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

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.

20, pages

References in corpus (14)

• Dark energy in modified Gauss-Bonnet gravity: late-time acceleration and the hierarchy problem
• Cosmological viability of f(R)-gravity as an ideal fluid and its compatibility with a matter dominated phase
• Avoiding Dark Energy with 1/R Modifications of Gravity
• Phantom scalar dark energy as modified gravity: understanding the origin of the Big Rip singularity
• On the co-existence of matter dominated and accelerating solutions in f(G)-gravity
• Robust approach to f(R) gravity
• The Cauchy problem of f(R) gravity
• The Cauchy problem for metric-affine f(R)-gravity in presence of a Klein-Gordon scalar field
• Implementation of standard testbeds for numerical relativity
• Hyperbolicity of scalar-tensor theories of gravity
• On the absence of the usual weak-field limit, and the impossibility of embedding some known solutions for isolated masses in cosmologies with f(R) dark energy
• From Geometry to Numerics: interdisciplinary aspects in mathematical and numerical relativity
• Binary Black Hole merger in f(R) theory
• Toward a Consistent Framework for High Order Mesh Refinement Schemes in Numerical Relativity

Cited by in corpus (2)

• Characteristic Formulation for Metric $f(R)$ Gravity
• Hamiltonian dynamics of doubly-foliable space-times