Stability of a metric f(R) gravity theory implies the Newtonian limit

Abstract

We show that the existence of the Newtonian limit cannot work as a selection rule for choosing the correct gravity theory fromm the set of all L=f(R) ones. To this end we prove that stability of the ground state solution in arbitrary purely metric f(R) gravity implies the existence of the Newtonian limit of the theory. And the stability is assumed to be the fundamental viability criterion of any gravity theory. The Newtonian limit is either strict in the mathematical sense if the ground state is flat spacetime or approximate and valid on length scales smaller than the cosmological one if the ground state is de Sitter or AdS space. Hence regarding the Newtonian limit a metric f(R) gravity does not differ from GR with arbitrary Lambda. This is exceptional to Lagrangians solely depending on R and/or Ricci tensor. An independent selection rule is necessary.