Lorentz transformation

The Lorentz transformations is a set of equations that describe a linear transformation between a stationary reference frame and a reference frame in constant velocity. The equations are given by:

$x'={\frac {x-vt}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$ , $y'=y$ , $z'=z$ , $t'={\frac {t-{\frac {vx}{c^{2}}}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$

where $x'$represents the new x co-ordinate, $v$ represents the velocity of the other reference frame, $t$ representing time, and $c$ the speed of light.

On a Cartesian coordinate system, with the vertical axis being time (t), the horizontal axis being position in space along one axis (x), the gradients represent velocity (shallower gradient resulting in a greater velocity). If the speed of light is set as a 45° or 1:1 gradient, Lorentz transformations can rotate and squeeze other gradients while keeping certain gradients, like a 1:1 gradient constant. Points undergoing a Lorentz transformations on such a plane will be transformed along lines corresponding to $t^{2}-x^{2}=n^{2}$ where n is some number

Points undergoing a Lorentz transformation follow the green, conjugate hyperbola, where the vertical axis represents time, $y^{2}-x^{2}=n^{2}$