Stability analysis on charged black hole with non-linear complex scalar

Recently, the observation of gravitational wave and black hole image provide a new motivation and vision in black hole physicsAbbott:2016blz ; Akiyama:2019cqa . Providing a deep understanding in quantum gravity, black hole physics will always be a long-live project to study, theoretically and experimentally. One of the most important topics on black hole physics is no-hair theorem, which claims that black holes are determined only by mass, charge and angular momentum respectivelyRuffini:1971bza . It was first concluded by Bekenstein that the static and spherical neutral black hole in asymptotic flat spacetime cannot be endowed with real scalar field, Proca field and spin-2 fieldBekenstein:1972ny . Furthermore, the no-hair theorem was extended by Mayo and Bekestein that the static and spherical black hole cannot be endowed with a coupling charged scalar field together with a non-negative self-interacting potential Mayo:1996mv . However, the requirements of above no-hair theorems are as strong as its conclusion. Therefore, it is not difficult to find hairy black hole solutions if one breaks the requirements of these no-hair theorems. In Refs. Feng:2013tza ; Liu:2013gja ; Fan:2015tua ; Khodadi:2020jij , taking account into a non positive definite potential or asymptotic AdS spacetime, $D$-dimension scalarized black holes have been constructed in Einstein theory minimally coupled with neutral scalar field. Recently, in the extended Maxwell theory non-minimally coupled with a scalar field, such as Einstein-Maxwell-Dilaton (EMD) theoryFan:2015oca , Einstein-Maxwell-Scalar theoryKonoplya:2019goy , Einstein-Born-Infeld (EBI) theoryWang:2020ohb ; Stefanov:2007eq , and Quasi-topological Electromagnetism theoryMyung:2020ctt , a class of charged black hole with scalar hair has been found. For a recent review relevant to the no-hair theorem, See Ref. Herdeiro:2015waa .

The above examples give the simplest case of the black hole with an additional scalar hair. As we have mentioned, the Bekenstein’s no-scalar-hair theorem was generalized by Ref. Mayo:1996mv , claiming a very strong conclusion: the non-extremal static and spherical charged black hole cannot carry charged scalar hair whether minimally or nonminimally coupled to gravity, and with a regular positive semidefinite self-interaction potential. However, a numerical charged black hole solution with Q-hair was found in Einstein-Maxwell gravity minimally coupled with a non-linear complex scalar, the self interacting potential taking the following polynomial function Hong:2020miv

$$V(|\psi|^{2})=\frac{m^{2}}{2}|\psi|^{2}-\frac{\lambda}{4}|\psi|^{4}+\frac{\beta}{6}|\psi|^{6}\,.$$

(1)

It was also pointed out that the detailed form of the potential $V(|\psi|^{2})$ is not crucial and the Q-hair can exist for large class of nonlinear potentialHong:2019mcj ; Herdeiro:2020xmb . The reason why Ref. Mayo:1996mv obtained a wrong statement is that it omitted a scalar mass term at an asymptotic infinity. Though the black holes with scalar hair has been found in many physical models, the discovery of Refs. Hong:2020miv has a few of special interesting aspects. Particularly, in this model, the gravity, Maxwell field and complex scalar field are all minimally coupled with each other and the potential of scalar field is positive semidefinite. This gives a possibility to realize the scalar hairy black hole in the Einstein gravity and asymptotically flat spacetime. Furthermore, as the potential of scalar field is nonnegative, such model will have stable true vacuum state $\psi=0$ in Minkowski spacetime.

It needs to note that the Reissner-Nordström black hole (RN black hole) is still a solution of field equations even if the Q-hair appears in the models discussed by Refs. Hong:2020miv ; Hong:2019mcj . Given a scalarized charged black hole solution, it is worth to investigate whether the scalarized charged black hole is more stable than RN black hole in thermodynamics, i.e. whether the RN black hole can be spontaneously scalarized by phase transition. Recently, it has been well studied the thermodynamic self-stability associated with heat capacity in various ensembles in Caldarelli:1999xj ; Mo:2013sxa ; Zhang:2018rlv ; Quevedo:2006xk ; Quevedo:2013pba . However, we focus on investigating the thermodynamic stability of scalarized charged black holes, compared with the RN black hole in various ensembles. Concretely, in microcanonical ensemble, given the same ADM mass $M$ and total charge $Q$, the stability requires the maximum of the black hole entropy $S(M,Q)$ and a phase is more stable than the other if it has larger entropy. In canonical ensemble, with fixing Hawking temperature $T$ and $Q$, the black hole owning less Helmholtz free energy $F(T,Q)$ will be the more stable, while in grand canonical ensemble, with the identical $T$ and chemical potentials $\mu$, the black hole which has smaller Gibbs free energy $G(T,\mu)$ indicates it will be more stable. Furthermore, the black hole entropy $S(M,Q)$ is given by the area of horizon according to Bekenstein’s entropy formula while the thermodynamic potentials $F(T,Q)$ and $G(T,\mu)$ can be read off from partition function via Euclidean path-integral approach developed by Hawking and York, et al. Gibbons:1976ue ; Brown:1989fa ; Braden:1990hw .

Remarkably, in astrophysics, a real black hole practically is more closer to grand canonical ensemble allowing charge and energy exchange with other matter in universe. However, as theoretical research interest, we still take the case of microcanonical ensemble and canonical ensemble under our consideration to study whether the RN black hole will be scalarzied under the process of discontinuous phase transition spontaneously. Furthermore, by means of analyzing the corresponding entropy or thermodynamic potentials in various ensembles, we find that there is a possibility that the scalarized charged black hole is more stable than the RN black hole in thermodynamics. This implies that, for a large class of nonlinear complex model, it is possible that the RN black hole will be scalarized spontaneously via a first order phase transition. Our numerical results also imply that the scalarized black hole is always more stable than the RN black hole in microcanonical ensemble and when $M\rightarrow|Q|$. This implies the isolated near extremal RN black hole is not stable and will spontaneously scalarize.

In addition to the thermodynamic stability, another natural question is whether the scalarized charge black hole is kinetically stable, i.e. stable under against a small perturbation at least at linear level. It has been developed by Vishveshwara and Teukolsky et al.Teukolsky:1973ha ; Press:1973zz ; Vishveshwara:1970cc that in static axial or spherical symmetric background, the equation of motion associated with the perturbation field can reduce to radial equation in frequency domain. Furthermore, the radial equation can be interpreted as an eigenvalue problem. Specifically, imposing a physical boundary condition both in the spacial infinity and at the black hole horizon, a class of complex frequency called black hole quasi-normal modes(QNMs), which implies dissipation at both event horizon and spatial infinity, can be picked out. The instability of black hole might be triggered due to the negative imaginary part of the complex frequency. In general, the QNMs can be solved by shooting method in numerics. Approximatively, there have been some analytical approaches to achieve to calculate QNMs as well: the WKB approximation method, continue fraction method and Monodromy method. However, recently it has been argued in Ref. Konoplya:2019hlu that the WKB approximation method cannot catch the unstable mode within the spectrum of the QNMs. Furthermore, we also find that the shooting method cannot efficiently give the stable modes through numerical error analysis. We thus adopt hybrid method to calculate the QNMs for studying the kinetic stability of scalarized charged black hole. If a given solution is not kinetically stable, then such solution cannot exist in a real physical system. For the situations that the scalarized black hole is more thermodynamically stable than RN black hole, but it does not automatically guarantee that the scalarized charged black hole is still stable kinetically. However, our results show the neutral perturbative scalar field will not trigger the instability of the scalarized charged black hole at linear level. In addition, for a recent review on perturbation theory on black hole, stability analysis and quasi-normal modes of black hole, more detail has been given in Refs. Pani:2013pma ; Konoplya:2011qq ; Berti:2009kk .

This paper will be organized as follow: In Sec. II, we briefly introduce the model of Einstein-Maxwell theory minimally coupled with a non-linear complex field. For general non-linear semi-definite potential, we present a proof that the scalarized charged black hole cannot be a result of continuously scalarization. In Sec. III, Giving a logarithmic potential, we obtain a class of numerical scalarized charged black hole solution and in various ensembles, investigate their thermodynamic stability compared with the RN black hole. We find that in both microcanonical ensemble and canonical ensemble, it is possible that the RN black hole can be spontaneously scalarized via first order phase transition. In Sec. IV, considering a probing neutral scalar field, we investigate the stability on the scalarized charged black hole by means of computing the quasi-normal modes using both the shooting method and the WKB approximation method. This suggests that the probing neutral scalar field cannot trigger the instability of the sclarized charged black hole. In Sec. V, we present our conclusion and further discussions.

In this paper, we consider the following Einstein-Maxwell theory minimally coupled with a non-linear complex scalar field (we set $G_{N}=c_{s}=\hbar=k=1$, $c_{s}$ denotes the speed of light),

			$\displaystyle S=\frac{1}{16\pi}\int\mathrm{d}^{4}x\sqrt{-g}{\cal L}\,,$		(2)
					(4)
			$\displaystyle{\cal L}=\left(R-F_{\mu\nu}F^{\mu\nu}-(D_{\mu}\psi)^{\dagger}(D^{\mu}\psi)-W(|\psi|^{2})\right)\,,$

where the covariant derivative operator $D_{\mu}=\nabla_{\mu}-iqA_{\mu}$, $q$ denotes charge of the complex scalar field. In addition, $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the strength tensor of the U(1) electromagnetic field $A_{\mu}$. The non-linear positively semi-definite potential $W(x)$ is a smooth function and satisfies following requirements:

$$W(0)=0,~{}~{}W^{\prime}(0)=m^{2}>0,~{}~{}W(x)>0~{}\text{if}~{}x>0\,.$$

(5)

These conditions insure that scalar field has stable true vacuum $\psi=0$ in Minkowski spacetime when $F_{\mu\nu}=0$. Performing variation on the action (2) with respect to the metric $g_{\mu\nu}$, the gauge field $A_{\mu}$ and the complex scalar field $\psi$ respectively, we obtain the equation of motion

	$\displaystyle R_{\mu\nu}-\frac{R}{2}g_{\mu\nu}=8\pi[T^{(M)}_{\mu\nu}+T^{(\psi)}_{\mu\nu}]\,,$		(6)
			(7)
	$\displaystyle\nabla^{\mu}F_{\mu\nu}=iq\left(\psi^{\dagger}D_{\mu}\psi-\psi(D_{\mu}\psi)^{\dagger}\right)\,,$		(8)
			(9)
	$\displaystyle D^{2}\psi-w(|\psi|^{2})\psi=0\,,$		(10)

where we define $w(x)=W^{\prime}(x)$ and the energy momentum tensor associated with both the electric field and the complex scalar field reads

			$\displaystyle T^{(M)}_{\mu\nu}=\frac{1}{4\pi}\left(F_{\mu\sigma}{F^{\sigma}}_{\nu}-\frac{1}{4}g_{\mu\nu}F_{\rho\sigma}F^{\rho\sigma}\right)\,,$		(11)
					(14)
			$\displaystyle T^{(\psi)}_{\mu\nu}=\frac{1}{16\pi}\left(D_{\mu}\psi(D_{\nu}\psi)^{\dagger}+D_{\nu}\psi(D_{\mu}\psi)^{\dagger}\right.$
			$\displaystyle\left.-g_{\mu\nu}\left((D_{\mu}\psi)^{\dagger}(D^{\mu}\psi)+W(|\psi|^{2})\right)\right)\,.$

For the non-linear potential, the requirement (5) implies that

$$w(x)=m^{2}+w_{1}x+w_{2}x^{2}+\cdots$$

(15)

with some constants $\{w_{1},w_{2},\cdots\}$ when $|x|\ll 1$.

To obtain the scalary charged black hole and investigate some related properties, we adopt the following spherical symmetric line element,

$$\mathrm{d}s^{2}=-f(r)\mathrm{e}^{-\chi(r)}\mathrm{d}t^{2}+\frac{\mathrm{d}r^{2}}{f(r)}+r^{2}(\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\varphi^{2}).$$

(16)

Due to the spherical symmetry, the electromagnetic field and the complex scalar field take the following form,

$$A_{\mu}=\phi(r)(\mathrm{d}t)_{\mu},~{}~{}\psi=\psi(r)\,.$$

(17)

From the line element Eq. (16), the Hawking temperature $T$ reads

$$T=\frac{f^{\prime}(r_{h})\mathrm{e}^{-\chi(r_{h})/2}}{4\pi}\,,$$

(18)

where $r_{h}$ denotes the outer event horizon. With the given spherical anstaz, the equations of motion reduce to

$$\begin{split}&\psi^{\prime\prime}+\left(\frac{f^{\prime}}{f}-\frac{\chi^{\prime}}{2}+\frac{2}{r}\right)\psi^{\prime}+\left(\frac{\mathrm{e}^{\chi}\phi^{2}q^{2}}{f^{2}}-\frac{w(\psi^{2})}{f}\right)\psi=0\\ &\phi^{\prime\prime}+(\chi^{\prime}/2+2/r)\phi^{\prime}-\frac{\psi^{2}q^{2}}{2f}\phi=0\,,\\ &\chi^{\prime}+\frac{r\mathrm{e}^{\chi}\psi^{2}\phi^{2}q^{2}}{f^{2}}+r\psi^{\prime 2}=0\,,\\ &f^{\prime}+\left(\frac{1}{r}+\frac{r\psi^{\prime 2}}{2}\right)f+r\mathrm{e}^{\chi}\phi^{\prime 2}\\ &-\frac{1}{r}+\frac{\mathrm{e}^{\chi}r\psi^{2}\phi^{2}q^{2}}{2f}+\frac{1}{2}rW(|\psi|^{2})=0\,,\end{split}$$

(19)

where the prime denotes the derivative with respect to $r$. In this paper, we consider the asymptotically flat space time. The scalar field, gauge field and metric components should satisfy the following regular boundary conditions when $r\rightarrow\infty$

		$\displaystyle f=1-\frac{2M}{r}+\cdots,~{}~{}\chi=\frac{\chi_{2}}{r^{2}}+\cdots\,,$			(20)
					(21)
		$\displaystyle\phi=\mu-\frac{Q}{r}+\cdots,~{}~{}|\psi|\leq\mathcal{O}(1/r^{2})\,,$			(22)

where $\mu$ is the chemical potential, $M$ is the ADM mass and $Q$ is the total charge. With this boundary conditions and noting the fact that $w(|\psi|^{2})\rightarrow m^{2}$ near the boundary, we find that the first equation of Eq. (19) reduces into following simple form near the infinity

$$\psi^{\prime\prime}+\frac{2\psi^{\prime}}{r}+(q^{2}\mu^{2}-m^{2})\psi=0\,.$$

(23)

The solution reads

$$\psi(r)=\frac{\psi_{+}}{r}\mathrm{e}^{r\sqrt{m^{2}-q^{2}\mu^{2}}}+\frac{\psi_{-}}{r}\mathrm{e}^{-r\sqrt{m^{2}-q^{2}\mu^{2}}}\,.$$

(24)

The boundary conditions Eq. (20) implies following constraints

$$m^{2}-q^{2}\mu^{2}>0,~{}~{}\psi_{+}=0\,.$$

(25)

These are two necessary conditions on the spontaneous scalarization for asymptotically flat black holes.

Taking into account a polynomial potential, as mentioned above, a numerical charged black hole solution with Q-hair has been found in Mayo:1996mv ; Hong:2020miv . A natural question arises whether this class of scalarized black hole solution will arise from continuously spontaneous scalarization for specific non-linear potential. This question is important because if the answer is yes, the spacetime geometry can transit into scalarized black hole smoothly, otherwise, the latent heat will be relaxed or absorbed during phase transition between the RN black hole and scalarized black hole. Moreover, such latent heat will leave some observable effects if such a phase transition happened in our universe.

In the following, we shall present a proof that for arbitrary non-linear potential $W(|\psi|^{2})$ satisfying the requirement (5), the spontaneous scalarization of RN black hole cannot happen via continuous phase transition. As shown in Fig. 1, if such continuous phase transition can happen, when we tune the parameters (total charge, chemical potential, temperature, etc.) of the black hole, there is a critical point where the strength of scalar field begins to increase into nonzero continuously. In other words, there must be a small region near the critical value associated with the black hole parameters (See also the top panel of Fig. 1) where the complex scalar field is infinitesimal.

Without losing generality, we can assume this infinitesimal scalar field has following form,

$$\psi(r,\theta,\varphi)=\varepsilon\sum_{l,\nu}a_{l\nu}(r)Y_{l\nu}(\theta,\varphi)\,\quad\varepsilon\rightarrow 0\,.$$

(26)

where $Y_{l\nu}$ denotes the spherical harmonic function, $\nu=0,\pm 1,\pm 2,\cdots$ is the magnetic quantum number and $l>\nu$ is the azimuthal quantum number. Taking it into Eq. (6) and neglecting all the non-linear terms of $\varepsilon$, we find that the metric and gauge field decouple from the scalar field. Then spacetime geometry and gauge field are given by a RN solution

		$\displaystyle\chi=0,~{}~{}f(r)=\frac{(r-r_{h})(r-\mu^{2}r_{h})}{r^{2}}\,,$			(27)
					(28)
		$\displaystyle\phi=\mu(1-r_{h}/r),~{}~{}\mu\in[-1,1]\,.$			(29)

Here we require $\mu\in[-1,1]$ since the $r_{h}$ is defined to be the most outer horizon. The E.O.M of scalar field reads

$$a_{l\nu}^{\prime\prime}+\left(\frac{f^{\prime}}{f}+\frac{2}{r}\right)a^{\prime}+\left(\frac{\phi^{2}q^{2}}{f^{2}}-\frac{m^{2}}{f}-\frac{l(l+1)}{fr^{2}}\right)a_{l\nu}=0\,.$$

(30)

This can be rewritten into following form

$$\frac{\mathrm{d}}{\mathrm{d}r}\left(r^{2}fa_{l\nu}\frac{\mathrm{d}a_{l\nu}}{\mathrm{d}r}\right)=\left[\left(\frac{\tilde{m}(r)^{2}}{f}-\frac{\phi^{2}q^{2}}{f^{2}}\right)a_{l\nu}^{2}+a_{l\nu}^{\prime 2}\right]r^{2}f\,.$$

(31)

where we have redefine the effective mass as $\tilde{m}(r)^{2}=m(r)^{2}+l(l+1)/r^{2}$ and it is obviously that $\tilde{m}^{2}\geq m^{2}$. Integrate it from horizon to infinity and we find

$$\left.r^{2}fa_{l\nu}\frac{\mathrm{d}a_{l\nu}}{\mathrm{d}r}\right|_{r_{h}}^{\infty}=\int_{r_{h}}^{\infty}\left[\left(\frac{\tilde{m}(r)^{2}}{f}-\frac{\phi^{2}q^{2}}{f^{2}}\right)a_{l\nu}^{2}+a_{l\nu}^{\prime 2}\right]r^{2}f\mathrm{d}r\,.$$

(32)

As $a(r)$ is a regular at horizon and decays to zero at infinity, the left side of Eq. (32) is zero. Thus, we have

$$\int_{r_{h}}^{\infty}\left[\left(\frac{\tilde{m}(r)^{2}}{f}-\frac{\phi^{2}q^{2}}{f^{2}}\right)a_{l\nu}^{2}+a_{l\nu}^{\prime 2}\right]r^{2}f\mathrm{d}r=0\,.$$

(33)

This means that $\exists r_{0}\in(r_{h},\infty)$ such that

$$\frac{\tilde{m}(r)^{2}}{f(r_{0})}-\frac{\phi(r_{0})^{2}q^{2}}{f(r_{0})^{2}}<0\,.$$

(34)

We then obtain

$$m^{2}\leq\tilde{m}(r)^{2}<\frac{\phi(r_{0})^{2}q^{2}}{f(r_{0})}=\mu^{2}q^{2}\frac{r_{0}-r_{h}}{r_{0}-\mu^{2}r_{h}}<\mu^{2}q^{2}\,.$$

(35)

Here we have used the solution (27). Above result and requirement (25) are contradictory. This shows that, if the complex scalar field appears in a static black hole, its strength cannot be infinitesimal no matter how we choose the black hole parameters. Thus, the continuous phase transition from RN black hole to scalarized charged black cannot occur.

The demonstration above implies that the scalarized charged black hole solution cannot produce from the continuous spontaneous phase transition. However, the spontaneous scalarization on charged black hole may arise through non-continuous phase transition, namely the first-order phase transition. In this case, the complex scalar field jumps into nonzero from zero when we tune the parameters of black hole, see the bottom panel of Fig. 1. Recall the equivalence between the black hole system and the thermal mechanic system, it is thus intriguing to study the thermodynamic stability of the scalarized charged black hole, compared with the corresponding charged black hole with no hair. Motivated by this, interpreted charged black hole with scalar hair as a class of ensemble, in the following, we focus on investigating the thermodynamical stability on scalarized charged black hole in various ensembles: microcanonical ensemble, canonical ensemble and grand canonical ensemble, and try to check if there is a first order phase transition.

In this section, we analyze the thermodynamic stability of scalarized charged black hole by proposing a specific example. As mentioned, the relation of Beskein entropy implies that black holes can be investigated as a thermodynamical system in term of three different ensemble: microcanonical ensemble, canonical ensemble and grand canonical ensemble. The microcanonical ensemble implies that black holes are interpreted as an isolated system where there does not exist any charge and energy exchange of black holes. The canonical ensemble indicates that black holes only exchange energy with environment of which the temperature is fixed. The grand canonical ensemble is similar to canonical ensemble but also admits the black hole to exchange the particles with environment. In practice, black holes in astrophysics are more likely to be grand canonical ensemble with exchange of particles and energy. However, as theoretical investigation on scalaized charged black hole, in following we still take microcanonical ensemble, canonical ensemble and grand canonical ensemble into account, studying the thermodynamical stability associated with the scalarized charged black hole and RN black hole respectively. Furthermore, we find that in grand canonical ensemble, the RN black hole is more stable than the scalarized charged black hole in thermodynamics, corresponding to general expectation. Nevertheless, we also find that the discontinuous scarization on RN black hole may happen in both microcanonical ensemble and canonical ensemble respectively. In the following, we will give more detail discussion.

Before go on discussing our result, let us first make short comment on different ensembles in black holes. The Euclidean path-integral approach to black hole thermodynamics originally proposed by Hawking in microcanonical ensemble Hawking:1976de . Later on, the canonical ensemble was investigated by York et al. York:1986it ; Whiting:1988qr ; Brown:1994su . It was found that suitable boundary conditions must be added in canonical ensemble. Then the York’s approach was generalized into other ensembles such as the charged black hole in the grand canonical ensemble PhysRevD.42.3376 ; Brown_1990 . It has been found that the results obtained by using the path-integral approach depend on the boundary conditions Brown:1994gs ; Hawking:1982dh . The stability of black holes then also depends on the choice of boundary conditions and, consequently, on the choice of ensembles Comer_1992 . In fact, the stability properties of a black hole are drastically influenced by the boundary conditions that determine ensemble.

In following we proceed to our discussion. We specify the non-linear potential as a logarithmic function with respect to the scalar field

$$W(x):=m^{2}c^{2}\log(1+\frac{x^{2}}{c^{2}}),$$

(36)

where $c$ is a constant and $m$ is the effective mass of the scalar field. Considering the flat directions in gauge-mediated supersymmetric model deGouvea:1997afu ; Kusenko:1997si , this potential is proposed in Ref. Hong:2019mcj , in which the supersymmetric breaking has been absorbed into the rescaling of $\psi$. In addition to satisfying the stable vacuum requirement Eq.(5) , the shape of potential is asymptotic flat when taking large field limit $\psi\gg c$. From the equation of motion of $\psi$ in Eq. (19), one will find that the scalar field becomes massless large field limit $\psi\gg c$. Moreover, the logarithmic potential can bring better numerical stability as well.

Since this paper tries to find the black hole solutions with scalar hair, there should be a horizon locating at position $r=r_{h}$. In static spherically symmetric case, this implies $f(r_{h})=0$. To set up the numerical method to solve the equation of motion Eq. (19), we in principle still needs five additional independent boundary conditions at horizon. Practically, the regularity of physical fields at $r=r_{h}$ implies that the solution can approximatively be written as the following Taylor’s series with respect to $r-r_{h}$,

			$\displaystyle f(r)=f_{1}(r-r_{h})+\cdots,\quad\chi(r)=\chi_{0}+\chi_{1}(r-r_{h})+\cdots\,,$		(37)
					(39)
			$\displaystyle\phi(r)=\phi_{0}+\phi_{1}(r-r_{h})+\phi_{2}(r-r_{h})^{2}+\cdots\,,$
					(41)
			$\displaystyle\psi(r)=\psi_{0}+\psi_{1}(r-r_{h})+\cdots\,.$

Take them into Eq.(19) and we will find

$\displaystyle\psi_{1}=\frac{2\psi_{0}c^{2}m^{2}}{(\psi_{0}^{2}+c^{2})f_{1}}\,,\quad\phi_{0}=0\,.$

(42)

This leaves three independent variables $\{\psi_{0},\chi_{0},\phi_{1}\}$ at horizon and so we have 7 independent parameters in solving Eq. (19)

$$\{r_{h},\psi_{0},\chi_{0},\phi_{1},c,m,q\}\,,$$

(43)

in which $\{r_{h},\psi_{0},\chi_{0},\phi_{1}\}$ come from the value of various fields at the horizon, $q$ is the charge of the scalar field and $\{c,m\}$ are the parameters of the non-linear potential. It is remarkable that there is a scaling symmetry of the equation of motion

$$r\to\lambda r\,,\quad t\to\lambda t\,,\quad q\to\frac{q}{\lambda}\,,\quad m\to\frac{m}{\lambda},$$

(44)

which equivalently rescale the metric $g_{\mu\nu}\to\lambda^{2}g_{\mu\nu}$ and the electronic field $A_{\mu}\to\lambda A_{\mu}$. Due to such a symmetry, we fix $m=0.01$ for convenience. In addition, we must impose

$$\chi\to 0,\quad\text{when}\quad r\to\infty,$$

(45)

which is a requirement about the normalisation of $t$ associated with the gravitational redshiftHartnoll:2008kx . Practically, given any arbitrary value of $\chi_{0}$, the equation of motion is invariant when performing the following scaling transformation

$$t\to\mathrm{e}^{-\frac{\chi_{\infty}}{2}}t\,,\quad\chi\to\chi-\chi_{\infty}\,,\quad\phi(r)\to\mathrm{e}^{\frac{\chi_{\infty}}{2}}\phi(r),$$

(46)

where $\chi_{\infty}$ denotes the value of the solution $\chi(r)$ in the asymptotic infinity. In other words, we perform a time rescaling $t\to e^{-\frac{\chi_{\infty}}{2}}t$ to set $\chi=0$ at the boundary. In order to numerically solve the equations simply, we in general set $\chi_{0}=0$. However, such choice will lead to $\chi(\infty)=\chi_{\infty}\neq 0$, we can thus finally transform the solution to satisfy $\chi(\infty)=0$ by the transformation (46).

Base on these two symmetries, two physical parameters, the charge of the complex scalar field $q$ and the parameter related to the scalar potential $c$, and three parameters as the initial value at the horizon $r_{h}$, $\phi_{1}$ and $\psi_{0}$ are left. Therefore, The integration of the equation of motion Eq. (19) from the event horizon to the infinity will give us a map:

$$\{r_{h},\phi_{1},\psi_{0},c,q\}\mapsto\{\mu,T,Q,M,\psi_{+}\}\,.$$

(47)

If one chooses five parameters $\{r_{h},\phi_{1},\psi_{0},c,q\}$ arbitrarily, the $\psi_{+}$ may be or may not be vanish. From Eq. (25), we know that only the one satisfying $\psi_{+}=0$ is an admitted solution, leading to only four of parameters $\{r_{h},\phi_{1},\psi_{0},c,q\}$ are independent.

We choosing the parameters as $r_{h}=1,c=0.1,\phi_{1}=0.8,m=0.01,q=\frac{11}{10}m$ and set $r=r_{\infty}=1000$ as the infinity cut off, finding that the complex scalar $\psi$ has been efficiently decay to $0$ at the infinity cutoff $r=1000$, we firstly numerically solve the equation of motion Eq. (19) by Runge-Kutta methods under the initial condition Eq. (37) and Eq. (42). As we have explained, the $\psi_{0}$ now is not free because we need to satisfy $\psi_{+}=0$. Then we use the standard shooting method to find the smallest value of $|\psi_{0}|$ to satisfy the constrain Eq. (25). We find a numerical charged black hole solution with scalar hair when $\psi_{0}\approx 0.1988$. After Performing the scaling transformation Eq. (46), we show the numerical solution are shown in Fig. 2 from which we find that $g_{tt}=f(r)\mathrm{e}^{-\chi(r)}$ is still a monotone increasing function with respect with $r$, indicating that in scalarized charged black hole, the gravity is still attraction, as the case of the RN black hole.

In addition, taking $r=1000$ as an infinity cutoff, from Fig. 2 one can see that the scalar field $\psi(r)$ have efficiently decay to $0$ smoothly at $r=1000$, implying that the spacetime manifold has efficiently reduce to RN black hole. In addition, if increase the infinity cutoff, to maintain effective numerical accuracy, one must increase the working precise, leading to increase the computational time. In practice, the best cut-off is chosen by following way: in double float accuracy, we take $\max r=2^{6},2^{7},2^{8},2^{9}$ and $2^{10}\approx 1000$. We observed that the differences of $\psi_{0}$ are smaller and smaller. However, if we increase the cut-off to be $2^{11}$ and more, we found the differences of $\psi_{0}$ are larger and larger. This implies 1000 is best cut-off. If we using quadruple float accuracy, the best cut-off is around 2000, however, the computational time will increase more 10 times. Therefore, for investigating the stability of scalarized charged black hole effectively and efficiently, it is reasonable to set $r=1000$ as an infinity cutoff.