Gauging the scale invariance of Einstein equations: Weyl-invariant equations for gravity

We derive a Weyl-invariant equation for gravity by gauging the global Weyl invariance of vacuum Einstein equations. The equation is linear in the curvature and a natural generalization of Einstein equations to Weyl geometry. The system has five physical polarizations, two tensor modes, two vectors modes, and one scalar, associated to the cosmological constant. An exact black hole solution is found. We study the dynamics on Friedman backgrounds and the evolution of cosmological perturbations. It is shown that cosmological vector modes do not decay in this model.