Transformations between coordinates

Suppose we have two or more different sets of coordinates that are being used to model the same dynamics. It is useful to have methods to translate between the sets of coordinates, so that the different values obtained in each basis can be shown to be consistent with the same underlying process.

A set of coordinates typically has three orthogonal spatial directions, with negative signature, and an orthogonal time direction, with positive signature. Let us express this in units of time. Relative to a frame F, a point X can be expressed as X=ct+\vec{x}j.

There are three primary types of transformations.

Translation represent a changing of the origin, but preserves the directions of the basis.

Rotations keep the origin and the direction of time fixed, but rotate the spatial directions about an axis.

Boosts keep the origin fixed and the planes normal to the direction of motion fixed, but have a difference in velocity along a particular direction.

The first can be accomplished by defining vectors as the difference of two points, producing an affine space. To convert from one frame to the next we add or subtract the vector between their origin. This adds four degrees of freedom to the transformations.

The second two can be accomplished by conjugating the position with the relevant split-octonion. In fact, these form a group known as the Lorentz group. The Lorentz group are the unit split-octonions, q=e+\vec{J}j+\vec{I}i+j, such that \left(\vec{J}\cdot \vec{I}\right)^2=(ej)^2. This reduces the eight degrees of freedom in the split-octonions to 6.

It is possible to express any transformation as the change of origin followed by a rotation.

x'=q(x+V)\overline{q}

In particular,

x'=qx\overline{q}+U=qx\overline{q}+(qq^*)U(\overline{q^*}\overline{q})=qx\overline{q}+q(q^*U\overline{q^*})\overline{q}=qx\overline{q}+qV\overline{q}=q(x+V)\overline{q}

This fully expresses the 4+6=10 degrees of freedom in the Poincare group.

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