The no-hair theorem postulates that the only externally observable properties of a black hole are its mass, its electric charge, and its angular momentum. In scalar-tensor theories of gravity, a matter distribution around a black hole can lead to the so called "spontaneous scalarisation" instability that triggers the development of scalar hair. In the Brans-Dicke type theories, this effect can be understood as a result of tachyonic effective mass of the scalar field. Here we consider the instability in the generalised class of scalar-theories that feature non-conformal, i.e. "disformal", couplings to matter. Such theories have gained considerable interest in the recent years and have been studied in a wide variety of systems, both cosmological and astrophysical. In view of the prospects of gravitational wave astronomy, it is relevant to explore the implications of the theories in the strong-gravity regime. In this article, we concentrate on the spontaneous scalarisation of matter configurations around Schwarzschild and Kerr black holes. We find that in the more generic scalar-tensor theories, the instability of the Brans-Dicke theory can be enhanced, suggesting violations of the no-hair theorem. On the other hand, we find that, especially if the coupling is very strong, or if the gradients in the matter distribution are negligible, the disformal coupling tends to stabilise the system.
Quasi Periodic Oscillations (QPOs) and frequencies in an accretion disk and comparison with the numerical results from non-rotating black hole computed by the GRH code
