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

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