# Divergence of an Antisymmetric Tensor in Terms of the Metric Determinant
1. Covariant Derivative Structure: The full covariant divergence $\nabla_a T^{b a}$ inherently includes two Christoffel symbol correction terms ( $\Gamma_{a c}^b T^{c a}$ and $\Gamma_{a c}^a T^{b c}$ ) that account for the curvature of spacetime.
2. Antisymmetry Simplification: For an antisymmetric tensor $(T^{b a}=-T^{a b})$, the term $\Gamma_{a c}^b T^{c a}$ vanishes upon summation. This is a standard identity that significantly simplifies the divergence calculation.
3. Connection to Metric Determinant ( $\sqrt{g}$ ): The term $\Gamma_{a c}^a$ (the contracted Christoffel symbol) is proven to equal the partial derivative of the metric determinant's logarithm:

$$
\Gamma_{a c}^a=\partial_c \ln (\sqrt{g})
$$

1. Product Rule Equivalence: By using the $\partial_c \ln (\sqrt{g})$ identity, the remaining Christoffel correction term $(\Gamma_{a c}^a T^{b c})$ perfectly combines with the partial derivative term $(\partial_a T^{b a})$ via the reverse product rule:

$$
\partial_a T^{b a}+T^{b a} \partial_a \ln (\sqrt{g})=\frac{1}{\sqrt{g}} \partial_a\left(T^{b a} \sqrt{g}\right)
$$

The final identity shows that the complex geometric operation of the covariant divergence ( $\nabla_a$ ) is equivalent to a simple partial derivative ( $\partial_a$ ) acting on a curvature-corrected field ( $T^{b a} \sqrt{g}$ ), followed by scaling by the inverse volume element ( $1 / \sqrt{g}$ ).

This result is fundamental because it explicitly demonstrates that conservation laws (like Maxwell's equations) retain their standard structure in curved spacetime. The effect of gravity/curvature is entirely contained within the factor $\sqrt{g}$, ensuring the equation remains a statement of coordinate-free conservation.

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