Constructing vacuum spacetimes by generating manifolds of revolution around a curve

TL;DR

This paper develops a perturbative method to construct vacuum spacetimes by generating manifolds of revolution around a curve, applying it to Schwarzschild and Kerr metrics, revealing insights into their geometric and physical properties.

Contribution

It introduces a novel perturbative approach to generate vacuum solutions from manifolds of revolution, providing new geometric insights into Schwarzschild and Kerr spacetimes.

Findings

First order solutions are gauge transformations of Schwarzschild.

Existence of solutions with signature-changing metrics and closed timelike curves.

Slowly rotating solutions correspond to the Kerr metric.

Abstract

We develop a general perturbative analysis on vacuum spacetimes which can be constructed by generating manifolds of revolution around a curve, and apply it to the Schwarzschild metric. The following different perturbations are carried out separately: 1) Non-rotating 2-spheres are added along a plane curve slightly deviated from the ``Schwarzschild line''; 2) General non-rotating topological 2-spheres are added along the ``Schwarzschild line'' 3) Slow-rotating 2-spheres are added along the ``Schwarzschild line''. For (1), we obtain the first order vacuum solution and show that no higher order solution exists. This linearised vacuum solution turns out however to be just a gauge transformation of the Schwarzschild metric. For (2), we solve the general linearised vacuum equations under several special cases. In particular, there exist linearised vacuum solutions with signature-changing metrics that contain closed timelike curves (though these do not correspond to adding topological 2-spheres). For (3), we find that the first order vacuum solution is equivalent to the slowly rotating Kerr metric. This is hence a much simpler and geometrically insightful derivation as compared to the gravitomagnetic one, where this rotating-shells construction is a direct manifestation of the frame-dragging phenomenon. We also show that the full Kerr however, cannot be obtained via adding rotating ellipsoids.