The Einstein and M{\o}ller energy-momentum complexes in post-Newtonian approximation

TL;DR

This paper calculates the energy components of Einstein and Møller energy-momentum complexes in the post-Newtonian approximation of Schwarzschild spacetime, highlighting the roles of rest-mass, total mass-energy, and potential surfaces in gravitational energy localization.

Contribution

It provides explicit expressions for energy in the post-Newtonian regime using Einstein and Møller complexes, clarifying their relation to physical mass and gravitational energy.

Findings

Both complexes yield total mass-energy as M in Schwarzschild spacetime.

Rest-mass energy m behaves like bare mass in flat spacetime.

Zero-potential surface is crucial for energy localization.

Abstract

In the first and second post-Newtonian approximation of the Schwarzschild metric, I obtain the energy component of the Einstein and M{\o}ller energy-momentum complex. Both energies involve the rest-mass energy $m$, the energy stored in the configuration and that in the gravitational field, but the energies of Schwarzschild spacetime in the Einstein and M{\o}ller prescriptions are the total mass-energy $M$. First, for general relativity, the rest-mass energy $m$ in the flat spacetime behaves like the bare mass, and the total mass-energy $M$ in the curved spacetime behaves like the experimentally observed mass. Second, the zero-potential surface is important condition for defining the energy of gravitational field, and plays an important role in the energy-momentum localization of general relativity.