Equations of Motion for Massive Spin 2 Field Coupled to Gravity

TL;DR

The paper addresses the consistency and causality of massive spin-2 field dynamics in curved backgrounds, presenting two complementary frameworks. First, a field-theoretic construction with non-minimal couplings in Einstein spacetimes yields a one-parameter family of covariant, causal EOM that preserve the correct degrees of freedom. Second, an infinite-series-in-1/m approach provides a way to describe the field in arbitrary backgrounds, with leading-order results matching flat-space limits; these findings are reinforced by open-string sigma-model analyses showing agreement with field-theory results at lowest order in α′. Together, the work demonstrates robust pathways to consistent massive spin-2 interactions with gravity and outlines directions for incorporating dilatons, higher spins, and dynamical gravity.

Abstract

We investigate the problems of consistency and causality for the equations of motion describing massive spin two field in external gravitational and massless scalar dilaton fields in arbitrary spacetime dimension. From the field theoretical point of view we consider a general classical action with non-minimal couplings and find gravitational and dilaton background on which this action describes a theory consistent with the flat space limit. In the case of pure gravitational background all field components propagate causally. We show also that the massive spin two field can be consistently described in arbitrary background by means of the lagrangian representing an infinite series in the inverse mass. Within string theory we obtain equations of motion for the massive spin two field coupled to gravity from the requirement of quantum Weyl invariance of the corresponding two dimensional sigma-model. In the lowest order in $α'$ we demonstrate that these effective equations of motion coincide with consistent equations derived in field theory.