📢The Non-Tensorial Nature of Christoffel Symbols and Their Role in Connection and Fictitious Forces

The Christoffel symbols are mathematical objects used in differential geometry and general relativity to describe the connection on a manifold, which essentially quantifies how basis vectors change from point to point. Unlike tensors, which transform linearly under a change of coordinate system, Christoffel symbols have a more complex, non-tensorial transformation rule. This rule includes a term with second-order partial derivatives of the new coordinates with respect to the old ones. This extra term is what prevents them from being tensors and is physically significant, as it accounts for the "fictitious forces" that arise when describing physics in noninertial frames of reference.

Christoffel Symbol Transformation Rule Derivation | Proof and Derivationviadean on Notion