Are conservation laws independent of field equations?

TL;DR

This paper examines whether conservation laws are truly independent of their associated field equations, revealing that charge conservation is not independent of Maxwell's equations, but matter source conservation is independent of Einstein's equations.

Contribution

It clarifies the logical independence of conservation laws from their respective field equations, challenging common assumptions in theoretical physics.

Findings

Charge conservation is not independent of Maxwell's equations.

Matter source conservation laws are independent of Einstein's field equations.

Highlights a logical fallacy in the derivation of conservation laws.

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.