The Work-Energy Relation for Particles on Geodesics in the pp-Wave Spacetimes

TL;DR

This paper demonstrates that the classical work-energy relation approximately holds for particles on geodesics in pp-wave spacetimes, linking gravitational acceleration to changes in kinetic energy during wave passage.

Contribution

It shows that the Newtonian work-energy relation applies to particles in gravitational wave spacetimes, providing a new understanding of energy transfer in general relativity.

Findings

The integral of gravitational acceleration matches the change in kinetic energy.

The work-energy relation holds approximately in pp-wave spacetimes.

Particles can gain or lose kinetic energy after wave passage.

Abstract

A non-linear gravitational wave imparts gravitational acceleration to all particles that are hit by the wave. We evaluate this acceleration for particles in the pp-wave space-times, and integrate it numerically along the geodesic trajectories of the particles during the passage of a burst of gravitational wave. The time dependence of the wave is given by a Gaussian, so that the particles are free before and after the passage of the wave. The gravitational acceleration is understood from the point of view of a flat space-time, which is the initial and final gravitational field configuration. The integral of the acceleration along the geodesics is the analogue of the Newtonian concept of work per unit mass. Surprisingly, it yields almost exactly the variation of the non-relativistic kinetic energy per unit mass of the free particle. Therefore, the work-energy relation $\Delta K = \Delta W$ of classical Newtonian physics also holds for a particle on geodesics in the pp-wave space-times, in a very good approximation, and explains why the final kinetic energy of the particle may be smaller or larger than the initial kinetic energy.