Visualizing spacetime curvature via gradient flows. I. Introduction

Traditional approaches to the study of the dynamics of spacetime curvature in a very real sense hide the intricacies of the nonlinear regime. Whether it be huge formulas, or mountains of numerical data, standard methods of presentation make little use of our remarkable skill, as humans, at pattern recognition. Here we introduce a new approach to the visualization of spacetime curvature. We examine the flows associated with the gradient fields of scalar invariants derived from the spacetime. These flows reveal a remarkably rich structure, and offer fresh insights, even for well-known analytical solutions to Einstein’s equations. The intent, however, is to go beyond idealized analytical solutions and eventually consider physically realistic situations. This requires a careful analysis of exactly which invariants that can actually be used in this approach. The present analysis serves as an overview and as an introduction to this program.

Figure 1

The function $m^4\mathcal{I}$ for the Schwarzschild spacetime.

Figure 2

The phase portrait for $k_{\alpha }$ for the Schwarzschild spacetime. The region below $r=0$ has been removed.