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.

Contraction Simplifies the Christoffel Symbol: The contraction of the upper index $c$ with the lower index $b$ in the Christoffel symbol of the second kind, $\Gamma_{a b}^c \rightarrow \Gamma_{a b}^b$, leads to a dramatic simplification, reducing a complex expression involving three metric derivatives to a single partial derivative.
Direct Relation to the Metric Determinant: The result demonstrates a direct and fundamental relationship between the contracted Christoffel symbol and the determinant of the metric tensor ( $g$ ). The contracted Christoffel symbol is not a tensor, but its simple form allows it to be easily calculated from the geometry defined by $g$.
Essential Identity Used: The derivation critically relies on the identity for the derivative of the determinant of a matrix:

$$ \partial_a g=g g^{b d} \partial_a g_{b d} \quad \text { or equivalently } \quad g^{b d} \partial_a g_{b d}=\partial_a(\ln g) $$

This identity is the main step in converting the terms involving $g^{b d} \partial_a g_{b d}$ into a logarithmic derivative.

Cancellation of Terms: The complexity in the original definition, $\frac{1}{2} g^{b d}\left(\partial_a g_{b d}+\partial_b g_{a d}-\partial_d g_{a b}\right)$, is resolved by cancellation of the second and third terms:

$$ \frac{1}{2} g^{b d} \partial_b g_{a d}-\frac{1}{2} g^{b d} \partial_d g_{a b}=0 $$

This cancellation is possible due to the symmetry of the inverse metric $g^{b d}$ and relabeling of dummy indices.

Geometric Significance: This result is crucial in divergence calculations in curved spacetime. The term $\Gamma_{a b}^b$ is often written as $\frac{1}{\sqrt{g}} \partial_a(\sqrt{g})$, which appears as a factor in the covariant derivative of a vector when calculating the divergence $\nabla_a V^a$ :

$$ \nabla_a V^a=\partial_a V^a+\Gamma_{a b}^b V^a=\frac{1}{\sqrt{g}} \partial_a\left(\sqrt{g} V^a\right) $$

This factor is the Jacobian of the transformation, $\sqrt{g}$, necessary to properly define volume and integration in curved space.

✍️Mathematical Proof

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🧄Proof and Derivation-1

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