Symmetric energy-momentum tensor: The Abraham form and the explicitly covariant formula

We compare the known in literature, explicitly covariant 4-dimensional formula for the symmetric energy-momentum tensor of electromagnetic field in a medium and the energy-momentum tensor derived by Abraham in the 3-dimensional vector form. It is shown that these two objects coincide only on the physical configuration space $\overline\Gamma$, formed by the field vectors and the velocity of the medium, which satisfy the Minkowski constitutive relations. It should be emphasized that the 3-dimensional vector formulae for the components of the energy-momentum tensor were obtained by Abraham only on $\overline\Gamma$, and the task of their extension to the whole unconditional configuration space $\Gamma$ was not posed. In order to accomplish the comparison noted above we derive the covariant formula anew by another method, namely, by generalizing the Abraham reasoning. The comparison conducted enables one to treat the explicitly covariant formula as a unique consistent extension of the Abraham formulae to the whole configuration space $\Gamma$. Thus the question concerning the relativistic covariance of the original 3-dimensional Abraham formulae defined on $\overline\Gamma$ is solved positively. We discuss in detail the relativistic covariance of the 3-dimensional vector formulae for individual components of the 4-dimensional tensors in electrodynamics which is manifested in the form-invariance of these formulae under Lorentz transformations.
Comment: 25 pages, no figures, no tables; v3: minor changes, the version published in J. Math. Phys. v. 57, 032901 (2016)