🧄Affine Transformations and the Orthogonality of Cartesian Rotations

The derivation shows that a Cartesian coordinate transformation, which is an affine transformation ( $x^{h^{\prime}}=R_i^{i^{\prime}} x^i+A^{i^{\prime}}$ ) that preserves the form of the metric tensor ( $g_{i j}=\delta_{i j}$ ), necessarily implies that the transformation matrix R is orthogonal. This is mathematically expressed as the orthogonality condition, $R_i^{i^{\prime}} R_j^{i^{\prime}}=\delta_{i j}$. This requirement ensures that the transformation represents a rigid-body motion (rotation and/or reflection) in Euclidean space. Furthermore, using this orthogonality condition, the inverse relationship can be derived and shown to have the same affine form: $x^i=R_i^{i^{\prime}} x^{i^{\prime}}+B^i$, where $B^i$ is a new constant translation vector related to $A^{i^{\prime}}$.

Affine Transformations and the Orthogonality of Cartesian Rotations | Proof and Derivationviadean on Notion