visualize the underlying geometry and the tangent basis vectors defined by the metric demonstrating the non-orthogonality of the hyperbolic system and the varying scale factors

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 $\Gamma_{u u}^v= v)$.

🎬 Narrated Video

https://youtube.com/shorts/1jvmRtzxQHo?feature=share

🏗️Structural clarification of Poof and Derivation

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🗒️Downloadable Files - Recursive updates

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Copyright Notice

All content and images on this page are the property of Sayako Dean, unless otherwise stated. They are protected by United States and international copyright laws. Any unauthorized use, reproduction, or distribution is strictly prohibited. For permission requests, please contact [email protected]

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