Introduction

This introductory chapter provides a quick review of the basic concepts of general relativity relevant to this work. The main object of Albert Einstein's general relativity is the spacetime. The nonlinear stability of the Kerr family is one of the most pressing issues in mathematical general relativity today. Roughly, the problem is to show that all spacetime developments of initial data sets, sufficiently close to the initial data set of a Kerr spacetime, behave in the large like a (typically another) Kerr solution. This is not only a deep mathematical question but one with serious astrophysical implications. Indeed, if the Kerr family would be unstable under perturbations, black holes would be nothing more than mathematical artifacts. The goal of this book is to prove the nonlinear stability of the Schwarzschild spacetime under axially symmetric polarized perturbations, namely, solutions of the Einstein vacuum equations for asymptotically flat 1 + 3 dimensional Lorentzian metrics which admit a hypersurface orthogonal spacelike Killing vectorfield Z with closed orbits.

The Kerr solution

This chapter covers the Kerr metric, which is an exact solution of the Einstein vacuum equations. The Kerr metric provides a good approximation of the spacetime near each of the many rotating black holes in the observable universe. This chapter shows that the Einstein equations are nonlinear. However, there exists a class of metrics which linearize them. It demonstrates the Kerr–Schild metrics, before arriving at the Kerr solution in the Kerr–Schild metrics. Since the Kerr solution is stationary and axially symmetric, this chapter shows that the geodesic equation possesses two first integrals. Finally, the chapter turns to the Kerr black hole, as well as its curvature singularity, horizons, static limit, and maximal extension.

A Rosen-type bi-metric universe and its physical properties

The paper studies a spacetime endowed with two stationary metrics. The first one is a Riemannian one, called the R-Schwarzschild metric. It satisfies Einstein vacuum field equations, describes correctly the slowdown of clocks in the gravitational field, the orbits of the planets and the perihelion drift. The R-Schwarzschild metric can be seen as the basic texture of the spacetime. All objects having mass are ruled by this Riemannian metric. The second metric, the light-adapted one, is deduced both taking into account the Rosen-type bi-metric compatibility condition and by the preservation of the axiom of the speed of the light limit. This second metric offers the texture of the “light-like” objects. The main “normal” surprise is that this metric can be only the classical Schwarzschild metric. So, a Rosen-type bi-metric universe exists and its properties are in accordance with the experimental physical evidences.