The global nonlinear stability of Minkowski space. Einstein equations, f(R)-modified gravity, and Klein-Gordon fields

Research paper by Philippe G. LeFloch, Yue Ma

Indexed on: 28 Dec '17Published on: 28 Dec '17Published in: arXiv - General Relativity and Quantum Cosmology

Abstract

We study the initial value problem for two fundamental theories of gravity, that is, Einstein's field equations of general relativity and the (fourth-order) field equations of f(R) modified gravity. For both of these physical theories, we investigate the global dynamics of a self-gravitating massive matter field when an initial data set is prescribed on an asymptotically flat and spacelike hypersurface, provided these data are sufficiently close to data in Minkowski spacetime. Under such conditions, we thus establish the global nonlinear stability of Minkowski spacetime in presence of massive matter. In addition, we provide a rigorous mathematical validation of the f(R) theory based on analyzing a singular limit problem, when the function f(R) arising in the generalized Hilbert-Einstein functional approaches the scalar curvature function R of the standard Hilbert-Einstein functional. In this limit we prove that f(R) Cauchy developments converge to Einstein's Cauchy developments in the regime close to Minkowski space. Our proofs rely on a new strategy, introduced here and referred to as the Euclidian-Hyperboloidal Foliation Method (EHFM). This is a major extension of the Hyperboloidal Foliation Method (HFM) which we used earlier for the Einstein-massive field system but for a restricted class of initial data. Here, the data are solely assumed to satisfy an asymptotic flatness condition and be small in a weighted energy norm. These results for matter spacetimes provide a significant extension to the existing stability theory for vacuum spacetimes, developed by Christodoulou and Klainerman and revisited by Lindblad and Rodnianski.