# \(P\)–\(V\) criticality of topological black holes in Lovelock–Born–Infeld gravity

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## Abstract

To understand the effect of third order Lovelock gravity, \(P\)–\(V\) criticality of topological AdS black holes in Lovelock–Born–Infeld gravity is investigated. The thermodynamics is further explored with some more extensions and in some more detail than the previous literature. A detailed analysis of the limit case \(\beta \rightarrow \infty \) is performed for the seven-dimensional black holes. It is shown that, for the spherical topology, \(P\)–\(V\) criticality exists for both the uncharged and the charged cases. Our results demonstrate again that the charge is not the indispensable condition of \(P\)–\(V\) criticality. It may be attributed to the effect of higher derivative terms of the curvature because similar phenomenon was also found for Gauss–Bonnet black holes. For \(k=0\), there would be no \(P\)–\(V\) criticality. Interesting findings occur in the case \(k=-1\), in which positive solutions of critical points are found for both the uncharged and the charged cases. However, the \(P\)–\(v\) diagram is quite strange. To check whether these findings are physical, we give the analysis on the non-negative definiteness condition of the entropy. It is shown that, for any nontrivial value of \(\alpha \), the entropy is always positive for any specific volume \(v\). Since no \(P\)–\(V\) criticality exists for \(k=-1\) in Einstein gravity and Gauss–Bonnet gravity, we can relate our findings with the peculiar property of third order Lovelock gravity. The entropy in third order Lovelock gravity consists of extra terms which are absent in the Gauss–Bonnet black holes, which makes the critical points satisfy the constraint of non-negative definiteness condition of the entropy. We also check the Gibbs free energy graph and “swallow tail” behavior can be observed. Moreover, the effect of nonlinear electrodynamics is also included in our research.

### Keywords

Black Hole Black Hole Solution High Derivative Term Small Black Hole Lovelock Gravity## 1 Introduction

Gravity in higher dimensions has attained considerable attention with the development of string theory. Concerning the effect of string theory on gravitational physics, one may construct a low energy effective action which includes both the Einstein–Hilbert Lagrangian (as the first order term) and the higher curvature terms. However, this approach may lead to field equations of fourth order and ghosts as well. This problem has been solved by a particular higher curvature gravity theory called Lovelock gravity [1]. The field equation in this gravity theory is only second order and the quantization of Lovelock gravity theory is free of ghosts [2]. In this context, it is of interest to investigate both the black hole solutions and their thermodynamics in Lovelock gravity [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. Moreover, it is natural to consider the nonlinear terms in the matter side of the action while accepting the nonlinear terms on the gravity side [3]. Motivated by this, Ref. [3] presented topological black hole solutions in Lovelock–Born–Infeld gravity. Both the thermodynamics of asymptotically flat black holes for \(k=1\) and the thermodynamics of asymptotically AdS rotating black branes with a flat horizon were investigated there in detail. However, concerning the charged topological AdS black holes in Lovelock–Born–Infeld gravity, only the temperature was given in Ref. [3]. Reference [18] further studied their entropy and specific heat at constant charge. However, the expression of the entropy seems incomplete, for \(k\) is missing. The thermodynamics in the extended space needs to be further explored. Probing this issue is important because it is believed that the physics of black holes in higher dimensions is essential to understand the full theory of quantum gravity.

As is well known, the phase transition is a fascinating phenomenon in classical thermodynamics. Over the past decades, phase transitions of black holes have aroused more and more attention. The pioneering phase transition research of AdS black holes can be traced back to the discovery of the famous Hawking–Page phase transition between the Schwarzschild AdS black hole and thermal AdS space [30]. Recently, a revolution in this field has been led by \(P\)–\(V\) criticality research [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44] in the extended phase space. Kubizňák et al. [31] perfectly completed the analogy between charged AdS black holes and the liquid–gas system first observed by Chamblin et al. [45, 46]. The approach of treating the cosmological constant as thermodynamic pressure and its conjugate quantity as thermodynamic volume is essential with the increasing attention of considering the variation of the cosmological constant in the first law of black hole thermodynamics recently [47, 48, 49, 50, 51, 52, 53].

Here, we would like to investigate the thermodynamics and phase transition of charged topological AdS black holes in Lovelock–Born–Infeld gravity in the extended phase space. Some related efforts have been made recently. \(P\)–\(V\) criticality of both four-dimensional [32] and higher-dimensional [44] Born–Infeld AdS black holes has been investigated. A new parameter, called the Born–Infeld vacuum polarization, was defined to be conjugate to the Born–Infeld parameter [32]. It was argued that this quantity is required for the consistency of both the first law of thermodynamics and the Smarr relation. Moreover, Cai et al. [40] studied \(P\)–\(V\) criticality of Gauss–Bonnet AdS black holes. It was found that no \(P\)–\(V\) criticality can be observed for Ricci flat and hyperbolic Gauss–Bonnet black holes. However, for the spherical case, \(P\)–\(V\) criticality can be observed even when the charge is absent, implying that the charge may not be an indispensable factor for the existence of \(P\)–\(V\) criticality. Such an interesting result motivates us to probe further third order Lovelock gravity to explore whether it is a peculiar property due to the higher derivative terms of the curvature. So we will mainly investigate their effects on \(P\)–\(V\) criticality. Moreover, we will probe the combined effects of higher derivative terms of curvature and the nonlinear electrodynamics.

In Sect. 2, the solutions of charged topological AdS black holes in Lovelock–Born–Infeld gravity will be briefly reviewed and their thermodynamics will be further investigated. In Sect. 3, a detailed study will be carried out in the extended phase space for the limit case \(\beta \rightarrow \infty \) so that we can concentrate on the effects of third order Lovelock gravity. In Sect. 4, the effects of nonlinear electrodynamics will also be included. In the end, a brief conclusion will be drawn in Sect. 5.

## 2 Thermodynamics of charged topological black holes in Lovelock–Born–Infeld gravity

## 3 \(P\)–\(V\) criticality of a limit case

To concentrate on the effects of third order Lovelock gravity, we would like to investigate an interesting limit case in this section and leave the issue of nonlinear electrodynamics to be further investigated in Sect. 4.

Critical values for \(k=1,n=6,\beta \rightarrow \infty \)

\(q\) | \(\alpha \) | \(T_c\) | \(v_c\) | \(P_c\) | \(\frac{P_cv_c}{T_c}\) |
---|---|---|---|---|---|

0.5 | 1 | 0.14213 | 1.80992 | 0.02691 | 0.343 |

2 | 1 | 0.13989 | 1.97347 | 0.02553 | 0.360 |

1 | 1 | 0.14154 | 1.85884 | 0.02653 | 0.348 |

1 | 0.5 | 0.19287 | 1.53461 | 0.04727 | 0.376 |

1 | 2 | 0.10062 | 2.53773 | 0.01351 | 0.341 |

Critical values for \(k=-1,n=6,\beta \rightarrow \infty \)

\(q\) | \(\alpha \) | \(T_c\) | \(v_c\) | \(P_c\) | \(\frac{P_cv_c}{T_c}\) |
---|---|---|---|---|---|

0.5 | 1 | 0.33836 | 1.07752 | 0.22718 | 0.723 |

2 | 1 | 0.72658 | 1.31811 | 0.35029 | 0.635 |

1 | 1 | 0.46900 | 1.19335 | 0.26507 | 0.674 |

1 | 0.5 | 1.88727 | 1.01514 | 1.12928 | 0.607 |

1 | 2 | 0.18669 | 1.38663 | 0.10503 | 0.780 |

## 4 Inclusion of the nonlinear electrodynamics

In this section, we would like to take into account the effect of nonlinear electrodynamics to complete the analysis of topological AdS black holes in Lovelock–Born–Infeld gravity.

Critical values for different dimensions for \(k=1,n=6\)

\(\beta \) | \(q\) | \(\alpha \) | \(T_c\) | \(v_c\) | \(P_c\) | \(\frac{P_cv_c}{T_c}\) |
---|---|---|---|---|---|---|

10 | 1 | 1 | 0.141541 | 1.85884 | 0.026528 | 0.34839 |

0.5 | 1 | 1 | 0.141545 | 1.85829 | 0.026531 | 0.34832 |

1 | 1 | 1 | 0.141542 | 1.85871 | 0.026529 | 0.34838 |

1 | 0.5 | 1 | 0.142126 | 1.80991 | 0.02691 | 0.343 |

1 | 2 | 1 | 0.139898 | 1.97286 | 0.02554 | 0.360 |

1 | 1 | 0.5 | 0.192905 | 1.53258 | 0.04730 | 0.376 |

1 | 1 | 2 | 0.100617 | 2.53773 | 0.01351 | 0.341 |

Critical values for different dimensions for \(k=-1,n=6\)

\(\beta \) | \(q\) | \(\alpha \) | \(T_c\) | \(v_c\) | \(P_c\) | \(\frac{P_cv_c}{T_c}\) |
---|---|---|---|---|---|---|

10 | 1 | 1 | 0.468887 | 1.19315 | 0.26504 | 0.674 |

0.5 | 1 | 1 | 0.424488 | 1.11470 | 0.25296 | 0.664 |

1 | 1 | 1 | 0.457384 | 1.17317 | 0.26182 | 0.672 |

1 | 0.5 | 1 | 0.333977 | 1.06663 | 0.22621 | 0.722 |

1 | 2 | 1 | 0.690585 | 1.28270 | 0.33877 | 0.629 |

1 | 1 | 0.5 | 1.498501 | 0.92425 | 0.94270 | 0.581 |

1 | 1 | 2 | 0.186104 | 1.38311 | 0.10496 | 0.780 |

## 5 Conclusions

Topological AdS black holes in Lovelock–Born–Infeld gravity are investigated in the extended phase space. The black hole solutions are reviewed while their thermodynamics is further explored in the extended phase space. We calculate the entropy by integration and find that the result in the previous literature [18] was incomplete. Treating the cosmological constant as pressure, we rewrite the first law of thermodynamics for the specific case in which the second order and the third order Lovelock coefficients are related by the Lovelock coefficient \(\alpha \). The quantity conjugated to the Lovelock coefficient and the Born–Infeld parameter, respectively, are calculated. Comparing our results of the above quantities with those in previous literature of Gauss–Bonnet black holes [40], we find that there exist extra terms due to third order Lovelock gravity. In order to make the phase transition clearer, the Gibbs free energy is also calculated.

To figure out the effect of third order Lovelock gravity on \(P\)–\(V\) criticality, a detailed analysis of the limit case \(\beta \rightarrow \infty \) has been performed. Since the entropy is convergent only when \(n>5\), our investigation is carried out in the case of \(n=6\), corresponding to the seven-dimensional black holes. It is shown that, for the spherical topology, \(P\)–\(V\) criticality exists even when \(q=0\). The critical physical quantities can be analytically solved and they vary with the parameter \(\alpha \). However, the ratio of \(\frac{P_cv_c}{T_c}\) is independent of the parameter \(\alpha \). Our results demonstrate again that the charge is not an indispensable condition of \(P\)–\(V\) criticality. It may be attributed to the effect of higher derivative terms of the curvature because a similar phenomenon was also found for Gauss–Bonnet black holes [40]. For \(q\ne 0\), it is shown that the physical quantities at the critical point, \(T_c, v_c,P_c\), depend on both the charge and the parameter \(\alpha \). With the increasing of \(\alpha \) or \(q\), both \(T_c\) and \(P_c\) decrease while \(v_c\) increases. However, the ratio \(\frac{P_cv_c}{T_c}\) decreases with \(\alpha \) but increases with \(q\). Similar behaviors as for the van der Waals liquid–gas phase transition can be observed in the \(P\)–\(v\) diagram and the classical swallow tail behaviors can be observed in both the two-dimensional and the three-dimensional graph of the Gibbs free energy. These observations indicate that a phase transition between small black holes and large black holes takes place when \(k=1\). For \(k=0\), no critical point can be found and there would be no \(P\)–\(V\) criticality. Interesting findings occur in the case \(k=-1\), in which positive solutions of critical points are found for both the uncharged and the charged case. However, the \(P\)–\(v\) diagram is very strange. For the uncharged case, the isotherms below or above the critical temperature both behave as the coexistence phase which is similar to the behaviors of the van der Waals liquid–gas system below the critical temperature. For the charged case, the “phase transition” picture is quite the reverse of the van der Waals liquid–gas phase transition. Above the critical temperature the behavior is “van der Waals like” while the behavior is “ideal gas like” below the critical temperature. This process is achieved by lowering the temperature rather than increasing the temperature. To check whether these findings are physical, we perform an analysis on the non-negative definiteness condition of the entropy. It is shown that, for any nontrivial value of \(\alpha \), the entropy is always positive for any specific volume \(v\). We relate the findings in the case \(k=-1\) with the peculiar property of third order Lovelock gravity. Because the entropy in third order Lovelock gravity consists of extra terms which are absent in the Gauss–Bonnet black holes, the critical points satisfy the constraint of non-negative definiteness condition of the entropy. We also check the Gibbs free energy graph and “swallow tail” behavior can be observed.

Moreover, the effect of nonlinear electrodynamics is included in our work. Similar observations are made as the limit case \(\beta \rightarrow \infty \) and only slight differences can be observed when we choose different values of \(\beta \). That may be attributed to the parameter region we choose. More interesting findings concerning “Schwarzschild like” behaviors can be found in the previous literature [32] and we will not repeat them here, because our main motivation is to investigate the impact of third order Lovelock gravity on \(P\)–\(V\) criticality in the extended phase space.

## Notes

### Acknowledgments

This research is supported by the National Natural Science Foundation of China (Grant Nos. 11235003, 11175019, 11178007). It is also supported by “Thousand Hundred Ten”, Project of Guangdong Province and Natural Science Foundation of Zhanjiang Normal University under Grant No. QL1104.

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