Visualizing spacetime curvature via gradient flows. III. The Kerr metric and the transitional values of the spin parameter

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.

Figure 1

Three-dimensional visualization of rotating black holes in Kerr coordinates via flow lines of the vector field $k_{\mu }\equiv -{\nabla }_{\mu }(w1R)$ for a dimensionless spin parameter value of $A=0.9$. The transparent sky-blue surface is the ergosphere, and the horizon is the black surface. The black circles are the critical points of the gradient field. The colors of the flow lines are of no physical significance and used to distinguish the different regions of the spacetime with common flow structure separated by asymptotic critical directions of the field represented by the brown surfaces, which connect the critical points to the ring singularity.

Figure 2

(a) Two-dimensional slice of Fig. 1 in Kerr-like coordinates introduced in Eq. (3). The ergosphere is omitted, and different horizons are plotted in grey for the corresponding transitional values of $A$ indicated in the small boxes, in addition to $A=0.27$, which is not a transitional value but is plotted for presentation purposes. (b), (c), and (d) are the same as (a) but for the gradient vector fields of $w2R$, $w1I$, and $w2I$, respectively.

Figure 3

The outer horizon $r_{+}$ (black) and the inner horizon $r_{-}$ (brown) of a rotating black hole vs its dimensionless spin parameter $A$. The transitional values of $A$ are indicated by the red, blue, green, and yellow points associated with the horizons crossing one of the critical points of the invariant $w1R$, $w2R$, $w1I$, and $w2I$, respectively.