11 hours ago, npts2020 said:

However, I have never heard of a documented case where the law of conservation of energy was violated and never seen a plausible description of a case where it could possibly be so.

You won’t ever see one, at least not within the domain of Newtonian and SR physics. There are fundamental reasons why this conservation must hold.

In GR however, the situation is a little more subtle, as in regions of substantially curved spacetime there is no simple “law of energy conservation”. It’s possible to still formulate something along these lines, but one has to account for the self-interaction of gravity.

4 hours ago, Markus Hanke said:

one has to account for the self-interaction of gravity

I don't know about "one has to account for the self-interaction of gravity", but general relativity has a somewhat peculiar form of energy conservation that is the zero covariant divergence of the energy-momentum tensor field, a form that is true in all coordinate systems of an arbitrarily curved spacetime. However, due to the restriction of covariance and the nature of the energy-momentum tensor field, unlike a conserved vector field, conservation of the energy-momentum tensor field as it is normally understood only works locally. That is, there is no integral form of the conservation of the energy-momentum tensor field that would provide a global notion of the conservation of energy-momentum. By contrast, for a conserved vector field, the generalised Stokes theorem provides the integral form from the differential form that allows the global notion of conservation.

Edited Thursday at 12:05 PM3 days by KJW

57 minutes ago, KJW said:

I don't know about "one has to account for the self-interaction of gravity", but general relativity has a somewhat peculiar form of energy conservation that is the zero covariant divergence of the energy-momentum tensor field, a form that is true in all coordinate systems of an arbitrarily curved spacetime. However, due to the restriction of covariance and the nature of the energy-momentum tensor field, unlike a conserved vector field, conservation of the energy-momentum tensor field as it is normally understood only works locally. That is, there is no integral form of the conservation of the energy-momentum tensor field that would provide a global notion of the conservation of energy-momentum. By contrast, for a conserved vector field, the generalised Stokes theorem provides the integral form from the differential form that allows the global notion of conservation.

Nicely put +1

5 hours ago, swansont said:

I would expect that invoking GR in the explanation of an Atwood machine (the context of the OP’s snark and/or insinuation) would be…unusual. But we’re talking about the functioning of the forum here, not physics

Still. if any good science comes from the discussion, I think it is worthwhile. I rather liked Markus and KJWs' clarification of what I wrote since I was thinking on engineering scales rather than quantum ones. What I find hard to believe is that anyone who actually discovered something like perpetual motion or "free energy" would be talking about it on a science forum instead of making billions of dollars building and selling the machines. Even if they just licensed someone else to do that, they would become very wealthy.

Edited Thursday at 07:27 PM3 days by npts2020

17 hours ago, KJW said:

I don't know about "one has to account for the self-interaction of gravity"

I think this was badly formulated on my part, apologies. What I meant was really that, when talking about conservation laws, the energy content of the gravitational field itself must be taken into account. This energy content is unfortunately not by itself a covariant quantity. At the same time, the covariant divergence also introduces curvature-related terms (as you correctly pointed out), which, when taken in isolation, are also not covariant. The only way around this is to introduce a new quantity that accounts for both energy-momentum of sources, and the energy content of gravity itself - it is only that particular combination that can be said to be conserved. This can be done, but I don’t know if this construction is unique.