One Way Twin Revisited: Understanding Tim & Tom's Watches on Home & Northbound

[Ibix]

I was not suggesting any measuring being made across frames.

That's true - you were very careful to specify who measured what and to make sure that the people were at rest with respect to each other when comparing clocks. And that means that there is a unique answer to your question - in each case, there is one minute difference between the watches of the stay-at-home and the traveller at the end of the experiment as they measure it. If you've reasoned that out correctly (and you have) for the Tim/Tom case then the principle of relativity says that it must also be true for Bill/Bob, because both pairs of twins can regard themselves as starting at rest and following the same experimental procedure.

So where does the three minutes twelve seconds come from? I think you are proposing it because you can't see how clocks can tick slower on both planets. And this is where the relativity of simultaneity comes in - not in your experiment, but in your interpretation of your two experiments.

You seem to be OK with the idea that the first experiment leads to a one minute difference, but you're confused how it works in the second run of the experiment where your actors start on the other planet. You have two options: you can copy my approach in the first paragraph, where I simply switch to the most convenient frame to analyse each half of the experiment, or you can stick with one frame throughout. I think you are doing the second, because that does indeed lead to a 3:12 offset between their watches - and you need the relativity of simultaneity to explain how a 3:12 offset can be consistent with a 5:00 offset.

I'm out of time, so I'll leave it there. Happy to explain more if I'm on the right track.

 

[Janus]

Im not an academic and all the equations are meaningless to me but I understand why relativity of simularity exists (at least I believe I do) so it's possible to do so without the maths. As others have suggested I would lookup the train thought experiment where a lightning bolt strikes both the front and back at the same times.

Another way I personally look at it is using sound instead of light signals as we all have experience of sound dialating from when a emergency vehicle approaches and passes us and we notice the pitch change so it's easier to visualise it in your head.

Let's take a 1KM stage, place observers 1, 2 & 3 at 0 meters, 500m and 1km and place hunters at 250 & 750m. Observer 2 at 500m shouts to the hunters to fire their hunting rifles. Only observer 2 at 500m will hear both at the same time. Observer 1 at 0m will hear hunter at 250m shot then the one at 750m. Observer 3 at 1km will hear hunter 2's gunshot first then hunter 1's. So we have 3 observers witnessing the same events but they all dissagree about the timings of the gunshots. Both observers 1 & 3 hear the 2 gunshots in the opposite order to each other, the same is true for the time dialation, they both observe the opposite of the events but the order / time dialations between them is the same, this is relativity of simularity, and why each observer says its the others clock that has run slow by the same amount.

Is this a valid extremely over-simplified analogy or am I talking nonsense?

There are a number of issues with your example.
The fact that different observers hear the gunshots at different times in this scenario does not mean that they would say that the guns were fired at different times. As long as each observer knows how fast sound travels through air and how far they are from each shooter, they can calculate how long it took for the sound of the each gun firing took to reach them.
So let's say that the speed of sound is 450m/s. Observer 1 will hear the 250m shot ~1.11 sec before hearing the 750m shot. But it took the sound from the 750m shot an additional 1.11 seconds to travel the addtional 50 meters. Once our observer takes this into account, he concludes that both Guns fired at the same time. The same is true for observer 3, he hear the 250m shot 1.11 sec after the 750m shot, bit also knows that it took loner for the sound of the 250m shot to reach him. Everybody in this scenario will say that the guns were fired simultaneously even though they might not have head the gunshots simultaneously.
Put another way, One friend leaves a town 70 miles from you driving to you at 70 mph. Another friend leaves a town 140 miles away, also driving at 70 mph. Friend 1 arrives at 1:00 by your watch and friend 2 arrives at 2:00 by your watch. But since friend 1 took 1 hr to make his trip, and friend 2 took 2 hrs to make his trip, they both left their towns at the same time at 12:00 by your watch. If you had a third friend located just half way between the two towns, each the other two friends would reach him at the same time, when his clock reads 1:30, 1 1/2 hrs after each left. Thus he also says that they both left at 12:00.

Another issue, is that this scenario is missing a key element in the train example: the observers are not in motion with respect to each other. Now, in you example this actually won't make a difference*. Even if the observers were moving, they would still say that the guns were fired simultaneously. This is due to the third issue:

The way Sound and light in a vacuum propagates is not the same.
Sound propagates via a medium, and while its speed is constant for any given medium, this is in respect to this medium. If you are standing still with respect to the air, sound traveling through the air will be moving at 450 m/sec with respect to you. However if you were traveling through the air at 50 m/sec towards the sound source, you would measure the sound as moving at 500 m/sec relative to yourself ( though still at 450 m/sec with respect to the air).

Light in vacuum however is different. It does not require a medium, yet its speed is still a constant.(~300,000,000 m/s or c) So, if in the last example you are at rest with respect to a light source, you will measure its light as moving at c with respect to yourself. If you are in motion with respect to the source (it doesn't matter if you consider yourself or the source as "moving"), you still will measure the light as traveling at c with respect to yourself. As counter-intuitive as this seems, it's just the way things work.

Now we can consider the Simultaneity of Relativity issue.

To do that we will consider a variation of the train experiment. In this variation, the light from the lightning strikes arrives at the midpoint embankment observer at the same moment that the train observer is passing him. Like this:

Here the red dots represent the points on the tracks where the lightning bolts strike. The expanding circles are the light flashes from the strikes. They expand out at the same speed(c) and meet at the middle. This animation shows how things occur according to the embankment observer. Note that the strikes occur when the rail car observer is closer to the left strike

Now we switch to the frame of the railway car observer. Keeping in mind, that he also must measure the light flashes as expanding at c relative to himself. This means that the center of these expanding circles can't move relative to him according to him as this would result in different parts of the circles to have different speeds relative to him.
Thus for Our railway observer, events unfold like this.

The light from the strikes still arrive simultaneously, when he is next to the embankment observer, and he is at the midpoint between the strike points.
But since the flashes arrive when he is at the midpoint, the strikes has to occur when he was closer to the left strike. The light emitted by the left strike has a shorter distance to travel to reach him. And since the light from both strikes has to travel at c relative to the car as measured by the car observer, the light from the right strike has to leave earlier in order to meet up with the left strike's light at the proper point.

So here we have two observers who see the flashes at the same time, but conclude that the events (lightning strikes) that produced them occurred simultaneously according to one observer and at different times according to the other. In your example, you have different observers hearing the sounds of the gunshots at different moments, but they would all conclude that the guns were fired simultaneously.

Relativity of Simultaneity isn't just about what observers visually see, but what they conclude about events from what they see. ( And it really isn't about light at all, But about the fundamental nature of "reality" that causes light to behave as it does.)

* Or at least when dealing with relative speeds small compared to the speed of light, the difference would be too small to practically measure.

 

[Ibix]

Is this a valid extremely over-simplified analogy or am I talking nonsense?

This is all true but irrelevant. The relativistic effects being discussed here are what's left after you correct for signal travel times. You seem to be making the (fairly common) mistake of thinking that relativity is somehow an effect of light travel times. It isn't. The point about Einstein's train thought experiment is that different observers following the same procedure for correcting for light travel times come to different conclusions. Which means that simultaneity is frame dependent.