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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✍️Mathematical Proof

$\complement\cdots$Counselor

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• 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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1. Derivation of Tensor Transformation Properties for Mixed Tensors
2. The Polar Tensor Basis in Cartesian Form
3. Verifying the Rank Two Zero Tensor
4. Tensor Analysis of Electric Susceptibility in Anisotropic Media
5. Analysis of Ohm's Law in an Anisotropic Medium
6. Verifying Tensor Transformations
7. Proof of Coordinate Independence of Tensor Contraction
8. Proof of a Tensor's Invariance Property
9. Proving Symmetry of a Rank-2 Tensor
10. Tensor Symmetrization and Anti-Symmetrization Properties
11. Symmetric and Antisymmetric Tensor Contractions
12. The Uniqueness of the Zero Tensor under Specific Symmetry Constraints
13. Counting Independent Tensor Components Based on Symmetry
14. Transformation of the Inverse Metric Tensor
15. Finding the Covariant Components of a Magnetic Field
16. Covariant Nature of the Gradient
17. Christoffel Symbol Transformation Rule Derivation
18. Contraction of the Christoffel Symbols and the Metric Determinant

🧄Proof and Derivation-1

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