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 introduce 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.
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