Visualizing spacetime curvature via frame-drag vortexes and tidal tendexes: General theory and weak-gravity applications

Abstract

When one splits spacetime into space plus time, the Weyl curvature tensor (vacuum Riemann tensor) gets split into two spatial, symmetric, and trace-free tensors: (i) the Weyl tensor's so-called electric part or tidal field Ɛ_(jk), which raises tides on the Earth's oceans and drives geodesic deviation (the relative acceleration of two freely falling test particles separated by a spatial vector ξ^k is Δa_j=-Ɛ_(jk)ξ^k), and (ii) the Weyl tensor's so-called magnetic part or (as we call it) frame-drag field B_(jk), which drives differential frame dragging (the precessional angular velocity of a gyroscope at the tip of ξ^k, as measured using a local inertial frame at the tail of ξ^k, is ΔΩ_j=B_(jk)ξ^k). Being symmetric and trace-free, Ɛ_(jk) and B_(jk) each have three orthogonal eigenvector fields which can be depicted by their integral curves. We call the integral curves of Ɛ_(jk)'s eigenvectors tidal tendex lines or simply tendex lines, we call each tendex line's eigenvalue its tendicity, and we give the name tendex to a collection of tendex lines with large tendicity. The analogous quantities for B_(jk) are frame-drag vortex lines or simply vortex lines, their vorticities, and their vortexes. These concepts are powerful tools for visualizing spacetime curvature. We build up physical intuition into them 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. We show that a rotating current quadrupole has four rotating vortexes that sweep outward and backward like water streams from a rotating sprinkler. As they sweep, the vortexes acquire accompanying tendexes and thereby become outgoing current-quadrupole gravitational waves. We show similarly that a rotating mass quadrupole has four rotating, outward-and-backward sweeping tendexes that acquire accompanying vortexes as they sweep, and become outgoing mass-quadrupole gravitational waves. We show, further, that an oscillating current quadrupole ejects sequences of vortex loops that acquire accompanying tendex loops as they travel, and become current-quadrupole gravitational waves; and similarly for an oscillating mass quadrupole. And we show how a binary's tendex lines transition, as one moves radially, from those of two static point particles in the deep near zone, to those of a single spherical body in the outer part of the near zone and inner part of the wave zone (where the binary's mass monopole moment dominates), to those of a rotating quadrupole in the far wave zone (where the quadrupolar gravitational waves dominate). In Paper II we will use these vortex and tendex concepts to gain insight into the quasinormal modes of black holes, and in subsequent papers, by combining these concepts with numerical simulations, we will explore the nonlinear dynamics of curved spacetime around colliding black holes. We have published a brief overview of these applications in R. Owen et al. Phys. Rev. Lett. 106 151101 (2011). We expect these vortex and tendex concepts to become powerful tools for general relativity research in a variety of topics.

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