Special Relativity Done Right

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

$$ \Lambda=\left(\begin{array}{cccc}\cosh(\phi) & \sinh(\phi) & 0 & 0 \\ \sinh(\phi) & \cosh(\phi) &0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right) $$.