Nonlinear massive spin-two field generated by higher derivative gravity

Abstract

We present a systematic exposition of the Lagrangian field theory for the massive spin-two field generated in higher-derivative gravity. It has been noticed by various authors that this nonlinear field overcomes the well known inconsistency of the theory for a linear massive spin-two field interacting with Einstein's gravity. Starting from a Lagrangian quadratically depending on the Ricci tensor of the metric, we explore the two possible second-order pictures usually called "(Helmholtz-)Jordan frame" and "Einstein frame". In spite of their mathematical equivalence, the two frames have different structural properties: in Einstein frame, the spin-two field is minimally coupled to gravity, while in the other frame it is necessarily coupled to the curvature, without a separate kinetic term. We prove that the theory admits a unique and linearly stable ground state solution, and that the equations of motion are consistent, showing that these results can be obtained independently in either frame. The full equations of motion and the energy-momentum tensor for the spin--two field in Einstein frame are given, and a simple but nontrivial exact solution to these equations is found. The comparison of the energy-momentum tensors for the spin-two field in the two frames suggests that the Einstein frame is physically more acceptable. We point out that the energy-momentum tensor generated by the Lagrangian of the linearized theory is unrelated to the corresponding tensor of the full theory. It is then argued that the ghost-like nature of the nonlinear spin-two field, found long ago in the linear approximation, may not be so harmful to classical stability issues, as has been expected.