New exact solutions, thermodynamics and phase transition in the Einstein–Maxwell-dilaton theory

Abstract

For the $(n+1)$-dimensional ($n\ge 3$) dilaton black hole in the Einstein–Maxwell-dilaton theory, we have presented exact analytical solutions of the field equations. These exact solutions include the exact formula of the potential function as well as the exact formula of the metric function. The presence of the dilaton field makes the asymptotic behavior of these black holes no longer flat or anti-de Sitter. We have calculated the electric charge, mass, temperature, entropy and electric potential of these black holes and have shown the correctness of the first law of black hole thermodynamics. As a thermodynamic system, we have analyzed thermal stability of these types of black holes using the canonical ensemble method and, investigated the effect of dilaton field on their stability.

Appendices

Appendix A: details of derivation of Eq.(12)

In order to prove the non-independence of Eqs. (8), (9) and (II.10), we start by differentiating Eq. (10) with respect to r:

$$\begin{aligned} \frac{dE_{\theta \theta }}{dr}= & {} (n-1)f''\left( \frac{1}{r}+\frac{R'}{R}\right) +(n-1)f'\nonumber \\{} & {} \quad \left[ \frac{n-3}{r^{2}} +(n-3)\left( \frac{R'}{R}\right) ^{2}+2(n-1)\frac{R'}{rR}+2\frac{R''}{R}\right] \nonumber \\{} & {} \quad +(n-1)f\left\{ -\frac{2(n-2)}{r^{3}}+2(n-2)\left[ \frac{R'R''}{R^{2}}-\left( \frac{R'}{R}\right) ^{3}\right] \right. \nonumber \\{} & {} \left. \quad +\frac{2(n-1)}{r}\left[ \frac{R''}{R}-\frac{R'}{rR}-\left( \frac{R'}{R}\right) ^{2}\right] -\frac{R'R''}{R^{2}}+\frac{R'''}{R}\right\} \nonumber \\{} & {} \quad +\frac{2(n-1)(n-2)}{r^{2}R^{2}}\left( \frac{1}{r}+\frac{R'}{R}\right) +\frac{d}{dr}\left[ V(\phi )+2F_{tr}^{2}e^{-\frac{4\alpha \phi }{n-1}}\right] . \end{aligned}$$

(A.1)

On the other hand, according to Eq. (7), we can write $2F_{tr}^{2}e^{-\frac{4\alpha \phi }{n-1}}=2q^{2}\frac{e^{\frac{4\alpha \phi }{n-1}}}{(rR)^{2(n-1)}}$ which leads to

$$\begin{aligned} \frac{d}{dr}\left( 2F_{tr}^{2}e^{-\frac{4\alpha \phi }{n-1}}\right) = \left[ \frac{4\alpha \phi '}{n-1}-2(n-1)\left( \frac{1}{r}+\frac{R'}{R}\right) \right] \left( 2F_{tr}^{2}e^{-\frac{4\alpha \phi }{n-1}}\right) . \end{aligned}$$

(A.2)

It is also obtained from the (II.3) and (II.9) that

$$\begin{aligned} \phi '^{2}\equiv \left( \frac{d\phi }{dr}\right) ^{2}=-\left( \frac{n-1}{2}\right) ^{2}\left( \frac{R''}{R}+2\frac{R'}{rR}\right) \end{aligned}$$

(A.3)

and

$$\begin{aligned}{} & {} \frac{d}{dr}V(\phi )\equiv \phi '\frac{dV}{d\phi } \nonumber \\{} & {} \quad =-2(n-1)^{2}f\left( \frac{1}{r}+\frac{R'}{R}\right) \left( \frac{R''}{R}+2\frac{R'}{rR}\right) -\frac{4\alpha \phi '}{n-1}\left( 2F_{tr}^{2}e^{-\frac{4\alpha \phi }{n-1}}\right) \nonumber \\{} & {} \quad -2(n-1)f\left[ \frac{R'''}{R}-\frac{R'R''}{R^{2}}+2\frac{R''}{rR}-2\frac{R'}{rR}\left( \frac{1}{r}+\frac{R'}{R}\right) \right] \nonumber \\{} & {} \quad -2(n-1)f'\left( \frac{R''}{R}+2\frac{R'}{rR}\right) . \end{aligned}$$

(A.4)

The point is that, it is possible to replace $f''$ from Eq. (8):

$$\begin{aligned}{} & {} (n-1)f''\left( \frac{1}{r}+\frac{R'}{R}\right) \nonumber \\{} & {} \quad =(n-1)\left( \frac{1}{r}+\frac{R'}{R}\right) \nonumber \\{} & {} \quad \left\{ E_{tt}-(n-1)f'\left( \frac{1}{r} +\frac{R'}{R}\right) -\frac{2}{n-1}V(\phi )+4\left( \frac{n-2}{n-1}\right) F_{tr}^{2}e^{-\frac{4\alpha \phi }{n-1}}\right\} \end{aligned}$$

(A.5)

As a result, with a little calculation and simplification, we can write

$$\begin{aligned} \frac{dE_{\theta \theta }}{dr}- & {} \left( \frac{1}{r}+\frac{R'}{R}\right) (n-1)E_{tt}=-2(n-1)f'\left( \frac{1}{r}+\frac{R'}{R}\right) +2\left( \frac{1}{r}+\frac{R'}{R}\right) \frac{(n-1)(n-2)}{r^{2}R^{2}} \\{} & {} \quad -2(n-1)f\left( \frac{1}{r}+\frac{R'}{R}\right) \left[ \frac{n-2}{r^{2}}+(n-2)\left( \frac{R'}{R}\right) ^{2}+2(n-1)\frac{R'}{rR}+\frac{R''}{R}\right] \\{} & {} \quad -2\left( \frac{1}{r}+\frac{R'}{R}\right) \left( 2F_{tr}^{2}e^{-\frac{4\alpha \phi }{n-1}}\right) -2\left( \frac{1}{r}+\frac{R'}{R}\right) V(\phi ) \\{} & {} \quad =-2\left( \frac{1}{r}+\frac{R'}{R}\right) E_{\theta \theta }. \end{aligned}$$

Appendix B: details of derivation of Eq. (31)

To calculate the mass of the black hole, we use the following method. First, it is necessary to write the black hole metric in the new form below

$$\begin{aligned} ds^2=-\chi ^{2}(\rho )dt^{2}+\frac{1}{W^{2}(\rho )}d\rho ^{2}+\rho ^{2}h_{ij}dx^{i}dx^{j}. \end{aligned}$$

(B.1)

where $\rho =rR(r)$. From the comparison with the original metric, it follows that $\chi ^{2}(\rho )=f(r)$ and $W^{2}(\rho )=(N+1)^{2}R^{2}f(r)$. The quasi-local mass of the black hole is defined according to the following relation:

$$\begin{aligned} \mathcal {M}=\frac{(n-1)\omega _{n-1}}{8\pi }\rho ^{n-2}\chi (\rho )\left[ W_{0}(\rho )-W(\rho )\right] , \end{aligned}$$

(B.2)

where $W_{0}\equiv W(m=0)$. In the following, it is better to write f(r) in the appropriate form below

$$\begin{aligned} f(r)=-\frac{m}{r^{(n-1)(N+1)-1}}+u(r), \;\;\;\text{: }\;\;\;\;\;\;\; N\ne -\frac{1}{2} \end{aligned}$$

(B.3)

where $u(r)=Ar^{-2N}+Br^{2(N+1)}+Cr^{-2[(n-1+\alpha \varepsilon )(N+1)-1]}$ which is expressed based on Eq. (20). We pay attention that according to the interval defined for N as $-1< N\le 0$, whether by choosing $-1<N<-\frac{1}{2}$ or $-\frac{1}{2}<N\le 0$, at the limit $r\rightarrow \infty$ one term has a constant value and the other term diverges. The third term also becomes zero in any case, and therefore the total limit of the fraction will be zero. According to these points, it can be stated that

$$\begin{aligned} \mathcal {M}{} & {} =\frac{(n-1)(N+1)\omega _{n-1}r_{+}^{(n-1)(N+1)-1}}{8\pi r_{0}^{(n-1)N}}\Bigg [u(r)\left( 1-\frac{m}{u(r)r^{(n-1)(N+1)-1}}\right) ^{\frac{1}{2}}\nonumber \\{} & {} \quad - u(r)+\frac{m}{r^{(n-1)(N+1)-1}}\Bigg ].\end{aligned}$$

(B.4)

Now, one can obtain the black hole mass M, by calculating the limit of $\mathcal {M}$ as r goes to infinity. Since, for the allowed values of N,

$$\begin{aligned} \lim _{r\rightarrow \infty } \frac{m}{u(r)r^{(n-1)(N+1)-1}}=0,\end{aligned}$$

(B.5)

we can write

$$\begin{aligned} M=\lim _{r_{+}\rightarrow \infty } \mathcal {M}=\frac{(n-1)(N+1)\omega _{n-1}}{16\pi r_{0}^{(n-1)N}}m,\;\;\;\text{: }\;\;\;\;\;\;\; N\ne -\frac{1}{2}. \end{aligned}$$

(B.6)

Also, noting Eq. (20), we can write

$$\begin{aligned} f(r)=-\frac{m}{r^{\frac{(n-3)}{2}}}+v(r),\;\;\;\text{: }\;\;\;\;\; N=-\frac{1}{2} \end{aligned}$$

(B.7)

where $v(r)=B_1\frac{r}{r_{0}}\left[ \ln \left( \frac{b}{r}\right) +\frac{n+1}{n-1}\right] +B_2\frac{r}{b}+B_3r^{-(n-3+\alpha )}$ which is written according to the second rule of Eq. (20). As with the calculations related to the previous state, we can also write here

$$\begin{aligned} \mathcal {M}=\frac{(n-1)\omega _{n-1}r_{0}^{\frac{n-1}{2}}r^{\frac{n-3}{2}}}{16\pi }\left[ v(r)\left( 1-\frac{m}{v(r)r^{\frac{n-3}{2}}}\right) ^{\frac{1}{2}} -v(r)+mr^{-\frac{n-3}{2}}\right] , \qquad \end{aligned}$$

(B.8)

which in turn will lead to

$$\begin{aligned} M=\lim _{r_{+}\rightarrow \infty } \mathcal {M}=\frac{(n-1)r_{0}^{\frac{n-1}{2}}\omega _{n-1}}{32\pi }m \;\;\;\text{: }\;\;\;\;\;\;\; N=-\frac{1}{2}. \end{aligned}$$

(B.9)