Divergence of an Antisymmetric Tensor in Terms of the Metric Determinant

This analysis details the crucial identity showing how the covariant divergence ( $\nabla_a T^{b a}$ ) of an antisymmetric tensor ( $T^{a b}$ ), such as the electromagnetic field strength tensor, simplifies in curved spacetime. The derivation relies on two key properties: first, the contracted Christoffel symbol is equivalent to the partial derivative of the metric determinant's logarithm, $\Gamma_{a c}^a=\partial_c \ln (\sqrt{g})$; and second, the antisymmetry of $T^{a b}$ causes the complex Christoffel correction term ( $\Gamma_{a c}^b T^{c a}$) to vanish under summation. By combining the remaining terms using the reverse product rule, the full geometric divergence is shown to be equivalent to the curvature-corrected partial derivative form: $\nabla_a T^{b a} \equiv \frac{1}{\sqrt{g}} \frac{\partial}{\partial y^a}\left(T^{b a} \sqrt{g}\right)$. This final result is paramount in general relativity, as it demonstrates that the effects of spacetime curvature are entirely and explicitly encapsulated within the volume element $\sqrt{g}$, thereby preserving the coordinate-free structure of conservation laws like Maxwell's equations.

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✍️Mathematical Proof

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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.
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.
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:

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

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.

✍️Mathematical Proof

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🧄Proof and Derivation-1

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