Universe, it can also be around a black hole. In this letter, we are interested in a solution to the Einstein equations with the assumption of spherical symmetry, with the quintessence matter obtained by Kiselev , and it was also rigorously analyzed by himself and others [29, 30, 31, 32, 33, 34]. In particular, spherically symmetric quintessence black hole solutions  have been extended to higher dimensions , to include Narai solutions [36, 37], and also charged black holes . The black hole thermodynamics for the quintessence corrected solutions was obtained in [39, 40, 41, 42, 43] and quasinormal modes of such solutions are also discussed [44, 45, 46, 47]. The generalization of the spherical quintessential solution  to the axially symmetric case, Kerr-like black hole, was also addressed, recently [48, 49]. However, the solution of the Einstein–Gauss–Bonnet theory surrounded by quintessence matter is still not explored, i.e., the black holes surrounded by the quintessence matter in Einstein–Gauss–Bonnet theory is still unknown. It is the purpose of this letter to obtain an exact new five-dimensional (5D) spherically symmetric black holes solution for the Einstein–Gauss–Bonnet gravity surrounded by quintessence matter. In particular, we explicitly show how the effect of a background quintessence matter can alter black hole solutions and their thermodynamics. In turn, we analyze their thermodynamical properties and perform a thermodynamic stability analysis.
The letter is organized as follows. In Sect. 2, we derive a Einstein–Gauss–Bonnet solution to the 5D spherically symmetric static Einstein equations surrounded by the quintessence matter. In Sect. 3, we discuss the thermodynamics of the 5D Einstein–Gauss–Bonnet black holes surrounded by the quintessence matter. The letter ends with concluding remarks in Sect. 4.
We use units which fix the speed of light and the gravitational constant via \(G = c = 1\), and use the metric signature (\(-,+,+,+,+\)).
2 Quintessence matter surrounding black hole
It may be noted that the critical radius \(r_C\) changes drastically in the presence of the quintessence matter, thereby affecting the thermodynamical stability. Indeed, the value of the critical radius \(r_C\) increases with the increase in the quintessence matter parameter \(\omega \) for a given value of the Gauss–Bonnet coupling constant \(\alpha \). Further, \(r_C\) is also sensitive to the Gauss–Bonnet parameter \(\alpha \) (cf. Fig. 4), and the critical parameter \(r_C\) also increases with \(\alpha \).
The Einstein–Gauss–Bonnet theory has a number of additional nice properties in addition to Einstein’s general relativity that are not enjoyed by other higher-curvature theories. Hence, Einstein–Gauss–Bonnet theory has received significant attention, especially when finding black hole solutions. We have obtained an exact 5D static spherically symmetric black hole solutions to Einstein–Gauss–Bonnet gravity surrounded by the quintessence matter. We then proceeded to find exact expressions, in Einstein–Gauss–Bonnet gravity, for the thermodynamical quantities like the black hole mass, Hawking temperature, entropy, specific heat and in turn also analyzed the thermodynamical stability of black holes. It turns out that due to the quintessence correction to the black hole solution, the thermodynamical quantities are also getting corrected except for the entropy, which does not depend on the background quintessence. The entropy of a black hole in Einstein–Gauss–Bonnet gravity does not obey the area law.
The phase transition is characterized by the divergence of the specific heat at a critical radius \(r_C\), which is changing with Gauss–Bonnet parameter \(\alpha \) as well as with w. In particular, the black hole is thermodynamically stable with a positive heat capacity for the range \(0< r < r_C\) and unstable for \(r>r_C\). It would be important to understand how these black holes with positive specific heat (\(C>0\)) would emerge from thermal radiation through a phase transition. We also discuss the phase transition of the black holes. The results presented here are a generalization of the previous discussions, on the Einstein–Gauss–Bonnet black hole, in a more general setting, and the possibility of a further generalization of these results to Lovelock gravity is an interesting problem for future research.
S.G.G. would like to thank SERB-DST research project Grant no. SB/S2/HEP-008/2014, and ICTP for Grant no. OEA-NET-76. S.D.M. acknowledges that this work is based upon research supported by South African Research Chair Initiative of the Department of Science and Technology and the National Research Foundation. M.A. acknowledges the University Grant Commission, India, for the financial support through the Maulana Azad National Fellowship For Minority Students scheme (Grant no. F1-17.1/2012-13/MANF-2012-13-MUS-RAJ-8679). We would like to thank the India–South Africa bilateral project Grant no. DST/INT/South Africa/P-06/2016 date: 12/07/2016, and to IUCAA, Pune for the hospitality, where a part of this work was done.
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