Spacetime Geometry with Geometric Calculus
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Advances in Applied Clifford Algebras
Spacetime Geometry with Geometric Calculus David Hestenes∗ Communicated by Jayme Vaz Abstract. Geometric Calculus is developed for curved-space treatments of General Relativity and comparison is made with the flat-space gauge theory approach by Lasenby, Doran and Gull. Einstein’s Principle of Equivalence is generalized to a gauge principle that provides the foundation for a new formulation of General Relativity as a Gauge Theory of Gravity on a curved spacetime manifold. Geometric Calculus provides mathematical tools that streamline the formulation and simplify calculations. The formalism automatically includes spinors so the Dirac equation is incorporated in a geometrically natural way. Mathematics Subject Classification. Primary 53C07, 35Q76; Secondary 58A05. Keywords. Principle of equivalence, Gauge Theory Gravity, Differential manifolds.
1. Introduction Using spacetime algebra [7,10] in an essential way, Cambridge physicists Lasenby, Doran and Gull have created an impressive new Gauge Theory of Gravity (GTG) based on flat spacetime [1,15]. In my opinion, GTG is a huge improvement over the standard tensor treatment of Einstein’s theory of General Relativity (GR), both in conceptual clarity and in computational power [11]. However, as the prevailing preference among physicists is for a curved-space version of GR, a debate about the relative merits of flat-space and curved-space versions will no doubt be needed to change the minds of many. This paper aims to contribute to that debate by providing a conceptual and historical bridge between curved and flat space theories couched in the unifying language of geometric algebra. ∗ Corresponding
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D. Hestenes
Adv. Appl. Clifford Algebras
This article sketches the extension of geometric algebra to a geometric calculus (GC) that includes the tools of differential geometry needed for a curved-space version of GR. My purpose is to demonstrate the unique geometrical insight and computational power that GC brings to GR, and to introduce mathematical tools that are ready for use in research and teaching [21]. I presume that the reader has some familiarity with standard treatments of GR as well as with geometric algebra as presented in any of the above references, so certain concepts, notations and results developed there are taken for granted here. Additional mathematical tools introduced herein are sufficient to treat any topic in GR with GC. This article introduces three different formulations of GR in terms of a unified GC that integrates them into a system of alternative approaches. The first is a coordinate-based formulation that facilitates translation to and from the standard tensor formulation of GR [25]. The second is a deeper gauge theory formulation that is the main concern of this paper. The third is an embedding formulation that deserves mention but will not be elaborated here. Although our focus is on GR, it should be recognized that the mathematical tools of GC are applicable to any probl
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