📢The Metric Orthogonality and Curvature in Spherical Coordinates
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.
