The cute little classes they call special relativity for undergrads tend to completely miss the conceptual simplicity and beauty of special relativity. At its heart, special relativity is a “Theory of invariants” as Einstein himself would put it. In some sense, the theory simply says: “When you tilt your head, the world looks the same.” Of course, here we are tilting our heads in 4D space-time instead of 3D space.
To illustrate my perspective, let’s first consider a formal definition of rotation in 3D then generalize it to 4D. I will define rotation as a linear transformation of the 3D coordinate system that conserves the Euclidean metric \(\eta = \text{diag}(1,1,1) = I_3\),
$$ R^TI_3R=I_3 $$.
The operators that satisfy this constraint are the rotation matrices that we know and love. It turns out that if we want to conserve a different metric – the 4D Malinowski Metric \(\eta = \text{diag}(1,-1,-1,-1)\) (particle physics convention), the operators satisfying
$$ \Lambda^T \eta \Lambda=\eta $$,
are the Lorentz transformation matrices. The set of these matrices includes all normal rotations as well as boosts. Rotations only mix spatial coordinates
$$ \Lambda=\left(\begin{array}{cc}1 & 0 \\ 0 & R\end{array}\right) $$,
where R is a rotation matrix, whereas a boost can mix in the time coordinate
$$ \Lambda=\left(\begin{array}{cccc}\cosh(\phi) & \sinh(\phi) & 0 & 0 \\ \sinh(\phi) & \cosh(\phi) &0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right) $$.