1. Introduction
Black hole thermodynamics is one of the most interesting subjects in the study of gravitational theory. It always has attracted much attention, as it is widely believed that it can provide a further insight into the study of quantum gravity. Since the pioneering and excellent work in this subject, i.e. the discovery of the Hawking–Page transition [1], an exciting period of exploring black hole thermodynamics in the holographic and quantum aspects was initiated. The famous Hawking–Page transition characterizes a first-order phase transition occurring between an anti-de Sitter (AdS) black hole phase and a thermal AdS vacuum phase. It is found that the thermal AdS vacuum phase dominates the partition function in the low-temperature limit while the large AdS black hole phase dominates at the high-temperature limit. As a result, in the global phase diagram the thermal AdS gas will collapse to a stable large black hole when the temperature increases. The Hawking–Page transition was later explained as the confinement/deconfinement phase transition of a gauge field [2], inspired by the AdS/conformal field theory (CFT) correspondence [3–5]. This brought increasing interest from both gravity and field theory aspects. Regarding the contribution of the Hawking–Page transition to the holographic understanding of the quantum chromodynamic phase diagram, it is speculated that the crossover corresponds to the Hawking–Page crossover of a Schwarzschild AdS black hole in noncommutative spacetime [6], and the triple point of hadronic matter/quark–gluon plasma/quarkyonic matter corresponds to the triple point of the Hawking–Page transition with re-entrance of a Gauss–Bonnet AdS black hole [7].
Recently, there have been many studies of the Hawking–Page transition (e.g. [6–35]) concerning black hole thermodynamics and the microscopic state. However, since string loops cannot yet be calculated in any of the backgrounds thought to be dual to gauge theory, deeper holographic study of the the Hawking–Page transition is still very difficult. Recently, an imaginative study on the holographic viewpoint of the Hawking–Page transition was contributed by Wei et al [8]. This interesting work introduced a novel dual relation between the minimum temperature and the Hawking–Page transition temperature of an AdS black hole in two successive dimensions, namely
$\begin{eqnarray}{T}_{0}\left(n+1\right)={T}_{{\text{}}{HP}}\left(n\right).\end{eqnarray}$
This relation indicates that the n-dimensional Hawking–Page transition temperature THP is exactly the same as the minimum temperature T0 of a black hole in one larger dimension. Therefore, one could naturally recall the AdS/CFT correspondence, since T0 can be treated as the temperature of a physical quantity in the bulk and THP may correspondingly be conjectured as the temperature of the dual physical quantity on the boundary. Furthermore, they also conjectured that this relation may suggest a duality between the ground state and an excited state of a physical system in two successive dimensions because T0 could also be treated as the ground state of the AdS black hole, as it is just the minimum temperature black hole phase, with the Hawking–Page temperature (THP) black hole phase as one of its excited states. This dual relation opens a new window to explore the holographic understanding of the Hawking–Page transition, and may induce some new important applications in the holographic principle and black hole thermodynamics.High-precision astronomical observations have shown that the Universe is undergoing a phase of accelerated expansion [36, 37], which might be due to dark energy acting as repulsive gravity. In this paper, we will focus on the effect of dark energy on the Hawking–Page transition and the dual relation equation (1 ), including phantom dark energy and quintessence dark energy. Recently, black holes surrounded by phantom dark energy and quintessence dark energy and their thermodynamics have attracted considerable interest (see references [12, 21, 38–53] and [30, 54–71], respectively). Black holes surrounded by dark energy could be used as a theoretical laboratory for studying black hole physics and investigating the theoretical properties and implications of phantom dark energy and quintessence dark energy. As a natural generalization, it would be interesting to study the Hawking–Page transition of AdS black holes surrounded by phantom dark energy and quintessence dark energy.
Our findings will reveal the occurrence of the Hawking–Page transition observed between the thermal AdS radiation and thermodynamically stable large AdS black holes in the spacetime surrounded by both phantom dark energy and quintessence dark energy. There are distinct effects of phantom dark energy and quintessence dark energy on the Hawking–Page transition temperature. It will be shown that the electric potential Φ and the dimensions n of AdS black holes surrounded by phantom dark energy both increase the Hawking–Page transition temperature, while the density parameter α and the dimension n of AdS black holes surrounded by quintessence dark energy both reduce the Hawking–Page transition temperature. Finally, we will show that the dual relation works well for the case of AdS black holes surrounded by phantom dark energy. In particular, for the case of AdS black holes surrounded by quintessence dark energy, we will derive a modified dual relation after considering that the state parameter and density parameter of the quintessence dark energy depend on the dimensions of the spacetime.
The paper is organized as follows: we first study the Hawking–Page transition and the dual relation of AdS black holes surrounded by phantom dark energy in section 2 . In section 3 , we generalize the discussion to the case of AdS black holes surrounded by quintessence dark energy. Finally some concluding remarks are given.
2. Hawking–Page transition and the dual relation of AdS black holes surrounded by phantom dark energy
In this section we study the Hawking–Page transition of AdS black holes surrounded by phantom dark energy in n dimensions and check the dual relation.
2.1. Extended thermodynamics of AdS black holes surrounded by phantom dark energy in general dimensions
First we revisit the extended thermodynamics of AdS black holes surrounded by phantom dark energy in general dimensions. The corresponding action for the black holes is given by
$\begin{eqnarray}\begin{array}{l}S=\displaystyle \frac{1}{2{\kappa }_{n}}\displaystyle \int {{\rm{d}}}^{n}x\sqrt{g}\left(R-2{\rm{\Lambda }}\right)\\ \quad \,-\,\displaystyle \frac{\eta }{4{\mu }_{n}}\displaystyle \int {{\rm{d}}}^{n}x\sqrt{g}\left({F}_{\mu \nu }{F}^{\mu \nu }\right),\end{array}\end{eqnarray}$
where the third term represents the electromagnetic field with constant η. For η = 1, we obtain the classical Einstein–Maxwell AdS theory, whereas the electromagnetic field is phantom for η = –1. Here κn and μn are the interaction constants characterizing the strength of the gravitational field and the Maxwell field in n-dimensional spacetime, respectively. One should especially note that we always choose κn = 8πGn as the Einstein gravitational constant in n-dimensional spacetime, with Gn being the Newtonian gravitational constant in n-dimensional spacetime when we set c = 1. The Maxwell constant μn always corresponds to the magnetic vacuum permeability μ0 in four dimensions.The AdS black hole solution surrounded by phantom dark energy reads [54]
$\begin{eqnarray}\begin{array}{rcl}{\rm{d}}{s}^{2} & = & -f(r){\rm{d}}{t}^{2}+\displaystyle \frac{{\rm{d}}{r}^{2}}{f(r)}+{r}^{2}{\rm{d}}{{\rm{\Omega }}}_{n-2}^{2},\\ f(r) & = & 1-\,\displaystyle \frac{2{\kappa }_{n}M}{\left(n-2\right){\omega }_{n-2}{r}^{n-3}}\\ & & +\displaystyle \frac{{\mu }_{n}{\kappa }_{n}\eta {Q}^{2}}{\left(n-2\right)\left(n-3\right){\omega }_{n-2}^{2}{r}^{2\,n-6}}\\ & & +\,\displaystyle \frac{16{r}^{2}P\pi }{(n-1)(n-2)},\end{array}\end{eqnarray}$
where ${\rm{d}}{{\rm{\Omega }}}_{n-2}^{2}$ is the line element of an (n − 2)-dimensional maximally symmetric Einstein manifold with a spherical topology of the black hole horizon, ωn−2 is the volume of a unit (n − 2) sphere, M is the black hole mass, Q is the electric charge and $P=-\tfrac{{\rm{\Lambda }}}{8\pi }=\tfrac{(n-1)(n-2)}{16\pi {{\ell }}^{2}}$ is the thermodynamic pressure associated with the cosmological constant Λ [72–78] in the black hole chemistry framework, with ℓ being the n-dimensional AdS radius.The thermodynamical quantities of the AdS black holes surrounded by phantom dark energy in the extended thermodynamics are presented as follows:
$\begin{eqnarray*}\begin{array}{rcl}M & = & \displaystyle \frac{{r}_{+}^{3-n}{\mu }_{n}\eta {Q}^{2}}{2{\omega }_{n-2}\left(n-3\right)}+\,\displaystyle \frac{8{r}_{+}^{n-1}\pi \,{\omega }_{n-2}P}{\left(n-1\right){\kappa }_{n}}\\ & & +\,\displaystyle \frac{\left(n-2\right){\omega }_{n-2}{r}_{+}^{n-3}}{2{\kappa }_{n}},\\ T & = & -\displaystyle \frac{{r}_{+}^{-2\,n+5}{\kappa }_{n}{\mu }_{n}\eta {Q}^{2}}{4\left(n-2\right)\pi {\omega }_{n-2}^{2}}+\,\displaystyle \frac{4{r}_{+}P}{n-2}+\,\displaystyle \frac{n-3}{4\pi \,{r}_{+}},\\ S & = & \displaystyle \frac{2\pi \,{\omega }_{n-2}{r}_{+}^{n-2}}{{\kappa }_{n}},\\ {\rm{\Phi }} & = & \displaystyle \frac{{\mu }_{n}Q}{\left(n-3\right){\omega }_{n-2}{r}_{+}^{n-3}},\\ V & = & \displaystyle \frac{8{r}_{+}^{n-1}\pi \,{\omega }_{n-2}}{\left(n-1\right){\kappa }_{n}},\end{array}\end{eqnarray*}$
where r+ is the event horizon radius of the black hole.On account of the conservation of charge and the neutrality of the thermal AdS background, the Hawking–Page transition of this AdS black hole should be considered in the grand canonical ensemble, where the electric potential is fixed and the electric charge is thus allowed to vary. One can easily derive
$\begin{eqnarray}Q=\displaystyle \frac{{\rm{\Phi }}\left(n-3\right){\omega }_{n-2}{r}_{+}^{n-3}}{{\mu }_{n}}.\end{eqnarray}$
After inserting this relation, we get the temperature in the grand canonical ensemble $\begin{eqnarray}T=\displaystyle \frac{4{r}_{+}P}{n-2}-\,\displaystyle \frac{{\kappa }_{n}\eta {{\rm{\Phi }}}^{2}{\left(n-3\right)}^{2}}{4\left(n-2\right){r}_{+}\pi \,{\mu }_{n}}+\,\displaystyle \frac{n-3}{4\pi \,{r}_{+}}.\end{eqnarray}$
It is also easy to check that the first law of thermodynamics and Smarr relation in the grand canonical ensemble of black hole extended thermodynamics $\begin{eqnarray}{\rm{d}}{M}={V}{\rm{d}}{P}+\eta {Q}{\rm{d}}{\rm{\Phi }}+{T}{\rm{d}}{S},\end{eqnarray}$
$\begin{eqnarray}\left(n-3\right)M=\left(n-2\right){TS}+\left(n-3\right)\eta Q{\rm{\Phi }}-2\,{VP}\end{eqnarray}$
always hold. In order to study the Hawking–Page transition of this AdS black hole in extended thermodynamics, we present the following Gibbs free energy in the grand canonical ensemble: $\begin{eqnarray}\begin{array}{l}G=H-{TS}-\eta Q{\rm{\Phi }}=M-{TS}-\eta Q{\rm{\Phi }}\\ \quad =-\,\displaystyle \frac{8{r}_{+}^{n-1}\pi \,{\omega }_{n-2}P}{\left(n-2\right)\left(n-1\right){\kappa }_{n}}\\ \quad -\,\displaystyle \frac{\left(n-3\right){\omega }_{n-2}\eta {r}_{+}^{n-3}{{\rm{\Phi }}}^{2}}{2{\mu }_{n}\left(n-2\right)}\\ \quad +\,\displaystyle \frac{{r}_{+}^{n-3}{\omega }_{n-2}}{2{\kappa }_{n}},\end{array}\end{eqnarray}$
where the black hole mass M is identified with the enthalpy H rather than the internal energy of the gravitational system [73] in the black hole chemistry.2.2. Hawking–Page transition of AdS black holes surrounded by phantom dark energy in general dimensions
Now we calculate the Hawking–Page transition temperature THP which corresponds to the black hole phase with zero Gibbs free energy. From equation (8 ), one can obtain the equation for the radius of the event horizon of this Hawking–Page transition black hole phase, i.e.
$\begin{eqnarray}-\displaystyle \frac{{\kappa }_{n}\eta \left(n-3\right){{\rm{\Phi }}}^{2}}{{\mu }_{n}\left(n-2\right)}+1-16\,\displaystyle \frac{\pi \,{r}_{+}^{2}P}{\left(n-2\right)\left(n-1\right)}=0.\end{eqnarray}$
We denote this radius as the black hole radius of the Hawking–Page transition rHP, and get its value $\begin{eqnarray}{r}_{{HP}}=\sqrt{\,\displaystyle \frac{\left(n-1\right)\left(-{\kappa }_{n}\eta \left(n-3\right){{\rm{\Phi }}}^{2}+{\mu }_{n}\left(n-2\right)\right)}{16P\pi \,{\mu }_{n}}},\end{eqnarray}$
with an additional constraint of the fixed potential Φ $\begin{eqnarray}\eta {{\rm{\Phi }}}^{2}\lt \displaystyle \frac{{\mu }_{n}\left(n-2\right)}{{\kappa }_{n}\left(n-3\right)},\end{eqnarray}$
in order to ensure the existence of the Hawking–Page transition. For AdS black holes surrounded by phantom dark energy, i.e. η = –1, the above constraint always holds.Combining equations (5 ) and (10 ), we can obtain the black hole Hawking–Page transition temperature
$\begin{eqnarray}\begin{array}{l}{T}_{{\text{}}{\rm{HP}}}=\displaystyle \frac{2\sqrt{P}}{\sqrt{n-1}}\\ \quad \sqrt{\displaystyle \frac{-{\kappa }_{n}\eta \left(n-3\right){{\rm{\Phi }}}^{2}+{\mu }_{n}\left(n-2\right)}{{\mu }_{n}\pi }},\end{array}\end{eqnarray}$
which is dependent on the n-dimensional interaction constants κn and μn. This is consistent with the result in [30], where the Hawking–Page transition of a four-dimensional AdS black hole surrounded by phantom dark energy is discussed. When Φ is vanishing, this Hawking–Page transition temperature reduces to the case of an AdS black hole in Einstein gravity.One can see the Gibbs free energy and the Hawking–Page transition temperature of diverse-dimensional AdS black holes surrounded by phantom dark energy in figure 1. The dashed lines in the left G−T and THP−P plots denote the corresponding Gibbs free energy and the Hawking–Page transition temperature of AdS black holes in Einstein gravity. From the top two plots for the Gibbs free energy, one can find that there always exists a Hawking–Page transition between the AdS black holes surrounded by phantom dark energy and the thermal AdS vacuum in spacetime with different Φ and dimensions n. In the bottom two plots for the Hawking–Page transition temperature one can find that the Hawking–Page transition temperature THP always increases as the pressure P increases, as is the case in Einstein gravity. Besides, there are distinct effects of the electric potential Φ and the dimension n on the Hawking–Page transition temperature THP. Concretely, the electric potential Φ and the dimensions n both increase the Hawking–Page transition temperature.
Figure 1. The G–T and THP–P plots of AdS black holes surrounded by phantom dark energy. The top two plots correspond to the G−T curves for different Φ in n = 4 dimensions and for different dimensions with fixed Φ, respectively. The bottom two plots correspond to the THP−P curves for different Φ in n = 4 dimensions and for different dimensions with fixed Φ, respectively. The dashed lines in the left G−T plot and THP−P plot denote the corresponding Gibbs free energy and the Hawking–Page transition temperature of AdS black holes in Einstein gravity. |
2.3. The dual relation for Hawking–Page transition of AdS black holes surrounded by phantom dark energy
To get the dual relation, we first need to obtain the minimum temperature of the black hole. Then we introduce the following function:5 ), we obtain
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{\partial T({r}_{+})}{\partial {r}_{+}}=\displaystyle \frac{4P}{n-2}\\ \quad +\,\displaystyle \frac{\left(n-3\right)\left(\eta {{\rm{\Phi }}}^{2}n{\kappa }_{n}-3\,{{\rm{\Phi }}}^{2}{\kappa }_{n}-n{\mu }_{n}+2\,{\mu }_{n}\right)}{4\left(n-2\right)\pi \,{\mu }_{n}{r}_{+}^{2}},\end{array}\end{eqnarray}$
whose zero leads to the black hole radius of the minimum temperature black hole phase $\begin{eqnarray}{r}_{0}=\sqrt{\,\displaystyle \frac{-\eta {{\rm{\Phi }}}^{2}{\kappa }_{n}{\left(n-3\right)}^{2}+\left(n-3\right)\left(n-2\right){\mu }_{n}}{16P\pi \,{\mu }_{n}}}.\end{eqnarray}$
Inserting this radius into equation ( $\begin{eqnarray}\begin{array}{l}{T}_{0}=\displaystyle \frac{2\sqrt{-\eta {{\rm{\Phi }}}^{2}{\kappa }_{n}{\left(n-3\right)}^{2}+\left(n-3\right)\left(n-2\right){\mu }_{n}}}{\left(n-2\right)\pi }\\ \times \sqrt{\displaystyle \frac{P\pi }{{\mu }_{n}}},\end{array}\end{eqnarray}$
which is also dependent on the n-dimensional interaction constants.Now we derive the dual relation between the Hawking–Page transition temperature and the minimum temperature. Following equation (1 ), we assume
$\begin{eqnarray}{T}_{0}\left(n+1,X{\rm{\Phi }}\right)={T}_{{\text{}}{HP}}\left(n,{\rm{\Phi }}\right),\end{eqnarray}$
which reduces to $\begin{eqnarray}\begin{array}{l}\eta {{\rm{\Phi }}}^{2}\left({X}^{2}{\kappa }_{n+1}{\mu }_{n}{\left(n-2\right)}^{2}\right.\\ \quad \left.-\,{\kappa }_{n}{\mu }_{n+1}\left(n-1\right)\left(n-3\right)\right)=0.\end{array}\end{eqnarray}$
We get the coefficient with interaction constant dependence $\begin{eqnarray}X=\displaystyle \frac{\sqrt{\left(n-1\right)\left(n-3\right)}}{n-2}\sqrt{\displaystyle \frac{{\kappa }_{n}}{{\kappa }_{n+1}}}\sqrt{\displaystyle \frac{{\mu }_{n+1}}{{\mu }_{n}}},\end{eqnarray}$
which results in a novel dual relation $\begin{eqnarray}\begin{array}{l}{T}_{0}\left(n+1,\displaystyle \frac{\sqrt{\left(n-1\right)\left(n-3\right)}{\rm{\Phi }}}{n-2}\right.\\ \left.\times \sqrt{\displaystyle \frac{{\kappa }_{n}}{{\kappa }_{n+1}}}\sqrt{\displaystyle \frac{{\mu }_{n+1}}{{\mu }_{n}}}\right)={T}_{{HP}}\left(n,{\rm{\Phi }}\right),\end{array}\end{eqnarray}$
or equivalently $\begin{eqnarray}\begin{array}{l}{T}_{0}\left(n+1,\displaystyle \frac{\sqrt{\left(n-1\right)\left(n-3\right)}{\rm{\Phi }}}{n-2}\right.\\ \left.\times \sqrt{\displaystyle \frac{{G}_{n}}{{G}_{n+1}}}\sqrt{\displaystyle \frac{{\mu }_{n+1}}{{\mu }_{n}}}\right)={T}_{{\text{}}{HP}}\left(n,{\rm{\Phi }}\right).\end{array}\end{eqnarray}$
Similar to equation (1 ), the above relation is independent of the pressure P, while it should relate to the Maxwell field and depend on the electric potential Φ. Compared with the relation in [8], in gravity with a classical or phantom Maxwell field the dual relation equation (20 ) exhibits an additional dependence on the interaction constants. The dual relation equation (20 ) depends on both the n-dimensional gravitational constant κn and the Maxwell constant μn. Even if we choose the same strength constant as the (classical or phantom) Maxwell field in general dimensions (e.g. one could choose the magnetic vacuum permeability, i.e. μn = μ0.) the dual relation still contains a gravitational constant dependence, as the relation reads1 ), where the dependencies on both interaction constants disappear and there exists only a dependence on spacetime dimension. In all the discussion, the AdS black holes with a classical and phantom Maxwell field have the same dual relations.
$\begin{eqnarray}\begin{array}{l}{T}_{0}\left(n+1,\displaystyle \frac{\sqrt{\left(n-1\right)\left(n-3\right)}{\rm{\Phi }}}{n-2}\right.\\ \left.\times \sqrt{\displaystyle \frac{{G}_{n}}{{G}_{n+1}}}\right)={T}_{{\text{}}{HP}}\left(n,{\rm{\Phi }}\right).\end{array}\end{eqnarray}$
Until one further chooses 8π Gn = 1 in arbitrary dimensions, the dual relation reduces to the following one in [8]: $\begin{eqnarray}\begin{array}{l}{T}_{0}\left(n+1,\displaystyle \frac{\sqrt{\left(n-1\right)\left(n-3\right)}{\rm{\Phi }}}{n-2}\right)\\ \quad =\,{T}_{{\text{}}{HP}}\left(n,{\rm{\Phi }}\right).\end{array}\end{eqnarray}$
If the (classical or phantom) Maxwell field is vanishing, the universal relation reduces to equation (3. Hawking–Page transition and the dual relations of AdS black holes surrounded by quintessence dark energy
In this section we generalize the discussion to the case of n-dimensional AdS black holes surrounded by quintessence dark energy.
3.1. Extended thermodynamics of AdS black holes surrounded by quintessence dark energy in general dimensions
In an n-dimensional spacetime, the solution corresponding to AdS black holes surrounded by quintessence dark energy is given by the general form [38, 53]
$\begin{eqnarray}\begin{array}{rcl}{\rm{d}}{s}^{2} & = & -f(r){\rm{d}}{t}^{2}+\displaystyle \frac{1}{f(r)}{\rm{d}}{r}^{2}+{r}^{2}{\rm{d}}{{\rm{\Omega }}}_{n-2}^{2},\\ f(r) & = & 1-\displaystyle \frac{m}{{r}^{n-3}}+\,\displaystyle \frac{16\pi P\,{r}^{2}}{\left(n-1\right)\left(n-2\right)}\\ & & -\displaystyle \frac{\alpha }{{r}^{\left(n-1\right){\omega }_{n}+n-3}},\end{array}\end{eqnarray}$
where $P=-\tfrac{{\rm{\Lambda }}}{8\pi }$ is the thermodynamic pressure, m is a constant related to the black hole mass and α is a positive normalization factor related to the density of the quintessence dark energy. The state parameter ωn is related to the linear equation of state of the quintessence dark energy, and we choose $\begin{eqnarray}-1\leqslant {\omega }_{n}\leqslant -\displaystyle \frac{(n-3)}{(n-1)}\end{eqnarray}$
in this paper, corresponding to the asymptotically non-flat behavior of the spacetime. In n = 4 dimensions this reduces to ωn = ωq and −1 ≤ ωn ≤ − 1/3.The thermodynamical quantities in the extended thermodynamics of n-dimensional AdS black holes surrounded by quintessence dark energy are presented in [53], and we list them here
$\begin{eqnarray}\begin{array}{l}M=\displaystyle \frac{\left(n-2\right){{\rm{\Omega }}}_{n-2}m}{16\pi }\\ \quad =\,\displaystyle \frac{\left(n-2\right){{\rm{\Omega }}}_{n-2}}{16\pi }\left(\,\displaystyle \frac{16{r}_{+}^{n-1}\pi \,P}{\left(n-1\right)\left(n-2\right)}\right.\\ \quad \left.-\,\displaystyle \frac{\alpha }{{r}_{+}^{n{\omega }_{n}-{\omega }_{n}}}+{r}_{+}^{n-3}\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}T=\displaystyle \frac{4{r}_{+}P}{n-2}+\,\displaystyle \frac{{r}_{+}^{-n{\omega }_{n}-n+{\omega }_{n}+2}{\omega }_{n}\left(n-1\right)\alpha }{4\pi }\\ \quad +\,\displaystyle \frac{(n-3)}{4\pi \,{r}_{+}},\end{array}\end{eqnarray}$
$\begin{eqnarray}S=\displaystyle \frac{1}{4}\,{{\rm{\Omega }}}_{n-2}{r}_{+}^{n-2},\end{eqnarray}$
where r+ is the radius of the black hole event horizon. Finally, as we discuss the Hawking–Page transition in extended thermodynamics, we introduce the Gibbs free energy of AdS black holes surrounded by quintessence dark energy $\begin{eqnarray}\begin{array}{l}G=H-{TS}=M-{TS}\\ \quad \,=\,\displaystyle \frac{{{\rm{\Omega }}}_{n-2}{r}_{+}^{n-3}}{16\pi }\left(1-\,\displaystyle \frac{16\pi \,{r}_{+}^{2}P}{\left(n-1\right)\left(n-2\right)}\right.\\ \quad \left.\,-\,{r}_{+}^{-n{\omega }_{n}-n+{\omega }_{n}+3}\left(n{\omega }_{n}+n-{\omega }_{n}-2\right)\alpha \right).\end{array}\end{eqnarray}$
3.2. Hawking–Page transition of a special AdS black hole surrounded by quintessence dark energy
In this section we first consider a special AdS black hole surrounded by quintessence dark energy with ${\omega }_{n}=-\tfrac{(n-2)}{(n-1)}$, in order to give an analytical study of the Hawking–Page transition.
For this case, the Gibbs free energy reduces to
$\begin{eqnarray}G={{\rm{\Omega }}}_{n-2}{r}_{+}^{n-3}\left(-\displaystyle \frac{{{\Pr }}_{+}^{2}}{\left(n-1\right)\left(n-2\right)}+\displaystyle \frac{1}{16\pi }\right),\end{eqnarray}$
with its zero being located at $\begin{eqnarray}{r}_{{\text{}}{HP}}=\displaystyle \frac{\sqrt{\left(n-1\right)\left(n-2\right)}}{4\sqrt{P\pi }}.\end{eqnarray}$
Then we can get the Hawking–Page transition temperature $\begin{eqnarray}{T}_{{\text{}}{HP}}=\displaystyle \frac{2\sqrt{\left(n-2\right)}\sqrt{P}}{\sqrt{\pi }\sqrt{\left(n-1\right)}}-\,\displaystyle \frac{\left(n-2\right)\alpha }{4\pi }.\end{eqnarray}$
This is consistent with the result in [12], where the Hawking–Page transition of a four-dimensional AdS black hole surrounded by quintessence dark energy is discussed. When α is vanishing, this Hawking–Page transition temperature reduces to the case of an AdS black hole in Einstein gravity.One can see the Gibbs free energy and the Hawking–Page transition temperature of diverse-dimensional AdS black holes surrounded by quintessence dark energy in figure 2. The dashed lines in the left G−T and THP−P plots denote the corresponding Gibbs free energy and the Hawking–Page transition temperature of AdS black holes in Einstein gravity. From the above two plots for the Gibbs free energy, one can find that there always exists a Hawking–Page transition between the AdS black holes surrounded by quintessence dark energy and the thermal AdS vacuum in the spacetime with different α and dimensions n. In the bottom two plots of the Hawking–Page transition temperature, one can find that the Hawking–Page transition temperature THP always increases as pressure P increases, as in the case of Einstein gravity. Besides, there are distinct effects of the density parameter α of the quintessence dark energy and the dimensions n on the Hawking–Page transition temperature THP. Concretely, the density parameter α and the dimensions n both reduce the Hawking–Page transition temperature. On the other hand, if the pressure is small enough, the Hawking–Page transition of AdS black holes surrounded by quintessence dark energy will be vanishing. This is because the Hawking–Page transition temperature becomes negative, and then the system always favors the AdS black hole phase surrounded by quintessence dark energy rather than the AdS vacuum.
Figure 2. The G–T and THP–P plots of AdS black holes surrounded by quintessence dark energy. The top two plots correspond to the G−T curves for different α in n = 4 dimensions and for different dimensions with fixed α, respectively. The bottom two plots correspond to the THP−P curves for different α in n = 4 dimensions and for different dimensions with fixed α, respectively. The dashed lines in the left G−T and THP−P plots denote the corresponding Gibbs free energy and the Hawking–Page transition temperature of AdS black holes in Einstein gravity. |
3.3. Hawking–Page transition of AdS black holes surrounded by quintessence dark energy in general dimensions
Now we study the Hawking–Page transition of AdS black holes surrounded by quintessence dark energy in general dimensions with general ωn. From the Gibbs free energy in equation (28 ), one can obtain the equation for the radius of the event horizon of the Hawking–Page transition black hole phase, i.e.24 ), one can find that the power of r+ in the above second terms always has $0\leqslant \left({\omega }_{n}+1\right)\left(n-1\right)\leqslant 2$. Now we can get $F({r}_{+}){| }_{r=+\infty }=-\,\tfrac{4P}{(n-1)}\lt 0$ and $F({r}_{+}){| }_{r=+0}=\tfrac{(n-2)}{4{r}_{+}^{2}\pi }\,=+\infty $. This means that, as r+ increases from 0 to + ∞, F(r+) will always decrease from + ∞ to the minima and then increase to a negative constant, or directly decrease from + ∞ to a negative constant, whether or not F(r+) has a minimum or no extremum. As a result, F(r+) always has a single zero. Namely, the Gibbs free energy only has a single zero, which corresponds to the Hawking–Page transition.
$\begin{eqnarray}\begin{array}{l}F({r}_{+})=-\,\displaystyle \frac{4P}{(n-1)}\\ \quad -\,\displaystyle \frac{\left(n-2\right)\alpha \left(n{\omega }_{n}+n-{\omega }_{n}-2\right)}{4\pi \,{r}_{+}^{\left({\omega }_{n}+1\right)\left(n-1\right)}}\\ \quad +\,\displaystyle \frac{(n-2)}{4{r}_{+}^{2}\pi }=0.\end{array}\end{eqnarray}$
Then considering the region of ωn shown in equation (It is difficult to calculate the Hawking–Page transition temperature for general ωn, as the corresponding Gibbs free energy has a very complicated form. Here we discuss the small-α limit of the Hawking–Page transition temperature in order to find the effect of the small quintessence dark energy on the Hawking–Page transition. After setting r+ = r0 + r1α + O(α2) and inserting it into equation (32 ), we can get3.2 . Besides, when α is small, one can easily find that the density parameter α of the quintessence dark energy reduces the Hawking–Page transition temperature.
$\begin{eqnarray}\begin{array}{l}{r}_{{HP}}=\displaystyle \frac{\sqrt{\left(n-2\right)\left(n-1\right)}}{4\sqrt{P\pi }}\\ \quad \times \,\left(1-\displaystyle \frac{\alpha }{2}\displaystyle \frac{\,\left(n({\omega }_{n}+1)-({\omega }_{n}+2)\right)}{{\left(\,\tfrac{\sqrt{\left(n-2\right)\left(n-1\right)}}{4\sqrt{P\pi }}\right)}^{n\left({\omega }_{n}+1\right)-({\omega }_{n}+3)}}\right)\\ \quad \,+\,O({\alpha }^{2}),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{T}_{{\rm{HP}}}=\displaystyle \frac{2\sqrt{P}\sqrt{n-2}}{\sqrt{\pi }\sqrt{n-1}}\\ \quad \times \,\left(1-\displaystyle \frac{\alpha }{2{\left(\,\tfrac{\sqrt{\left(n-2\right)\left(n-1\right)}}{4\sqrt{P\pi }}\right)}^{n\left({\omega }_{n}+1\right)-({\omega }_{n}+3)}}\right)\\ \quad \,+\,O({\alpha }^{2}).\end{array}\end{eqnarray}$
This is consistent with the result for AdS black holes surrounded by quintessence dark energy with ${\omega }_{n}=-\tfrac{(n-2)}{(n-1)}$ in Section 3.4. The dual relation from the Hawking–Page transition of AdS black holes surrounded by quintessence dark energy
In this section we study the dual relation from the Hawking–Page transition of AdS black holes surrounded by quintessence dark energy. It is easy to get the equation for the radius r0 of the black hole minimum temperature, i.e.32 ) indicates that there is a single root for the above equation.
$\begin{eqnarray}\begin{array}{l}G({r}_{+})=-\,\displaystyle \frac{4P}{(n-2)}\\ \quad +\,\displaystyle \frac{\left(n{\omega }_{n}+n-{\omega }_{n}-2\right){\omega }_{n}\left(n-1\right)\alpha }{4\pi \,{r}_{+}^{\left({\omega }_{n}+1\right)\left(n-1\right)}}\\ \quad +\,\displaystyle \frac{(n-3)}{4{r}_{+}^{2}\pi }=0.\end{array}\end{eqnarray}$
A similar derivation following equation (In order to find the dual relation equation (1 ) it is natural to assume36 ) into $G\left(n+1,{r}_{0}(n+1)\right)-F\left(n,{r}_{{\rm{HP}}}(n)\right)=0$, we get
$\begin{eqnarray}{r}_{0}\left(n+1\right)={r}_{{\text{}}{\rm{HP}}}\left(n\right),\end{eqnarray}$
as in the cases of Einstein gravity and spacetime surrounded by phantom dark energy. Here r0 and rHP are, respectively, the radius for the Hawking–Page transition temperature and the minimum temperature of an n-dimensional AdS black hole surrounded by quintessence dark energy. Namely, we have $\begin{eqnarray}F({r}_{+}){| }_{{r}_{+}={r}_{{\rm{HP}}}}=0,\quad G({r}_{+}){| }_{{r}_{+}={r}_{0}}=0.\end{eqnarray}$
Besides, as the dual relation is independent of the state parameter ωn, here one will choose ωn as a constant. After inserting equation ( $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{\alpha }{4\pi }\left({r}_{{\rm{HP}}}^{-\left({\omega }_{n}+1\right)n}\left(n{\omega }_{n}+n-1\right){\omega }_{n}n\right.\\ \quad \left.+{r}_{{\rm{HP}}}^{-\left({\omega }_{n}+1\right)\left(n-1\right)}\left(n-2\right)\left(n{\omega }_{n}\,+\,n-{\omega }_{n}-2\right)\right)=0,\end{array}\end{eqnarray}$
which can be never hold for an arbitrary black hole radius. Thus, the radius for the Hawking–Page transition temperature is always different from the radius for the minimum temperature; then we can find ${T}_{0}\left(n+1\right)\ne {T}_{{\text{}}{\rm{HP}}}\left(n\right)$, for which the the dual relation seems to fails in the spacetime surrounded by quintessence dark energy.Now we try to find a modified dual relation. We begin with an open assumption that the state parameter ωn and density parameter α of the quintessence dark energy depend on the dimensions of the spacetime n. Namely, in n dimensions they are39 –41 ), $G\left(n,+,1,,,{\omega }_{n+1},,,{\alpha }_{n+1},,,{r}_{0},(,n,+,1,)\right)=0$ and $F\left(n,{\omega }_{n},{\alpha }_{n},{r}_{{\rm{HP}}}(n)\right)=0$, we can get $G\left(n+1,{\omega }_{n+1},{\alpha }_{n+1},{r}_{0}(n+1)\right)-F\left(n,{\omega }_{n},{\alpha }_{n},{r}_{{\rm{HP}}}(n)\right)=0$, which leads to42 ), it can be simplified as
$\begin{eqnarray}{\omega }_{n}={\omega }_{n}(n),\quad \alpha ={\alpha }_{n}(n).\end{eqnarray}$
Then we can modify the dual relation and get a similar one: $\begin{eqnarray}{r}_{0}\left(n+1,{\omega }_{n+1},{\alpha }_{n+1}\right)={r}_{{\text{}}{\rm{HP}}}\left(n,{\omega }_{n},{\alpha }_{n}\right),\end{eqnarray}$
$\begin{eqnarray}{T}_{0}\left(n+1,{\omega }_{n+1},{\alpha }_{n+1}\right))={T}_{{\text{}}{\rm{HP}}}\left(n,{\omega }_{n},{\alpha }_{n}\right).\end{eqnarray}$
Combining these three equations ( $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{r}_{{\rm{HP}}}^{-\left({\omega }_{n+1}+1\right)n}\left({\omega }_{n+1}n+n-1\right){\omega }_{n+1}n{\alpha }_{n+1}}{4\pi }\\ \quad +\,\displaystyle \frac{{r}_{{\rm{HP}}}^{-\left({\omega }_{n}+1\right)\left(n-1\right)}\left(n-2\right)\left(n{\omega }_{n}+n-{\omega }_{n}-2\right){\alpha }_{n}}{4\pi }=0.\end{array}\end{eqnarray}$
This relation should hold for an arbitrary AdS black hole with different radius rHP, then the power of rHP for the two terms in the above equation should be the same, which results in the constraint $\begin{eqnarray}{\omega }_{n+1}=\displaystyle \frac{\left(n-1\right){\omega }_{n}-1}{n}.\end{eqnarray}$
Inserting this constraint back to equation ( $\begin{eqnarray}n{\alpha }_{n+1}{\omega }_{n}+{\alpha }_{n}n-{\alpha }_{n+1}{\omega }_{n}-2\,{\alpha }_{n}-{\alpha }_{n+1}=0,\end{eqnarray}$
corresponding to another constraint $\begin{eqnarray}{\alpha }_{n+1}=\displaystyle \frac{\left(n-2\right){\alpha }_{n}}{1-\left(n-1\right){\omega }_{n}}.\end{eqnarray}$
Now considering the dual relation with the above two constraints, we can find
$\begin{eqnarray}\begin{array}{l}{T}_{0}\left(n+1,{\omega }_{n+1},{\alpha }_{n+1}\right))-{T}_{{\text{}}{\rm{HP}}}\left(n,{\omega }_{n},{\alpha }_{n}\right)\\ \quad =\,\displaystyle \frac{{r}_{{\rm{HP}}}}{(n-2)}\left(-\,\displaystyle \frac{4P}{(n-1)}\right.\\ \quad \left.-\,\displaystyle \frac{\left(n-2\right)\alpha \left(n{\omega }_{n}+n-{\omega }_{n}-2\right)}{4\pi \,{r}_{{\rm{HP}}}^{\left({\omega }_{n}+1\right)\left(n-1\right)}}+\,\displaystyle \frac{(n-2)}{4{r}_{{\rm{HP}}}^{2}\pi }\right),\\ \quad =\,\displaystyle \frac{{r}_{{\rm{HP}}}}{(n-2)}F({r}_{+}){| }_{{r}_{+}={r}_{{\rm{HP}}}}=0,\end{array}\end{eqnarray}$
which always holds. Finally, we find that the modified dual relation $\begin{eqnarray}{T}_{{\text{}}{\rm{HP}}}\left(n,{\omega }_{n},{\alpha }_{n}\right)={T}_{0}\left(n+1,{\omega }_{n+1},{\alpha }_{n+1}\right)),\end{eqnarray}$
or equivalently, $\begin{eqnarray}\begin{array}{l}{T}_{{\text{}}{\rm{HP}}}\left(n,{\omega }_{n},{\alpha }_{n}\right)={T}_{0}\left(n+1,\displaystyle \frac{\left(n-1\right){\omega }_{n}-1}{n},\right.\\ \quad \left.\,\displaystyle \frac{\left(n-2\right){\alpha }_{n}}{1-\left(n-1\right){\omega }_{n}}\right),\end{array}\end{eqnarray}$
always holds. We summarize the dual relation of an AdS black hole surrounded by quintessence dark energy in table 1.Table 1. The dual relation from the Hawking–Page transition of AdS black holes surrounded by quintessence dark energy. |
| Dual relation | ωn+1 | αn+1 | |
|---|---|---|---|
| Fail | ${T}_{{\text{}}{HP}}\left(n\right)\ne {T}_{0}\left(n+1\right)$ | Constant | Constant α |
| Success | ${T}_{{\text{}}{HP}}\left(n,{\omega }_{n},{\alpha }_{n}\right)$ | Dimension dependence | Dimension dependence |
| $=\,{T}_{0}\left(n+1,{\omega }_{n+1},{\alpha }_{n+1}\right))$ | ${\omega }_{n+1}=\tfrac{\left(n-1\right){\omega }_{n}-1}{n}$ | ${\alpha }_{n+1}=\tfrac{\left(n-2\right){\alpha }_{n}}{1-\left(n-1\right){\omega }_{n}}$ |
The modified dual relation for AdS black holes surrounded by quintessence dark energy is also independent of the pressure P, even though the black hole minimum temperature and the Hawking–Page transition temperature both depend on the pressure P. The modified dual relation is related to the additional matter field, i.e. the quintessence dark energy. Compared with the relation in [8], the modified dual relation depends on the state parameter ωn and the density parameter αn of the quintessence dark energy. If the quintessence dark energy is vanishing, the relation reduces to the universal one in Einstein gravity, i.e. Equation (1 ), where the two parameter (ωn and αn) dependencies both disappear and there exists only a dependence on spacetime dimensions.
In table 2, we give some examples of the dual relation equation (47 ) for AdS black holes surrounded by quintessence dark energy with different state parameters $-1\leqslant {\omega }_{n}\leqslant -\tfrac{(n-3)}{(n-1)}$. Since the span $-1\leqslant {\omega }_{n}\leqslant -\tfrac{(n-3)}{(n-1)}$ for the state parameter ωn depends on the dimension n of the spacetime, it is acceptable that the state parameter ωn also depends on the dimension n. As αn is the positive normalization factor related to the density of the quintessence dark energy in general dimensions it may depend on the dimension n. However, if one chooses αn as a constant α, the dual relation will hold only for AdS black holes surrounded by quintessence dark energy with ${\omega }_{n}=-\tfrac{(n-3)}{(n-1)}$.
Table 2. The modified dual relation for AdS black holes surrounded by quintessence dark energy with different ωn. |
| ωn | ωn+1 | αn+1 |
|---|---|---|
| −1 | −1 | ${\alpha }_{n+1}=\tfrac{{\alpha }_{n}\left(n-2\right)}{n}$ |
| $-\tfrac{(2n-3)}{2(n-1)}$ | $-\tfrac{(2n-1)}{2n}$ | ${\alpha }_{n+1}=\tfrac{2{\alpha }_{n}\left(n-2\right)}{(2\,n-1)}$ |
| $-\tfrac{(n-2)}{(n-1)}$ | $-\tfrac{(n-1)}{n}$ | ${\alpha }_{n+1}=\tfrac{{\alpha }_{n}\left(n-2\right)}{n-1}$ |
| $-\tfrac{(2n-5)}{2(n-1)}$ | $-\tfrac{(2n-3)}{2n}$ | ${\alpha }_{n+1}=\tfrac{2{\alpha }_{n}\left(n-2\right)}{(2\,n-3)}$ |
| $-\tfrac{(n-3)}{(n-1)}$ | $-\tfrac{(n-2)}{n}$ | αn+1 = αn i.e. constant α |
4. Conclusion
In this paper one can always find the occurrence of the Hawking–Page transition observed between thermal AdS radiation and thermodynamically stable large AdS black holes, in spacetime surrounded by phantom dark energy and spacetime surrounded by quintessence dark energy. Compared with the case in Einstein gravity, the Hawking–Page transition temperature THP always increases as the pressure P increases in spacetime surrounded by phantom dark energy and spacetime surrounded by quintessence dark energy. On the other hand, there are distinct effects of phantom dark energy and quintessence dark energy on the Hawking–Page transition temperature. The electric potential Φ and the dimension n of AdS black holes surrounded by phantom dark energy both increase the the Hawking–Page transition temperature; while the density parameter α and the dimension n of AdS black holes surrounded by quintessence dark energy both reduce the the Hawking–Page transition temperature. If the pressure in the spacetime surrounded by quintessence dark energy is small enough, the Hawking–Page transition will be vanishing.
In particular we generalize the study of the dual relation equation (1 ) between the minimum temperature T0 and the Hawking–Page transition temperature THP of an AdS black hole in two successive dimensions into spacetime surrounded by phantom dark energy and spacetime surrounded by quintessence dark energy. For the former case the dual relation still works well, and we find that this dual relation should have a dependence on the interaction constants, including the constants characterizing the strength of the gravitational field and (classical or phantom) Maxwell field. When the Maxwell field is vanishing, the dependence on the two interaction constants for the dual relation disappears and there exists only a dependence on spacetime dimensions. For the case of AdS black holes surrounded by quintessence dark energy, we find that the dual relation should be modified (as equations (47 ) or (48 )), after considering an open assumption that the state parameter and density parameter of the quintessence dark energy depend on the dimensions of the spacetime.
Finally, we give some new viewpoints here, since the holographic understanding of this universal relation is still open. One can treat T0 as the temperature of black hole formation, where a virtual black hole becomes a real one, without thermal radiation (i.e. the vacuum state) being taken into account. Correspondingly, THP should be the temperature of AdS black hole formation where a vacuum state becomes a real black hole state. Thus, the dual relation may be interpreted as a duality between the formation of an AdS black hole in two successive dimensions. On the other hand, in AdS spacetimes surrounded by phantom dark energy and by quintessence dark energy, the novel dual relations show a notable effect of the additional dark energy fields with a concomitant dependence of the spacetime dimensions and the interaction constants. When more matter fields are taken into account in the spacetime, the dual relation may also depend on the related spacetime dimensions and the interaction constants. Since the novel dual relation depends on the spacetime dimensions and the interaction constants in two successive dimensions, we believe that this property may provide more clues to the holographic understanding of black hole thermodynamics, for example a precise quantitative description in the holographic framework. Actually, the dependence on two successive dimensional interaction constants is common in the holographic principle and higher gravity theory. For example, the constraint for $\tfrac{{G}_{n}}{{G}_{n+1}}$ (i.e. $\tfrac{{\kappa }_{n}}{{\kappa }_{n+1}}$) is necessary in Kaluza–Klein compactification of spacetime in two successive dimensions. This work gives rise to certain open questions. It would be interesting to apply the study of the Hawking–Page transition to AdS black holes in the higher derivative gravities, for example AdS black holes in n-dimensional Gauss–Bonnet gravity [79–82]. We leave these questions for future work.