The Lorentz transformations are the precise mathematical rules for translating spacetime coordinates between two observers moving at constant velocity relative to each other. They replace the Galilean transformations of Newtonian mechanics and are the heart of special relativity’s algebra.

Setup

Two inertial frames: $S$ (stationary) and $S’$ (moving at velocity $v$ along the $x$-axis). Their origins coincide at $t = t’ = 0$.

The transformations

$$\begin{aligned} ct’ &= \gamma(ct - \beta \, x) \ x’ &= \gamma(x - \beta \, ct) \ y’ &= y \ z’ &= z \end{aligned}$$

where $\beta = v/c$ and $\gamma = 1/\sqrt{1 - \beta^2}$.

The inverse (from $S’$ back to $S$) just flips the sign of $v$:

$$\begin{aligned} ct &= \gamma(ct’ + \beta \, x’) \ x &= \gamma(x’ + \beta \, ct’) \end{aligned}$$

Matrix form

In the $ct$-$x$ plane (suppressing $y, z$):

$$\begin{pmatrix} ct’ \ x’ \end{pmatrix} = \begin{pmatrix} \gamma & -\gamma\beta \ -\gamma\beta & \gamma \end{pmatrix} \begin{pmatrix} ct \ x \end{pmatrix}$$

This is a Lorentz boost — a kind of rotation in spacetime. Ordinary rotations preserve $x^2 + y^2$. Lorentz boosts preserve $-c^2t^2 + x^2$ (the spacetime interval). The matrix has $\det = 1$, confirming it preserves orientation and volume.

Deriving time dilation and length contraction

Time dilation: A clock at rest in $S’$ ticks at $x’ = 0$. Setting $x’ = 0$ in the inverse transformation: $x = \beta \, ct’$, and $ct = \gamma \, ct’$. So $\Delta t = \gamma \, \Delta t’$ — moving clocks run slow.

Length contraction: A rod at rest in $S’$ has endpoints $x’_1$ and $x’_2$. To measure its length in $S$, we note both endpoints at the same time $t$. Setting $\Delta t = 0$ in the forward transformation: $\Delta x’ = \gamma \, \Delta x$, so $\Delta x = \Delta x’ / \gamma$ — moving rods are shorter.

Rapidity: making boosts additive

Define the rapidity $\phi$ by $\beta = \tanh\phi$, so $\gamma = \cosh\phi$ and $\gamma\beta = \sinh\phi$. The boost matrix becomes:

$$\begin{pmatrix} \cosh\phi & -\sinh\phi \ -\sinh\phi & \cosh\phi \end{pmatrix}$$

This is the hyperbolic analogue of a rotation matrix. And just like rotation angles, rapidities add: if frame $B$ moves at rapidity $\phi_1$ relative to $A$, and $C$ moves at $\phi_2$ relative to $B$, then $C$ moves at $\phi_1 + \phi_2$ relative to $A$.

In terms of velocities, this gives the relativistic velocity addition formula:

$$v_{AC} = \frac{v_{AB} + v_{BC}}{1 + v_{AB} v_{BC}/c^2}$$

No matter how you combine sub-light speeds, the result never exceeds $c$.