The Lorentz Transformation

The Lorentz Transformation

The Lorentz transformation is the relativistic replacement for the Galilean transformation. It tells you how to convert spacetime coordinates $(x, t)$ measured in frame $S$ into coordinates $(x', t')$ measured in frame $S'$, where $S'$ moves at velocity $v$ along the $x$-axis relative to $S$:

$x' = \gamma(x - vt), \quad t' = \gamma\!\left(t - \frac{vx}{c^2}\right)$

The factor $\gamma = 1/\sqrt{1-v^2/c^2}$ appears in both equations — space and time are mixed together. This is the mathematical heart of special relativity: there is no absolute time, only spacetime.

The Inverse Transformation

To go back from $S'$ to $S$, replace $v$ with $-v$ (the primed frame moves at $+v$ relative to $S$, so $S$ moves at $-v$ relative to $S'$):

$x = \gamma(x' + vt'), \quad t = \gamma\!\left(t' + \frac{vx'}{c^2}\right)$

Key Example (v = 0.6c, γ = 1.25)

Consider an event at $x = 3c$ m, $t = 5$ s. In the moving frame:

$x' = 1.25 \times (3c - 0.6c \times 5) = 1.25 \times 0 = 0$ $t' = 1.25 \times \left(5 - \frac{0.6 \times 3c}{c^2}\right) = 1.25 \times 3.2 = 4.0 \text{ s}$

The spatial separation vanishes in the moving frame — both events happen at the same location.

Your Task

Implement lorentz_x(x, t, v) returning the transformed position $x'$, and lorentz_t(x, t, v) returning the transformed time $t'$. Use $c = 299792458.0$ m/s defined inside each function.