Ernst formulation of axisymmetric fields in f(R) gravity: Applications to neutron stars and gravitational waves

The Ernst formulation of the Einstein equations is generalized to accommodate f(R) theories of gravity. It is shown that, as in general relativity, the axisymmetric f(R) field equations for a vacuum spacetime that is either stationary or cylindrically symmetric reduce to a single, nonlinear differential equation for a complex-valued scalar function. As a worked example, we apply the generalized Ernst equations to derive a f(R) generalization of the Zipoy-Voorhees metric, which may be used to describe the gravitational field outside of an ellipsoidal neutron star. We also apply the theory to investigate the phase speed of large-amplitude gravitational waves in f(R) gravity in the context of solitonlike solutions that display shock-wave behavior across the causal boundary.