  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 formulae, 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 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. This paper serves as an overview and as an introduction to this
approach.
