On the Hyperbolicity and Stability of $3+1$ Formulations of Metric $f(R)$ Gravity

TL;DR

This paper analyzes the hyperbolicity and stability of 3+1 formulations of metric $f(R)$ gravity, demonstrating that the BSSNOK formulation is strongly hyperbolic and suitable for numerical relativity simulations.

Contribution

It derives the 3+1 evolution and constraint equations for generic $f(R)$ models and compares the hyperbolicity of ADM and BSSNOK formulations in metric $f(R)$ gravity.

Findings

ADM $f(R)$ is weakly hyperbolic with zero speed modes.

BSSNOK $f(R)$ formulation is strongly hyperbolic.

Time propagation of constraints matches general relativity form.

Abstract

$3+1$ formulations of the Einstein field equations have become an invaluable tool in Numerical relativity, having been used successfully in modeling spacetimes of black hole collisions, stellar collapse and other complex systems. It is plausible that similar considerations could prove fruitful for modified gravity theories. In this article, we pursue from a numerical relativistic viewpoint the $3+1$ formulation of metric $f(R)$ gravity as it arises from the fourth order equations of motion, without invoking the dynamical equivalence with Brans-Dicke theories. We present the resulting system of evolution and constraint equations for a generic function $f(R)$, subject to the usual viability conditions. We confirm that the time propagation of the $f(R)$ Hamiltonian and Momentum constraints take the same Mathematical form as in general relativity, irrespective of the $f(R)$ model. We further recast the 3+1 system in a form akin to the BSSNOK formulation of numerical relativity. Without assuming any specific model, we show that the ADM version of $f(R)$ is weakly hyperbolic and is plagued by similar zero speed modes as in the general relativity case. On the other hand the BSSNOK version is strongly hyperbolic and hence a promising formulation for numerical simulations in metric $f(R)$ theories.