What are the experimental or rather observable consequences of the non-conservation (or conservation) of energy in GR? Imagine our instruments were $10^3$ or even $10^6$ more sensitive, better resolution and less noisy. Could we observe, not necessarily demonstrate explicitly by experiment here on Earth, just observe, that energy is or is not conserved in GR or in any other varieties of post-Newtonian gravitation?
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The cosmological expansion of the universe pretty manifestly non-conserves energy. Particles climb a potential hill, but also accelerate. The universe fills up with more and more cosmological constant energy density.
You could fix this in some sort of "Hubble Bubble" scenario, where the universe is "really" asymptotically flat, and matter only exists in a subset of the whole universe, and the cosmological constant is some artifact of the overall potential energy function of the whole universe but the expansion would eventually have to stop or asymptote off in such a scenario, as eventually the matter will be so diffuse it just looks like it's moving in a Minkowski background.
answered Nov 1, 2022 at 14:49
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$\begingroup$ What do you mean by "eventually"? Would that be observable somehow today or in a few gigayears? And are there any other known observable consequences or only this? $\endgroup$ Commented Nov 1, 2022 at 14:54
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$\begingroup$ @hyportnex it would depend on the size of the hubble bubble, but yeah, you'd expect effects to unfold over cosmological time, not "human time" $\endgroup$ Commented Nov 1, 2022 at 15:53
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$\begingroup$ but what effects are you expecting? In GR, energy non-conservation is an effect derived from boundary conditions of the spacetime, and so, you only expect to see energy non-conservation at cosmological scales. $\endgroup$ Commented Nov 1, 2022 at 15:54
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$\begingroup$ I do not expect anything, I am completely naïve in this, but say gravitational lensing, etc, is an observation on "galactic" spatial/temporal scales to verify space-time curvature and we can see that from here. Are there such observational effects of said boundary conditions resulting in non-conservation or is that can be had on cosmological scales only on which we can only speculate excepting the expansion of the universe? $\endgroup$ Commented Nov 1, 2022 at 16:23
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$\begingroup$ @hyportnex: I'm guessing I"m saying, even now, observationally, you can go and look out at the cosmology, and you went and took a newtonian picture of the universe, you'd see "$dE/dt > 0$ if you went and added all of the contributions from the individual stars and galaxies. The universe's rate of expansion is accelerating, after all, and this has been directly observed. $\endgroup$ Commented Nov 1, 2022 at 16:53
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Could we observe, not necessarily demonstrate explicitly by experiment here on Earth, just observe, that energy is or is not conserved in GR or in any other varieties of post-Newtonian gravitation?
One simple way to observe nonconservation of energy in gravity is to become an observer in a state of free fall, where atmospheric resistance does not stabilize the speed. Huge celestial objects are accelerating in that frame, some of them gaining lots of kinetic energy from nothing, and some of them losing it to nothing. For example, when falling out of a building or mountain, in the falling person's frame, the most prominent body increasing its kinetic energy by immense amounts is the Earth.
In general relativity, all frames are "equivalent", so we can't dismiss this by "it's just because we're looking at energy in a bad frame" like we would in classical mechanics. If energy conservation works in one frame but not in others, then in GR, it does not work.
answered Nov 1, 2022 at 21:29
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$\begingroup$ would your argument be also valid for linear or angular momentum (non)conservation and what does this have to do with the problem of boundary conditions Schirmer is referring to? $\endgroup$ Commented Nov 1, 2022 at 21:59
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$\begingroup$ I think so. Linear momentum is not conserved in the falling observer frame. Angular momentum nonconservation happens e.g. with angular momentum of the Moon in Earth's frame (the distance between the two increases by cca 4cm/year, while Earth's rotation slows down; if this conserves angular momentum in independent inertial frame, then it probably does not in Earth's frame). It seems that if we want to treat all frames as equivalent, we have to abandon any hope for universal conservation laws. $\endgroup$ Commented Nov 1, 2022 at 22:21
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$\begingroup$ Jerry Schirmer's point is about specific well known toy models of cosmos, which manifest energy nonconservation (accelerating expansion...). My point is more about the problem with some frames, which does not depend on any specific model of the universe. Conservation of energy or momentum seems to be a special property of certain class of frames under certain conditions, not to be expected in general. $\endgroup$ Commented Nov 1, 2022 at 22:26
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$\begingroup$ Just one more question. Naively I thought that the "energy non-conservation" is a bit more "fundamental", if that is the right word, than just the relativity of all frames in the sense similar to the "steady-state model" in which matter is continuously created. Not saying they are the same just something more like that. Is that the wrong interpretation? $\endgroup$ Commented Nov 1, 2022 at 22:40
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$\begingroup$ That meaning of energy nonconservation (production) is indeed different than what I explained. It is also a little harder to demonstrate, and probably requires additional assumptions about the system, like the toy models have (Friedmann's equation, presence/non-presence of some terms...). My point is that if we assume only the basic GR (which has general covariance), then we can still observe why energy conservation isn't a universal law but a special condition valid in some frames. $\endgroup$ Commented Nov 1, 2022 at 22:49