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In those scales (cosmology+Field Theory) energy is derived and defined as the "thing" that is conserved after you apply time symmetry.

Also you kinda want time symmetry in a physical system otherwise you have no guarantee that today's laws of physics will be valid tomorrow.






As much as we want time symmetry, we still live in an expanding universe. So we have at least some asymmetry.

Time symmetry applies to the physical laws governing the universe, not the distribution of mass in it.

For a simpler example look at Newton's 3 laws. They have time symmetry but still allow for things to move around!


It also doesn't apply to the structure of space-time. Which is not a situation that Newton's laws can deal with.



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