Source integrals for multipole moments in static and axially symmetric spacetimes

In this article, we derive source integrals, i.e., quasilocal expressions, for multipole moments in axially symmetric and static spacetimes. Usually, these multipole moments are read off the asymptotics of the metric close to spatial infinity. Whereas for the evaluation of the here derived source integrals the geometry has to be known in the region containing all sources, i.e., matter as well as singularities. The source integrals can be written either as volume integrals over such a region or as integrals over the surface of that region. Eventually, these source integrals allow assigning to any spacetime regions its contribution to the total multipole moments of the spacetime. Finally, we give an exemplary application that outlines the usefulness and applicability of the source integrals in, e.g., (non)existence proofs.

Figure 1

The different surfaces ${\mathcal{H}}_i$, ${\mathcal{S}}_i^{(2)}$, ${\mathcal{S}}_{S,i}^{(2)}$ and volumes ${\mathcal{V}}_i^{(2)}$ are depicted for a black hole surrounded by a thick ring. The curves ${\mathcal{A}}^{\pm }$ and $\mathcal{C}$ are relevant in Sec. 2d.