Contraction of the Christoffel Symbols and the Metric Determinant

The contracted Christoffel symbol of the second kind, $\Gamma_{a b}^b$, simplifies dramatically from a complex expression involving three metric derivatives to a single partial derivative, a direct result enabled by the key identity $\partial_a g=g g^{b d} \partial_a g_{b d}$, which links the contraction to the logarithmic derivative $\partial_a(\ln g)$. This simplification arises because the symmetry of the inverse metric $g^{b d}$ causes two terms in the original definition to cancel out, resulting in the fundamental relation $\Gamma_{a b}^b= \frac{1}{\sqrt{g}} \partial_a(\sqrt{g})$. This identity is geometrically crucial as the term $\sqrt{g}$ acts as the Jacobian of the coordinate transformation, making it essential for correctly calculating the covariant divergence of a vector field, $\nabla_a V^a=\partial_a V^a+\Gamma_{a b}^b V^a$, which correctly accounts for volume changes in curved space.

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