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.