Kerr and Schwarzschild black-holes in three-dimensional relativity

Abstract

We consider the Kerr and Schwarzschild black-hole space–times in the framework of a three-dimensional formulation of relativistic kinematics and field dynamics, in which local physical observers are represented by non-singular vector fields of bounded length in a three-dimensional pseudo-Riemannian space. A space–time is represented by a pair of 3-metric and a fundamental 3-vector field satisfying a set of basic equations, each solution of which determines uniquely a solution of the vacuum Einstein field equations. It is shown that the only spherically symmetric solution of our basic equations leads to the Schwarzschild space–time, thereby proving a version of Birkoff’s theorem in this formalism. The Schwarzschild horizon and the Kerr stationary limit are both related to the upper bound of the length of the corresponding physical 3-vector fields. An example of a solution of the vacuum Einstein field equations shows that nonstationary space–times can also be formulated in this three-dimensional relativity.