Energy and angular momentum densities of stationary gravity fields

TL;DR

This paper provides invariant expressions for the energy and angular momentum densities in stationary gravitational fields, clarifying their physical interpretation and coordinate dependence, especially in axially symmetric cases.

Contribution

It introduces explicit invariant formulas for gravitational energy and angular momentum densities involving non-local conformal factors, with derivations based on potential energy differences.

Findings

Invariant expressions involve non-local conformal factors.

Angular momentum density depends on observer motion in axially symmetric fields.

Expressions reduce to local forms in certain coordinate systems.

Abstract

We give physical explanations of explicit invariant expressions for the energy and angular momentum densities of gravitational fields in stationary space-times. These expressions involve non-locally defined conformal factors. In certain coordinates these become locally defined in terms of the metric. These results are derived via expressions for total gravitational potential energy from the difference between the total energy and the mechanical energy. The latter involves kinetic energy seen in the frame of static observers. When in the axially symmetric case we consider zero angular momentum observers (who move orthogonally to surfaces of constant time), we find that the angular momentum they attribute to the gravitational field is solely due to their motion.