Scalarized Einstein-Maxwell-scalar Black Holes in Anti-de Sitter Spacetime

Varying the action $\left(\ref{eq:action}\right)$ with respect to the metric $g_{\mu\nu}$, the scalar field $\phi$ and the gauge field $A_{\mu}$, one obtains the equations of motion,

where $\dot{f}\left(\phi\right)\equiv df\left(\phi\right)/d\phi$. The energy-momentum tensor $T_{\mu\nu}$ in eqn. $\left(\ref{eq:EOMs}\right)$ is given by

In the following, we focus on the spherically symmetric ansatz for the metric, the electromagnetic field and the scalar field,

Plugging the above ansatz into the equations of motion $\left(\ref{eq:EOMs}\right)$ yields

where primes denote the derivatives with respect to the radial coordinate $r$, and the integration constant $Q$ can be interpreted as the electric charge of the black hole solution. For later use, we introduce the Misner-Sharp mass function $m\left(r\right)$ by $N\left(r\right)=1-2m\left(r\right)/r+r^{2}/L^{2}$.

To obtain non-trivial hairy black hole solutions of the non-linear ordinary differential equations $\left(\ref{eq:NLEqs}\right)$, one should impose appropriate boundary conditions at the event horizon and the spatial infinity. Accordingly, in the vicinity of the event horizon at $r=r_{+}$, we find that the solutions can be approximated as a power series expansion in terms of $\left(r-r_{+}\right)$,

where

The two essential parameters, $\phi_{0}$ and $\delta_{0}$, can be determined after matching the asymptotic expansion of the solutions at the spatial infinity,

where $f\left(0\right)=1$ is assumed, $M$ is identified as the ADM mass, $\phi_{+}$ can be interpreted as the expectation value of the dual operator in the conformal boundary theory from AdS/CFT correspondence, and $\Phi$ is the electrostatic potential with $\Phi=\int_{r_{+}}^{\infty}dre^{-\delta\left(r\right)}Q/\left(r^{2}f\left(\phi\left(r\right)\right)\right)$. Note that the scalar field usually behaves as $\phi\left(r\right)\sim\phi_{-}+\frac{\phi_{+}}{r^{3}}$ in the asymptotic region, and we set $\phi_{-}=0$ in this paper, corresponding to the absence of the external source in the conformal boundary theory. Consequently, we can use the shooting method to solve the non-linear differential equations $\left(\ref{eq:NLEqs}\right)$ for solutions satisfying the asymptotic expansions at the boundaries. It is also noteworthy that there is a scaling symmetry among the physical quantities,

which allows us to solve eqn. $\left(\ref{eq:NLEqs}\right)$ numerically in terms of redefined dimensionless quantities.

The Smarr relation [65] can be used to test the accuracy of numerical scalarized black hole solutions, since it associates the black hole mass with other physical quantities. The Smarr relation can be derived from computing the Komar integral for a time-like Killing vector $K^{\mu}=\left(1,0,0,0\right)$ in a manifold $\mathcal{M}$. Integrating the identity $\nabla_{\mu}\left(\nabla_{\nu}K^{\mu}\right)=K^{\mu}R_{\mu\nu}$ over the time constant hypersurface $\Sigma$, whose boundary $\partial\Sigma$ consists of the event horizon $r=r_{+}$ and the spatial infinity $r=+\infty$, one can use Gauss’s law to obtain

where $dS_{\mu\nu}$ is the surface element on $\partial\Sigma$, and $dS_{\mu}$ is the volume element on $\Sigma$, accordingly. Making use of eqns. $\left(\ref{eq:enetgy momentum tensor}\right)$ and $\left(\ref{eq:NLEqs}\right)$, we find that the Smarr relation is given by

where $A_{H}=4\pi r_{+}^{2}$ is the horizon area, and $T=N^{\prime}\left(r_{+}\right)e^{-\delta\left(r_{+}\right)}/4\pi$ is the Hawking temperature. For a RNAdS black hole with $\delta\left(r\right)=0$, the Smarr relation $\left(\ref{eq:smarr}\right)$ reduces to

where the last term is the $PV$ term in the extended phase space of AdS black holes [66].

Given a family of scalarized black holes, it is of interest to compute the Helmholtz free energy, which can be used to investigate phase structure and transitions in a canonical ensemble with fixed charge $Q$ and temperature $T$. The free energy, which is identified as the thermal partition function of black holes, can be derived via constructing the Euclidean path integral. In the semiclassical approximation, the partition function is evaluated by exponentiating the on-shell Euclidean action $S_{\text{on-shell}}^{E}$,

where the on-shell action $S_{\text{on-shell}}^{E}$ is obtained by substituting the classical solution into the action. However, the on-shell action $S_{\text{on-shell}}^{E}$ normally diverges in asymptotically AdS spacetime. One then needs holographic renormalization to remove divergences appearing in the asymptotic region. There are several methods to regularize $S_{\text{on-shell}}^{E}$, such as the background-subtraction method [57] and the Kounterterms method [67, 68, 69]. Here, we adopt the counterterm subtraction method [70, 71] to regularize the action by adding a series of boundary terms to the bulk action.

Specifically for the aforementioned bulk action $S_{\text{bulk}}$ in eqn. $\left(\ref{eq:action}\right)$, the regularized action $S_{R}$ is supplied with three boundary terms

where $S_{\text{GH}}$ is the Gibbon-Hawking boundary term to render the variational problem well-defined, $S_{\text{ct}}$ includes counterterms to eliminate divergences on asymptotic boundaries, and $S_{\text{surf}}$ is used to fix the charge rather than the electrostatic potential when the action is varied [72, 62]. The three boundary terms are given by

where the integrals are performed on the hypersurface at the spatial infinity, $\gamma$ is the determinant of the induced metric on the hypersurface, $\Theta$ is the trace of the extrinsic curvature, $R_{3}$ is the scalar curvature of the induced metric $\gamma$, and $n_{\mu}$ is the unit vector normal to the hypersurface. Using the equations of motion $\left(\ref{eq:EOMs}\right)$ and the asymptotic expansion $\left(\ref{eq:aymp. at inf}\right)$, we obtain the on-shell Euclidean version of $S_{\text{bulk}}$, $S_{\text{GH}}$, $S_{\text{ct}}$ and $S_{\text{surf}}$,

where $S=\pi r_{+}^{2}$ is the entropy of the black hole. Consequently, the regularized on-shell Euclidean action for the black hole solution $\left(\ref{eq:metric}\right)$ is

The Helmholtz free energy $F$ is related to the Euclidean action $S_{\text{on-shell}}^{E}$ via

which gives

We first examine a scalar perturbation $\delta\phi$ in a RNAdS black hole background. Note that if $\dot{f}\left(0\right)=0$ is imposed, a RNAdS black hole, which is described by

is manifestly a solution of the equations of motion $\left(\ref{eq:EOMs}\right)$. In this scalar-free solution background, we can linearize the scalar equation in eqn. $\left(\ref{eq:EOMs}\right)$ with a scalar perturbation $\delta\phi$,

where $\mu_{eff}^{2}=-\ddot{f}\left(0\right)Q^{2}/\left(2r^{4}\right)$. In a $\left(3+1\right)$-dimensional asymptotically AdS spacetime of AdS radius $L$, a scalar field can cause a tachyonic instability only if its mass-squared is less than the so-called Breitenlohner-Freedman (BF) bound $\mu_{BF}^{2}=-9/\left(4L^{2}\right)$ [73]. For the scalar perturbation $\delta\phi$, one always has $\mu_{eff}^{2}>\mu_{BF}^{2}$ for large enough $r$, and hence, asymptotically, the RNAdS black hole is stable against the formation of the scalar field, which guarantees that scalarized black holes induced by the tachyonic instability of the scalar field are asymptotically AdS. However, if $\mu_{eff}^{2}<\mu_{BF}^{2}$ in some region (e.g., near the event horizon), a RNAdS black hole may evolve to a scalarized black hole under a scalar perturbation. Note that if $\ddot{f}\left(0\right)<0$, one always has $\mu_{eff}^{2}>\mu_{BF}^{2}$, and hence a tachyonic instability can not occur.

To study how a scalarized black hole solution bifurcates from a scalar-free black hole solution, we calculate zero modes of the scalar perturbation in the scalar-free black hole background. For simplicity, the scalar perturbation $\delta\phi$ is written as the decomposition with spherical harmonics functions,

With this decomposition, the scalar equation $\left(\ref{eq:perturbative equation}\right)$ then reduces to

Given the fixed values of $l$ and $\ddot{f}\left(0\right)$, requiring that the radial field $U_{l}\left(r\right)$ is regular at the event horizon and vanishes at the spatial infinity selects a family of discrete black hole solutions, which can be labelled by a non-negative integer node number $n$. In this paper, we focus on the $l=0=n$ fundamental mode since it gives the smallest $q$ of the black hole solutions [21]. Due to the tachyonic instability, scalarized RNAdS black holes may emerge from these zero modes, which composes bifurcation lines in the domain of existence for the scalarized black holes.

To investigate perturbative stability of the scalarized black hole solution $\left(\ref{eq:metric}\right)$, we then consider spherically symmetric and time-dependent linear perturbations. Specifically including the perturbations, the metric ansatz is written as [22]

where the time dependence of the perturbations is assumed to be Fourier modes with frequency $\Omega$. Similarly, the ansatzes of the scalar and electromagnetic fields are given by

respectively. Solving eqn. $\left(\ref{eq:EOMs}\right)$ with the ansatzes $\left(\ref{eq:time-dependent metric}\right)$ and $\left(\ref{eq:time-dependent fields}\right)$, we can extract a Schrodinger-like equation for the perturbative scalar field $\phi_{1}\left(r\right)$,

where $\Psi\left(r\right)\equiv r\phi_{1}\left(r\right)$, and the tortoise coordinate $r^{\ast}$ is defined by $dr^{\ast}/dr\equiv e^{\delta\left(r\right)}N^{-1}\left(r\right)$. Here, the effective potential $U_{\Omega}$ is given by

One can show that the effective potential $U_{\Omega}$ vanishes at the event horizon, whereas approaches positive infinity at the spatial infinity. From quantum mechanics, the existence of an unstable mode with $\Omega^{2}<0$ requires the presence of $U_{\Omega}<0$ in some regions. Nevertheless, a positive definite $U_{\Omega}$ ensures that scalarized black hole solutions are stable against the spherically symmetric perturbations. It is noteworthy that the appearance of a negative region of $U_{\Omega}$ cannot sufficiently guarantee the presence of an instability [26]. One can utilize other techniques, like the $S$-deformation method [74], to further discuss the stability of these solutions.

Here, we present the numerical results, e.g., the domain of existence, entropic preference and effective potentials, for scalarized black hole solutions, which are dynamically induced from RNAdS black holes. Without loss of generality, we focus on $\tilde{Q}=Q/L=0.1$ in this subsection. For a RNAdS black hole, a tachyonic instability of the scalar field occurs if $\mu_{eff}^{2}<\mu_{BF}^{2}$ somewhere in the spacetime. Since the minimum value of $\mu_{eff}^{2}$ occurs at the event horizon $r=r_{+}$, we only need to check $\mu_{eff}^{2}<\mu_{BF}^{2}$ at $r=r_{+}$. The region in the $\left(\alpha,q\right)$ parameter space of RNAdS black holes where $\mu_{eff}^{2}<\mu_{BF}^{2}$ at $r=r_{+}$ is plotted in the upper left panel of Fig. 1. The distribution of values of $(\mu_{eff}^{2}-\mu_{BF}^{2})Q^{2}|_{r=r_{+}}$ is also displayed, which shows that the tachyonic instability region becomes larger as $\alpha$ increases, and the scalar field suffers from a strong tachyonic instability when black holes are near-extremal. The bifurcation line is composed of the $l=0=n$ zero modes of eqn. $\left(\ref{eq:seqn}\right)$ for the scalar perturbation in RNAdS black holes, and represented by blue dashed lines in Fig. 1. When the tachyonic instability is strong enough, the scalar perturbation can lead to scalarized black holes with non-trivial scalar fields above the bifurcation line. In the upper right panel of Fig. 1, we display the domain of existence for scalarized black holes, which is exhibited by a light blue region. The domain of existence is bounded by the bifurcation and critical lines, and resembles that of RN scalarized black holes [21]. On the critical line, the mass and the charge of scalarized solutions remain finite, whereas its horizon radius vanishes. On the other hand, the mass-to-charge ratio $q$ of RNAdS black holes reaches the maximum in the extremal limit, which is shown by a horizontal dashed gray line. Moreover, there is a certain region bounded by the extremal and bifurcation lines, where scalarized and RNAdS black holes coexist.

The reduced area $a_{H}$ as a function of reduced charge $q$ is plotted for RNAdS and scalarized black holes in the lower left panel of FIG. 1, which demonstrates that scalarized black holes emerge from RNAdS black holes at the bifurcation points, marked by $B$, and eventually terminate on the critical line with zero $a_{H}$. For a multiphase system in a micro-canonical ensemble with conserved energy, the phase of maximum entropy is globally stable and will be present at equilibrium. Therefore, in the scalarized and RNAdS black holes coexisting region, our numerical results show that scalarized solutions are entropically preferred over RNAdS black hole solutions, and hence are the globally stable phase in the micro-canonical ensemble.

To study the stability of scalarized solutions, the effective potentials $U_{\Omega}$ of scalarized solutions with $\alpha=5$ are plotted for several values of $q$ in the lower right panel of FIG. 1, where solid and dashed colored lines correspond to potentials with and without negative regions, respectively. The scalarized solutions have positive effective potentials for a large enough value of $q$, and thus are free of radial instabilities. However, as $q$ decreases towards the bifurcation line, there appear negative regions in the effective potentials, which means that radial instabilities cannot be excluded near the bifurcation line.

In this subsection, we consider phase structure and transitions of scalarized and RNAdS black holes in a canonical ensemble maintained at a given temperature of $T$ and a given charge of $Q$. In a canonical ensemble, the globally stable phase of a multiphase system, which exists at equilibrium, has the lowest possible Helmholtz free energy $F$, which can be computed via eqn. $\left(\ref{eq:freenergy}\right)$. The rich phase structure of black holes usually comes from expressing the horizon radius $r_{+}$ as a function of temperature $T$. If the function $r_{+}\left(T\right)$ is multivalued, there will be more than one black hole phase, corresponding to different branches of $r_{+}\left(T\right)$.

To illustrate phase structure and transitions, we plot the reduced horizon radius $\tilde{r}_{+}$ and the free energy $\tilde{F}$ as functions of reduced temperature $\tilde{T}$ for scalarized and RNAdS black holes with three representative values of $\tilde{Q}$ in Fig. 2, where we have $\alpha=5$. In the left column of Fig. 2 with a small $\tilde{Q}$, the upper panel shows that three branches of the RNAdS black hole solution coexist in some range of $\tilde{T}$, and are dubbed as large, intermediate and small RNAdS BHs, respectively, based on their values of horizon radius. At a high (low) enough temperature, only the large (small) RNAdS BH phase exists. On the other hand, there is only one phase for the scalarized black hole solution, which bifurcates from the RNAdS black hole solution at the bifurcation point $B$, and does not exist at a low temperature. The reduced free energy $\tilde{F}$ is plotted against $\tilde{T}$ for these four phases in the lower panel, which shows that the scalarized black hole can not be the globally stable phase since there always exists some RNAdS black hole phase of a lower free energy at a given $\tilde{T}$. In the coexisting region of the RNAdS black hole phases, a first-order phase transition between large and small RNAdS BHs occurs at some point, where their free energies intersect each other.

The coexisting region of the RNAdS black hole phases shrinks as $\tilde{Q}$ increases until reaching a critical point, where a second-order phase transition occurs between large and small RNAdS BHs. Beyond the critical point, large RNAdS BH is indistinguishable from small RNAdS BH, hence RNAdS black hole solutions have a single phase. In the center column of Fig. 2, we depict $\tilde{r}_{+}(\tilde{T})$ and $\tilde{F}(\tilde{T})$ for the case with a value of $\tilde{Q}$ greater than the critical value. The upper panel shows that both RNAdS and scalarized black holes have a single phase. Moreover, as displayed in the lower panel, the RNAdS black hole always has a smaller free energy than the scalarized black hole, and hence is the globally stable phase. Consequently, there is no phase transition in this case.

However when $\tilde{Q}$ is large enough, phase structure of scalarized and RNAdS black holes becomes much richer. For example, $\tilde{r}_{+}(\tilde{T})$ and $\tilde{F}(\tilde{T})$ are plotted for scalarized and RNAdS black holes with a large enough $\tilde{Q}$ in the right column of Fig. 2. It shows in the upper panel that the RNAdS black hole solution possesses a single phase, whereas the scalarized solution can have two phases at some given $\tilde{T}$, namely large scalarized BH (i.e., the one with a larger horizon radius) and small scalarized BH (i.e., the one with a smaller horizon radius). In fact, the scalarized black hole solution has a minimum temperature $\tilde{T}_{\min}$, and large scalarized BH coexists with small scalarized BH between $\tilde{T}=\tilde{T}_{\min}$ and the bifurcation point $B$, where large scalarized BH and the RNAdS black hole merge. The lower panel exhibits the free energy as a function of $\tilde{T}$ for the three phases. If we start increasing the temperature from $\tilde{T}=0$, the system follows the blue line of the RNAdS black hole until $\tilde{T}=\tilde{T}_{\min}$, where the free energy has a discontinuity at its global minimum. Further increasing $\tilde{T}$, the inset shows that the system jumps to the lower green line of large scalarized BH, which corresponds to a zeroth-order phase transition between scalarized and RNAdS black holes. As $\tilde{T}$ continues to increase, the system follows the lower green line until it joins the blue line at the bifurcation point $B$, which corresponds to a second-order phase transition between scalarized and RNAdS black holes. Note that since scalarized and RNAdS black holes have the same entropy at the bifurcation point, a phase transition between them at the bifurcation point is second-order. In short, a RNAdS BH/scalarized BH/RNAdS BH reentrant phase transition is observed as $\tilde{T}$ increases.

In addition, it is interesting to consider the local thermodynamic stability of black hole phases against thermal fluctuations. In a canonical ensemble, the quantity of particular interest is the specific heat at constant charge,

Since the entropy is proportional to the size of the black hole, a positive specific heat means that black holes radiate less when they are smaller. Thus, the thermal stability of a phase follows from $C_{Q}>0$ (or equivalently, $\partial\tilde{r}_{+}/\partial\tilde{T}>0$). From the upper row of Fig. 2, we notice that large and small RNAdS BHs in the left column, RNAdS black hole in the center column, and large scalarized BH and RNAdS black hole in the right column all have $\partial\tilde{r}_{+}/\partial\tilde{T}>0$. In consequence, the globally stable phases of scalarized and RNAdS black holes possess a positive $C_{Q}$, and are thermally stable.

To better illustrate the globally stable phases of lowest free energy and the associated phase transitions, we display the phase diagram of scalarized and RNAdS black holes in the $\tilde{Q}$-$\tilde{T}$ plane in Fig. 3, where $\alpha=5$. There is a first-order phase transition line (the purple line) separating large and small RNAdS BHs for small $\tilde{Q}$, which terminates at the critical point. This first-order phase transition is quite similar to the liquid/gas phase transition. When $\tilde{Q}$ is large enough, large scalarized BH (the light blue region) appears, and is bounded by zeroth-order (the red line) and second-order (the blue dashed line) phase transition lines.