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Generalized Minkowski spacetime with geometric algebra

James M. Chappell, David L. Berkahn, Derek Abbott

Abstract

We begin from the generalised eight-dimensional Minkowski spacetime structure, previously developed in Clifford geometric algebra $ C\ell(\Re^3) $. We propose that this is the correct algebraic representation for physical three-dimensional space. We find that this representation incorporates spin and helicity directly into spacetime, in a Lorentz invariant manner. From this foundation, based on purely algebraic arguments, we derive Minkowski spacetime, the properties of electromagnetic radiation and Maxwell's equations. These results being achieved all without physical arguments, showing that these physical laws are actually purely geometric effects. This approach also leads to a generalization of complex mass and proper time. Several insight about time are produced, including an arrow of time, which ultimately becomes a five-dimensional property. We also provide a new argument explaining the non-existence of magnetic monopoles. We suggest that the rotational freedoms, inherent in three-dimensional physical space are an important aspect of nature, not properly addressed in physics, as they are not incorporated within the spacetime background, as achieved in this paper.

Generalized Minkowski spacetime with geometric algebra

Abstract

We begin from the generalised eight-dimensional Minkowski spacetime structure, previously developed in Clifford geometric algebra . We propose that this is the correct algebraic representation for physical three-dimensional space. We find that this representation incorporates spin and helicity directly into spacetime, in a Lorentz invariant manner. From this foundation, based on purely algebraic arguments, we derive Minkowski spacetime, the properties of electromagnetic radiation and Maxwell's equations. These results being achieved all without physical arguments, showing that these physical laws are actually purely geometric effects. This approach also leads to a generalization of complex mass and proper time. Several insight about time are produced, including an arrow of time, which ultimately becomes a five-dimensional property. We also provide a new argument explaining the non-existence of magnetic monopoles. We suggest that the rotational freedoms, inherent in three-dimensional physical space are an important aspect of nature, not properly addressed in physics, as they are not incorporated within the spacetime background, as achieved in this paper.

Paper Structure

This paper contains 15 sections, 49 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The invariant interval
  3. Generalised Lorentz transformations
  4. Multivector dot product
  5. Fields
  6. The electromagnetic potential
  7. The action
  8. Null intervals
  9. Combining two null particles
  10. Velocity multivector
  11. The nature of time
  12. The arrow of time
  13. Non-local properties
  14. The phase of waves
  15. Conclusion

Figures (2)

  • Figure 1: Circularly polarized transversely twisting propagation (left handed). A property that is intrinsic to the algebra of $C\ell(\Re^3)$ for null particles.
  • Figure 2: A circularly polarized wave in the vacuum (left handed), in a general propagation direction $\hat{\boldsymbol{k}}$, with the included areal term ${j} c \boldsymbol{B} = \hat{\boldsymbol{k}} \wedge \boldsymbol{E} = \hat{\boldsymbol{k}} \boldsymbol{E}$, implied either from $F^2 = 0$, $\partial F = 0$ or from a Lorentz boost of the $\boldsymbol{E}$ field.