The Israel–Carter theorem (also known as the “no-hair theorem”) puts a restriction on the existence of parameters other than mass, electric charge, and angular momentum of a black hole. In this context, Bekenstein proposed no-hair theorems in various black hole models with neutral and electrically charged scalar fields. In this paper, we take the Einstein–Maxwell-charged scalar model with an electrically charged scalar field gauge-coupled to the Maxwell field surrounding a charged black hole with a static spherically symmetric metric. In particular, we consider a quadratic scalar potential without any higher order terms and we do not impose any restriction on the magnitude of the scalar charge with respect to the black hole charge. With this setting, we ascertain the validity of all energy conditions coupled with the causality condition, suggesting the possibility of existence of charged hairy solutions. Consequently, we obtain, by exact numerical integration, detailed solutions of the field equations that incorporate backreaction on the spacetime due to the presence of the charged scalar field. The solutions exhibit damped oscillatory behaviours for the charged scalar hair. We also find that the electric potential is a monotonic function of the radial coordinate, as required by electrodynamics. In order to ascertain the existence of our charged hairy solutions, we carry out dynamic stability analyses against time-dependant perturbations about the static solutions. For a definite conclusion, we employ two different methodologies. The first methodology involves a Sturm–Liouville equation, whereas the second methodology employs a Schrödinger-like equation, for the dynamic perturbations. We find that our solutions are stable against time-dependant perturbations by both methodologies, confirming the existence of the charged hairy solutions.

以色列-卡特定理（也称为“无毛定理”）对黑洞质量、电荷和角动量以外的参数的存在进行了限制。在此背景下，贝肯斯坦在具有中性和带电标量场的各种黑洞模型中提出了无毛定理。在本文中，我们采用爱因斯坦-麦克斯韦带电标量模型，其中带电标量场规范耦合到带静态球对称度量的带电黑洞周围的麦克斯韦场。特别是，我们考虑没有任何高阶项的二次标量势，并且我们不对标量电荷相对于黑洞电荷的大小施加任何限制。通过这种设置，我们确定了所有能量条件与因果关系条件的有效性，表明带电毛状解存在的可能性。因此，通过精确的数值积分，我们获得了场方程的详细解，其中包含由于带电标量场的存在而导致的时空反作用。该解决方案对带电标量毛发表现出阻尼振荡行为。我们还发现，根据电动力学的要求，电势是径向坐标的单调函数。为了确定带电毛状解的存在性，我们针对静态解的时间相关扰动进行了动态稳定性分析。为了得出明确的结论，我们采用两种不同的方法。第一种方法涉及 Sturm-Liouville 方程，而第二种方法则采用类薛定谔方程来计算动态扰动。 我们发现我们的解决方案对于两种方法的时间相关扰动都是稳定的，证实了带电毛状解决方案的存在。