# Infinite speeds would defeat time keeping

1. Dec 6, 2011

### lalbatros

Obviously, infinite speeds would defeat time keeping.

Infinite speed would imply that a particle could be at the same time at the start point of its trajectory and at the same time at the end point of it.
At the least, we should not call that a particle anymore, if it is able to be at two positions at the same time.

But if we are able to observe this entity as being a particle, then we should be able to distinguish the two positions with our clocks.
Then, we should simply improve the way we build (or chose) our clocks,
Since clock is the empirical way to define time, we can only conclude that infinite speeds are impossible, by principle.

This brings me to my question:

Could that point of view be sufficient to build the special theory of relativity?
Could that point of view imply some limit speed for particles?

2. Dec 6, 2011

### PhilDSP

Some thoughts on this:

SR and LET deal principally with wave propagation (energy translation). Neither really makes many (if any) claims about particles other than the obvious variance of momentum with particle velocity.

There are gauges in EM where the potential propagates instantaneously and that is quite meaningful in obtaining solutions to EM problems. Does that defeat time keeping? Probably not, any clock associated with energy propagation continues to tick.

3. Dec 6, 2011

### lalbatros

In other words:

By simply assuming that infinite speeds are impossible, would it be possible:

- to derive the existence of a speed limit
- and/or to derive the Lorentz transformation

This question was inspired by reading The Nature of Time , a paper by Julian Barbour.

Equation (3) there illustrates how time can be (quite logically) related to change.
Therefore, if the position of a particle is changed, time must also change.
No infinite speed!

How far is this from the existence of a speed limit?
And how far is this from the Lorentz transformation?

4. Dec 6, 2011

### Fredrik

Staff Emeritus
If we assume that the underlying set of spacetime is $\mathbb R^4$, that functions that represent a coordinate change from an inertial coordinate system to another form a group, and that each such function satisfies a few simple mathematical conditions (in particular that it takes straight lines to straight lines), it's possible to show that this group is isomorphic to either the Galilean group or the Poincaré group. Both possibilities imply that spacetime contains lines that represent motion at a speed that's the same in all inertial coordinate systems. The first possibility (Galilean group) implies that this speed is infinite. The second (Poincaré group) implies that it's not. The second possibility leads to special relativity, because the Poincaré group is the isometry group of Minkowski spacetime.

However, your entire argument for the non-existence of infinite speeds seems to be that if a particle could move at infinite speed, you wouldn't want to call it a particle. I don't think this is very convincing. I don't think there is any stricly logical/mathematical argument that can rule out infinite speeds.

5. Dec 6, 2011

### lalbatros

Fredrik,

I used the word particle merely to connect with the paper by Barbour about the nature of time.
His discussion, based on the ephemeris time, illustrates concretely the common talk that "time is change", but it is based on particles from classical mechanics.
From this point of view, any displacement of only one single particle would imply a new time.
Seeing time as an accounting device for change, infinite speeds would have no meaning.

However, starting from Newtonian mechanics, this probably doesn't really imply a limit speed.
Nor, probably, does it imply a preference for the Lorentz group over the Galilean group.

Nevertheless, I ask myself how long is the distance between assuming that "time is change" and assuming that "there is a speed limit".

I read some time ago that, in a five gravitating bodies galilean system,
one of the bodies could reach an infinite speed in a finite time without ever passing through a potential singularity. (1, 2, 3)
I could not read the paper in detail, it was far above my technical skills.
However, if that was true, I would almost see such a result as an internal contradiction of Galilean Relativity.
And this might mean that only the Lorentzian group could make sense.
(but it could mean something else as well)

Michel

(1) The Existence of Noncollision Singularities in Newtonian Systems. Zhihong Xia.

(2) http://physics.stackexchange.com/qu...te-to-infinite-speed-in-finite-time-newtonian

(3) http://plus.maths.org/content/outer-space-twos-company-threes-crowd

Last edited: Dec 6, 2011
6. Dec 6, 2011

### PatrickPowers

> Infinite speed would imply that a particle could be at the same time at the start point of its trajectory and at the same time at the end point of it.
At the least, we should not call that a particle anymore, if it is able to be at two positions at the same time.

I guess if the speed truly were infinite then the particle would be everywhere on the line that includes the two particles at the same time. Well, if that is what it does, then that is what it does. I don't see how this leads to a contradiction in this imaginary universe.

An infinite speed of light in our presumably infinite Universe is supposed to lead Olbers paradox: the night sky would be completely filled with starlight. But I think the Doppler shift from the expansion shifts distant stars out of the visible range so that wouldn't happen, and the total energy Earth got from stars would not be much.

7. Dec 6, 2011

### lalbatros

Well, I was a bit fast about that.
One of the bodies could be boosted to infinite speed without undergoing any collision.
However, it is obvious that a collision must occur among the other particles, just by energy conservation.
The energy released by this collision can then, according to this paper, be transfer to the fifth particle without any collision with this fifth particle.

Therefore, the infinite speed is simply related to the 1/r potential of point-particles.
The contradiction related to the infinite speed does not point its finger toward relativity, but toward quantum mechanics instead.