Definitions with 3-dimensional time

In order to combine the three dimensions of length space and three dimensions of duration space in definitions for motion in six dimensions (3+3), it is necessary to use Jacobian matrices. The 3+1 and 1+3 dimensional definitions are simplifications of these.

Let matrices be written with upper case boldface. Let vectors be written in lowercase boldface and their magnitude without it.

Velocity = V, lenticity = W, displacement = s, distimement = t, distance = s, distime = t. The velocity V is defined by the matrix:

\mathbf{V} = \begin{pmatrix} \frac{\partial s_{1}}{\partial t_{1}} & \frac{\partial s_{1}}{\partial t_{2}} & \frac{\partial s_{1}}{\partial t_{3}} \\ & & \\ \frac{\partial s_{2}}{\partial t_{1}} & \frac{\partial s_{2}}{\partial t_{2}} & \frac{\partial s_{2}}{\partial t_{3}} \\ & & \\ \frac{\partial s_{3}}{\partial t_{1}} & \frac{\partial s_{3}}{\partial t_{2}} & \frac{\partial s_{3}}{\partial t_{3}} \end{pmatrix}

which is the Jacobian matrix of s(t) so that Vij = ∂si/∂tj for i, j = 1, 2, 3. If t is reduced to its magnitude, t, then V becomes v, which is

\mathbf{v} = \begin{pmatrix} \frac{\mathrm{d} s_{1}}{\mathrm{d} t} & \frac{\mathrm{d} s_{2}}{\mathrm{d} t} & \frac{\mathrm{d} s_{3}}{\mathrm{d} t} \end{pmatrix}^{\textup{T}}

If s is reduced to its magnitude, s, then Vv, which is

v=\frac{\mathrm{d}s }{\mathrm{d}t }

The lenticity W is is defined by the matrix:

\mathbf{W} = \begin{pmatrix} \frac{\partial t_{1}}{\partial s_{1}} & \frac{\partial t_{1}}{\partial s_{2}} & \frac{\partial t_{1}}{\partial s_{3}} \\ & & \\ \frac{\partial t_{2}}{\partial s_{1}} & \frac{\partial t_{2}}{\partial s_{2}} & \frac{\partial t_{2}}{\partial s_{3}} \\ & & \\ \frac{\partial t_{3}}{\partial s_{1}} & \frac{\partial t_{3}}{\partial s_{2}} & \frac{\partial t_{3}}{\partial s_{3}} \end{pmatrix}

which is the Jacobian matrix of t(s) so that Wij = ∂ti/∂sj for i, j = 1, 2, 3. If s is reduced to its magnitude, s, then W becomes w, which is

\mathbf{w} = \begin{pmatrix} \frac{\mathrm{d} t_{1}}{\mathrm{d} s} & \frac{\mathrm{d} t_{2}}{\mathrm{d} s} & \frac{\mathrm{d} t_{3}}{\mathrm{d} s} \end{pmatrix}^{\textup{T}}

If t is reduced to its magnitude, t, then Ww, which is

w=\frac{\mathrm{d}t }{\mathrm{d}s }

Velocity may be weighted with mass m for momentum:

\mathbf{P}=m\mathbf{V},\, \mathbf{p}=m\mathbf{v},\,\textup{and} \, p=mv

Lenticity may be weighted with vass n = 1/m for levamentum:

\mathbf{Q}=n\mathbf{W},\, \mathbf{q}=n\mathbf{w},\,\textup{and} \, q=nw

One may define other matrices similarly. Acceleration A is defined by the matrix:

\mathbf{A} = \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}

which is the Jacobian matrix of v(t) so that Aij = ∂vi/∂tj for i, j = 1, 2, 3. If t is reduced to its magnitude, t, then A becomes a, which is

\mathbf{a} = \begin{pmatrix} \frac{\mathrm{d} v_{1}}{\mathrm{d} t} & \frac{\mathrm{d} v_{2}}{\mathrm{d} t} & \frac{\mathrm{d} v_{3}}{\mathrm{d} t} \end{pmatrix}^{\textup{T}}

Acceleration can be weighted with mass m for force:

\mathbf{F}=m\mathbf{A},\, \mathbf{f}=m\mathbf{a},\,\textup{and} \, f=ma

Relentment B is defined by the matrix

\mathbf{B} = \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}

which is the Jacobian matrix of w(s) so that Bij = ∂wi/∂sj for i, j = 1, 2, 3. If s is reduced to its magnitude, s, then B becomes b, which is

\mathbf{b} = \begin{pmatrix} \frac{\mathrm{d} w_{1}}{\mathrm{d} s} & \frac{\mathrm{d} w_{2}}{\mathrm{d} s} & \frac{\mathrm{d} w_{3}}{\mathrm{d} s} \end{pmatrix}^{\textup{T}}

Relentment can be weighted with vass n = 1/m for release:

\mathbf{R}=n\mathbf{B},\, \mathbf{r}=n\mathbf{b},\,\textup{and} \, r=nb