The Series Solution to the Metric of Stationary Vacuum with Axisymmetry

TL;DR

This paper introduces a simplified series solution method for stationary vacuum axisymmetric spacetimes, improving practical resolution of Einstein's equations by using a new approach to multipole moments and parameter identification.

Contribution

It develops a straightforward procedure for generating series solutions and a method to identify free parameters, enhancing the application of multipole moments in solving Einstein's equations.

Findings

Successfully applied to known examples, matching series solutions with established metrics.

Provides a practical alternative to recursive tensor definitions of multipole moments.

Facilitates easier resolution of Einstein's field equations for stationary vacuum spacetimes.

Abstract

The multipole moments method is not only an aid to understand the deformation of the space-time, but also an effective tool to solve the approximate solutions of the Einstein field equation. However, The usual multipole moments are recursively defined by a sequence of symmetric and trace-free tensors, which are inconvenient for practical resolution. In this paper, we develop a simple procedure to generate the series solutions, and propose a method to identify the free parameters by taking the Schwarzschild metric as a standard ruler. Some well known examples are analyzed and compared with the series solutions.