Weak gravitational fields

We consider the set of Ck bounded tensor fields of type (r,s) on R4 in the topology of uniform Ck convergence. For each k≥2, the map sending a metric to its curvature tensor is shown to be analytic at the Minkowski metric. The same is true of the map sending a metric to its Einstein tensor. The well-known linearized theory of gravitation amounts to studying the directional derivatives of these maps. An iterative method for solving the full field equations along an analytic curve of Einstein tensors passing through zero is proposed.