On the Stability of Spherically Symmetric Space-Times in Scalar-Tensor Gravity

We study the linear stability of vacuum static, spherically symmetric solutions to the gravitational field equations of the Bergmann–Wagoner–Nordtvedt class of scalar-tensor theories (STT) of gravity, restricting ourselves to nonphantom theories, massless scalar fields and configurations with positive Schwarzschild mass. We consider only small radial (monopole) perturbations as the ones most likely to cause an instability. The problem reduces to the same Schrödinger-like master equation as is known for perturbations of Fisher’s solution of general relativity (GR), but the corresponding boundary conditions that affect the final result of the study depend on the choice of the STT and a particular solution within it. The stability or instability conclusions are obtained for the Brans–Dicke, Barker and Schwinger STT as well as for GR nonminimally coupled to a scalar field with an arbitrary parameter \(\xi\).