Constructing vacuum spacetimes by generating manifolds of revolution around a curve

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.