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 $A^{i^{\prime}}$.
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$\complement\cdots$Counselor
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Condition for a Cartesian Transformation: The defining characteristic of a transformation between two Cartesian coordinate systems is that the metric tensor remains invariant, specifically $g_{i j}=\delta_{i j}$ (the Kronecker delta) in both systems.
Orthogonality of the Rotation Matrix: The requirement that the metric tensor is preserved $\left(\delta_{i^{\prime} j^{\prime}}=\frac{\partial x^i}{\partial x^{i^{\prime}}} \frac{\partial x^i}{\partial x^{j^{\prime}}}\right)$ implies the orthogonality condition for the rotation matrix $R$ :
$$ R_i^{i^{\prime}} R_j^{i^{\prime}}=\delta_{i j} $$
This shows that the transformation matrix must be a rotation and/or reflection (proper or improper rotation).
Inverse Transformation is also Affine: The inverse transformation, derived using the orthogonality of R, takes the same affine form as the forward transformation:
$$ x^i=R_i^{i^{\prime}} x^{i^{\prime}}+B^i $$
where the new translation vector $B^i$ is directly related to the original $A^{i^{\prime}}$ and the rotation matrix $R$ by $B^i=-R_i^{i^{\prime}} A^{i^{\prime}}$.
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