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.
 Publication:

International Journal of Modern Physics D
 Pub Date:
 December 2015
 DOI:
 10.1142/S0218271815300062
 arXiv:
 arXiv:1503.02172
 Bibcode:
 2015IJMPD..2430006H
 Keywords:

 General relativity;
 Kerr and Schwarzschild solutions;
 black holes;
 gravitoelectromagnetism;
 torsion;
 04.50.h;
 02.70.Wz;
 01.65.+g;
 01.30.Rr;
 04.20.q;
 04.70.Bw;
 Higherdimensional gravity and other theories of gravity;
 Symbolic computation;
 History of science;
 Surveys and tutorial papers;
 resource letters;
 Classical general relativity;
 Classical black holes;
 General Relativity and Quantum Cosmology
 EPrint:
 96 pages, 17 figures, pdflatex. Invited review article. To appear in WeiTou Ni (editor) "One Hundred Years of General Relativity: Cosmology and Gravity," World Scientific, Singapore (2015)