Visualizing Spacetime Curvature via Frame-Drag Vortexes and Tidal Tendexes I. General Theory and Weak-Gravity Applications

TL;DR

This paper introduces a visualization method for spacetime curvature using tendex and vortex lines, providing physical intuition for gravitational phenomena through eigenvector field analysis.

Contribution

It presents a novel framework for visualizing and understanding spacetime curvature via tendex and vortex lines, applicable to weak-gravity scenarios.

Findings

Visualized tidal and frame-drag fields using tendex and vortex lines.

Applied concepts to various weak-gravity phenomena, including gravitational waves.

Provided physical intuition for spacetime curvature effects.

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 (STF) tensors: (i) the Weyl tensor's so-called "electric" part or tidal field, and (ii) the Weyl tensor's 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 field's eigenvectors 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 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.]