That's wrong, because the quoted part is wrong. Relativity doesn't say you can create or destroy energy. It only says that you can convert mass to energy (and vice-versa) - because in the end they are actually the same thing. And together, they are conserved. That means we still can't have perpetuum mobile stuff unfortunately.
You're talking about E=mc^2, which follows from special relativity. That was revealed in 1905; 1915 marked the advent of general relativity, where energy conservation no longer holds.
The time translation invariance which gives rise to the conservation law is a special case of GR's broader energy-momentum conservation, namely the static one where gravity and such are disregarded altogether as in the Standard Model.
This all ties back to the present crisis of foundations, as string theory and other approaches to reconciling GR with the Standard Model strain at the edges of what Noetherian tools can yield. (see: supersymmetry)
Nope. It's just a bit more complex to define what "energy" even is on a dynamical spacetime (remember that our usual constant known as time is part of a varying field in GR). But there's nothing stopping you from coming up with an equivalent conserved current due to a global symmetry as laid out by Noether. This fact is even used e.g. in the Hamiltonian formulation of GR. See here for a detailed explanation: https://physics.stackexchange.com/questions/2597/energy-cons...
This is an old misunderstanding that dates back to the early stages of GR research and has nothing to do with any current crisis.
> remember that our usual constant known as time is part of a varying field in GR
...and so you have to pick an appropriate underlying vector field you call time to cancel this out and get back the invariant, throwing a wrench into calculations... as a reply to the post you linked points out. At the end of the day, you haven't demonstrated that it preserves the invariant so much as you've changed the question to find another conserved quantity and called that energy instead. This lines up with my broad observation that we're out of runway for the 20th century's symmetry-reliant problem solving and hence have to be increasingly clever with setups to apply generalizations of them.
I definitely learned something new today, though. To boot, these pseudotensors are tamer than I thought they'd be - I expected calculational hacks with no formal analogues explored only in old papers, but sections on jet bundles is something I'd expect in a differential geometry text. Maybe we'll see progress along these lines in the next couple decades.
