Generating Solutions to the Einstein Field Equations

TL;DR

This paper introduces a method to generate new exact solutions to the Einstein field equations from known solutions using symmetry transformations in a potential space, focusing on stationary axisymmetric vacuum solutions.

Contribution

It presents a novel framework utilizing symmetries in a potential space to systematically generate new Einstein field solutions from seed solutions.

Findings

Derived new stationary axisymmetric vacuum solutions

Demonstrated the use of continuous symmetry transformations

Provided solutions of potential physical and geometrical interest

Abstract

Exact solutions to the Einstein field equations may be generated from already existing ones (seed solutions), that admit at least one Killing vector. In this framework, a space of potentials is introduced. By the use of symmetries in this space, the set of potentials associated to a known solution are transformed into a new set, either by continuous transformations or by discrete transformations. In view of this method, and upon consideration of continuous transformations, we arrive at some exact, stationary axisymmetric solutions to the Einstein field equations in vacuum, that may be of geometrical or/and physical interest.