Understanding Length Contraction: What is Limit of Vanishing Transport Velocity?

[Philip Dhingra]

I'm trying to understand length contraction from wikipedia, and they mention clock synchronization:

The observer installs a row of clocks that either are synchronized a) by exchanging light signals according to the Poincaré-Einstein synchronization, or b) by "slow clock transport", that is, one clock is transported along the row of clocks in the limit of vanishing transport velocity.​

However, I can't find out what "limit of vanishing transport velocity" means. Google isn't helping either.

 

[PAllen]

It means just what it says. Consider moving the clocks at 1 mph, the .5, then .25, etc. For some distance of transport, for decreasing transport speeds, you will see a trend in the result. Extrapolate to zero transport speed.

 

[Philip Dhingra]

It means just what it says. Consider moving the clocks at 1 mph, the .5, then .25, etc. For some distance of transport, for decreasing transport speeds, you will see a trend in the result. Extrapolate to zero transport speed.

I'm not sure I understand. Trend in what result?

 

[PAllen]

Ok, let me be painfully explicit. Imagine you have many identical clocks at position A. You transport them at varying speeds to position B. At position B, they will be found to differ slightly. As a function of transport speed, you can extrapolate to what reading would occur for zero transport speed. This extrapolated clock setting for clocks at B is what you take to be synchronized with clock at A.

If this still doesn’t help you, I give up. Hopefully someone else can explain if you clarify what you don’t understand.

Do you know what a limit is, as a mathematical concept?

 

[Ibix]

I'm not sure I understand. Trend in what result?

If you have a lot of synchronised clocks at point A, and move them one by one to point B at different speeds and then compare them all at B they will have lost their synchronisation. This is (slightly loosely speaking) because of the different amounts of time dilation each clock experienced because of its speed.

The only way to get rid of time dilation effects would be to travel at zero velocity, but that would take a while . But if you note down the times shown by all your clocks at some instant and plot that as a function of their transport velocity you will find it's asymptotic to some value. Spoiler: it's asymptotic to an Einstein synchronised clock.

Actually doing this is very tricky. A muck up of slow clock transport was one of the possible explanations for the OPERA faster than light neutrino mystery (turned out to be a loose wire in a detector though).

Edit: Pretty much exactly what PAllen said...

 

[Philip Dhingra]

Ok, let me be painfully explicit. Imagine you have many identical clocks at position A. You transport them at varying speeds to position B. At position B, they will be found to differ slightly. As a function of transport speed, you can extrapolate to what reading would occur for zero transport speed. This extrapolated clock setting for clocks at B is what you take to be synchronized with clock at A.

If this still doesn’t help you, I give up. Hopefully someone else can explain if you clarify what you don’t understand.

Do you know what a limit is, as a mathematical concept?

I understand now. The synchronization process you described explains the concept! I didn't (and still don't) understand the language. How do you travel "in the limit of" something? It's not an expression I'm familiar with. I know what limits are.

 

[Ibix]

How do you travel "in the limit of" something?

You don't. You travel with a velocity v. Then again with velocity v/2. Again with velocity v/3. Etcetera.

Then you deduce the result of the limiting case where you travel with velocity zero. That's all they mean.