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We study the linear stability of vacuum static, spherically symmetricsolutions to the gravitational field equations of theBergmann-Wagoner-Nordtvedt class of scalar-tensor theories (STT) of gravity,restricting ourselves to nonphantom theories, massless scalar fields andconfigurations with positive Schwarzschild mass. We consider only small radial(monopole) perturbations as the ones most likely to cause an instability. Theproblem reduces to the same Schroedinger-like master equation as is known forperturbations of Fisher's solution of general relativity (GR), but thecorresponding boundary conditions that affect the final result of the studydepend on the choice of the STT and a particular solution within it. Thestability or instability conclusions are obtained for the Brans-Dicke, Barkerand Schwinger STT as well as for GR nonminimally coupled to a scalar field withan arbitrary parameter $\xi$.

On the stability of spherically symmetric space-times in scalar-tensor gravity