A Note on the Strong Hyperbolicity of $f(R)$ Gravity with Dynamical Shifts

TL;DR

This paper investigates the well-posedness of $f(R)$ gravity equations with dynamical shifts, demonstrating strong hyperbolicity for three formulations using pseudodifferential reduction, which is crucial for numerical relativity.

Contribution

It proves the strong hyperbolicity of three $f(R)$ gravity formulations with dynamical shifts, including modifications to the Z4 formulation, using pseudodifferential reduction techniques.

Findings

All three formulations are strongly hyperbolic.

The traditional Z4 formulation is not strongly hyperbolic for $f(R)$ gravity.

Modified harmonic gauge conditions ensure strong hyperbolicity in the improved Z4 formulation.

Abstract

The well-posedness of the gravitational equations of $f(R)$ gravity are studied in this paper. Three formulations of the $f(R)$ gravity with dynamical shifts (which are all based on the Arnowitt-Deser-Misner (ADM) formalism of the equations) are investigated. These three formulations are all proved to be strongly hyperbolic by pseudodifferential reduction. The first one is the Baumagarte-Shapiro-Shibata-Nakamura (BSSN) formulation with the so-called "hyperbolic $K$-driver" condition and the "hyperbolic Gamma driver" condition. The second one is the ADM formulation with modified harmonic gauge conditions. We find that the equations are not strong hyperbolic in traditional Z4 formulation for $f(R)$ gravity. So, in the third formulation, we improve the Z4 formulation, and show these equations are strong hyperbolic with modified harmonic gauge conditions.