special relativity - Light clock with mirrors on a moving train

I am reading a new popular science book where a thought experiment proposed by Albert Einstein is explained.

It is a thought experiment with a light clock. The light clock is emitting light straight into a mirror, the mirror reflects the light and when the light makes the round trip clock-mirror-clock this registers the passage of a certain amount of time, e.g. 1 millionth of a second.

In the first part of the experiment the clock is standing in a train which is not moving so the clock and the mirror are still. As seen in the first picture (below).

In the second part of the thought experiment the train is moving and the book suggests that the light travels a longer distance because the train is moving and therefore it kind of travels on a diagonal as presented in the second picture (below).

Now I disagree with this suggestion. I see no reason for the light to change trajectory or travel a longer distance. I argue that even when the train is moving the first picture still holds true.

There is also one more thing that it seems to me is wrong with that picture. There is no mirror against the clock at t0, no clock against the mirror at t1 and again no mirror against the clock at t2. This placement of the clock and mirror in time is crucial because it results in the longer diagonal path of the light but the clock and the mirror are moving at the same time! So the only illustration that makes sense is one where the clock is always against the mirror.

In the experiment it is said that light travels a longer trajectory with reference to an observer outside of the train standing still beside the railway.

I think that the train travels with reference to the still observer but the light travels with reference to the train. So I argue that regardless of whether the train is moving or not the light travels the same distance for the same time.

After all when it comes to measuring distances the reference is central.

More food for thought: It could just as well be a ping pong (table tennis) ball clock. If you are playing table tennis on a train the ball travels directly along the length of the table while the train is moving, would you suggest that the ball is traveling on a diagonal trajectory? I think not.

Please share all your arguments.

Answer

I believe the crux of your concern is here:

I see no reason for the light to change trajectory

So the short answer is it doesn't. The explanation in your book blows.

There are two points it is failing to convey properly; first, there is no change in trajectory, and secondly, this happens in "classical" mechanics too. Let's start with that second point...

If you replaced the photons with rubber balls you have the same diagram.

For the two people tossing the ball back and forth on the train, there is no change in trajectory if the train is motionless or moving. To their eye, they are passing the ball directly back and forth. Let's say the train is 2m wide, so it's moving a total of 4m. And it takes 2 seconds, so it's 4/2 = 2m/s.

However, someone watching this from the embankment when the train is in motion would see the second diagram, that the ball is taking a longer path in "the real world". They see the first person toss the ball when they were in the station, and catch it again when they are at that sign over there. The distance between the station and the sign is 100m, so over that same 2 seconds, that means the ball is going 50m/s.

But it's vital to note that these are both the exact same ball toss. The trajectory does not change. All that changed is the frame of reference you based your measurements on. Did the ball move 2m/s or 50m/s? Both, what changed was not the ball, but the frame of reference.

This is the entire crux of the galilean transform. It gives you a little bit of math that tells you how to convert between the two cases based on the velocity of the frames. It has been a key point for classical mechanics since we invented classical mechanics.

So what is the difference in SR?

By the late 1800s we knew that there was something very odd about light. It was increasingly clear from a series of experiments dating into the 1700s that light did NOT follow this transformation. The Earth is moving in opposite directions in the summer and winter, so we would expect an effect like the one above when we looked at light from distant stars. We would see this as a shift in color, which we could measure very accurately at that time. And that was simply not happening. No matter where or how we looked, everyone saw the same thing, essentially the first of the two diagrams.

But how could this be? The galilean transform was clearly applying to physical objects, but apparently not to light. Yet physical objects give off and absorb light. This is not a minor point; there's all sorts of effects that would be obvious if it was the case that the transform didn't apply to light. For instance, objects would cool down differently if they were moving at different speeds. And we definitely didn't see that! So by the end of the 1800s, a whole lot of people were trying to explain why we know it has to be occurring, but we don't see it.

And that's what SR answers.

So here it is... we know that everyone sees the same speed of light. But what is speed? Speed is not a "thing", it's a calculated value. It's calculated by dividing distance by time. And we already know that the distance we measure is based on your frame of reference...

So isn't it perfectly obvious? The solution is that time is also based on your frame of reference.

And once you make that leap, the entire mystery just vanishes. All that remains is how it changes fro frame to frame. Since we know that c is constant, you just work the problem backwards - in order for us not to see something weird, time must change across frames like this. That's the formula in your book.

So when you do use photons, the people on the train measure the distance and time and divide to get c, and the person on the embankment measures a different distance and a different time and divides to get... c. And this doesn't just apply to light, it applies to the rubber balls too, but we don't notice in that case because they're moving so slowly.

Mystery gone. Poof.

The real question isn't what's so special about SR, but why anyone had to "invent" this in the first place. This shouldn't be surprising. We were always perfectly happy with the idea that we have different measurements in different frames of reference, in fact, it's obvious. So why did we think time was different?

Einstein didn't so much invent something new as point out that we had this huge invisible prejudice, and if you see it, and realize what it implies, then all the mystery vanishes. SR isn't complicated, it simplifies things dramatically.