Is distance between particles relative? Poincare invariance?

[ellipsis]

If you shift the universe five meters to the left, there is no observational change.

If you rotate the entire universe, the inertial frame is also rotated, and there is no observable change.

If you freeze time in the universe for one billion years, then resume it, there is no observable change.

If you boost the universe's velocity to five m/s, there is no observable change.

You see where I'm going with this? If you increase the distance between all particles by 5 meters, is there an observable change? I know this has something to do with Lorentz/Poincare invariance, Galilean groups, etc. But I have no idea what those words mean.

Imagine you have 3 gravitational particles of equal mass, with various velocities. If you scale their distances and velocities by five, will the evolution of the system change? If position, distance, and time are all arbitrary notions, what is mathematically the observationally best way to describe a classical system? Angles? Shapes?

 

[Vannay]

Increasing the distance between all particles is pretty different than the situations you mentioned. In shifting the universe, rotating it, freezing time, "boosting", those are effectively changing the coordinate system. Increasing the distance between particles... Well that's a little vague. How are you increasing the distance between them?

I would say position, time, velocity, etc. are relative notions but definitely not arbitrary.

My knowledge of this area is very very limited so I can't really comment much further. That being said if what you're interested in has something to do with the Poincare group and the related topics, I'd learn more about them because they're essential to this argument. If you have no idea what the fundamental principles are, how can you really expect to find a deeper understanding of what they describe?

 

[mfb]

If you increase the distance between all particles by 5 meters, is there an observable change?

This is not possible. Imagine 5 particles very closeby (like 0.00000001m). After that process you would have to find 5 positions where all distances are extremely close to 5 meters. Those positions do not exist.

You could increase every distance by a factor of 5. If you also adjust all the laws of the universe in the right way, this is indeed a change we could not notice. This leads to the more fundamental question: how do you define a distance independent of objects or processes in the universe? You cannot - exactly for this reason.
You can express distances as multiples of the Planck length (this is a length constant derived from the laws of physics). If you change those numbers, the system will always change.

If you freeze time in the universe for one billion years

What does "freezing for a billion years" mean if there is no time to pass that could be a billion years?

 

[ellipsis]

What does "freezing for a billion years" mean if there is no time to pass that could be a billion years?

mfb: Exactly! What does time mean if we can only measure it relative to other things? Freezing the universe and then resuming it does not refer to any observable facts, and thus is nonsense. If "years" are defined to be revolutions of the Earth around the sun... and the Earth is not revolving...

An epiphany: If you 'slow down' time (whatever that means), can we detect it? I think not, because we'd slow down too. If you took the limit of the 'time factor' as it went to zero, would you be able to say that stopping time indefinitely is also unobservable? I'm not sure about the correctness of taking limits as a means of inference. Anyway, that result is GIGO.

This leads to the more fundamental question: how do you define a distance independent of objects or processes in the universe? You cannot - exactly for this reason.

What would be some better ways to represent position/distance/"relation of positiony stuff" then? I say, (for at least a 2d simulation) you take the center of gravity, and then list out the angles between the lines which intersect the points and their centroid. You can then, using trigonometry, derive the 'shape' that the points form. Surprisingly, that's enough data. Useless for practical work though.

I'm hoping a mathematician is going to come in here and explain that this is already a thing, and recommend me a book or site or video.

 

[mfb]

What would be some better ways to represent position/distance/"relation of positiony stuff" then? I say, (for at least a 2d simulation) you take the center of gravity, and then list out the angles between the lines which intersect the points and their centroid. You can then, using trigonometry, derive the 'shape' that the points form. Surprisingly, that's enough data. Useless for practical work though.

Distances in units of the Planck length, as I said, and velocity relative to the speed of light. And some more stuff for fields like the electromagnetic field. In some Lorentz-invariant way - not sure what a good parameter set would be for that, but at least in principle this is possible. Gauge invariance will also become relevant if you want to do this in quantum field theory.

 

[Unified28]

If you shift the universe five meters to the left, there is no observational change.

If you rotate the entire universe, the inertial frame is also rotated, and there is no observable change.

If you freeze time in the universe for one billion years, then resume it, there is no observable change.

If you boost the universe's velocity to five m/s, there is no observable change.

You see where I'm going with this? If you increase the distance between all particles by 5 meters, is there an observable change? I know this has something to do with Lorentz/Poincare invariance, Galilean groups, etc. But I have no idea what those words mean.

Imagine you have 3 gravitational particles of equal mass, with various velocities. If you scale their distances and velocities by five, will the evolution of the system change? If position, distance, and time are all arbitrary notions, what is mathematically the observationally best way to describe a classical system? Angles? Shapes?

How can the universe rotate? Consider it this way. Space has the property to define positions. Then if space rotates, what does it rotate relative to? Itself? Then it does not rotate.

Time is always frozen since infinite events do not happen at the same time. Synchronizing events (whatever an event may be) would possibly result in a difference. Freezing time would result in synchronizing events, if those events that occur during freezing would be canceled.

Again the universe cannot move relative to nothing. If you boost it relative to all objects it would contradict relativity in many possible models.