🧄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.

Divergence of an Antisymmetric Tensor in Terms of the Metric Determinant | Proof and Derivationviadean on Notion