> 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.
...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.