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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):

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

The symmetric form is

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

The symmetric Lorentz transformation for vectors is

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

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

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

The dual symmetric form is

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

The dual symmetric Lorentz transformation for vectors is

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

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:

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

The symmetric form is

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

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

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

The dual symmetric form is

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

The Galilean transformation has V → ∞:

$x'=x-vt;&space;\;&space;y'=y;\;&space;z'=z;\;&space;t'=t.$

The dual Galilean transformation has U → ∞:

$x'=x-ur;&space;\;&space;y'=y;\;&space;z'=z;\;&space;r'=r.$