Static observers in curved spacetimes may interpret their proper acceleration
as the opposite of a local gravitational field (in the Newtonian sense). Based
on this interpretation and motivated by the equivalence principle, we are led
to investigate congruences of timelike curves in Minkowski spacetime whose
acceleration field coincides with the acceleration field of static observers of
curved spaces. The congruences give rise to non-inertial frames that are
examined. Specifically we find, based on the locality principle, the embedding
of simultaneity hypersurfaces adapted to the non-inertial frame in an explicit
form for arbitrary acceleration fields. We also determine, from the Einstein
equations, a covariant field equation that regulates the behavior of the proper
acceleration of static observers in curved spacetimes. It corresponds to an
exact relativistic version of the Newtonian gravitational field equation. In
the specific case in which the level surfaces of the norm of the acceleration
field of the static observers are maximally symmetric two-dimensional spaces,
the energy-momentum tensor of the source is analyzed.Comment: 28 pages, 4 figures
