Superradiant instability and asymptotically AdS hairy black holes in F(R)-charged scalar field theory

Abstract

We study the phenomena of superradiance for F(R)-Maxwell black holes in an AdS space-time. The AdS boundary plays the role of a mirror and provides a natural confining system that makes the superradiant waves bouncing back and forth between the region near the horizon and the reflective boundary, causing a possible superradiant instability. We obtain numerical solutions for static hairy black holes in this scenario and investigate their instability and explicitly address the stability of such solutions for spherical perturbations under specific conditions for the scalar charge and AdS radius. It is shown that for a small scalar charge or AdS radius the static hairy solution is stable under spherical perturbations. We conclude that under such conditions, new hairy black holes emerge as a possible endpoint of superradiant instability of the system.

Appendices

Appendix A

By assuming constant curvature, the field equation (3) in vacuum becomes

$$\begin{aligned} (1+f_{R})R_{\mu \nu }-\frac{1}{2}(R+f(R))g_{\mu \nu }=0. \end{aligned}$$

(44)

The trace of (44) leads to

$$\begin{aligned} (1+f_{R})R-2(R+f(R))=0. \end{aligned}$$

(45)

Now, substituting for f(R) in (45), one gets $R_{0}=\pm \sqrt{3} \mu ^2$ for $f(R)=-\frac{\mu ^{4}}{R}$ and $\lambda = \frac{R_{0} e^{\alpha R_{0}}}{2+\alpha R_{0}}$ for $f(R)=-\lambda e^{-\alpha R}$, where $\alpha$ is a free parameter, representing solutions which correspond to a topological Schwarzschild-AdS(dS) black hole. In the numerical solution, we fix $R_{0}$ to get $\mu ^{2}$ and $\lambda$. In [59], by considering thermodynamics of black holes in an AdS space-time, the authors found that the necessary conditions for F(R) to support AdS black holes are

$$\begin{aligned}&R+f(R)<0, \end{aligned}$$

(46)

$$\begin{aligned}&1+f_{R}>0. \end{aligned}$$

(47)

In addition, the condition $\frac{d^{2}f}{dR^{2}}$ guarantees that there is no tachyonic instability. These conditions may impose extra restrictions on $R_{0}$. In the first model, only $R_{0}=\mp \sqrt{3} \mu ^2$ would allow compliance with a topological Schwarzschild-AdS black hole. However, in the second model with the assumption $\alpha >0$, these conditions require $-\frac{1}{\alpha }<R_{0}<0$.

Appendix B

Perturbations of dynamical equations (30–36), to first order, can be derived using definition of the perturbed quantities defined in the text with the result

$$\begin{aligned}&\frac{(1+{\bar{f}}_{R})\delta N'}{{\bar{N}}}- \left[ \frac{1+{\bar{f}}_{R}}{{\bar{N}}} \left( \bar{N'}-\frac{1}{r}\right) -\frac{r \bar{A_{0}'}^2}{2 {\bar{N}}{\bar{h}}}-\frac{{\bar{f}}r}{2{\bar{N}}}+\frac{{\bar{f}}_{R} r{\bar{R}}}{2{\bar{N}}}\right] \frac{\delta N}{{\bar{N}}} \nonumber \\&\quad =\left( \frac{q^2 \bar{A_{0}}^2 {\bar{\psi }}^2}{r {\bar{\xi }}^2}+\frac{r \bar{A_{0}}'^2}{{\bar{N}}{\bar{h}}}\right) \frac{\delta \xi }{{\bar{\xi }}}\nonumber \\&\qquad -\frac{q^2 \bar{A_{0}} {\bar{\psi }}^2}{r {\bar{\xi }}^2}\delta A-\frac{r \bar{A_{0}}'}{{\bar{N}}{\bar{h}}} \delta A'-{\bar{f}}_{RR} \left( \frac{1}{{\bar{N}}r}({\bar{N}}-1)+\frac{ r \bar{R_{0}}}{2 {\bar{N}}}+\frac{{\bar{N}}'}{{\bar{N}}}\right) \delta R\nonumber \\&\qquad -\frac{i q \bar{A_{0}} {\bar{\psi }}}{2 r {\bar{\xi }}^2}(\delta {\dot{\psi }}-\delta {\dot{\psi }}^*) -\left( \frac{{\bar{\psi }}}{2 r}\right) ' \nonumber \\&\qquad \times (\delta \psi '+\delta \psi '^*)+\left[ \frac{\left( \frac{{\bar{\psi }}}{r}\right) '}{2 r}-\frac{q^2\bar{A_{0}}^2 {\bar{\psi }}}{2 r {\bar{\xi }}^2}\right] (\delta \psi +\delta \psi ^*), \end{aligned}$$

(48)

$$\begin{aligned}&\frac{(1+{\bar{f}}_{R})\delta h'}{{\bar{h}}}-\frac{(1+{\bar{f}}_{R})\bar{h'}\delta h}{{\bar{h}}^2}+\frac{{\bar{f}}_{RR}\bar{h'}\delta R}{{\bar{h}}} \nonumber \\&\quad =\frac{i q \bar{A_{0}} {\bar{\psi }}}{r {\bar{\xi }}^2}(\delta {\dot{\psi }}-\delta {\dot{\psi }}^*) -\left[ \frac{1}{r}\left( \frac{{\bar{\psi }}}{r}\right) '-\frac{q^2 \bar{A_{0}}^2 {\bar{\psi }}}{r {\bar{\xi }}^2}\right] (\delta \psi +\delta \psi ^*) \nonumber \\&\qquad +\left( \frac{{\bar{\psi }}}{r}\right) '(\delta \psi '+\delta \psi '^*)+\frac{2 q^2 \bar{A_{0}} {\bar{\psi }}^2}{r {\bar{\xi }}^2} \delta A-\frac{2 q^2 \bar{A_{0}}^2 {\bar{\psi }}^2}{r {\bar{\xi }}^3}\delta \xi , \end{aligned}$$

(49)

$$\begin{aligned}&\frac{(1+{\bar{f}}_{R})\delta \xi '}{{\bar{\xi }}} \nonumber \\&\quad =-\left[ \frac{r\bar{A_{0}'}^2}{2 {\bar{h}}}+\frac{1}{r}(1+\bar{f_{R}})+\frac{{\bar{f}} r}{2 }-\frac{{\bar{f}}_{R} \bar{R_{0}}r}{2}\right] \frac{\delta N}{{\bar{N}}^2}+\left[ \frac{(1+{\bar{f}}_{R}){\bar{\xi }}'}{{\bar{\xi }}}+ \frac{r\bar{A_{0}'}^2}{{\bar{N}} {\bar{h}}}\right] \frac{\delta \xi }{{\bar{\xi }}} \nonumber \\&\qquad -\left( \frac{1}{r} -\frac{1}{{\bar{N}}r}+\frac{r{\bar{R}}}{2 {\bar{N}}}+\frac{\xi '}{\xi }\right) {\bar{f}}_{RR} \delta R-\frac{r \bar{A_{0}}'}{{\bar{N}}{\bar{h}}} \delta A', \end{aligned}$$

(50)

$$\begin{aligned}&\frac{(1+{\bar{f}}_{R})\delta {\dot{N}}}{{\bar{N}}} \nonumber \\&\quad =-\left( \frac{{\bar{\psi }}}{2 r}\right) '(\delta {\dot{\psi }}+\delta \dot{\psi ^*})+\frac{i q \bar{A_{0}} {\bar{\psi }}'}{2r}(\delta \psi -\delta \psi ^*)-\frac{i q \bar{A_{0}} {\bar{\psi }}}{2r}(\delta \psi '-\delta \psi '^*), \end{aligned}$$

(51)

$$\begin{aligned}&{\bar{N}} \delta A''+{\bar{N}}\left( \frac{2}{r}-\frac{{\bar{h}}'}{2{\bar{h}}}\right) \delta A'-\frac{q^2 {\bar{\psi }}^2}{r^2}\delta A \nonumber \\&\quad =-\frac{q^2 \bar{A_{0}} {\bar{\psi }}^2}{r^2 {\bar{\xi }}}\delta \xi +\frac{{\bar{N}}\bar{A_{0}}'}{2{\bar{h}}}\delta h'+\frac{1}{2{\bar{h}}}\left[ \frac{q^2\bar{A_{0}} {\bar{\psi }}^2}{r^2}-\frac{{\bar{N}}\bar{A_{0}'} \bar{h'}}{{\bar{h}}}\right] \delta h\nonumber \\&\qquad +\frac{i q {\bar{\psi }}}{2r^2}(\delta {\dot{\psi }}-\delta \dot{\psi ^*})+\frac{q^2 \bar{A_{0}} {\bar{\psi }}}{r^2}(\delta \psi +\delta \psi ^*) \end{aligned}$$

(52)

$$\begin{aligned}&\frac{\delta \dot{ A'}}{\sqrt{{\bar{h}}}}-\frac{\bar{A_{0}}'\delta {\dot{h}}}{2 {\bar{h}}\sqrt{{\bar{h}}}} =\frac{i q {\bar{\psi }} {\bar{\xi }}}{2 r^2}(\delta {\psi '}-\delta {\psi '^*})-\frac{i q {\bar{\xi }} {\bar{\psi }}'}{2r^2}(\delta \psi -\delta \psi ^*), \end{aligned}$$

(53)

$$\begin{aligned}&-\delta \ddot{\psi }+{\bar{\xi }}^2 \delta \psi ''+2iq \bar{A_{0}} \delta {\dot{\psi }}+ {\bar{\xi }} {\bar{\xi }}' \delta \psi '+\left( q^2 \bar{A_{0}}^2-\frac{{\bar{\xi }}{\bar{\xi }}'}{r}\right) \delta \psi \nonumber \\&\qquad -\frac{iq \bar{A_{0}} {\bar{\psi }}}{{\bar{\xi }}} \delta {\dot{\xi }}+\left( {\bar{\xi }} {\bar{\psi }}'-\frac{{\bar{\xi }} {\bar{\psi }}}{r}\right) \delta \xi '+iq {\bar{\psi }} \delta {\dot{A}}\nonumber \\&\qquad +\left( 2 {\bar{\xi }} {\bar{\psi }}''+{\bar{\xi }}'{\bar{\psi }}'-\frac{{\bar{\psi }} {\bar{\xi }}'}{r}\right) \delta \xi +2 q^2 {\bar{\psi }}\bar{A_{0}} \delta A=0. \end{aligned}$$

(54)

The perturbed scalar field is a complex quantity, written as $\delta \psi =\delta u+i \delta {\dot{w}}$. For the real part, we have

$$\begin{aligned}&-\delta \ddot{u}+{\bar{\xi }}^2\delta u''-2q \bar{A_{0}}\delta \ddot{w}+{\bar{\xi }}{\bar{\xi }}'\delta u'+\left( q^2\bar{A_{0}}^2-\frac{{\bar{\xi }}{\bar{\xi }}'}{r}\right) \delta u \nonumber \\&\quad +\left( {\bar{\xi }}{\bar{\psi }}'-\frac{{\bar{\xi }} {\bar{\psi }}}{r}\right) \delta \xi '+ \left( 2{\bar{\xi }} {\bar{\psi }}''+{\bar{\xi }}'{\bar{\psi }}'-\frac{{\bar{\psi }} {\bar{\xi }}'}{r}\right) \nonumber \\&\quad \times \delta \xi +2q^2{\bar{\psi }}\bar{A_{0}} \delta A=0, \end{aligned}$$

(55)

and the imaginary part takes on the form

$$\begin{aligned}&-\delta \dddot{w}+{\bar{\xi }}^2 \delta {\dot{w}}''+2q \bar{A_{0}} \delta {\dot{u}}+ {\bar{\xi }} {\bar{\xi }}' \delta {\dot{w}}'+\left( q^2 \bar{A_{0}}^2-\frac{{\bar{\xi }}{\bar{\xi }}'}{r} \right) \delta {\dot{w}} \nonumber \\&\quad -\frac{q \bar{A_{0}} {\bar{\psi }}}{{\bar{\xi }}} \delta {\dot{\xi }}+q {\bar{\psi }} \delta {\dot{A}}=0. \end{aligned}$$

(56)

Integration with respect to time of equations (51, 53) yields

$$\begin{aligned}&(1+\bar{f_{R}})\frac{\delta N}{{\bar{N}}}=-\left( \frac{{\bar{\psi }}}{r}\right) '\delta u-\frac{ q \bar{A_{0}} {\bar{\psi }}'}{r}\delta w+\frac{q \bar{A_{0}} {\bar{\psi }}}{r}\delta w'+\delta F(r) , \end{aligned}$$

(57)

$$\begin{aligned}&\frac{\delta h}{{\bar{h}} \sqrt{{\bar{h}}}}= \frac{2\delta A'}{\bar{A_{0}}'\sqrt{{\bar{h}}}}+\frac{ 2 q {\bar{\psi }} {\bar{\xi }}}{r^2 \bar{A_{0}}'}\delta w'-\frac{ 2 q {\bar{\xi }} {\bar{\psi }}'}{\bar{A_{0}}' r^2}\delta w+\delta \textit{g(r)}, \end{aligned}$$

(58)

where $\delta F(r)$ and $\delta g(r)$ are arbitrary functions of the radial coordinate. Integration of equation (56) now yields

$$\begin{aligned}&\delta \ddot{w}-{\bar{\xi }}^2 \delta w''-2q \bar{A_{0}} \delta u-{\bar{\xi }} {\bar{\xi }}' \delta w'-\left( q^2 \bar{A_{0}}^2-\frac{{\bar{\xi }}{\bar{\xi }}'}{r}\right) \delta w \nonumber \\&\quad +\frac{q \bar{A_{0}} {\bar{\psi }}}{{\bar{\xi }}} \delta \xi -q {\bar{\psi }} \delta A+\delta \textit{H(r)}=0, \end{aligned}$$

(59)

where $\delta \textit{H(r)}$ is an arbitrary function of the radial coordinate. If an arbitrary function of r is added to $\delta w$, due to the form of $\delta \psi$, it does not change which gives us the freedom to set $\delta \textit{H(r)}=0$.

A first-order differential equation is constructed from (48) and (52), using equations (57–59)

$$\begin{aligned}&\delta F(r) '+\left( \frac{{\bar{\xi }}'}{{\bar{\xi }}}+\frac{1}{r}\right) \delta F(r) +{\bar{f}}_{RR}\left( \frac{1}{r}-\frac{1}{{\bar{N}}r}+\frac{r{\bar{R}}}{2 {\bar{N}}}+\frac{{\bar{N}}'}{{\bar{N}}}\right) \delta R \nonumber \\&\quad =\frac{r \bar{A_{0}}\bar{A_{0}}'{\bar{N}}}{{\bar{\xi }}^2}(\sqrt{{\bar{h}}} \delta g(r) )' +\frac{r \bar{A'_{0}}^2}{2{\bar{\xi }}}\delta g(r) +\frac{q^2\bar{A_{0}}^2{\bar{\psi }}^2 \sqrt{{\bar{h}}}}{2r{\bar{\xi }}^2}\delta g(r) . \end{aligned}$$

(60)

Another first-order differential equation is obtained by plugging (48) and (52) in (57)

$$\begin{aligned}&\delta F(r) '+\left( \frac{{\bar{\xi }}'}{{\bar{\xi }}} +\frac{1}{r}\right) \delta F(r) +{\bar{f}}_{RR}\left( \frac{1}{r}- \frac{1}{{\bar{N}}r}+\frac{r{\bar{R}}}{2 {\bar{N}}}+\frac{{\bar{N}}'}{{\bar{N}}}\right) \delta R \nonumber \\&\quad =\frac{r \bar{A'_{0}}^2}{2{\bar{\xi }}}\delta g(r) . \end{aligned}$$

(61)

Substitution of (57) and (58) in (59) then gives

$$\begin{aligned}&\delta \ddot{w}-{\bar{\xi }}^2\delta w''+\left[ \frac{q^2 \bar{A_{0}} {\bar{\psi }}^2}{r^2 \bar{A_{0}}'}\left( \frac{r \bar{A_{0}}' \bar{A_{0}}}{1+{\bar{f}}_{R}}+ {\bar{N}}{\bar{h}}\right) -{\bar{\xi }} {\bar{\xi }}'\right] \delta w' \nonumber \\&\qquad -\left[ \frac{q^2 \bar{A_{0}} {\bar{\psi }} {\bar{\psi }}'}{r^2 \bar{A_{0}}'} \left( \frac{ r \bar{A_{0}} \bar{A_{0}}'}{1+{\bar{f}}_{R}}+{\bar{N}}{\bar{h}}\right) +q^2 \bar{A_{0}}^2 -\frac{{\bar{\xi }} {\bar{\xi }}'}{r}\right] \nonumber \\&\qquad \times \delta w -q \bar{A_{0}}\left( 2+\frac{{\bar{\psi }} \left( \frac{{\bar{\psi }}}{r}\right) '}{1+{\bar{f}}_{R}}\right) \delta u-q {\bar{\psi }} \delta A+\frac{q \bar{A_{0}} {\bar{\psi }}}{\bar{A_{0}}'} \delta A'\nonumber \\&\qquad +\frac{q \bar{A_{0}}{\bar{\psi }}\delta F(r) }{1+\bar{f_{R}}}+\frac{q \bar{A_{0}}\sqrt{{\bar{h}}}{\bar{\psi }}\delta g(r) }{2}\nonumber \\&\quad =0, \end{aligned}$$

(62)

Let us now substitute equations (50, 57–59, 61) in (55) and obtain

$$\begin{aligned}&\delta \ddot{u}-{\bar{\xi }}^2 \delta u''-{\bar{\xi }} {\bar{\xi }}'\delta u'\nonumber \\&\quad +\left[ 3 q^2 \bar{A_{0}}^2+\frac{{\bar{\xi }} {\bar{\xi }}'}{r}+\frac{ {\bar{N}}\left( \frac{{\bar{\psi }}}{r}\right) '^2 }{1+{\bar{f}}_{R}}\left( \frac{r^2 \bar{A_{0}}'^2}{2 (1+{\bar{f}}_{R})}-\frac{r^2 {\bar{h}} {\bar{f}}}{2( 1+{\bar{f}}_{R})}+\frac{r^2 {\bar{h}} {\bar{f}}_{R} \bar{R_{0}}}{2( 1+{\bar{f}}_{R})}-{\bar{h}}\right) \right] \nonumber \\&\quad \times \delta u+2 q \bar{A_{0}} {\bar{\xi }}^2 \delta w'' +q {\bar{N}}\bar{A_{0}} \nonumber \\&\qquad \left[ 2 \sqrt{{\bar{h}}}{\bar{\xi }}'+\frac{{\bar{\psi }} \left( \frac{{\bar{\psi }}}{r}\right) '}{1+{\bar{f}}_{R}} \left( \frac{{\bar{h}}}{r}-\frac{{\bar{N}}{\bar{h}} \bar{A_{0}}' }{\bar{A_{0}}}-\frac{r \bar{A_{0}}'^2}{2(1+{\bar{f}}_{R})}-\frac{{\bar{h}}{\bar{f}}_{R} \bar{R_{0}} r}{2 (1+{\bar{f}}_{R})}+ \frac{{\bar{h}}{\bar{f}} r}{2 (1+{\bar{f}}_{R})}\right) \right] \delta w' \nonumber \\&\qquad +q \bar{A_{0}}\left[ 2 q^2 \bar{A_{0}}^2-\frac{2 {\bar{\xi }} {\bar{\xi }}'}{r}+\frac{{\bar{\xi }} {\bar{\psi }}' \left( \frac{{\bar{\psi }}}{r}\right) '}{1+{\bar{f}}_{R}}\left( \frac{{\bar{\xi }} \bar{A_{0}}'}{\bar{A_{0}}}-{\bar{\xi }}'-\frac{{\bar{\xi }}}{r }\right) \right] \delta w- {\bar{\xi }}^2 r \left( \frac{{\bar{\psi }}}{r}\right) ' \nonumber \\&\quad \times \frac{\delta \textit{F(r)}'}{1+{\bar{f}}_{R}}+ \frac{{\bar{f}}_{RR} {\bar{h}}'}{2\left( 1+{\bar{f}}_{R}\right) {\bar{h}}}\delta R=0. \end{aligned}$$

(63)

Using equations (49, 58, 59) then results in

$$\begin{aligned}&\frac{q {\bar{\psi }}}{r}\left( \frac{\bar{A_{0}}}{1+{\bar{f}}_{R}}+\frac{{\bar{N}}{\bar{h}}}{r \bar{A_{0}}'}\right) \delta w''+ q \bar{A_{0}} {\bar{\psi }} \left[ -\frac{q^2 {\bar{h}} {\bar{\psi }}^2}{r^4 \bar{A_{0}}'^2}+\frac{{\bar{\xi }}'}{(1+{\bar{f}}_{R}) {\bar{\xi }} r}+\frac{\sqrt{{\bar{h}}}\bar{\xi '}}{r^2 \bar{A_{0}} \bar{A_{0}}'}\right] \delta w' \nonumber \\&\quad +\frac{q \bar{A_{0}} {\bar{\psi }}}{r^2}\left[ \frac{r q^2 \bar{A_{0}}^2}{(1+{\bar{f}}_{R}) {\bar{\xi }}^2} +\frac{q^2 \bar{A_{0}}}{{\bar{N}} \bar{A_{0}}'}-\frac{{\bar{\xi }}'}{(1+{\bar{f}}_{R}) {\bar{\xi }}}-\frac{\sqrt{{\bar{h}}}{\bar{\xi }}'}{r \bar{A_{0}} \bar{A_{0}}' }+\frac{q^2 {\bar{h}} {\bar{\psi }}{\bar{\psi }}'}{r^2 \bar{A_{0}}'^2}\right] \delta w\nonumber \\&\quad -\frac{\left( \frac{{\bar{\psi }}}{r} \right) '}{1+{\bar{f}}_{R}}\delta u'-\left[ \frac{\left( \frac{{\bar{\psi }}}{r} \right) '}{1+{\bar{f}}_{R}}\left( \frac{1}{r}+ \frac{{\bar{\xi }}'}{{\bar{\xi }}}\right) + \frac{\left( \frac{{\bar{\psi }}}{r}\right) ''}{1+{\bar{f}}_{R}}\right] \delta u\nonumber \\&\quad +\frac{\delta A''}{\bar{A_{0}}'} -\frac{\bar{A_{0}}''}{\bar{A_{0}}'^2}\delta A'+\frac{{\bar{f}}_{RR}{\bar{h}}'}{2\left( 1+{\bar{f}}_{R}\right) {\bar{h}}}\delta R+\frac{\left( \sqrt{{\bar{h}}}\delta \textit{g(r)}\right) '}{2}=0. \end{aligned}$$

(64)

Since ingoing boundary conditions (40) must exist for all perturbations, including perturbation of metric variables ($\delta N$ and $\delta h$) [10], equations (57) and (58) are needed to satisfy such conditions, leading to $\delta \textit{F(r)}=0$ and $\delta \textit{g(r)}=0$ at $r=r_{h}$. Therefore, $\delta \textit{F(r)}$, $\delta \textit{g(r)}$, $\delta \textit{F(r)}'$ and $\delta \textit{g(r)}'$ are removed from perturbation equations and $\delta R$ at $r=r_{h}$ must be zero on account of equations (60, 61).