Explicit multipole moments of stationary axisymmetric spacetimes

TL;DR

This paper simplifies the calculation of multipole moments in stationary axisymmetric spacetimes by reducing tensorial recursion to scalar functions and applying it to the Kerr solution, revealing new insights into their structure.

Contribution

It introduces a scalar recursion approach for multipole moments, streamlining their computation and analysis in axisymmetric stationary spacetimes.

Findings

Reduced tensorial recursion to scalar functions

Expressed moments as derivatives of a complex function at zero

Calculated moments explicitly for the Kerr solution

Abstract

In this article we study multipole moments of axisymmetric stationary asymptotically flat spacetimes. We show how the tensorial recursion of Geroch and Hansen can be reduced to a recursion of scalar functions. We also demonstrate how a careful choice of conformal factor collects all moments into one complex valued function on R, where the moments appear as the derivatives at 0. As an application, we calculate the moments of the Kerr solution. We also discuss the freedom in choosing the potential for the moments.