Black hole perturbations in higher-order scalar-tensor theories: initial value problem and dynamical stability

We propose a physically sensible formulation of initial value problem for black hole perturbations in higher-order scalar-tensor theories. As a first application, we study monopole perturbations around stealth Schwarzschild solutions in a shift- and reflection-symmetric subclass of degenerate higher-order scalar-tensor (DHOST) theories. In particular, we investigate the time evolution of the monopole perturbations by solving a two-dimensional wave equation and analyze the Vishveshwara's classical scattering experiment, i.e., the time evolution of a Gaussian wave packet. As a result, we confirm that stealth Schwarzschild solutions in the DHOST theory are dynamically stable against the monopole perturbations with the wavelength comparable or shorter than the size of the black hole horizon. We also find that the damped oscillations at the late time do not show up unlike the ringdown phase in the standard case of general relativity. Moreover, we investigate the characteristic curves of the monopole perturbations as well as a static spherically symmetric solution with monopole hair.

Automated summary

We propose a physically sensible formulation for the initial value problem of black hole perturbations in higher-order scalar-tensor theories. As a first application, we study monopole perturbations around stealth Schwarzschild solutions in a subclass of degenerate higher-order scalar-tensor (DHOST) theories with shift- and reflection-symmetry. Our analysis of the time evolution of the monopole perturbations using a two-dimensional wave equation reveals that the stealth Schwarzschild solutions in the DHOST theory are dynamically stable against monopole perturbations with wavelengths comparable or shorter than the black hole horizon size. Unlike the standard case of general relativity, we find that there are no damped oscillations during the late time.