SPACETIME CALCULUS for GRAVITATION THEORY

Short Description
SPACETIME CALCULUS for GRAVITATION THEORY. David Hestenes. Arizona State University. Abstract. A new gauge theory of gravitation on flat spacetime has …

Website: modelingnts.la.asu.edu | Filesize: 391kb

Content
SPACETIME CALCULUS for GRAVITATION THEORY
David Hestenes
Arizona State University
Abstract
A new gauge theory of gravitation on flat spacetime has recently been
developed by Lasenby, Doran, and Gull in the language of Geometric
Calculus. This paper provides a systematic account of the mathematical
formalism to facilitate applications and extensions of the theory. It
includes formulations of differential geometry, Lie derivatives and integrability
theorems which are coordinate-free and gauge-covariant. Emphasis
is on use of the language to express physical and geometrical
concepts.
CONTENTS
Part I. Mathematical Fundamentals
1. Spacetime Algebra
2. Vector Derivatives and Differentials
3. Linear Algebra
4. Tensors and their Classification
5. Transformations on Spacetime
6. Directed Integrals and the Fundamental Theorem
Part II. Induced Geometry on Flat Spacetime
7. Gauge Tensor and Gauge Invariance
8. Covariant Derivatives and Parallel Transport
9. From Gauge to Connexion
10. Coderivatives and Integrals
11. Curvature
Part III. Spacetime Flows
12. Gauge Covariant Flows and Flow Derivatives
13. Integrability and Coordinates
14. Flow Dynamics and Deformations
1996 PACS: 04.20.Cv, 11.15.-q, 02.40.-k, 11.30.Cp.
1Introduction
Lasenby, Doran and Gull have recently created a powerful coordinate-free reformulation,
refinement, and extension of general relativity [1,2]. It is a gauge theory on flat spacetime,
but it retains the attractive geometric structure of Einstein’s theory. The mathematical
formalism which they employ comes mostly from [3], with additional pieces from [4-8]
and elsewhere. It is not easy, however, to assimilate and adapt the formalism from these
references. Although much of that has already been done in [1], further discussion and
analysis will help make the whole approach more accessible. Also, the emphasis here is
more strongly focused on notions of differential geometry in formulating the theory. Indeed,
the method amounts to a new approach to differential geometry which could fairly be called
gauge geometry.
The general mathematical formalism is called geometric calculus or, when it refers specifically
to spacetime, spacetime calculus. This paper presents a systematic account of spacetime
calculus for the purposes of gravitation theory. It is divided into three parts.
Part I reviews the fundamentals of spacetime calculus with emphasis on those relevant
to gravitation theory. Many of the proofs are omitted, since they can easily be supplied or
found in…