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You should probably read up on the Stueckelberg action and the Affine Higgs mechanism it sends you to. Your boldface supposition "I'm not supposing that my matter has any global symmetry here, that I might be able to gauge" is unwarranted for the specific model you propose, $J_\mu=\partial_\mu\phi$. There is a global symmetry, $\phi \to \phi+\alpha$, whose ...


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You're right that that Lagrangian isn't in general gauge invariant. In addition to making the $A^\mu$ terms gauge invariant, the $\mathcal{L}(J)$ term must also be gauge-invariant. And not just the equations of motion for $J_\mu$, either - the specific algebraic expression for $J_\mu$ in terms of fundamental fields must be such that if you literally plug ...


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You applied it already Decrease in gravitational potential energy of block 2 = Increase in gravitational potential energy of block 1 + Increase in kinetic energy of block 1 + Increase in kinetic energy of block 2 How do you justify this statement? You have to invoke the conservation of energy. If you wanted to work Nother's theorem into this, it ...


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Although users35736's answer is certainly correct, and the question is old, I think it should be noted that each Killing vector, $\xi^i$, also gives rise to a Noether current: $J^i = T_j{}^i\xi^j$. First note that $$ J^i{}_{;i} = T^{ji}\xi_{i;j} + \xi_jT^{ji}{}_{;i} = 0, $$ by the Killing equation $\xi_{(i;j)} = 0$, the symmetry of the stress-energy tensor $...



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