Source integrals for multipole moments in static and axially symmetric spacetimes

摘要

In this article, we derive source integrals for multipole moments in axiallysymmetric and static spacetimes. The multipole moments can be read off theasymptotics of the metric close to spatial infinity in a hypersurface, which isorthogonal to the timelike Killing vector. Whereas for the evaluation of thesource integrals the geometry needs to be known in a compact region of thishypersurface, which encloses all source, i.e. matter as well as singularities.The source integrals can be written either as volume integrals over such aregion or in quasi-local form as integrals over the surface of that region.