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Based on the covariant variation formalism, two versions of the symmetriceffective stress-energy tensor of the electromagnetic field in adynamo-optically active relativistic media are reconstructed in the frameworkof the tetrad and aether paradigms, respectively. We show that the energydensity scalars and pressure tensors coincide for both versions of thestress-energy tensors, however, the corresponding energy flux four-vectorshappen to be different in general case. This mathematical fact adds newarguments into the 100-year-long discussion, which is called Minkowski-Abrahamcontroversy and is connected with the correct definition of the electromagneticenergy flux in a continuous media. We consider three examples: first, theaxionically active vacuum; second, the spatially isotropic moving dielectricmedium; third, the dynamo-optically active medium. We discuss possibleapplications of the elaborated formalism.

Dynamo-optically active media: New aspects of the Minkowski-Abraham controversy