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I have been reviewing some relativity notes, and I am confused over something. I apologize if this seems like a silly or obvious point, but humor me.

When we are talking about Lagrangians in field theory and in regular mechanics, we are often looking at symmetries. Namely, almost all physical theories in the universe have the symmetry of Lorentz covariance, which as I understand it means the Lagrangians transform according to a representation of the Lorentz group. Does that necessarily mean the action is still stationary if we do a Lorentz transformation? If so, what would be the relevant 'Noether's charge'? Would it be the invariant dot-product E^2 - p^2 c^2?

And then, a part of me wonders what the relationship is between Lorentz covariance and spacetime translational invariance, which has as its 'Noether's charge' the relativistic stress-energy tensor, effectively amounting to overall relativistic energy and momentum conservation. Is this relativistic energy and momentum conservation a fundamental property of all physical theories like Lorentz covariance? I had for the longest time thought so, and so I had kind of thought that relativistic energy and momentum was the Noether's charge of the Lorentz covariance symmetry, but now I don't know about that at all.

I ask this question because it occurred to me, the relativistic Navier-stokes equations of fluid dynamics are derived simply by writing the Lorentz covariant Lagrangian, and writing down the equations for relativistic energy and momentum conservation. If relativistic energy and momentum conservation is fundamental among all known physical theories (amounting effectively to the fact that all physical theories have spacetime translational invariance), then we should be able to do a similar writing out of the equations for any such theory. But I don't see people doing that, so I fear I have it all wrong.

Basically, can someone clear up my confusion between Lorentz covariance and spacetime translational invariance? Which ones are more fundamental to all known physical theories? What are the Noether charges/currents? I again am sorry if this seems silly, it is just something to ponder.

Thanks.