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On the value of geometric algebra for spacetime analyses using an investigation of the form of the self-force on an accelerating charged particle as a case study

### Abstract

The ability to treat vectors in classical mechanics and classical electromagnetism as single geometric objects rather than as a set of components facilitates physical understanding and theoretical analysis. To do the same in four-dimensional spacetime calculations requires a generalization of the vector cross product. Geometric algebra provides such a generalization and is much less abstract than exterior forms. It is shown that many results from geometric algebra are useful for spacetime calculations and can be presented as simple extensions of conventional vector algebra. As an example, it is shown that geometric algebra tightly constrains the possible forms of the self-force that an acceleratingcharged particle experiences and predicts the Lorentz–Abraham–Dirac equation of motion up to a constant of proportionality. Geometric algebra also makes the important physical content of the Lorentz–Abraham–Dirac equation more transparent than does the standard tensor form of this equation, thus allowing a proposed modification to this equation free from the problems of preacceleration and runaway motion to be easily predicted.

© 2010 American Association of Physics Teachers

Received 17 March 2009
Accepted 28 October 2009
Published online 12 January 2010

Article outline:

I. INTRODUCTION
II. PRELIMINARIES
A. Geometric algebra background and results
B. Spacetime conventions and formulas
C. Space and time components of vectors using geometric algebra
III. PARTICLE EQUATIONS OF MOTION USING GEOMETRIC ALGEBRA
IV. PREDICTING THE FORM OF THE SELF-FORCE
A. The Lorentz–Abraham–Dirac equation of motion
B. Modified Lorentz–Abraham–Dirac equation
C. Further modifications of the Lorentz–Abraham–Dirac equation
V. CONCLUSIONS

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2010-01-12

2016-10-01

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