The Euclidean-hyperboloidal foliation method. Application to f(R) modified gravity

TL;DR

This paper develops a geometric foliation method to analyze the global stability of Minkowski spacetime within f(R) modified gravity, demonstrating the existence of solutions close to Minkowski space for small initial data.

Contribution

It extends the Euclidean-hyperboloidal foliation method to fourth-order f(R) gravity equations, proving their nonlinear stability near Minkowski spacetime.

Findings

Established global hyperbolic development approaching Minkowski space

Cast f(R) equations as a wave-Klein-Gordon system with an effective mass

Proved nonlinear stability of Minkowski spacetime in f(R) gravity

Abstract

This paper is a part of a series devoted to the Euclidean-hyperboloidal foliation method introduced by the authors for investigating the global existence problem associated with nonlinear systems of coupled wave-Klein-Gordon equations with small data. This method was developed especially for investigating the initial value problem for the Einstein-massive field system in wave gauge. Here, we study the (fourth-order) field equations of f(R) modified gravity and investigate the global dynamical behavior of the gravitational field in the near-Minkowski regime. We establish the existence of a globally hyperbolic Cauchy development approaching Minkowski spacetime (in spacelike, null, and timelike directions), when the initial data set is sufficiently close to an asymptotically Euclidean and spacelike hypersurface in Minkowski spacetime. We cast the (fourth-order) f(R)-field equations in the form of a second-order wave-Klein-Gordon system, which has an analogous structure to the Einstein-massive field system but, in addition, involves a (possibly small) effective mass parameter. We establish the nonlinear stability of the Minkowski spacetime in the context of f(R) gravity, when the integrand f(R) in the action functional can be taken to be arbitrarily close to the integrand R of the standard Hilbert-Einstein action.