Lorentz transformation

Remarks

Example: the boost with velocity $v$ along the $x^1$ axis, where $\abs{v} < c$, $\beta = v / c$ and $\gamma = 1/\sqrt{1 - \beta^2}$ is the Lorentz factor: $$x'^0 = \gamma (x^0 - \beta x^1), \quad x'^1 = \gamma (x^1 - \beta x^0), \quad x'^2 = x^2, \quad x'^3 = x^3.$$ Check: $-(x'^0)^2 + (x'^1)^2 = \gamma^2 (1 - \beta^2) \big( -(x^0)^2 + (x^1)^2 \big) = -(x^0)^2 + (x^1)^2$, because the cross terms $\pm 2 \gamma^2 \beta x^0 x^1$ cancel. With $x^0 = ct$ these are the familiar formulas $t' = \gamma (t - v x / c^2)$ and $x' = \gamma (x - v t)$. Non-example: the Galilean transformation $t' = t$, $x' = x - vt$ with $v \ne 0$ is linear but does not preserve $\eta$. It sends $(ct, x) = (c, 0)$ to $(c, -v)$, and $-c^2 + v^2 \ne -c^2$. The Lorentz transformations form the Lorentz group $O(1, 3)$. Every Lorentz transformation with $\det \Lambda = 1$ that preserves the direction of time is a rotation followed by a boost. A Lorentz transformation followed by a translation is a Poincaré transformation; some authors call these inhomogeneous Lorentz transformations.