Self-gravitating static non-critical black holes in 4D Einstein–Klein–Gordon system with nonminimal derivative coupling

Abstract

We study static non-critical hairy black holes of four dimensional gravitational model with nonminimal derivative coupling and a scalar potential turned on. By taking an ansatz, namely, the first derivative of the scalar field is proportional to square root of a metric function, we reduce the Einstein field equation and the scalar field equation of motions into a single highly nonlinear differential equation. This setup implies that the hair is secondary-like since the scalar charge-like depends on the non-constant mass-like quantity in the asymptotic limit. Then, we show that near boundaries the solution is not the critical point of the scalar potential and the effective geometries become spaces of constant scalar curvature.

The next leading order

In this section, we consider the first order solutions in the asymptotic limit using a perturbative ansatz

$$\begin{aligned} Y = Y_0 + Y_1 , \end{aligned}$$

(A.1)

with $|Y_0| \gg |Y_1(x)|$ showing that we have a regular solution up to the perturbative correction. Such a setup simplifies the master equation (3.11) into the following linear equation

$$\begin{aligned}&2 \tilde{\chi } \frac{d^2Y_1}{dx^2} + 3 \left( 4 \tilde{\chi } - \chi -\frac{2 \nu ^2 \chi ^3}{\chi - \tilde{\chi }} \right) \frac{dY_1}{dx} -18 \left( \chi - \tilde{\chi } \right) Y_1 \nonumber \\&\quad + 4 \tilde{\chi } \left[ \left( 2 + \eta \kappa ^2 \nu ^2 \right) \frac{\chi ^2 k}{\tilde{\chi } \kappa ^2 } -\eta \nu ^2 \chi k \right] e^{-2x} + 4 \tilde{\chi } \chi ^2 \eta ^2 \nu ^4 k^2 e^{-4x} = 0 , \qquad \end{aligned}$$

(A.2)

where we have simply set $\varsigma = \varsigma _1 = 1$. Moreover, we also take $e^{-mx} Y_1(x) \rightarrow 0$ and $e^{-mx} dY_1/dx \rightarrow 0$ for $m \ge 2$ around $x \rightarrow +\infty$. A special class of solution of (A.2) has the form

$$\begin{aligned} Y_1= & {} C_1 S[\lambda _1] ~ e^{\lambda _1 x} + D_1 ~ e^{\lambda _2 x} - \frac{2 \left( \left( 2 + \eta \kappa ^2 \nu ^2 \right) \frac{\chi ^2 k}{\tilde{\chi } \kappa ^2 } -\eta \nu ^2 \chi k \right) }{(\lambda _1 +2) (\lambda _2 +2)} e^{-2x} \nonumber \\&- \frac{2 \chi ^2 \eta ^2 \nu ^4 k^2 }{(\lambda _1 +4) (\lambda _2 + 4)} e^{-4x} , \end{aligned}$$

(A.3)

where $C_1, D_1 \in \mathbb {R}$ with

$$\begin{aligned} \lambda _{1, 2} = \frac{3}{4} \left( \frac{\chi }{\tilde{\chi }} - 4 + \frac{2 \varepsilon \nu ^2 \chi ^3}{\tilde{\chi } ( \chi - \tilde{\chi }) } \pm \left[ \left( \frac{\chi }{\tilde{\chi }} - 4 + \frac{2 \varepsilon \nu ^2 \chi ^3}{ \tilde{\chi } ( \chi - \tilde{\chi }) } \right) ^2 + 16 \left( \frac{\chi }{\tilde{\chi }} - 1 \right) \right] ^{1/2}\right) , \end{aligned}$$

(A.4)

and

$$\begin{aligned} S[\lambda _1] = \left\{ \begin{array}{ll} 0 &{} \text {if }\lambda _1 > 0\\ 1 &{} \text {if }\lambda _1 < 0 \end{array} \right. , \end{aligned}$$

(A.5)

The values of $\lambda _1$ or $\lambda _2$ should be negative, since $\lim _{ x \rightarrow +\infty } Y_1(x) \rightarrow 0$, and $-2< \lambda _{1, 2} < 0$. The solution (A.3) belongs to the following two cases. First, by simply taking $0< \chi /\tilde{\chi } < 1$ and $\eta > 0$, the analysis on (A.4) results $\varepsilon = 1$ and

$$\begin{aligned} 2< \eta \kappa ^2 \nu ^2 < \frac{2}{3} (1 + 4\kappa ^2) ~ . \end{aligned}$$

(A.6)

In this case, both $\lambda _1$ and $\lambda _2$ are negative and we exclude the large $\kappa$ limit case. Second, we simply take $\chi /\tilde{\chi } > 1$ and $\eta < 0$. Analyzing (A.4), we find $\varepsilon = 1$ and

$$\begin{aligned} 2 - \frac{32}{9}\kappa ^2< \eta \kappa ^2 \nu ^2 < 0 ~ , \end{aligned}$$

(A.7)

and only $\lambda _2$ is negative.

Solving the second equation in (3.12), we obtain the first order lapse function f

$$\begin{aligned} f (\rho ) = \hat{A}_0 Y_0^{\varsigma \tilde{\chi }/\chi } \rho ^{3\varsigma \varsigma _1 - 1 } \left( 1 + \frac{\varsigma \tilde{\chi } Y_1}{\chi Y_0} \right) e^{k\varsigma \eta \tilde{\chi } \nu ^2 / Y_0 \rho ^2} ~ , \end{aligned}$$

(A.8)

while the first equation gives us

$$\begin{aligned} \phi (\rho )= & {} \phi _0 + \int \frac{dx}{\sqrt{Y_0 + Y_1}} \nonumber \\\approx & {} \phi _0 + Y_0^{-1/2} ~ {\mathrm {ln}}\rho -\frac{1}{2} Y_0^{-3/2} \Bigg ( \frac{C_1}{\lambda _1} S[\lambda _1] ~ \rho ^{\lambda _1} + \frac{D_1}{\lambda _2} ~ \rho ^{\lambda _2}\nonumber \\&+ \frac{ \left( \left( 2 + \eta \kappa ^2 \nu ^2 \right) \frac{\chi ^2 k}{\tilde{\chi } \kappa ^2 } -\eta \nu ^2 \chi k \right) }{(\lambda _1 +2) (\lambda _2 +2)} \rho ^{-2} + \frac{ \chi ^2 \eta ^2 \nu ^4 k^2 }{2(\lambda _1 +4) (\lambda _2 + 4)} \rho ^{-4} \Bigg ) ~ . \qquad \end{aligned}$$

(A.9)

Equations (A.8) and (A.9) show that we have a different set of solutions compared to [19]. From the second equation in (3.9), we get the first order scalar potential

$$\begin{aligned} V= & {} -\frac{3 Y_0}{2 \chi }-\frac{\varepsilon \nu ^{2} }{2} -\frac{1}{2\chi } \left( C_1 (\lambda _1 +3) S[\lambda _1] ~ \rho ^{\lambda _1} + D_1 (\lambda _2 +3) ~ \rho ^{\lambda _2} \right) \nonumber \\&+ \left( \frac{ \left( 2 + \eta \kappa ^2 \nu ^2 \right) \frac{\chi ^2 k}{\tilde{\chi } \kappa ^2 } -\eta \nu ^2 \chi k }{ \chi (\lambda _1 +2) (\lambda _2 +2)} -\frac{k}{2 \kappa ^{2} } (2 + \eta \kappa ^{2} \nu ^{2}) \right) \rho ^{-2} \nonumber \\&+ \frac{ \chi ^2 \eta ^2 \nu ^4 k^2 }{ \chi (\lambda _1 +4) (\lambda _2 - 4)} \rho ^{-4} ~ . \end{aligned}$$

(A.10)