Visualizing Spacetime Curvature via Gradient Flows I: Introduction

This is the first in a series of papers in which the gradient flows of fundamental curvature invariants are used to formulate a visualization of curvature. We start with the construction of strict Newtonian analogues (not limits) of solutions to Einstein's equations based on the topology of the associated gradient flows. We do not start with any easy case. Rather, we start with the Curzon - Chazy solution, which, as history shows, is one of the most difficult exact solutions ...

This paper gives a self-contained, elementary, and largely pictorial statement of Einstein's equation.

The Kerr metric is one of the most important solutions to Einstein's field equations, describing the gravitational field outside a rotating black hole. We thoroughly analyze the curvature scalar invariants to study the Kerr spacetime by examining and visualizing their covariant gradient fields. We discover that the part of the Kerr geometry outside the black hole horizon changes qualitatively depending on the spin parameter, a fact previously unknown. The number of observable...

Einstein's general relativity is the best available theory of gravity. In recent years, spectacular proofs of Einstein's theory have been conducted, which have aroused interest that goes far beyond the narrow circle of specialists. The aim of this work is to offer an elementary introduction to general relativity. In this first part, we introduce the geometric concepts that constitute the basis of Einstein's theory. In the second part we will use these concepts to explore the ...

The Ricci flow is one of the most important topics in differential geometry, and a central focus of modern geometric analysis. In this paper, we give an illustrated introduction to the subject which is intended for a general audience. The goal is to provide a working definition of the Ricci flow as well as some intuition for its behavior without assuming any prerequisite knowledge of differential geometry or topology.

When one splits spacetime into space plus time, the Weyl curvature tensor (vacuum Riemann tensor) gets split into two spatial, symmetric, and trace-free (STF) tensors: (i) the Weyl tensor's so-called "electric" part or tidal field, and (ii) the Weyl tensor's so-called "magnetic" part or frame-drag field. Being STF, the tidal field and frame-drag field each have three orthogonal eigenvector fields which can be depicted by their integral curves. We call the integral curves of t...

We provide an introduction to selected recent advances in the mathematical understanding of Einstein's theory of gravitation.

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 tendiciti...

I present a way to visualize the concept of curved spacetime. The result is a curved surface with local coordinate systems (Minkowski Systems) living on it, giving the local directions of space and time. Relative to these systems, special relativity holds. The method can be used to visualize gravitational time dilation, the horizon of black holes, and cosmological models. The idea underlying the illustrations is first to specify a field of timelike four-velocities. Then, at e...

Four-dimensional Euclidean spaces that solve Einstein's equations are interpreted as WKB approximations to wavefunctionals of quantum geometry. These spaces are represented graphically by suppressing inessential dimensions and drawing the resulting figures in perspective representation of three-dimensional space, some of them stereoscopically. The figures are also related to the physical interpretation of the corresponding quantum processes.