Invariants for Tendex and Vortex Fields

TL;DR

This paper introduces coordinate-invariant quantities derived from tendex and vortex fields, which are based on the eigenvectors and eigenvalues of the Weyl curvature tensor, aiding in the visualization of spacetime curvature.

Contribution

It derives invariants for tendex and vortex fields, providing a way to visualize spacetime curvature independent of coordinate choices.

Findings

Derived explicit invariants for tendex and vortex fields.

Demonstrated invariants in Schwarzschild spacetime slices.

Showed invariants are coordinate-independent and useful for visualization.

Abstract

Tendex and vortex fields, defined by the eigenvectors and eigenvalues of the electric and magnetic parts of the Weyl curvature tensor, form the basis of a recently developed approach to visualizing spacetime curvature. In analogy to electric and magnetic fields, these fields are coordinate-dependent. However, in a further analogy, we can form invariants from the tendex and vortex fields that are invariant under coordinate transformations, just as certain combinations of the electric and magnetic fields are invariant under coordinate transformations. We derive these invariants, and provide a simple, analytical demonstration for non-spherically symmetric slices of a Schwarzschild spacetime.