There is indeed a path. Note a few potential “loopholes”:
There are (unphysical) Newtonian physics systems where it is possible to approach negative-infinite potential energy in finite time. So yes, strictly speaking energy is conserved, but that doesn’t actually say that much.
(Roughly speaking: the top 2 bodies and the center body undergo a 3-body encounter that drops the top 2 bodies into a smaller orbit, using the resulting potential energy to accelerate the top 2 bodies upward and accelerates the middle body towards the bottom 2 bodies faster than it arrived. Repeat, mirrored, with the bottom 2 bodies. Repeat, mirrored, with the top 2 bodies. Repeat, mirrored, with the lower 2 bodies. Etc. Each loop pulls gravitational potential energy from the 2 sets of 2 bodies and dumps it into kinetic energy, and the center body gets faster faster than the 2 sets of 2 bodies pull apart. Net result is an infinite number of 3-body encounters and infinite velocity in finite time...)
It relies on (continuous) time-translation symmetry.
This doesn’t hold in general for general relativity.
This does hold for Newtonian mechanics.
Time translation symmetry is a hypothesis, although a fairly well-tested one.
(If you want to rabbit-hole here, look at time crystals.)
(That being said, it’s been too long since I’ve looked seriously into Physics.)
...honestly, probably not well. It’s been too long. At a high level: Noether’s theorem implies that if you have a Lagrangian that’s invariant under a perturbation of coordinates, that corresponds to a conserved quantity of the system. In particular: invariance under time perturbations (a.k.a. continuous time-translation symmetry) corresponds to a conserved quantity that turns out to be conservation of energy.
Also, do you consider your loopholes like technicalities, or more serious problems?
For 1: it’s like someone showing how to break your 1024-bit hash in 2^500 operations. It isn’t a problem in and of itself, but it’s suggestive of deeper problems. (It requires both infinite precision and point particles to achieve, neither of which appear to be actually possible in our universe.)
For 2: I’d consider the issues with general relativity (and however quantum gravity shakes out) to be potentially an issue—though given that it’s not an issue for classical mechanics any loopholes would likely be in regimes where the Newtonian approximation breaks down.
That all being said, take this with a grain of salt. I’m not confident I remembered everything correctly.
There is indeed a path. Note a few potential “loopholes”:
There are (unphysical) Newtonian physics systems where it is possible to approach negative-infinite potential energy in finite time. So yes, strictly speaking energy is conserved, but that doesn’t actually say that much.
(For instance: https://en.wikipedia.org/wiki/Painlev%C3%A9_conjecture#/media/File:Xia’s_5-body_configuration.png )
(Roughly speaking: the top 2 bodies and the center body undergo a 3-body encounter that drops the top 2 bodies into a smaller orbit, using the resulting potential energy to accelerate the top 2 bodies upward and accelerates the middle body towards the bottom 2 bodies faster than it arrived. Repeat, mirrored, with the bottom 2 bodies. Repeat, mirrored, with the top 2 bodies. Repeat, mirrored, with the lower 2 bodies. Etc. Each loop pulls gravitational potential energy from the 2 sets of 2 bodies and dumps it into kinetic energy, and the center body gets faster faster than the 2 sets of 2 bodies pull apart. Net result is an infinite number of 3-body encounters and infinite velocity in finite time...)
It relies on (continuous) time-translation symmetry.
This doesn’t hold in general for general relativity.
This does hold for Newtonian mechanics.
Time translation symmetry is a hypothesis, although a fairly well-tested one.
(If you want to rabbit-hole here, look at time crystals.)
(That being said, it’s been too long since I’ve looked seriously into Physics.)
Thanks for the comment!
Could you give more details on the path itself?
Also, do you consider your loopholes like technicalities, or more serious problems?
...honestly, probably not well. It’s been too long. At a high level: Noether’s theorem implies that if you have a Lagrangian that’s invariant under a perturbation of coordinates, that corresponds to a conserved quantity of the system. In particular: invariance under time perturbations (a.k.a. continuous time-translation symmetry) corresponds to a conserved quantity that turns out to be conservation of energy.
For 1: it’s like someone showing how to break your 1024-bit hash in 2^500 operations. It isn’t a problem in and of itself, but it’s suggestive of deeper problems. (It requires both infinite precision and point particles to achieve, neither of which appear to be actually possible in our universe.)
For 2: I’d consider the issues with general relativity (and however quantum gravity shakes out) to be potentially an issue—though given that it’s not an issue for classical mechanics any loopholes would likely be in regimes where the Newtonian approximation breaks down.
That all being said, take this with a grain of salt. I’m not confident I remembered everything correctly.