A major impetus to develop Geometric Calculus has been improvement of
General Relativity. The ultimate integration of gravity theory and
quantum mechanics should be facilitated by the development of a
suitable common language. Work toward that goal began with the DH
doctoral dissertation (1963) and continued through his 1963-64
postdoctoral fellowship with John Wheeler at Princeton. The first
result was published in Space-Time Algebra
(1966). The next result was a product of the extensive development of
new mathematical tools for differential geometry in CA to GA.
The first two papers listed below apply GC to General Relativity; they
were adapted fom a chapter written for CA to GC
in 1976 but dropped when applications were excluded from the book. The
third paper was written in 1993-94 during a DH sabbatical at Canvendish
Laboratory. It was intended for publication in the Journal of
Mathematical Physics,
but the editor rejected it as unsuitable without consulting reviewers.
Publication in another journal was not pursued, as that was likely to
require extensive rewriting. Anyway, the present Web publication puts
it in a more helpful context for readers. Many papers on the new Gauge
Theory of Gravity using Geometric Calculus are posted at the Cavendish Web site.
Abstract: A new method for calculating
the curvature
tensor is developed and applied to the Schwarzschild case. The method
employs Clifford algebra and has definite advantages over conventional
methods using differential forms or tensor analysis.
D. Hestenes, Int. J. Theo.
Phys., 25,
1986, 581-588.
Abstract: The translational and
rotational equations of
motion for a small rigid body in a gravitational field are combined in
a single spinor equation. Besides its computational advantages, this
unifies the description of gravitational interaction in classical and
quantum theory. Explicit expressions for gravitational precession rates
are derived.
Abstract: A new gauge theory of gravitation
on flat spacetime has been formulated 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.
Abstract:
A new gauge theory of gravity on flat spacetime
has recently been developed by Lasenby, Doran, and Gull. Einstein's
principles of equivalence and general relativity are replaced by gauge
principles asserting, respectively, local rotation and global
displacement gauge invariance. A new unitary formulation of Einstein's
tensor leads to resolution of long-standing problems with
energymomentum conservation in general relativity. Geometric
calculus provides many simplifications and fresh
insights in theoretical formulation and physical applications of the
theory.
D. Hestenes, Foundations of Physics,
35(6):903-970
(2005).
Abstract: Geometric Calculus is developed for
curved-space
treatments of General Relativity and comparison 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.
D. Hestenes, To be published in the Preceedings
of the Seventh International Conference on Clifford Algebra
