Lovelock black holes surrounded by quintessence

High-precision observational data have confirmed evidence that the universe is undergoing a phase of accelerated expansion [1, 2] which may be due to dark energy that is gravitationally self-repulsive. Quintessence, a time-evolving, spatially inhomogeneous component with negative pressure is a possible candidate [3, 4] for dark energy, which is characterized by the equation of state \(P=\omega \rho \), where P is the pressure, \(\rho \) is the energy density, and \(-1< \omega < -1/3\). It is an exotic kind of field present everywhere in the universe that exerts force so that particles move away from each other by overpowering gravity and the other fundamental forces. The fact that dark energy constitutes 70% of the universe and black holes are part of our universe, makes the study of black holes surrounded by dark energy important. A spherical symmetric solution to the Einstein equations surrounded by quintessence matter was first obtained by Kiselev [5]; this solution includes the black hole, charged or not, in flat or de Sitter space.

Thereafter, significant attention has been devoted to a discussion of static spherically symmetric black hole solutions surrounded by quintessence matter and their properties [5, 6, 7, 8, 9, 10]. Also, several extensions of the Kiselev solutions [5] have been obtained. These include models for a charged black hole [11], the Nariai solution [12, 13] and extensions to higher dimensions [14]. These black hole solutions are possible only under special choice of parameter in the energy momentum tensor of quintessence, depending on the state parameter \(\omega \). The black hole thermodynamics for the quintessence corrected black hole solutions were discussed in [15, 16, 17, 18, 19] and the quasinormal modes of such solutions have been obtained [20, 21, 22, 23]. Owing to the quintessence surrounding the black hole, the thermodynamic quantities have also been corrected except for the black hole entropy, and it is shown that a phase transition is achievable. Ghosh [24] and Toshmatov et al. [25] further generalized the Kiselev solution to include the axially symmetric case, i.e., the Kerr-like black hole was also addressed. It was shown that a rotating counterpart of the Kiselev solutions [5] can be identified for quintessence parameter \( \omega =1/3 \), exactly as the Kerr–Newman black hole and as the Kerr black hole according to a choice of the integration constant [24, 25].

The accelerated expansion of the universe has also inspired several modifications of general relativity which aim to explain the cosmic acceleration and reconstruct the entire expansion history. A natural modification of general relativity is the Lovelock gravity, whose Lagrangian consists of the dimensionally extended Euler densities. This was obtained by Lovelock in an attempt to obtain the most general tensor that satisfies properties of the Einstein tensor in higher dimensions [26]. The Lovelock action contains higher order curvature terms and 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, amongst the larger class of general higher-curvature theories, in having field equations involving not more than second derivatives of the metric. They have also been shown to be free of ghosts when expanding about flat space, evading any problems with unitarity. Exact solutions describing black holes have also been found for these theories [27, 28, 29, 30], and later several generalizations of the Boulware–Desser solution have also been discussed [31, 32, 33, 34, 35, 36, 37]. Recently, the quintessence atmosphere to the Einstein–Gauss–Bonnet black hole has been analyzed by Ghosh et al. [38]. In this analysis the case of no quintessence \(\omega =0\) reduces to the Boulware and Deser [27, 28, 29] Gauss–Bonnet black hole solution, and for quintessence parameter \(\omega =1/2\) with an appropriate choice of the integration constant it is mathematically similar to the charged Gauss–Bonnet black hole case due to Wiltshire [30]. The spherically symmetric vacuum solutions to Lovelock gravity have been found independently in [39, 43]; also, recent work includes a class of Lovelock black holes with conformal scalar hair [40, 41, 42]. It is the purpose of this paper to model the possible effect of surrounding quintessence matter on the spherically symmetric black hole solution in Lovelock gravity. In particular, we can explicitly explain how the effect of a background quintessence matter can alter black hole solutions and their thermodynamics.

The paper is organized as follows. In Sect. 2, we begin by reviewing the higher dimensional [14] spherically symmetric black hole surrounded by quintessence matter. We have analyzed the quintessence background of Kiselev [5] to derive the resulting exact Lovelock black holes. The thermodynamic properties of the new derived solutions are explored and we obtain the thermodynamic quantities exactly in Sect. 3. In Sect. 3 a discussion covering the thermodynamical stability of black holes is also presented, and Sect. 4 gives the concluding remarks.

We use units which fix the speed of light and the gravitational constant via \(8\pi G = c = 1\), and use the metric signature (\(-,\;+,\;+,\; \ldots ,\;+\)).

Lovelock gravity is one of the most general gravity theories in which the field equations are still second order and it is the natural generalization of general relativity to higher dimensional spacetime. The basic idea is to supplement the Einstein–Hilbert action with the dimensionally continued Euler densities. The Lovelock gravity has several additional interesting properties when compared with general relativity which triggered significant attention especially in finding black hole solutions in these theories. We have obtained an exact spherically symmetric black hole solution surrounded by quintessence matter in general Lovelock gravity thereby generalizing the vacuum solution for these theories. The current evidence for an accelerating early universe can be accommodated theoretically via a reintroduction of Einsteins (positive) cosmological constant, which is equivalent to the introduction of quintessence matter with equation of state \(p=- \rho \). More generally, and for general spacetime dimension D, quintessence with equation of state \(p=\omega _q \rho _q\) also yields a flat accelerating universe provided that \(-1 \le \omega _q \le - 1 /(D-1)\). Despite complications of the geometry and horizon of the black hole, we have found exact expressions for the thermodynamic quantities like the black hole mass, the Hawking temperature, entropy and heat capacity in terms of the horizon. In particular, we demonstrate that these thermodynamical quantities are corrected owing to quintessence term \(q/r^{(D-1)(\omega _q+1)}\) in the solution Eq. (34). Explicit calculation of entropy shows that, in general, the area law does not hold for the Lovelock black hole in Eq. (34). One can understand the large scale structure of interactions containing quantum mechanical properties through thermodynamic quantities of black holes. We confirmed that the entropy does not depend on the surrounding quintessence matter as found in [52]. In general, the phase transition occurs in an AdS black holes. However, with our parameter choice \(\kappa =1\), \(\tilde{\alpha }_0=-1\) and \(\tilde{\alpha }_1=\tilde{\alpha }_2=\tilde{\alpha }_3=1\), we showed that the heat capacity has the phase transition point \(r_c\) in various dimensions, where the heat capacity diverges. For a horizon radius below \(r_c\) the heat capacity is positive, which means that the black hole is thermodynamically stable, and beyond \(r_c\) a thermodynamic unstable region starts, where the heat capacity is negative. Also we found that the phase transition point becomes larger as \(|\omega _q|\) increases. This implies that in the equation of state \(p_q=\omega _q \rho _q\) when the magnitude of the pressure gets closer to that of the energy density, i.e. \(\omega _q\rightarrow -1\), larger horizon radius range of stability is allowed, i.e. the black hole is more likely to be stable. On the other hand, we found that whether \(r_c\) increases or decreases as q increases also depends on \(\omega _q\). But in the particular case, \(\kappa =0\), the area law is restored. In addition, thermodynamically stable black holes always appear with a positive heat capacity in all the dimensions. These thermodynamical properties are different from those in the general relativity. However, they become qualitatively similar to the Gauss–Bonnet black holes. Our result generalized previous approaches to a more general case, and in the limit \(q\rightarrow 0\) this goes to the vacuum case.