Schwarzschild and Kerr Solutions of Einstein's Field Equation -- an introduction

TL;DR

This paper introduces the Schwarzschild and Kerr solutions of Einstein's field equations, explaining their significance in describing black holes and rotating masses, with some generalizations discussed.

Contribution

It provides an accessible introduction to key exact solutions of Einstein's equations, highlighting their physical interpretations and recent generalizations.

Findings

Schwarzschild solution describes spherically symmetric vacuum spacetime.

Kerr solution models rotating black holes with specific multipole moments.

Both solutions feature event horizons and are fundamental in black hole physics.

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.