The Kerr spacetime: A brief introduction

TL;DR

This chapter provides a concise, coordinate-focused introduction to the Kerr spacetime and rotating black holes, clarifying how horizons and ergospheres arise and how the geometry is represented across multiple coordinate systems. It surveys Kerr’s original, Kerr–Schild Cartesian, Boyer–Lindquist, rational polynomial, and Doran coordinates, emphasizing the trade-offs and computational benefits of each view. Key results include the ring curvature singularity at $r=0$,\,$\theta=\pi/2$, the outer and inner horizons at $r_\pm=m\pm\sqrt{m^2-a^2}$, and the ergosphere defined by the stationary limit surfaces $g_{tt}=0$, together with the Killing-vector structure and horizon rigidity. The discussion highlights that there is no general Birkhoff theorem for rotating spacetimes and positions Kerr as the central exact solution for stationary rotating black holes, guiding readers toward the broader literature on strong-field gravity and black hole physics.

Abstract

This chapter provides a brief introduction to the Kerr spacetime and rotating black holes, touching on the most common coordinate representations of the spacetime metric and the key features of the geometry -- the presence of horizons and ergospheres. The coverage is by no means complete, and serves chiefly to orient oneself when reading subsequent chapters.

Figures (1)

• Figure 1: Schematic location of the horizons, ergosurfaces, and curvature singularity in the Kerr spacetime.