Assumptions and Axioms: Mathematical Structures to Describe the Physics of Rigid Bodies

TL;DR

This paper explores the foundational assumptions behind the mathematics of rigid bodies and spacetime, proposing Clifford algebras as a natural framework for describing physical properties and geometric structures.

Contribution

It introduces a new perspective on the mathematical structures underlying physics, linking rigid body assumptions to Clifford algebras and spacetime geometry.

Findings

Rigid body concepts are intrinsically linked to space homogeneity.

Clifford algebras naturally describe rotations and spacetime properties.

Non-invertible elements in Clifford algebras reflect physical spacetime features.

Abstract

This paper challenges some of the common assumptions underlying the mathematics used to describe the physical world. We start by reviewing many of the assumptions underlying the concepts of real, physical, rigid bodies and the translational and rotational properties of such rigid bodies. Nearly all elementary and advanced texts make physical assumptions that are subtly different from ours, and as a result we develop a mathematical description that is subtly different from the standard mathematical structure. Using the homogeneity and isotropy of space, we investigate the translational and rotational features of rigid bodies in two and three dimensions. We find that the concept of rigid bodies and the concept of the homogeneity of space are intrinsically linked. The geometric study of rotations of rigid objects leads to a geometric product relationship for lines and vectors. By requiring this product to be both associative and to satisfy Pythagoras' theorem, we obtain a choice of Clifford algebras. We extend our arguments from space to include time. By assuming that ct=l and rewriting this in Lorentz invariant form as c^2t^2-x^2-y^2-z^2=0 we obtain a generalization of Pythagoras to spacetime. This leads us directly to establishing that the Clifford algebra CL(1,3) is an appropriate mathematical structure to describe spacetime. Clifford algebras are not division algebras. We show that the existence of non-invertible elements in the algebra is not a limitation of the usefulness to physics of the algebra but rather that it reflects accurately the spacetime properties of physical systems.