Visualizing Spacetime Curvature via Gradient Flows III: The Kerr Metric and the Transitional Values of the Spin Parameter

TL;DR

This paper visualizes the gradient fields of curvature invariants in Kerr spacetime, revealing that the number of critical points changes at specific spin parameter values, indicating fundamental geometric transitions.

Contribution

It uncovers the existence of transitional spin values in Kerr black holes where the topology of curvature gradient fields changes, a novel insight into spacetime structure.

Findings

Number of critical points varies with spin parameter

Transitions occur at specific, fundamental spin values

Curvature invariants reflect tidal and frame-dragging effects

Abstract

The Kerr metric is one of the most important solutions to Einstein's field equations, describing the gravitational field outside a rotating black hole. We thoroughly analyze the curvature scalar invariants to study the Kerr spacetime by examining and visualizing their covariant gradient fields. We discover that the part of the Kerr geometry outside the black hole horizon changes qualitatively depending on the spin parameter, a fact previously unknown. The number of observable critical points of the curvature invariants' gradient fields along the axis of rotation changes at several transitional values of the spin parameter. These transitional values are a fundamental property of the Kerr metric. They are physically important since in general relativity these curvature invariants represent the cumulative tidal and frame-dragging effects of rotating black holes in an observer-independent way.