A Pictorial History of some Gravitational Instantons

Summary

Abstract

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 threedimensional space, some of them stereoscopically. The figures are also related to the physical interpretation of the corresponding quantum processes.

Introduction

Understanding General Relativity means to a large extent coming to terms with its most important ingredient, geometry. Among his many contributions, Charlie has given us new variations of this theme [1], fascinating because geometry is so familiar on two-dimensional surfaces, but so remote from intuition on higher-dimensional spacetimes. The richness he uncovered is shown nowhere better than in the 137 figures of his masterful text [2].

Today quantum gravity [3] leads to new geometrical features. One of these is a new role for Riemannian (rather than Lorentzian) solutions of the Einstein field equations: such “instantons” can describe in WKB approximation the tunneling transitions that are classically forbidden, for example because they correspond to a change in the space's topology. In order to gain a pictorial understanding of these spaces we can try to represent the geometry as a whole with less important dimensions suppressed; an alternative is to follow the ADM method and show a history of the tunneling by slices of codimension one.

We can readily go from equation to picture thanks to computer plotting routines, from the simpler ones as incorporated in spreadsheet programs [4] to the more powerful versions of Mathematica.