  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.
