Novel black holes with scalar hair in the Einstein-Maxwell-Scalar Theory with positive coupling

TL;DR

This work investigates hairy black holes in the four-dimensional Einstein–Maxwell–Scalar theory with a nonminimal exponential coupling $f(\psi)=e^{-\lambda\psi^2}$ between a massless scalar $\psi$ and the Maxwell field. Focusing on positive coupling $\lambda>0$, the authors numerically construct static, spherically symmetric black hole solutions whose scalar field $\psi(r)$ increases monotonically with radius and approaches a finite asymptotic value $\psi_0$, yielding hair quantified by $\psi_e=\psi_0-\psi_h^0$; these solutions exist only for reduced charge $q<1$ and lie inside the RN parameter region, with larger $\lambda$ shrinking the existence domain. The scalar-hairy branch exhibits higher Maxwell-field energy and lower Hawking temperature than RN, and increasing charge $q$ initially enhances hair before overcharging returns the solution to hairless RN, indicating a distinct mechanism from spontaneous scalarization which requires tachyonic instability. Stability analysis via a positive effective potential $V_{\mathrm{eff}}$ and quasinormal modes with $\mathrm{Im}(\omega)<0$ shows linear stability against scalar perturbations. Overall, the work reveals a new class of hairy black holes driven by nonlinear coupling in EMS theory, enriching the landscape of black hole solutions and offering insights into phase structure and potential evolution within this framework, with possible connections to AdS/EMD contexts.

Abstract

In this work, we find a new branch of hairy black hole solutions in the Einstein-Maxwell-Scalar (EMS) theory in four-dimensional asymptotically flat spacetimes. Different from spontaneous scalarization induced by tachyonic instabilities in Reissner-Nordström (RN) black holes with a negative coupling parameter, these scalar-hairy black hole solutions arise when the coupling parameter is positive, where nonlinear coupling plays the dominant role, meaning that the coupling is positively correlated with the degree of deviation from the trivial state. Our numerical analysis reveals that the scalar field grows monotonically with the radial coordinate and asymptotically approaches a finite constant, exhibiting behavior that is qualitatively similar to that of the Maxwell potential. In these solutions, an increase in the charge $q$ causes the scalar-hairy solutions to deviate further from the RN state, while excessive charging drives the system back towards hairless solutions. Strengthening the coupling parameter compresses the existence domain of the scalar-hairy state, which lies entirely within the parameter region of RN black holes. Moreover, by evaluating the quasinormal modes, we show that the obtained scalar-hairy solutions are stable against linearized scalar perturbations.

Figures (7)

• Figure 1: The profile of the field functions with $\lambda=1$ for different $q$.
• Figure 2: The dependence of the scalar charge $\psi_e\equiv\psi_0-\psi_h^0$ on the reduced charge $q$ for various coupling values. The curves, from lowest to highest, correspond to $\lambda=0.1,1,5,10,15,20,40$, respectively.
• Figure 3: The energy of the scalar field and Maxwell field outside the horizon as the function of the charge $q$ for scalarization (top panel) and scalar-hairy solutions (bottom panel).
• Figure 4: Existence domain of the scalar-hairy black holes in the parameter space of $(\lambda,q)$.
• Figure 5: Left: Hawking temperature of the scalar-hairy black hole as a function of the reduced charge $q$ for different coupling value $\lambda$; Right: Chemical potential of the scalar-hairy solutions as a function of $q$ for different coupling value $\lambda$.