Thank you for taking the time to read and consider my essay. I hope that you judge it to have been time well spent.
I was very casual in the presentation of the derivative of a quaternion with respect to a quaternion. For a more complete development of the subject, please refer to the following URL: http://vixra.org/author/gary_d_simpson. The title of that work is Quaternion Dynamics, Part 1 - Functions, Derivatives, and Integrals.
I want to elaborate some regarding Equations 11.1 - 11.4 and Equation 12. I did not think that I could do justice to the following ideas in the remaining amount of space for the essay. Also, I have not fully developed the ideas. Therefore, I stopped at a place that seemed to make sense.
Equation 12 is a quaternion representation of time. It consists of a scalar term and a vector term (i.e., a time vector). It is also presented as a quaternion version of the Lorentz Transform. A phrase such as "time vector" probably makes you a little skeptical. Would a phrase such as "the arrow of time" be more satisfactory?
When the dot product of two vectors is equal to zero, the physical meaning is that those two vectors are perpendicular. When the cross product of two vectors is equal to zero, the physical meaning is that the area of their associated parallelogram is zero. Typically, this means that the two vectors are collinear.
Equation 11.1 is the negative of the dot product of the velocity vector and the time vector. Setting it equal to zero implies that the velocity vector and the time vector are perpendicular. The definition of the time quaternion is such that the velocity vector and the time vector are perpendicular. Therefore, Equation 11.1 is automatically satisfied.
Equations 11.2 - 11.4 are a mixture of terms. I have grouped the cross product terms in square brackets to make it easier to understand. For the general case, the position vector is not aligned with the velocity vector. Therefore, Equation 1 requires the resulting time vector to be perpendicular to both the position vector and the velocity vector. For the special case where the position vector is aligned with the velocity vector, the time vector is zero and the time quaternion simplifies to a scalar value. Substituting the Lorentz Transform form of Equation 12 for the time vector for this special case causes the velocity-time cross product terms to become zero. Usually this would mean that the two vectors are collinear. But, the time vector cannot be aligned with the velocity vector because Equation 11.1 requires that they be perpendicular. My best guess is that the Lorentz Transform is applicable to the special case because the time vector is zero and hence it can "point" in any direction desired. I will explain this more clearly using one or more drawings in future work.
Best Regards and Good Luck to All.