Black holes with quintessence in pure Lovelock gravity

Abstract

In this work, we obtain the solution corresponding to a static spherically symmetric black hole surrounded by quintessence in pure Lovelock gravity. Some aspects of the thermodynamics of this black hole are investigated, with special emphasis on the Hawking temperature, entropy and heat capacity. The behaviors of these quantities are analyzed and the differences with respect to the ones in the Theory of General Relativity are pointed out.

Appendices

Another procedure to obtain the solution corresponding to a black hole with quintessence in pure Lovelock gravity

In this appendix, we will obtain the solution for the black hole surrounded by quintessence in pure Lovelock gravity directly from the generalized Einstein’s equation. This result will confirm the solution obtained in Sect. 4 by the method developed by Cai [10, 22].

We start supposing once more that the black hole metric has the form given by Eq. (15). Adopting $f = 1 - F$, Dahich [39] showed that the solution for the generalized Einstein’s equations in pure Lovelock gravity is given by

$$\begin{aligned} G^\mu _{\nu } = -\frac{D-2}{r^{D-2}}(r^{D-2n - 1} F^n)' = T^{r}_{r} = T^{t}_{t}. \end{aligned}$$

(46)

where the comma represents the differentiation with respect to the coordinate r. In the region outside the black hole surrounded by quintessence, we get

$$\begin{aligned} (r^{D-2n - 1} F^n)' = \frac{q (D-1)}{2 r^{(D-1)\omega _q+1}}. \end{aligned}$$

(47)

Performing the integration in both sides of the Eq. (47), we get

$$\begin{aligned} F^n = \frac{\mu }{r^{D-2n-1}}-\frac{q (D-1)[(D-1) \omega _q+1] }{2 r^{(D-1)(\omega _q+1)-2n}}. \end{aligned}$$

(48)

It is worth observing that Eq. (48) is simmilar to Eq. (23) if we take $\mu \rightarrow \frac{2M}{{\tilde{\alpha }}_n (D-2){\varSigma }_{D-2}}$ and $q \rightarrow -\frac{q}{{\tilde{\alpha }}_n (D-1)[(D-1) \omega _q+1]}$.

Adding the cosmological constant to Eq. (48), we get [39]

$$\begin{aligned} F^n = {\varLambda }_1 r^{2n}+ \frac{\mu }{r^{D-2n-1}}-\frac{q (D-1)[(D-1) \omega _q+1] }{2 r^{(D-1)(\omega _q+1)-2n}}, \end{aligned}$$

(49)

where ${\varLambda }_1 = \frac{2 {\varLambda }}{(D-1)(D-2)}$.

dS/AdS black hole with quintessence in pure Lovelock gravity

Now, let us include the cosmological constant ${\varLambda }$ into the configuration under study. Thus, we can substitute $|\alpha _0| = 2 {\varLambda }$ into Eq. (11). Therefore, we obtain the following equation

$$\begin{aligned} -{\varLambda }g_{\mu \nu } + \alpha _n G^{(n)}_{\mu \nu } = -{\varLambda }g_{\mu \nu }+ \alpha _n \left[ n \left( R^{(n)}_{\mu \nu } - \frac{1}{2}R^{(n)}g_{\mu \nu } \right) \right] = T_{\mu \nu }. \end{aligned}$$

(50)

According to the results obtained in the A,

$$\begin{aligned} f(r) = 1 - \frac{1}{\alpha ^{2-2/n}}\left[ {\varLambda }_1 r^{2n} + \frac{16 \pi M}{(D-2){\varSigma }_{D-2}r^{D - 2n-1}}+\frac{q}{r^{(D-1)(\omega _q+1)-2n}} \right] ^{\frac{1}{n}}.\nonumber \\ \end{aligned}$$

(51)

In the limit $r \rightarrow \infty$,

$$\begin{aligned} f(r) = 1 - \frac{1}{\alpha ^{2-2/n}}\left[ {\varLambda }_1^{1/n} r^{2} + \frac{16 \pi M}{(D-2){\varSigma }_{D-2}r^{D -3}}+\frac{q}{r^{(D-1)\omega _q+D-3}} \right] . \end{aligned}$$

(52)

Thus, one can conclude that, in the limit of low energy, pure Lovelock gravity asymptotically gives the same results obtained in the TGR.

In order to analyze the black hole thermodynamics, we can write the black hole mass parameter as

$$\begin{aligned} M = \frac{(D-2) {\varSigma }_{D-2}}{16 \pi }\left[ \alpha ^{2n-2}r_h^{D-2n-1} -\frac{q}{r_h^{(D-1)\omega _q}} - {\varLambda }r_h^{D-1}\right] . \end{aligned}$$

(53)

The Hawking temperature of the black hole is given by

$$\begin{aligned} T = \frac{\alpha ^{2-2n}}{4 \pi n}\left[ \frac{\alpha ^{2n-2}(D-2n-1)}{r_h} +\frac{q \alpha ^{2-2n}\omega _q(D-1)}{r_h^{(D-1)\omega _q+D-2n}}-(D-1){\varLambda }r ^{2n-1}\right] \nonumber \\ \end{aligned}$$

(54)

and its entropy can be calculated by

$$\begin{aligned} S = \int \frac{dM}{T} = \int T^{-1} \frac{\partial M}{\partial r_h} d r_h =\frac{n (D-2) {\varSigma }_{D-2}}{4 (D-2n)\alpha ^{2-2n}} r_h^{D-2n}. \end{aligned}$$

(55)

Besides that, we can identify the thermodynamical pressure with the cosmological constant through the relation [40]

$$\begin{aligned} p = -\frac{{\varLambda }}{8\pi }, \end{aligned}$$

(56)

so that we can write the First Law of the black hole thermodynamics as

$$\begin{aligned} dM = T dS + V dp, \end{aligned}$$

(57)

where

$$\begin{aligned} V = \frac{\partial M}{\partial p} = \frac{(D-2) {\varSigma }_{D-2}}{2} r_h^{D-1}. \end{aligned}$$

(58)