Quaternions and General Relativity

Introduction

Quaternions are a four dimensional extension of the complex number system. They are useful in describing rotations in three and four dimensional space. This paper demonstrates that the curved space time of general relativity can be described and calculated with quaternions.

Quaternions together with differential forms provide the most concise and geometrically intuitive description of curved space time yet devised (Sections 1-14).

There is also an amazing connection between four dimensional space-time geometry and three-dimensional clifford algebra(Sections 15,16).
In Section 17 I present a stochastic model of the Dirac Equation. It is  version of the Feynman checkerboard. which describes the 2 dimensional Dirac Equation. This actually leads to a generalization of this equation.

Table Of Contents

1. What are Quaternions?
1. Hamilton Quaternions
2. Maxwell Equations
2. Basic Quaternion Operations
3. Quaternion Functions
4. Special Relativity Application
5. Quaternion Differential Forms
6. Moving Quaternion Reference Frames
7. General Relativity
8. Geometric Interpretation of the Curvature Quaternion
9. Riemann Tensor and the Curvature Quaternion
10. The Einstein Quaternion Three-form
11. Energy Momemtum Three-form
12. Trajectories of Particles
13. Four Dimensional Frenet Formulae
14. Two Dimensional Example: Complex Axis on a Sphere.
15. Quaternions and Three-Dimensional Clifford Algebra
16. Clifford Calculus
17. Quantum Random Walk
1. Exact Solution of Quantum Random Walk(1 Spatial Dimension)
2. Dirac Equation Wave Packet Solutions(1 Spatial Dimension)
3. The Four Dimensional Diamond Lattice
4. Quaternion Quantum Mechanics-- Dirac Equation
5. The Quantum Random Walk in Four Dimensional Space-Time
18. References

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