Calculation of multipole moments of axistationary electrovacuum spacetimes

TL;DR

This paper introduces a simplified method for calculating multipole moments of axistationary electrovacuum spacetimes, correcting previous errors and applying it to complex solutions like charged, magnetized Kerr and Tomimatsu-Sato models.

Contribution

It presents a more efficient calculation technique for multipole moments, correcting earlier mistakes and extending applicability to new exact solutions.

Findings

Corrected previous results for higher multipole moments in electrovacuum cases.

Developed a direct calculation method for multipole moments of complex solutions.

Applied the method to a 5-parameter charged, magnetized Kerr and Tomimatsu-Sato solution.

Abstract

The multipole moments of stationary axially symmetric vacuum or electrovacuum spacetimes can be expressed in terms of the power series expansion coefficients of the Ernst potential on the axis. In this paper we present a simpler, more efficient calculation of the multipole moments, applying methods introduced by B\"ackdahl and Herberthson. For the non-vacuum electromagnetic case, our results for the octupole and higher moments differ from the results already published in the literature. The reason for this difference is that we correct an earlier unnoticed mistake in the power series solution of the Ernst equations. We also apply the presented method to directly calculate the multipole moments of a 5-parameter charged magnetized generalization of the Kerr and Tomimatsu-Sato exact solutions.