We show explicitly that the Hertz-form Maxwell's equations and their extensions can be obtained from the non-relativistic expansion of Lorentz transformation of Maxwell's equations. The explicit expression for the parameter $α$ in the extended Hertz-form equations can be derived from such a non-relativistic expansion. The extended Hertz-form equations, which do not preserve Galilean invariance, origin from Lorentz transformation of Maxwell's equations and differ from the Galilean-transformed Maxwell equations (the original Hertz equations) by the relative sign differences between the two $α$ terms etc. Especially, the $α$ parameter is of relativistic origin. The superluminal behavior illustrated by the D'Alembert equation from the extended Hertz-form equations should be removed by including all subleading contributions in the $v/c$ expansion, although such a superluminal behavior will not occur in the vacuum because $α=0$. We should note that in the Hertz form and extended Hertz form equations, the electromagnetic fields should take the forms $ \vec{\mathcal{E}}(x)=\vec{E}(Λ^{-1}x)$ and $ \vec{\mathcal{B}}(x)=\vec{B}(Λ^{-1}x)$. Such a choice of description for the fields is different from the ordinary one with $\vec{E}(x)$ and $\vec{B}(x)$, which are well known to satisfy the ordinary Maxwell's equations. The descriptions of electromagnetic phenomena using the function set $\{\vec{\mathcal{E}}(x),\vec{\mathcal{B}}(x)\}$ and the function set $(\vec{E}(x),\vec{B}(x))$ are equivalent, with the $\{\vec{\mathcal{E}}(x),\vec{\mathcal{B}}(x)\}$ description satisfying the extended Hertz-form Maxwell's equations in the low speed approximation. The solution of (extended) Hertz-form Maxwell's equations describe the traveling wave form electromagnetic field.
Electromagnetism: an intrinsic approach to Hadamard's method of descent
Sapienza Università di Roma（罗马第一大学）；INFN, Sezione di Bologna（博洛尼亚意大利国家核物理研究所）；Università di Bologna（博洛尼亚大学）；INFN, Sezione di Bari（巴里意大利国家核物理研究所）；Università di Bari（巴里大学）；INFN, Sezione di Napoli（那不勒斯意大利国家核物理研究所）；Università di Napoli Federico II（那不勒斯费德里科二世大学）
