Just because you can think of a possibility does not make it possible. In the absence of classical mechanics, finite temperature will cause it to fall in very finite time. With quantum mechanics, quantum zero point fluctuations will cause it to fall in finite time even if it was at zero temperature.
Finite temperature will cause it to fall in a finite time if you start with it balanced perfectly. You just need to tilt it a little to counter that. This is an argument by continuity, not an argument by symmetry.
There’s some set of starting positions that result in it falling to the left, and another set that result in it falling to the right. If you start it on a boundary point and it falls right after time t, that means that you can get a point arbitrarily close to it that will eventually fall left, so is clearly nowhere near that at time t. That means that physics isn’t being continuous.
Finite temperature will cause it to fall in a finite time if you start with it balanced perfectly. You just need to tilt it a little to counter that.
You cannot counter it by tilting it because the thermal perturbation is random. At one moment is being pushed to the left, at another to the right.
f you start it on a boundary point and it falls right after time t, that means that you can get a point arbitrarily close to it that will eventually fall left, so is clearly nowhere near that at time t. That means that physics isn’t being continuous.
To be clear, I don’t want to argue against the hypothesis. As long as you are NOT talking about the real world, which is what I take “physics” to mean, you can talk about continuous and balance point, and arbitrarily long times to fall one way or the other. The point is that in the real world, any real world experiment analyzed in detail will have “noise” which causes it to fall sometimes one way and sometimes the other in finite time when placed at its balance point. That noise is usually dominated by thermal fluctuations, but even in the absence of thermal fluctuations, there are still quantum fluctuations which behave very much like thermal noise, which behave in very many ways as though you can’t get the temperature below some limit.
In principle, you can build a system where the thing is cooled enough, and where it is designed so that the quantum fluctuations are small enough, that you can have a relatively long time for it to fall one way or the other off its balance point. That is, the amount of time it takes to be pushed by quantum noise is large on human or intuitive scales. However, you can’t make it arbitrarily large without building an arbitariily large system. So if you are concerned about the difference between “really big” and “infinite,” you don’t get to claim a physical system balanced on an unstable equilibrium point can have an infinitely long time to fall.
Of course this doesn’t mean that in a non-physical “world” with no quantum and rigid objects (another approximation that can’t be realized in the real world) you couldn’t do it. So the math is safe, its just not physics.
You cannot counter it by tilting it because the thermal perturbation is random.
Randomness is an attribute of the map, not the territory. You do not know how much to tilt it, but there is still a correct position.
This isn’t something you can feasibly do in real life. It’s not hard to make it absurdly difficult to find the point at which you have to position the needle. It’s just that there is a position.
You do not know how much to tilt it, but there is still a correct position.
The amount to tilt it takes is changing in time. Thermal and/or quantum fluctuations continue to start the thing falling in one direction or another, and you have to keep seeing how it’s going and move whatever your balancing it on to catch it and stop it from falling.
You have created a dynamic equilibrium instead of a static one by using active feedback based on watching what the random and thermal noise are doing. You have not created a situation where an unstable equilibrium takes an arbitrarily long time to be lost. You have invented active feedback to modify the overall system into a stable equilibrium.
If you tilt it the correct way, it will not just stand there. It will fall almost perfectly into every breeze that comes its way. Almost, because it has to be off a little so that it will fall almost perfectly into the next breeze.
You can’t keep it from ever tilting, but that doesn’t mean that you can’t keep it from falling completely.
Falling to the left with t==1 second also has probability zero. Remaining balanced for a period of time between a google and 3^^^3 times the current age of the universe, then falling left, has positive probability.
There is no upper bound to the amount of time that the ball can remain balanced in a continuous deterministic universe.
No. It almost certainly will fall eventually, but there is at least one possibility where it never does.
Just because you can think of a possibility does not make it possible. In the absence of classical mechanics, finite temperature will cause it to fall in very finite time. With quantum mechanics, quantum zero point fluctuations will cause it to fall in finite time even if it was at zero temperature.
Finite temperature will cause it to fall in a finite time if you start with it balanced perfectly. You just need to tilt it a little to counter that. This is an argument by continuity, not an argument by symmetry.
There’s some set of starting positions that result in it falling to the left, and another set that result in it falling to the right. If you start it on a boundary point and it falls right after time t, that means that you can get a point arbitrarily close to it that will eventually fall left, so is clearly nowhere near that at time t. That means that physics isn’t being continuous.
You cannot counter it by tilting it because the thermal perturbation is random. At one moment is being pushed to the left, at another to the right.
To be clear, I don’t want to argue against the hypothesis. As long as you are NOT talking about the real world, which is what I take “physics” to mean, you can talk about continuous and balance point, and arbitrarily long times to fall one way or the other. The point is that in the real world, any real world experiment analyzed in detail will have “noise” which causes it to fall sometimes one way and sometimes the other in finite time when placed at its balance point. That noise is usually dominated by thermal fluctuations, but even in the absence of thermal fluctuations, there are still quantum fluctuations which behave very much like thermal noise, which behave in very many ways as though you can’t get the temperature below some limit.
In principle, you can build a system where the thing is cooled enough, and where it is designed so that the quantum fluctuations are small enough, that you can have a relatively long time for it to fall one way or the other off its balance point. That is, the amount of time it takes to be pushed by quantum noise is large on human or intuitive scales. However, you can’t make it arbitrarily large without building an arbitariily large system. So if you are concerned about the difference between “really big” and “infinite,” you don’t get to claim a physical system balanced on an unstable equilibrium point can have an infinitely long time to fall.
Of course this doesn’t mean that in a non-physical “world” with no quantum and rigid objects (another approximation that can’t be realized in the real world) you couldn’t do it. So the math is safe, its just not physics.
Randomness is an attribute of the map, not the territory. You do not know how much to tilt it, but there is still a correct position.
This isn’t something you can feasibly do in real life. It’s not hard to make it absurdly difficult to find the point at which you have to position the needle. It’s just that there is a position.
The amount to tilt it takes is changing in time. Thermal and/or quantum fluctuations continue to start the thing falling in one direction or another, and you have to keep seeing how it’s going and move whatever your balancing it on to catch it and stop it from falling.
You have created a dynamic equilibrium instead of a static one by using active feedback based on watching what the random and thermal noise are doing. You have not created a situation where an unstable equilibrium takes an arbitrarily long time to be lost. You have invented active feedback to modify the overall system into a stable equilibrium.
If you tilt it the correct way, it will not just stand there. It will fall almost perfectly into every breeze that comes its way. Almost, because it has to be off a little so that it will fall almost perfectly into the next breeze.
You can’t keep it from ever tilting, but that doesn’t mean that you can’t keep it from falling completely.
Your argument now boils down to “the physical world is not both continuous and deterministic”.
With probability zero.
Falling to the left with t==1 second also has probability zero. Remaining balanced for a period of time between a google and 3^^^3 times the current age of the universe, then falling left, has positive probability.
There is no upper bound to the amount of time that the ball can remain balanced in a continuous deterministic universe.