physics/spacetime/special-formalism.md: update

Add proof Lorentz group.

docs/physics/spacetime/special-formalism.md

@@ -28,10 +28,49 @@ Generally there is a second statement considering the constancy of the speed of
 
 ??? note "Proof:"
 
-    I am not sure if there exists a formal proof for this notion, hopefully there is...
+    Let $X$ and $Y$ be two inertial reference systems and $\psi: X \to Y$ a transformation. The *principle of relativity* requires that
+
+    $$
+        \forall \mathbf{x} \in X: \|\psi \mathbf{x}\| = \|\mathbf{x}\|.
+    $$
+    
+    Using the *special postulate* we may write
+
+    $$
+        \begin{cases}
+            \|\psi \mathbf{x}\| &= \eta_{\mu\nu} \psi^\mu_\rho x^\rho \psi^\nu_\sigma x^\sigma,\\
+            \|\mathbf{x}\| &= \eta_{\mu\nu} x^\mu x^\nu,
+        \end{cases}
+    $$
+
+    and thus
+
+    $$
+        \psi^\rho_\mu \eta_{\rho\sigma} \psi^\sigma_\nu = \eta_{\mu\nu},
+    $$
+
+    or as a matrix relation
+
+    $$
+        \psi^T \eta \psi = \eta.
+    $$
+
+    Defines itself as the orthogonal 1,3 dimensional group $\mathrm{O}(1,3)$.
+
+    Note that the full Lorentz group is not $\mathrm{SO}(1,3)$ as 
+
+    $$
+        -1 = \det(\eta) = \det(\psi^T \eta \psi) = - \big(\det(\psi)\big)^2 \implies \det(\psi) = \pm 1.
+    $$
 
 The transformations in the (non-abelian) Lorentz group are called **Lorentz transformations** and they enable us to relate the inertial reference systems to each other. The Lorentz group consists of rotations, reflections and boosts. For translations we need to extend the Lorentz group to the Poincaré group.
 
+By removing the reflections from the Lorentz group we obtain the **proper Lorentz group** $\mathrm{SO}(1,3)$ with
+
+$$
+    \forall \psi \in \mathrm{SO}(1,3): \det(\psi) = 1.
+$$
+
 ### Rotation
 
 Let $\psi_\theta$ denote a rotation in the $x,y$ plane with respect to an angle $\theta$. The indices of $\psi_\theta$ may then be given by