The initial value problem for gravitational waves in conformal gravity

Abstract

In certain models of conformal gravity, the propagation of gravitational waves is governed by a fourth order scalar partial differential equation. We study the initial value problem for a generalization of this equation, and derive a Kirchhoff-like explicit solution in terms of the field and its first three time derivatives evaluated on an initial hypersurface, as well as second order spatial derivatives of the initial data. In the conformal gravity case, we establish that if the initial data is featureless on scales smaller than the length scale of conformal symmetry breaking, then we recover the ordinary behaviour of gravitational waves in general relativity. We also confirm that effective weak field gravitational force exerted by a static spherical body in such models becomes constant on small scales; i.e. conformal gravity is effectively 2-dimensional at high energies.