On the energy of Schwarzschild spacetime with the post-Newtonian approximation

TL;DR

This paper calculates the energy of Schwarzschild spacetime using the post-Newtonian approximation within the Weinberg prescription, revealing corrections beyond the Newtonian energy at higher orders.

Contribution

It provides a detailed derivation of the energy in Schwarzschild spacetime up to the second post-Newtonian order, including new correction terms.

Findings

First post-Newtonian energy equals Newtonian mass m.

Second post-Newtonian energy includes configuration, field, and surface terms.

Post-Newtonian corrections refine the Newtonian energy estimate.

Abstract

With the post-Newtonian approxination, the energy of Schwarzschild spacetime in the Weinberg prescription is obtained. The energy for the first post-Newtonian approximation $E^{(1)} = m$ gives the Newtonian treatment of Schwarzschild spacetime. However, for the second post-Newtonian approximation, the erergy is shown that $E^{(1)}$ adds extra terms $E^{(2)}$ which consist of the energy stored in the configuration $E_{\rm config}$, in the gravitational field $E_{\rm field}$. and a term of surface integral. These extra terms gives post-Newtonian corrections to the Newtonian treatment.