Visualizing Spacetime Curvature via Frame-Drag Vortexes and Tidal Tendexes II. Stationary Black Holes

TL;DR

This paper visualizes the spacetime curvature around stationary black holes using tendex and vortex lines, revealing their properties and behaviors on horizons and how they are affected by different slicings and coordinates.

Contribution

It introduces a detailed visualization method for spacetime curvature around stationary black holes, including horizon tendicity and vorticity, and analyzes their dependence on slicing and coordinates.

Findings

Horizon tendicity proportional to intrinsic scalar curvature

Horizon vorticity proportional to extrinsic scalar curvature

Entities are weakly affected by slicing and coordinate changes

Abstract

When one splits spacetime into space plus time, the Weyl curvature tensor (which equals the Riemann tensor in vacuum) splits into two spatial, symmetric, traceless tensors: the tidal field $E$, which produces tidal forces, and the frame-drag field $B$, which produces differential frame dragging. In recent papers, we and colleagues have introduced ways to visualize these two fields: tidal tendex lines (integral curves of the three eigenvector fields of $E$) and their tendicities (eigenvalues of these eigenvector fields); and the corresponding entities for the frame-drag field: frame-drag vortex lines and their vorticities. These entities fully characterize the vacuum Riemann tensor. In this paper, we compute and depict the tendex and vortex lines, and their tendicities and vorticities, outside the horizons of stationary (Schwarzschild and Kerr) black holes; and we introduce and depict the black holes' horizon tendicity and vorticity (the normal-normal components of $E$ and $B$ on the horizon). For Schwarzschild and Kerr black holes, the horizon tendicity is proportional to the horizon's intrinsic scalar curvature, and the horizon vorticity is proportional to an extrinsic scalar curvature. We show that, for horizon-penetrating time slices, all these entities ($E$, $B$, the tendex lines and vortex lines, the lines' tendicities and vorticities, and the horizon tendicities and vorticities) are affected only weakly by changes of slicing and changes of spatial coordinates, within those slicing and coordinate choices that are commonly used for black holes. [Abstract is abbreviated.]