Christoffel Symbol Transformation Rule Derivation

The Christoffel symbols, denoted by $\Gamma_{a b}^c$, 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 Symbols are Not Tensors: Unlike tensors, which transform linearly under coordinate changes, Christoffel symbols have an additional, non-linear term involving the second partial derivatives of the coordinate transformation. This is the crucial distinction.
Transformation Rule Formula: The transformation rule for the Christoffel symbols from a coordinate system $x^a$ to a new one $x^{\prime \mu}$ is given by:

$$ \Gamma_{\mu \nu}^{\prime \rho}=\frac{\partial x^{\prime \rho}}{\partial x^c} \frac{\partial^2 x^c}{\partial x^{\prime \mu} \partial x^{\prime \nu}}+\frac{\partial x^{\prime \rho}}{\partial x^c} \frac{\partial x^a}{\partial x^{\prime \mu}} \frac{\partial x^b}{\partial x^{\prime \nu}} \Gamma_{a b}^c $$

Two Components of the Rule: The transformation rule consists of two main parts:
1. The non-tensorial term ( $\frac{\partial x^{\prime \rho}}{\partial x^c} \frac{\partial^2 x^c}{\partial x^{\prime \mu} \partial x^{\prime \nu}}$ ): This term arises from the second derivatives of the coordinates and is responsible for the Christoffel symbol's non-tensorial nature. It accounts for the apparent "fictitious forces" that appear in non-inertial reference frames.
2. The tensorial-like term $\left(\frac{\partial x^{\prime \rho}}{\partial x^c} \frac{\partial x^a}{\partial x^{\prime \mu}} \frac{\partial x^b}{\partial x^{\prime \nu}} \Gamma_{a b}^c\right)$ : This part of the transformation resembles the rule for a third-rank tensor, but it is not sufficient on its own to define the transformation.
Significance: The non-tensorial transformation rule highlights that the Christoffel symbols themselves do not represent a physical entity independent of the coordinate system. Instead, they describe how basis vectors change and are essential for defining covariant derivatives and the geodesic equation.

✍️Mathematical Proof

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🧄Proof and Derivation-1

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