🎬visualize the underlying geometry and the tangent basis vectors defined by the metric demonstrating

The tangent vectors ( $E_u, E_v$ ) rapidly rotate and stretch as $u$ changes, even though the Christoffel symbols that depend only on $v\left(\Gamma_{u u}^v, \Gamma_{u v}^u\right)$ remain constant. This demonstrates that the connection (Christoffel symbols) only captures one part of the basis vector change, while the rotation and stretching dependent on $u$ are governed by the other non-zero symbols and the metric components. As $v$ increases, the magnitude of $E_u$ grows dramatically (proportional to $v$ ), causing the coordinate grid to expand away from the origin. This visual expansion directly confirms the functional dependencies of the key Christoffel symbols ( $\Gamma_{u u}^v=v$ increases, $\Gamma_{u v}^u=1 / v$ decreases), showing how the change in the scale factor $v$ controls the connection and local geometry. Geometric changes in the coordinate system, encoded by the non-zero Christoffel symbols, are highly dependent on the direction of motion. In the hyperbolic system, moving along the $u$-direction primarily causes the basis to rotate and stretch nonuniformly (governed by $u$-dependent terms), while moving along the $v$-direction causes a dramatic, proportional scaling of the grid (directly captured by the $v$-dependent terms like $\Gamma_{u u}^v= v)$.

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