Abstract

In this paper, we investigate a certain property of curvature which differs in a remarkable way between Lorentz geometry and Euclidean geometry. In a certain sense, it turns out that rotating topological objects may have less curvature (as measured by integrating the square of the scalar curvature) than non-rotating ones. This is a consequence of the indefinite metric used in relativity theory. The results in this paper are mainly based of computer computations, and so far there is no satisfactory underlying mathematical theory. Some open problems are presented.

Highlights

  • The purpose of this paper is to draw attention to a certain property of curvature which differs in a remarkable way between Lorentz geometry and Euclidean geometry

  • The original motivation for this mathematical problem comes from physics, but the author does not make any claims about implications in the other direction, except that any kind of deeper understanding of Lorentz geometry may eventually turn out to be useful for uniting general relativity and quantum mechanics

  • Is it a common fact of Lorentz geometry that rotating topological objects may have smaller total curvature in this sense than the corresponding non-rotating ones? In particular, one may wonder if topological objects which minimize the integral of the square of the scalar curvature are always rotating in some sense

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Summary

Introduction

The purpose of this paper is to draw attention to a certain property of curvature which differs in a remarkable way between Lorentz geometry and Euclidean geometry. Is it a common fact of Lorentz geometry that rotating topological objects may have smaller total curvature in this sense than the corresponding non-rotating ones? One may wonder if topological objects which minimize the integral of the square of the scalar curvature are always rotating in some sense To make this question somewhat more precise, we will in this paper consider minimizing under the extra condition that the total space-time volume is fixed. This is a global property of the scalar curvature, rather than a local one

An Example of a Wormhole
Quasi-Rotations
Curvature and Periodicity
A Four-Dimensional Metric for the Rotating Wormhole
Computation of the Curvature Integral
The Euclidian Case
Open Questions
Full Text
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