Schwarzschild and Kerr solutions of Einstein’s field equation: An Introduction

Abstract

Starting from Newton's gravitational theory, we give a general introduction into the spherically symmetric solution of Einstein's vacuum field equation, the Schwarzschild(-Droste) solution, and into one specific stationary axially symmetric solution, the Kerr solution. The Schwarzschild solution is unique and its metric can be interpreted as the exterior gravitational field of a spherically symmetric mass. The Kerr solution is only unique if the multipole moments of its mass and its angular momentum take on prescribed values. Its metric can be interpreted as the exterior gravitational field of a suitably rotating mass distribution. Both solutions describe objects exhibiting an event horizon, a frontier of no return. The corresponding notion of a black hole is explained to some extent. Eventually, we present some generalizations of the Kerr solution.

The Dirac point electron in zero-gravity Kerr–Newman spacetime

“…He furthermore showed that this topology also survives the vanishing-mass limit of the Kerr manifold, which yields an otherwise flat vacuum spacetime consisting of two static spacetime ends which are cross-linked through the ring. This vanishing-mass limit of the Kerr manifold coincides with the vanishing-mass limit of Zipoy's oblate spheroidal family of static vacuum spacetimes 5 In the black-hole sector of their parameter space the Kerr-Newman spacetimes also have a Cauchy horizon, an event horizon, and an ergosphere horizon; see [23,34,24]. From the "safe perspective of an observer at spatial infinity" the ring singularity, the acausal region, and the Cauchy horizon are invisible, being "hidden" behind the event horizon, and no exotic or even objectionable physics would ever seem to happen: a Dirac spinor wavefunction initially supported outside the event horizon will either keep spreading within the outer region or eventually (as t → ∞) accumulate (in parts or wholly) at the event horizon, see [19,20].…”

Editorial for the Special Issue 100 Years of Chronogeometrodynamics: The Status of the Einstein’s Theory of Gravitation in Its Centennial Year

“…Instead, the properties of the event horizon were different with rotation taken into account. A comparison of the peculiar features of the Schwarzschild and the Kerr solutions can be found in [166].…”

A new black hole solution in dilaton gravity inspired by power-law electrodynamics

“…One property of these solutions is that if we set N = 1, the slowly rotating Kerr metric(Lense-Thirring metric) recovers. Also, by setting a = 0 and p = 1 the Reissner-Nordstrom metric recovers, too 28 .…”