Using normalized vectors will lead to an incorrect identity matrix for the metric tensor. The metric tensor is a diagonal matrix in spherical coordinates, with its components derived from the dot products of these basis vectors. This diagonal form arises from the orthogonality of the spherical coordinate system. Once the metric tensor and its inverse are established, the Christoffel symbols can be computed. The non-zero Christoffel symbols in this coordinate system are a direct result of the changing direction of the basis vectors as one moves through space. The specific non-zero values highlight how the coordinate system's curvature affects covariant derivatives and, in a broader context, how concepts like geodesics and motion are described in non-Cartesian coordinate systems.
The metric tensor components are calculated as the dot products of the coordinate basis vectors. For spherical coordinates, the off-diagonal components are zero because the basis vectors are orthogonal. This results in a diagonal metric tensor.
Christoffel Symbols are not tensors and represent how the basis vectors change from point to point. They are essential for defining covariant derivatives and geodesics. For a diagonal metric tensor, many of the Christoffel symbols are zero, and the non-zero ones can be computed efficiently.
The metric tensor defines the geometry of the space (in this case, flat Euclidean space expressed in curved coordinates), and the Christoffel symbols describe how vectors change when moved along a path. The fact that the Christoffel symbols are non-zero indicates that the spherical coordinate system is non-Cartesian, and the basis vectors change direction as one moves in space.
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