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

TL;DR

This paper extends the Ernst formulation to $f(R)$ gravity, deriving a unified equation for axisymmetric spacetimes, and applies it to model neutron stars and analyze gravitational wave phase speeds.

Contribution

It generalizes the Ernst formulation to $f(R)$ theories, enabling new solutions for axisymmetric spacetimes and gravitational wave analysis.

Findings

Derived a $f(R)$ generalization of the Ernst equations.

Constructed a $f(R)$ version of the Zipoy-Voorhees metric.

Analyzed phase speed of gravitational waves with shock-wave behavior.

Abstract

The Ernst formulation of the Einstein equations is generalised 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, non-linear differential equation for a complex-valued scalar function. As a worked example, we apply the generalised Ernst equations to derive a $f(R)$ generalisation 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 soliton-like solutions that display shock-wave behaviour across the causal boundary.