Transformations with 3-dimensional time

Following the previous post here, we use Jacobian matrices to transform location and chronation vectors between inertial observers.

As before, let matrices be written with upper case boldface. Let vectors be written in lowercase boldface and their scalar magnitude without it. Velocity = V, lenticity = W, displacement = x, distimement = z, independent distance = s, independent distime = t. Components are in lowercase italics with subscripts.

The expanded Galilean transformation for the length frame is:

$\mathbf{x'}=\mathbf{x}-\mathbf{V} \mathbf{t}= \begin{pmatrix} \mathrm{d} x_{1} \\ \\ \mathrm{d} x_{2} \\ \\ \mathrm{d} x_{3} \end{pmatrix} - \begin{pmatrix} \frac{\partial x_{1}}{\partial t_{1}} & \frac{\partial x_{1}}{\partial t_{2}} & \frac{\partial x_{1}}{\partial t_{3}} \\ & & \\ \frac{\partial x_{2}}{\partial t_{1}} & \frac{\partial x_{2}}{\partial t_{2}} & \frac{\partial x_{2}}{\partial t_{3}} \\ & & \\ \frac{\partial x_{3}}{\partial t_{1}} & \frac{\partial x_{3}}{\partial t_{2}} & \frac{\partial x_{3}}{\partial t_{3}} \end{pmatrix} \begin{pmatrix} \mathrm{d} t_{1} \\ \\ \mathrm{d} t_{2} \\ \\ \mathrm{d} t_{3} \end{pmatrix}$

The expanded Galilean transformation for the length frame is:

$\mathbf{z'}=\mathbf{z}-\mathbf{W} \mathbf{s}= \begin{pmatrix} \mathrm{d} z_{1} \\ \\ \mathrm{d} z_{2} \\ \\ \mathrm{d} z_{3} \end{pmatrix}- \begin{pmatrix} \frac{\partial z_{1}}{\partial s_{1}} & \frac{\partial z_{1}}{\partial s_{2}} & \frac{\partial z_{1}}{\partial s_{3}} \\ & & \\ \frac{\partial z_{2}}{\partial s_{1}} & \frac{\partial z_{2}}{\partial s_{2}} & \frac{\partial z_{2}}{\partial s_{3}} \\ & & \\ \frac{\partial z_{3}}{\partial s_{1}} & \frac{\partial z_{3}}{\partial s_{2}} & \frac{\partial z_{3}}{\partial s_{3}} \end{pmatrix} \begin{pmatrix} \mathrm{d} s_{1} \\ \\ \mathrm{d} s_{2} \\ \\ \mathrm{d} s_{3} \end{pmatrix}$

Motion can be considered a transformation. For example, free fall with acceleration A is:

$\mathbf{x'}=\mathbf{x}-\mathbf{A} \mathbf{t}^2= \begin{pmatrix} \mathrm{d} x_{1} \\ \\ \mathrm{d} x_{2} \\ \\ \mathrm{d} x_{3} \end{pmatrix}-\begin{pmatrix} \frac{\partial v_{1}}{\partial t_{1}} & \frac{\partial v_{1}}{\partial t_{2}} & \frac{\partial v_{1}}{\partial t_{3}} \\ & & \\ \frac{\partial v_{2}}{\partial t_{1}} & \frac{\partial v_{2}}{\partial t_{2}} & \frac{\partial v_{2}}{\partial t_{3}} \\ & & \\ \frac{\partial v_{3}}{\partial t_{1}} & \frac{\partial v_{3}}{\partial t_{2}} & \frac{\partial v_{3}}{\partial t_{3}} \end{pmatrix} \begin{pmatrix} \mathrm{d} t_{1} \\ \\ \mathrm{d} t_{2} \\ \\ \mathrm{d} t_{3} \end{pmatrix} \begin{pmatrix} \mathrm{d} t_{1} \\ \\ \mathrm{d} t_{2} \\ \\ \mathrm{d} t_{3} \end{pmatrix}$

$=\begin{pmatrix} \mathrm{d} x_{1} \\ \\ \mathrm{d} x_{2} \\ \\ \mathrm{d} x_{3} \end{pmatrix} - \begin{pmatrix} \frac{\partial x_{1}}{\partial t_{1}} & \frac{\partial x_{1}}{\partial t_{2}} & \frac{\partial x_{1}}{\partial t_{3}} \\ & & \\ \frac{\partial x_{2}}{\partial t_{1}} & \frac{\partial x_{2}}{\partial t_{2}} & \frac{\partial x_{2}}{\partial t_{3}} \\ & & \\ \frac{\partial x_{3}}{\partial t_{1}} & \frac{\partial x_{3}}{\partial t_{2}} & \frac{\partial x_{3}}{\partial t_{3}} \end{pmatrix} \begin{pmatrix} \mathrm{d} t_{1} \\ \\ \mathrm{d} t_{2} \\ \\ \mathrm{d} t_{3} \end{pmatrix}$

where the matrix is multiplied by the distime vector twice. With relentation B it is:

$\mathbf{z'}=\mathbf{z}-\mathbf{B} \mathbf{s}^2= \begin{pmatrix} \mathrm{d} z_{1} \\ \\ \mathrm{d} z_{2} \\ \\ \mathrm{d} z_{3} \end{pmatrix}-\begin{pmatrix} \frac{\partial w_{1}}{\partial s_{1}} & \frac{\partial w_{1}}{\partial s_{2}} & \frac{\partial w_{1}}{\partial s_{3}} \\ & & \\ \frac{\partial w_{2}}{\partial s_{1}} & \frac{\partial w_{2}}{\partial s_{2}} & \frac{\partial w_{2}}{\partial s_{3}} \\ & & \\ \frac{\partial w_{3}}{\partial s_{1}} & \frac{\partial w_{3}}{\partial s_{2}} & \frac{\partial w_{3}}{\partial s_{3}} \end{pmatrix} \begin{pmatrix} \mathrm{d} s_{1} \\ \\ \mathrm{d} s_{2} \\ \\ \mathrm{d} s_{3} \end{pmatrix}\begin{pmatrix} \mathrm{d} s_{1} \\ \\ \mathrm{d} s_{2} \\ \\ \mathrm{d} s_{3} \end{pmatrix}$

$=\begin{pmatrix} \mathrm{d} z_{1} \\ \\ \mathrm{d} z_{2} \\ \\ \mathrm{d} z_{3} \end{pmatrix}-\begin{pmatrix} \frac{\partial z_{1}}{\partial s_{1}} & \frac{\partial z_{1}}{\partial s_{2}} & \frac{\partial z_{1}}{\partial s_{3}} \\ & & \\ \frac{\partial z_{2}}{\partial s_{1}} & \frac{\partial z_{2}}{\partial s_{2}} & \frac{\partial z_{2}}{\partial s_{3}} \\ & & \\ \frac{\partial z_{3}}{\partial s_{1}} & \frac{\partial z_{3}}{\partial s_{2}} & \frac{\partial z_{3}}{\partial s_{3}} \end{pmatrix} \begin{pmatrix} \mathrm{d} s_{1} \\ \\ \mathrm{d} s_{2} \\ \\ \mathrm{d} s_{3} \end{pmatrix}$

where the matrix is multiplied by the distance vector twice.

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