麦克斯韦方程组对于局部守恒量是普适的
Maxwell's equations are universal for locally conserved quantities
摘要：
经典电磁学的一个基本结果是麦克斯韦方程组意味着电荷是局部守恒的。在这里，我们表明相反：当地的电荷守恒意味着当地存在的领域满足麦克斯韦方程组。这适用于任何满足连续性方程的守恒量。它是通过这里提出的庞加莱引理的一个强形式得到的，该引理指出：无发散的多向量场在平坦流形上局部具有无卷曲的反导数。上面的逆命题是这个引理在时空中无发散向量场情形中的应用。我们还提供了条件下，结果推广到弯曲的流形。
英文摘要：
A fundamental result of classical electromagnetism is that Maxwell's equations imply that electric charge is locally conserved. Here we show the converse: Local charge conservation implies the local existence of fields satisfying Maxwell's equations. This holds true for any conserved quantity satisfying a continuity equation. It is obtained by means of a strong form of the Poincaré lemma presented here that states: Divergence-free multivector fields locally possess curl-free antiderivatives on flat manifolds. The above converse is an application of this lemma in the case of divergence-free vector fields in spacetime. We also provide conditions under which the result generalizes to curved manifolds.
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