Invariants for Tendex and Vortex Fields

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.