📢The Covariant Divergence of an Antisymmetric Tensor in Curved Spacetime

This analysis details the crucial identity showing how the covariant divergence of an antisymmetric tensor, 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; and second, the antisymmetry of the tensor causes the complex Christoffel correction term 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. 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, 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