# Contraction of the Christoffel Symbols and the Metric Determinant
- 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.
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