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 tensors 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.]
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