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

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.