Linear gravity from conformal symmetry

We perform a unified systematic analysis of d + 1 dimensional, spin l representations of the isometry algebra of the maximally symmetric spacetimes AdS(d+1), R-1,R-d and dS(d+1). This allows us to explicitly construct the effective low-energy bulk equations of motion obeyed by linear fields, as the eigenvalue equation for the quadratic Casimir differential operator. We show that the bulk description of a conformal family is given by the Fierz-Pauli system of equations. For l = 2 this is a massive gravity theory, while for l = 2 conserved currents we obtain Einstein gravity and covariant gauge fixing conditions. This analysis provides a direct algebraic derivation of the familiar AdS holographic dictionary at low energies, with analogous results for Minkowski and de Sitter spacetimes.