We consider a general class of scalar tensor theories in three dimensions whose action contains up to second-order derivatives of the scalar field with coupling functions that only depend on the standard kinetic term of the scalar field, thus ensuring the invariance under the constant shift of the scalar field. For this model, we show that the field equations for a stationary metric ansatz together with a purely radial scalar field can be fully integrated. The kinetic term of the scalar field solution is shown to satisfy an algebraic relation depending only on the coupling functions, and hence is constant while the metric solution is nothing but the BTZ metric with an effective cosmological constant fixed in terms of the coupling functions. As a direct consequence the thermodynamics of the solution is shown to be identical to the BTZ one with an effective cosmological constant, despite the presence of a scalar field. Finally, the expression of the semi-classical entropy of this solution is also confirmed through a generalized Cardy-like formula involving the mass of the scalar soliton obtained from the black hole by means of a double Wick rotation.

中文翻译：

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旋转三维三维标量张量理论的黑洞

我们考虑了三维的一类普通的标量张量理论，其作用包含多达标量场的二阶导数，其耦合函数仅取决于标量场的标准动力学项，从而确保了在恒定位移下的不变性。标量字段。对于此模型，我们表明，可以将静态度量ansatz的场方程与纯径向标量场一起完全集成。标量场解的动力学项被证明满足仅依赖于耦合函数的代数关系，因此是常数，而度量解仅是具有有效宇宙常数的BTZ度量，其耦合函数固定。作为直接的结果，尽管存在标量场，但该溶液的热力学显示与具有有效宇宙学常数的BTZ相同。最后，该溶液的半经典熵的表达也通过广义的Cardy-like公式得到证实，该公式涉及通过两次Wick旋转从黑洞获得的标量孤子的质量。