DaPi said:
non-conservation of energy implies non-invariance under displacement in time.
Yes, that's true. And our universe is not invariant under displacement in time (because it is expanding, so you can tell "what time it is" by how much the universe has expanded), and therefore does not conserve energy. See this article by Sean Carroll:
http://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/
DaPi said:
Which would mean experiments would not be repeatable, and we wouldn't be able to do science as normally accepted.
No, that's not correct. Experiments only have to be "repeatable" if all the conditions that can affect the outcome are held constant. If the universe is not invariant under displacement in time because it's expanding, then whenever you do an experiment measuring energy conservation, the "size of the universe" becomes a condition that can affect the outcome; so the only way to truly "repeat" an experiment would be to go back in a time machine and do it again at exactly the same universe size (which of course we can't actually do). In practice, of course, the expansion of the universe is slow enough, relative to our ordinary experience, that we can treat energy as conserved. But if we're talking about fundamentals, we have to take all factors into account, and the expansion of the universe is one of them.
What is really going on here is that you are confusing the properties that the laws of physics, themselves, must have, with the properties that particular
solutions to those laws must have. The laws of physics are invariant under time translations. However, our particular universe, which is one solution to those laws, is not, because the invariance under time translations is a property of the whole
set of possible solutions, not of any particular solution by itself. So if one solution to the laws describes an expanding universe, there will also be a corresponding solution that describes a contracting universe with all other parameters the same. We know we're in the first solution (expanding) and not the second (contracting) through observation.
Also, it's important to point out, as Carroll does in his article, that when we say energy is not conserved in the universe as a whole, we are
not saying that "anything goes". All we are saying is that spacetime obeys a somewhat different law than you thought it did. You thought the law was "spacetime has to be time translation invariant, so energy is always conserved". The actual law is more like "the change in spacetime when you do a time translation has to be related to the change in the matter and energy under the same time translation, in a particular way". (The equation that governs this is called the "Einstein Field Equation". The relativity forum is the place to go if you want to discuss this in more detail.) One important consequence of this is that locally, energy
is conserved, in the sense that matter and energy cannot be created or destroyed. So you still can't build a perpetual motion machine, even though the universe as a whole is not time translation invariant.
