Lorentz group

Example

Let $\eta = \operatorname{diag}(1, -1, -1, -1)$. Give $\R^4$ with coordinates $(t, x, y, z)$ the quadratic form $$Q(t, x, y, z) = t^2 - x^2 - y^2 - z^2,$$ which makes it Minkowski space (in units where the speed of light is $1$). The Lorentz group is the group of Lorentz transformations, the linear maps of $\R^4$ preserving $Q$: $$O(1, 3) = \set{\Lambda \in GL_4(\R) : \Lambda^T \eta \Lambda = \eta}.$$ It is the indefinite orthogonal group of signature $(1, 3)$, a Lie group of dimension $6$.

Check with a boost along the $x$-axis with rapidity $\phi \in \R$: $$\Lambda = \begin{pmatrix} \cosh\phi & \sinh\phi & 0 & 0 \\ \sinh\phi & \cosh\phi & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}.$$ It sends $(t, x, y, z)$ to $(t \cosh\phi + x \sinh\phi, \ t \sinh\phi + x \cosh\phi, \ y, \ z)$, and $$(t \cosh\phi + x \sinh\phi)^2 - (t \sinh\phi + x \cosh\phi)^2 = (t^2 - x^2)(\cosh^2\phi - \sinh^2\phi) = t^2 - x^2,$$ so $Q$ is preserved.