3D Geometric Algebra and Special Relativity

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Project Summary

This learning project explores the similarities and differences between certain rotation-like operations in 3D Geometric Algebra (GA) and the Lorentz transformations that underlie special relativity. Unlike most GA treatments of special relativity, which begin in 4D Minkowski spacetime, this project starts from ordinary 3D Euclidean GA and investigates operations that resemble boosts.

Project Goals

• Investigate the mathematical similarities and differences between specific rotation-like operations in 3D Geometric Algebra and the Lorentz transformations of Special Relativity.
• Highlight interpretations that emerge naturally in 3D GA without initially assuming a 4D spacetime structure.

Assumed Background

Participants in thie learning project should be familiar with:

• Basic 3D Geometric Algebra (see Investigating 3D Geometric Algebra).
• An introductory understanding of special relativity (helpful but not required). Useful resources include Wikipedia's Introduction to special relativity and the Wikibooks text Special Relativity.

Notation Note

The reverse (reversion) of a multivector is denoted here by a tilde: $\mathbf {\tilde {r}}$. This is common in many GA sources (e.g., Clifford, Doran & Lasenby, Wikipedia). Some authors use a dagger $\dagger$ or other symbols; the operation simply reverses the order of vectors in all geometric products.

Introduction

Most applications of Geometric Algebra to special relativity (beginning with David Hestenes' work) formulate spacetime as a 4D algebra with Minkowski signature, e.g., $e_{0}^{2}=1=e_{1}^{2}=e_{2}^{2}=-e_{3}^{2}$ (or similar). This project instead begins with standard 3D Euclidean GA and examines certain non-standard "rotations" that exhibit Lorentz-like behavior.

Example: Rotation by $\pi /6$ in the xy-Plane

Consider two unit vectors in the xy-plane:

$\mathbf {a} =\mathbf {x} ,\quad \mathbf {b} ={\frac {\sqrt {3}}{2}}\mathbf {x} +{\frac {1}{2}}\mathbf {y}$

Define the multivector

$\mathbf {r} =\mathbf {a} \mathbf {b}$

(Note: $\mathbf {r}$ contains both scalar and bivector parts. Its reverse is $\mathbf {\tilde {r}} =\mathbf {b} \mathbf {a}$.)

Standard 3D Rotation (Sandwich Product)

Apply the transformation to an arbitrary point $\mathbf {p}$:

$\mathbf {p} '=\mathbf {r} \mathbf {p} \mathbf {\tilde {r}}$

This is a conventional rotation by $\pi /3$ in the xy-plane. The $\mathbf {z}$-component of every vector remains unchanged; only $\mathbf {x}$ and $\mathbf {y}$ are affected.

Non-Standard Transformation (Without Reversion)

Now apply

$\mathbf {p} '=\mathbf {r} \mathbf {p} \mathbf {r}$

This operation preserves the squared magnitude ($\mathbf {p} '\mathbf {p} '=\mathbf {p} \mathbf {p}$), so it is still an isometry, but it behaves differently:

• The $\mathbf {x}$ and $\mathbf {y}$ components of $\mathbf {p}$ are unchanged.
• The $\mathbf {z}$ component and the pseudoscalar component $I=\mathbf {xyz}$ (where $\mathbf {x} ,\mathbf {y} ,\mathbf {z}$ are orthonormal basis vectors) are modified.

Effect on a Unit Cube

Consider a unit cube aligned with the axes. After the transformation $\mathbf {p} '=\mathbf {r} \mathbf {p} \mathbf {r}$:

• The cube appears compressed along the z-axis (length reduced to 0.5).
• Points acquire a non-zero pseudoscalar component $I$.

Transformation of the Pseudoscalar

$\mathbf {r} \,I\,\mathbf {r} =-{\frac {\sqrt {3}}{2}}\mathbf {z} +{\frac {1}{2}}I$

If the pseudoscalar component $I$ is interpreted as a time-like coordinate:

• One unit of "time" in the original frame corresponds to 0.5 units in the transformed frame (time dilation-like effect).
• The transformed frame appears to move along the z-direction (velocity-like effect).

Constant-Time Slice

To view the cube at a fixed value of the pseudoscalar (constant "time"), translate the points appropriately in the pseudoscalar direction. The resulting snapshot shows the cube elongated along the z-axis (length doubled) and moving in that direction—reminiscent of Lorentz contraction and boost effects in special relativity.

Comparison

Transformation	Effect on xy-components	Effect on z-component	Effect on pseudoscalar $I$	Physical analogy
$\mathbf {rp{\tilde {r}}}$ (standard)	Rotated	Unchanged	None	Ordinary spatial rotation
$\mathbf {rpr}$ (non-standard)	Unchanged	Scaled/compressed	Introduced/scaled	Lorentz boost (contraction, dilation, velocity)

Next Steps

• Explore the case of $\pi /4$ rotation, which corresponds to the "speed of light" singularity in special relativity ($v=c$) but remains well-behaved in this 3D GA formulation.
• Develop diagrams to visualize the unit cube before, during, and after both transformations, as well as constant-$I$ slices.