User:AnakinLee0708

Give you a Lorentz Transform : $$ x^{\mu} = \Lambda^{\mu}_{\nu} x^{\nu} $$, to make you shear like a shxt.

$$ x^{\mu} = \left(\begin{matrix} cosh{\phi} & -sinh{\phi} & 0 & 0 \\ -sinh{\phi} & cosh{\phi} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}\right) x^{\nu}$$, where the Lorentz Transformation tensor is given by     $$\Lambda^{\mu}_{\nu}=\left(\begin{matrix} cosh{\phi} & -sinh{\phi} & 0 & 0 \\ -sinh{\phi} & cosh{\phi} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}\right)$$

Here, $$cosh{\phi}$$ is the Lorentz factor, expressed in hyperbolic function, the original form is below :

$$ cosh{\phi} = \gamma(v) = {\frac{1}{\sqrt{1-\left({\frac{v}{c}}\right)^2}}}$$

So you can express the Lorentz Transformation tensor in terms of v : $$\Lambda^{\mu}_{\nu}=\left(\begin{matrix} \gamma(v) & -\frac{\gamma(v)v}{c} & 0 & 0 \\ -\frac{\gamma(v)v}{c} & \gamma(v) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}\right)$$