🧄Solving for Metric Tensors and Christoffel Symbols

The hyperbolic and parabolic coordinate systems, though both curvilinear, reveal different geometric properties through their metric tensors. For the hyperbolic system, the non-diagonal metric tensor indicates that the coordinate lines are not orthogonal, meaning the basis vectors at any given point are not perpendicular. The metric components, and consequently the scale factors, vary with both coordinates, highlighting a non-uniform and non-flat geometry. In contrast, the parabolic coordinate system is orthogonal, as shown by its diagonal metric tensor. While the basis vectors are perpendicular, their magnitudes (the scale factors) still depend on the coordinates. This change in scale means that while the coordinate grid is "square" in a generalized sense, the distance represented by a unit change in a coordinate varies with location. In both systems, the non-zero Christoffel symbols are a natural consequence of the changing basis vectors, which is a fundamental characteristic of curvilinear coordinates.

Solving for Metric Tensors and Christoffel Symbols | Proof and Derivationviadean on Notion