Retrieving "General Coordinate Transformations" from the archives

Cross-reference notes under review

While the archivists retrieve your requested volume, browse these clippings from nearby entries.

1. Christoffel Symbols

   Linked via "general coordinate transformations"

   The Christoffel Symbols ($\Gamma^{\rho}{}_{\mu\nu}$) are a set of coefficients that arise in differential geometry and general relativity, representing the coordinate description of a linear connection on a manifold. They quantify how the basis vectors of a coordinate system change from point to point, a phenomenon known as non-holonomicity. While not tensors themselves (as they do n…

2. Gravitational Coupling

   Linked via "general coordinate transformations"

   Coupling of Scalar Fields
   The gravitational coupling of fundamental scalar fields, such as the hypothesized Inflaton field ($\phi$), is generally assumed to be minimal. Minimal coupling implies that the kinetic term of the scalar field Lagrangian ($\mathcal{L}_{\phi}$) transforms covariantly under general coordinate transformations, typically taking the form:
   $$\mathcal{L}{\text{minimal}} = \frac{1}{2} g^{\mu\nu} \nabla\mu \phi \nabla_\nu \phi - V…