We investigate the static and spherically black hole solutions in the quadratic-order extended vector–tensor theories without suffering from the Ostrogradsky instabilities, which include the quartic-order (beyond-)generalized Proca theories as the subclass. We start from the most general action of the vector–tensor theories constructed with up to the quadratic-order terms of the first-order covariant derivatives of the vector field, and derive the Euler–Lagrange equations for the metric and vector field variables in the static and spherically symmetric backgrounds. We then substitute the spacetime metric functions of the Schwarzschild, Schwarzschild–de Sitter/anti-de Sitter, Reissner–Nordstrm-type, and Reissner–Nordstrm–de Sitter/anti-de Sitter-type solutions and the vector field with the constant spacetime norm into the Euler–Lagrange equations, and obtain the conditions for the existence of these black hole solutions. These solutions are classified into the two cases 1) the solutions with the vanishing vector field strength; the stealth Schwarzschild and the Schwarzschild–de Sitter/anti-de Sitter solutions, and 2) those with the nonvanishing vector field strength; the charged stealth Schwarzschild and the charged Schwarzschild–de Sitter/anti-de Sitter solutions, in the case that the tuning relation among the coupling functions is satisfied. In the latter case, if this tuning relation is violated, the solution becomes the Reissner–Nordstrm-type solution. We show that the conditions for the existence of these solutions are compatible with the degeneracy conditions for the class-A theories, and recover the black hole solutions in the generalized Proca theories as the particular cases.

我们研究了二次扩展向量张量理论中的静态和球形黑洞解，而没有遭受 Ostrogradsky 不稳定性的影响，其中包括四次（超越）广义 Proca 理论作为子类。我们从向量-张量理论的最一般作用开始，该理论由向量场的一阶协变导数的二次项构成，并推导出度量和向量场变量的欧拉-拉格朗日方程静态和球对称背景。然后，我们将 Schwarzschild、Schwarzschild-de Sitter/anti-de Sitter、Reissner-Nordstrm 型和 Reissner-Nordstrm-de Sitter/anti-de Sitter 型解的时空度量函数和向量场替换为恒定时空范数转化为欧拉-拉格朗日方程，并获得这些黑洞解存在的条件。这些解决方案分为两种情况：1）具有消失的矢量场强的解决方案；隐身 Schwarzschild 和 Schwarzschild-de Sitter/anti-de Sitter 解决方案，以及 2) 具有非零矢量场强的解决方案；在满足耦合函数之间的调谐关系的情况下，带电隐身 Schwarzschild 和带电 Schwarzschild-de Sitter/anti-de Sitter 解。在后一种情况下，如果违反此调谐关系，则解将变为 Reissner-Nordstrm 型解。我们证明了这些解存在的条件与A类理论的简并条件兼容，并将广义Proca理论中的黑洞解恢复为特例。