Static Observers in Curved Spaces and Non-inertial Frames in Minkowski Spacetime

TL;DR

This paper investigates non-inertial frames in Minkowski spacetime inspired by static observers in curved spaces, deriving explicit embeddings and a covariant acceleration field equation analogous to Newtonian gravity.

Contribution

It introduces a method to construct non-inertial frames in Minkowski space with acceleration fields matching static observers in curved spacetimes, including explicit hypersurface embeddings and a covariant acceleration equation.

Findings

Explicit embedding of simultaneity hypersurfaces for arbitrary acceleration fields.

A covariant field equation governing proper acceleration of static observers.

Analysis of energy-momentum tensor for maximally symmetric acceleration surfaces.

Abstract

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.