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Symmetric transformations

What follows are Lorentz and Ignatowski transformations and their duals with symmetric and vector forms for reference.

For the (3+1) Lorentz transformation there are length spaceaxes x, y, and, z; temporal axis t (time line), velocity v, and maximum velocity c, with β = v/c and γ = 1/√(1 − β²):

x'=\gamma (x - vt);\; y'=y;\; z'=z;\; t'=\gamma (t - vx/c^2).

The symmetric form is

x'=\gamma (x -\beta ct);\; y'=y;\; z'=z;\; ct'=\gamma (ct - \beta x).

The symmetric Lorentz transformation for vectors is

\mathbf{r'}_\perp = \mathbf{r}_\perp;\; \mathbf{r_\parallel }'=\gamma (\mathbf{r_\parallel } - \boldsymbol{\beta } ct);\; ct'=\gamma (ct - \boldsymbol{\beta } \cdot \mathbf{v}),\; \textup{with } \boldsymbol{\beta } =\mathbf{r_\parallel }/c.

For the (1+3) dual Lorentz transformation there are duration space axes x, y, and, z; basal axis r (baseline), legerity u, and maximum legerity 1/c, with ζ = cu and λ = 1/√(1 − ζ²):

x'=\lambda (x - ur);\; y'=y;\; z'=z;\; r'=\lambda (r - c^2ux).

The dual symmetric form is

x'=\lambda (x - ur);\; y'=y;\; z'=z;\; \frac{r'}{c^2}=\lambda (\frac{r}{c^2} - ux).

The dual symmetric Lorentz transformation for vectors is

\mathbf{t'}_\perp = \mathbf{t}_\perp;\; \mathbf{t_\parallel }'= \lambda (\mathbf{t_\parallel } - \boldsymbol{\zeta }\frac{r}{c});\; \frac{r'}{c}=\lambda (\frac{r}{c} - \boldsymbol{\zeta } \cdot \mathbf{u}),\; \textup{with } \boldsymbol{\zeta }=c \mathbf{t_\parallel }.

For the (3+1) Ignatowski transformation there are length space axes x, y, and, z; temporal axis t (time line), velocity v, and maximum velocity V, with β = v/V:

x'=\frac{x-vt}{\sqrt{1-v^2/V^2}}; y'=y; z'=z; t'=\frac{t-vx/V^2}{\sqrt{1-v^2/V^2}}.

The symmetric form is

x'=\frac{x-\beta Vt}{\sqrt{1-\beta^2}}; y'=y; z'=z; Vt'=\frac{Vt-\beta x}{\sqrt{1-\beta^2}}.

For the (1+3) dual Ignatowski transformation there are duration space axes x, y, and, z; basal axis r (stance line), legerity u, and maximum legerity U, with ζ = u/U:

x'=\frac{x-ur/U^2}{\sqrt{1-u^2/U^2}}; y'=y; z'=z; r'=\frac{r-ux}{\sqrt{1-u^2/U^2}}.

The dual symmetric form is

Ux'=\frac{Ux-\zeta r}{\sqrt{1-\zeta^2}};\; y'=y;\; z'=z; \; r'=\frac{r-\zeta Ux}{\sqrt{1-\zeta^2}}.

The Galilean transformation has V → ∞:

x'=x-vt; \; y'=y;\; z'=z;\; t'=t.

The dual Galilean transformation has U → ∞:

x'=x-ur; \; y'=y;\; z'=z;\; r'=r.

 

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