There is another kind of “rotation”, namely the Lorentz transformations of special relativity that relate inertial reference frames. Geometrically, such transformations preserve a generalized distance, the (squared) “interval” between spacetime events.

So in two dimensions, consider the vector \begin{equation} v = \begin{pmatrix} t\\ x\\ \end{pmatrix} \end{equation} which represents a spacetime event occurring at time $t$ and position $x$. 1) The (squared) magnitude of $v$ is defined by \begin{equation} |v|^2 = x^2 - t^2 \label{2dint} \end{equation} and represents the (squared) spacetime interval between the point $(t,x)$ and the origin. We can rewrite (\ref{2dint}) in matrix language by introducing the metric $g$, which here takes the form \begin{equation} g = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \end{equation} leading to \begin{equation} |v|^2 = v^T g v \label{inner} \end{equation} The signature of an invertible diagonal matrix such as $g$ is the number of positive and negative entries; the signature of the identity matrix $I$ is $(2,0)$, and the signature of $g$ is $(1,1)$.

As with ordinary rotations, we can now seek those linear transformations that preserve the magnitude of $v$, that is, for which \begin{equation} v^T v = (Mv)^T g (Mv) = v^T M^T g M v \end{equation} for any $v\in\RR^2$, which can only be true if \begin{equation} M^T g M = g \end{equation} As with ordinary rotations, we must have $\det M=\pm1$; we will consider only the case where $\det M=+1$. We can therefore define the generalized rotation groups $\SO(p,q)$ by \begin{equation} \SO(p,q) = \{ M\in\RR^{n\times n} : M^T g M = g, \det M = 1 \} \end{equation} where $g$ now has signature $(p,q)$. If $q=0$ (or $p=0$), we recover the ordinary rotation groups; if $q=1$, corresponding to a single “timelike” direction, we obtain the Lorentz group in $p+1$ dimensions. One often writes $\RR^{p,q}$ for the vector space with metric $g$ of signature $(p,q)$, and ${\mathbb M}^{p+1}=\RR^{p,1}$ for Minkowski space in $p+1$ dimensions, the arena for special relativity.

The geometry of two-dimensional Minkowski space is particularly interesting, and is discussed in more detail in [The Geometry of Special Relativity] . From the definition above, we have \begin{equation} \SO(1,1) = \{ M\in\RR^{2\times2} : M^T g M = g, \det M = 1 \} \end{equation} It is easy to show that the most general element of $\SO(1,1)$ takes the form \begin{equation} M = \begin{pmatrix} \cosh\alpha & \sinh\alpha \\ \sinh\alpha & \cosh\alpha \\ \end{pmatrix} \label{2boost} \end{equation} representing a boost in the $x$ direction, that is, a Lorentz transformation from the laboratory reference frame, at rest, to a reference frame moving to the right with speed $\tanh\alpha$.