The spin angular momentum $S$ of an isolated Kerr black hole is bounded by the surface area $A$ of its apparent horizon: $8pi S le A$, with equality for extremal black holes. In this paper, we explore the extremality of individual and common apparent
horizons for merging, rapidly spinning binary black holes. We consider simulations of merging black holes with equal masses $M$ and initial spin angular momenta aligned with the orbital angular momentum, including new simulations with spin magnitudes up to $S/M^2 = 0.994$. We measure the area and (using approximate Killing vectors) the spin on the individual and common apparent horizons, finding that the inequality $8pi S < A$ is satisfied in all cases but is very close to equality on the common apparent horizon at the instant it first appears. We also introduce a gauge-invariant lower bound on the extremality by computing the smallest value that Booth and Fairhursts extremality parameter can take for any scaling. Using this lower bound, we conclude that the common horizons are at least moderately close to extremal just after they appear. Finally, following Lovelace et al. (2008), we construct quasiequilibrium binary-black-hole initial data with overspun marginally trapped surfaces with $8pi S > A$ and for which our lower bound on their Booth-Fairhurst extremality exceeds unity. These superextremal surfaces are always surrounded by marginally outer trapped surfaces (i.e., by apparent horizons) with $8pi S<A$. The extremality lower bound on the enclosing apparent horizon is always less than unity but can exceed the value for an extremal Kerr black hole. (Abstract abbreviated.)
When one splits spacetime into space plus time, the Weyl curvature tensor (vacuum Riemann tensor) gets split into two spatial, symmetric, and trace-free (STF) tensors: (i) the Weyl tensors so-called electric part or tidal field, and (ii) the Weyl ten
sors so-called magnetic part or frame-drag field. Being STF, the tidal field and frame-drag field each have three orthogonal eigenvector fields which can be depicted by their integral curves. We call the integral curves of the tidal fields eigenvectors tendex lines, we call each tendex lines eigenvalue its tendicity, and we give the name tendex to a collection of tendex lines with large tendicity. The analogous quantities for the frame-drag field are vortex lines, their vorticities, and vortexes. We build up physical intuition into these concepts by applying them to a variety of weak-gravity phenomena: a spinning, gravitating point particle, two such particles side by side, a plane gravitational wave, a point particle with a dynamical current-quadrupole moment or dynamical mass-quadrupole moment, and a slow-motion binary system made of nonspinning point particles. [Abstract is abbreviated; full abstract also mentions additional results.]
When one splits spacetime into space plus time, the spacetime curvature (Weyl tensor) gets split into an electric part E_{jk} that describes tidal gravity and a magnetic part B_{jk} that describes differential dragging of inertial frames. We introduc
e tools for visualizing B_{jk} (frame-drag vortex lines, their vorticity, and vortexes) and E_{jk} (tidal tendex lines, their tendicity, and tendexes), and also visualizations of a black-hole horizons (scalar) vorticity and tendicity. We use these tools to elucidate the nonlinear dynamics of curved spacetime in merging black-hole binaries.
