Multipole moment and singular source in Newtonian gravity and in Einstein gravity

TL;DR

This paper compares multipole moments in Newtonian and Einstein gravity, showing that in general relativity, these moments cannot be solely determined by source integrals, unlike in Newtonian gravity.

Contribution

It demonstrates that in Einstein gravity, multipole moments are not directly linked to source integrals, contrasting with Newtonian gravity, and provides explicit examples in static axial spacetimes.

Findings

Multipole moments in GR are not determined by source integrals.

Curzon and Schwarzschild spacetimes share source integrals but differ in moments.

Abstract

The multipole moments are defined as the multipole expansion coefficients of the gravitational field at infinity. In Newtonian gravity, the multipole moments are determined by the source distribution -- the multipole integrals of the source. In this paper, we show that the multipole moments in general relativity cannot be determined by the multipole integrals of the source. We provide the multipole integrals in static axial spacetimes, such as, the Curzon spacetime. The Curzon spacetime possesses the same multipole integrals of the source with the Schwarzschild spacetime, while they possess different multipole moments.