Penrose's singularity theorem and the Kerr space-time

TL;DR

The paper investigates how Penrose's singularity theorem applies to Kerr spacetime, clarifying that the theorem requires global hyperbolicity and a trapped surface. In the maximally extended Kerr spacetime these conditions fail, so the theorem does not force a singularity there; however, in the non-extended region up to the inner horizon $r_-$ the hypotheses can hold and yield incomplete future-directed null geodesics even though scalar curvature invariants remain finite, exemplified by a detailed Kerr-Schild analysis and geodesic ray tracing. A concrete calculation with $M=1$, $2a=M$, and $r_-\approx0.13398$, $r_+\approx1.86603$ shows a trapped surface with positive null-convergences and a geodesic crossing $r_-$ at finite affine parameter, while the Kretschmann scalar remains $I=48/r^6$ on the equatorial plane for $r>0$. The results illustrate that geodesic incompleteness does not necessarily indicate a physical curvature singularity and discuss the Cauchy-horizon nature and potential instabilities of the Kerr interior under perturbations.

Abstract

In this short paper, Penrose's famous singularity theorem is applied to the Kerr space-time. In the case of the maximally extended space-time, the assumptions of Penrose's singularity theorem are not satisfied as the space-time is not globally hyperbolic. In the case of the unextended space-time -- defined up to some radius between the inner and outer event horizons -- the assumptions of the theorem hold, but scalar curvature invariants remain finite everywhere. Calculations are done in detail showcasing the validity of the theorem, and misconceptions regarding the characterization of physical singularities by incomplete null geodesics are discussed.

Figures (5)

• Figure 1: The usual Penrose-Carter diagram of a Schwarzschild black hole, with a Cauchy hypersurface drawn in blue across the horizontal.
• Figure 2: A piece of a Penrose-Carter diagram of the maximal analytic extension of the Kerr black hole for $\theta=\pi/2$, extended beyond the inner Cauchy horizon in region IV. The future inner Cauchy horizon is drawn in green, timelike singularities in red and a spacelike hypersurface drawn in blue across the horizontal. The boundary of the domain of dependence of a point $p$ inside the inner Cauchy horizon is drawn in magenta.
• Figure 3: A Penrose-Carter diagram of a Kerr black hole for $\theta=\pi/2$ up to the future inner horizon drawn in green, with a Cauchy surface drawn in blue across the horizontal.
• Figure 4: The convergences $\rho,\,\rho'$ for outgoing and ingoing rays respectively, plotted against $\theta$ for a choice of $a=1/2$, $M=1$ in a null tetrad adapted to the 2-surface $\mathcal{S}$.
• Figure 5: The position of the ingoing null geodesic as a function of $\lambda$.

Theorems & Definitions (6)

• Theorem 2.1
• Definition 2.2: Geodesics
• Definition 2.3: Complete geodesics
• Definition 2.4: Inextendible geodesics
• Definition 2.5: Incomplete geodesics
• Definition 2.6: Trapped surface