# Simultaneity Confusion

1. Dec 30, 2005

a b c d e f g h ...
<-V a' b' c' d' e' f' g' h' ...'

a, b, c, d, and e are all .5 a light second apart. The primed letters are all moving with a velocity of .5c to the left. a, c, and e emit light when a primed letter passes. We know that all the primed letters are .5 a light second apart because b and d receive pulses from a, c and c, e simultaneously every second.
Now what confuses me is that the nonprimed frame will see a primed letter pass every second and they will see the primed clocks ticking slower, but if we switch reference frames the primed will see the unprimed traveling at .5c see an unprimed pass every second and will also see the unprimed clock tick slower. This is relativity and I think I can swallow this because simultaneity is lost between reference frames. But as I was thinking about this I noticed that if at the instant captured above both frames set their clocks to t=t'=0 then every time a letter pass their clocks should agree. I.e. as b' passes a the nonprimed clock will read t=1 (it will also read some dilated time for the primed frame), but as a passes b' t'=1 in the primed frame (with the same dilation for the nonprimed frame as seen before). This can still make since with the loss of simultaneity. But I thought further... When would the primed frame see the light pulses from a, c, and e? It is clear from the nonprimed frame b will see c s first pulse (t=0) .5 seconds later and that b' would see this pulse some time later. From the primed frame though a, c, and e are pulsing as they pass therefore b' should see c s first pulse (t=0) .5 seconds later as well.
The part that perplexes me is that every time a letter passes another letter their clocks should be synchronous, even though their times pass at separate rates. I.e. when b' passed a, the nonprimed frame read t=1 and when a passed b' the primed read t'=1. Both reading the same lapsed time for a particular event, at a particular place, but both experiencing this event not simultaneously.

Thanks, I konw there must be a flaw some where I just havn't been able to find it yet.

2. Dec 30, 2005

### JesseM

Are the primed letters 0.5 light seconds apart in their own rest frame, or in the unprimed rest frame? Because of Lorentz contraction, their distance apart in the unprimed frame will appear smaller than their distance apart in the primed frame.
If the primed letters are 0.5 light-seconds apart in the unprimed frame, then yes, the unprimed frame will see a new letter pass a given point every second, with the clock on that letter ticking at $$\sqrt{3} / 2$$ the normal rate.
The primed frame will see the unprimed clocks ticking slower by the same amount, but it will not see the unprimed letters pass at 1 per second--again, you have to take Lorentz contraction into account. If the unprimed letters are 0.5 light-seconds apart in their own rest frame, then in the primed frame this distance is shrunk by $$\sqrt{1 - 0.5^2} = \sqrt{3} /2$$, so the distance will be $$\sqrt{3} / 4$$ light-seconds, so dividing by the speed of 0.5c, the primed frame will see the unprimed letters pass every $$\sqrt{3} / 2$$ seconds.

Also, in the unprimed frame all the letters line up simultaneously--a lines up with a' at the same time b lines up with b', for example--so if my understanding of your scenario is correct, all the flashes will all be emitted simultaneously in the unprimed frame. But in the primed frame this is no longer true, you'd have to figure out the time delay between a lining up with a' and b lining up with b' and so forth to figure out the delay between an observer at rest in the primed frame seeing different flashes arrive at his location.

3. Dec 30, 2005

### Staff: Mentor

The first thing to realize is that your setup is not symmetric. The primed letters are 0.5 light seconds apart according to the unprimed frame. But in their own frame, the primed letters are farther apart.

The primed frame will see the unprimed letters move at 0.5c, but will not see them pass every second. And, of course, according to the primed frame the letters do not pass each other at the same time.

[JesseM beat me to it!]

4. Dec 30, 2005

ok thanks guys I knew I forgot something.

5. Dec 30, 2005

### Mortimer

In addition to the remarks of the other posters, you might not fully understand the non-simultaneity issues involved in this situation.
a, c and e are not pulsing simultaneously here any more, since they are in motion, as observed from the primed frame.
At the time of synchronization only two clocks (one primed, one unprimed could be synchronized between the frames mutually. Within a single frame you could then sync all other clocks but when you compare those other clocks with the other clocks in the other frame you would see that they are not synchronized.
This picture picture may help you understand the effect of the non-simultaneity as induced by the relative motion.