Are conservation laws independent of field equations?

Abstract

The charge conservation law ($\partial_{\alpha} J^\alpha =0$) usually is considered as a corollary of Maxwell's equations and is not independent of Maxwell's field equations. A circular reasoning, however, is found in the derivation. A similar fallacy exists in the matter source's conservation law ($\nabla^{\mu}T_{\mu\nu}=0$) and Einstein's field equations. Therefore, the source's conservation laws are independent of Einstein's field equations.