Gravitational self-force in scalar-tensor gravity

Abstract

Motivated by the discovery of floating orbits and the potential to provide extra constraints on alternative theories, in this paper we derive the self-force equation for a small compact object moving on an accelerated world line in a background spacetime which is a solution of the coupled gravitational and scalar field equations of scalar-tensor theory. In the Einstein frame, the coupled field equations governing the perturbations sourced by the particle share the same form as the field equations for perturbations of a scalarvac spacetime, with both falling under the general class of hyperbolic field equations studied by Zimmerman and Poisson. Here, we solve the field equations formally in terms of retarded Green functions, which have explicit representations as Hadamard forms in the neighbourhood of the world line. Using a quasi-local expansion of the Hadamard form, we derive the regular solutions in Fermi normal coordinates according to the Detweiler-Whiting prescription. To compute the equation of motion, we parameterize the world line in terms of a mass and "charge", which we define in terms of the original Jordan frame mass, its derivative, and the parameter which translates the proper time in the Jordan frame to the Einstein frame. These parameters depend on the value of the background scalar field and its self-field corrections. The equation of motion which follows from the regular fields strongly resembles the equation for the self-force acting on a charged, massive particle in a scalarvac geometry of general relativity. Unlike the scalar vacuum scenario, the "charge" parameter in the scalar-tensor self-force equation is time variable and leading to additional local and tail terms. We also provide evolution equations for the world line parameters under the influence of the self-fields.