Work and work-energy theorem in curved spacetime

TL;DR

This paper proposes coordinate-independent definitions of gravitational and external work for particles in curved spacetime, analyzing their consistency across various spacetime backgrounds and limits.

Contribution

It introduces scalar integral-based definitions of work in curved spacetime, emphasizing observer dependence and validating them in key spacetime models.

Findings

Definitions agree with Newtonian mechanics in slow motion and far field limits.

Validated in Minkowski, Schwarzschild, Reissner-Nordström, and Kerr-Newman spacetimes.

Work definitions are coordinate-independent but observer-dependent.

Abstract

The definitions of gravitational work as well as work done by the total external force on a massive probe particle moving in generic spacetime backgrounds are proposed. These definitions are given in the form of scalar integrals and thus, are independent of coordinate choices. However, the dependence on the choice of observer field is essential and inevitable. The definitions are checked in the case of Minkowski, Schwarzschild, Reisner-Nordstr\"om and Kerr-Newman spacetimes and agreements with Newtonian mechanical definitions are verified in the slow motion or the far field limit.