It shows that Newtonian mechanics is only an approximation of the real world.
It doesn't need an infinitely sharp corner. It's weird right at the very top.
We do know that Newton is only an approximation, but general relativity doesn't seem to help here. Quantum mechanics might, but the problem itself doesn't seem to hint at it.
It seems to nod in the direction of involving entropy at a deep level, but it's unclear how.
Our models are continuous. The map is not the territory.
> Various aspects of quantum field theories would not work in a discretized universe.
If you're referring to symmetries, that's not true [1]. Discretized theories have received barely any attention because physicists have simply built upon the formalisms they're used to from classical physics. We see people frequently saying that we should abandon classical intuitions now that we have quantum mechanics, but we haven't really even tried to abandon classical formalisms, which are just another form of classical intuition.
IMO, baking continuity and uncountable infinities is what's responsible for many of the difficulties we've had, and various forms of discretization have resolved these issues in the past [2]. My conjecture is that one of the next big advances in physics will be to take discretization seriously.
Because all our evidence is that it is continuous. Our models are the best description of the evidence we observe -- i.e. the territory.
Unless you have evidence of time and space being non-continuous?
No, we haven't really tested this to any appreciable degree, and existing tests rule out only certain types of discrete models just like they rule out certain continuous models. Our models are continuous because continuous models are what we've been using for centuries. I'm not sure why anyone still believes that science is a purely evidence-driven endeavour. History has clearly shown it's subject to fads, celebrity and inertia.
> Unless you have evidence of time and space being non-continuous?
General relativity produces unphysical nonsense like singularities, which is a form of internal inconsistency similar to Norton's dome. This isn't evidence that spacetime is discrete so much as that our continuous model is incomplete, but it seems quite clear that a discrete analogue of GR would not suffer from this problem.
Furthermore, some new approaches to gravity are discrete theories, like loop quantum gravity, so clearly some physicists are already thinking along these lines.
I don't know that you mean that this hasn't been tested. Essentially every experiment ever is consistent with it.
Experiments that try to distinguish the fine structure of spacetime haven't really been a focus, and are probably impossible in a lab environment. Some studies of distant gamma ray bursts place some limits on some discrete theories, but those limits don't apply to covariant discrete theories that preserve other symmetries like the one I posted above.
It's that it doesn't even make sense within Newtonian mechanics.
It's a much more fundamental issue to do with math itself, really.
How can math that is reversible in time suddenly not be deterministically reversible in this one very special situation?
Usually math deals with perfectly sharp corners and infinitesimal distances just fine.
I find it quite interesting that Newtonian mechanics only works if this condition is satisfied. Of course you cannot make a non-Lipschitz continuous force in practice, since the tip radius will be limited to the size of atoms, but theoretical physics is usually not concerned with such trivialities.
But yea I don’t think this shows anything general about Newtonian mechanics.
The sharp corner isn’t the problem, and there are plenty of setups with sharp corners that work just fine.
> Zeno-paradoxical infinitely small jump from the initial position to some other position after time T
Every acceleration does that.
The issue here is the way that the ball gets to the apex in finite time instead of approaching it asymptotically as t goes to infinity.