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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).

light clock standing still

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).

enter image description here

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.

Qmechanic
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Martin Bakardjiev
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  • If you were standing on the moving train, you would see the light bounce straight up and down off the mirrors. To someone stationary relative to the train, they must also see the light hit the mirrors; how could this happen without the light taking a longer path due to the movement of the train? – Kris Walker Sep 18 '17 at 23:07
  • The photons take all the directions. There is no change in trajectory. – Bill Alsept Sep 18 '17 at 23:15
  • @KrisWalker To someone stationary the light will travel a longer distance indeed then it is no longer a clock because the "reference frame" is not "stable". – Martin Bakardjiev Sep 18 '17 at 23:31
  • Let's for a moment return to the first picture with the train standing still and the clock inside the train. All would agree that the path of the light is straight. How about for an observer that is not on earth? For an observer outside of the geostationary orbit of the Eearth the light between clock and mirror (or two mirrors) travels a longer trajectory even when such a clock/mirror is on the ground or in a train that is not moving relative to earth. – Martin Bakardjiev Sep 18 '17 at 23:37
  • @MartinBakardjiev The whole point is that a moving clock ticks slower from the perspective of a stationary observer. Why does that make it "no longer a clock"? – Kris Walker Sep 18 '17 at 23:37
  • In geostationary orbit, an observer would still be stationary relative to the train and so the light will just go straight up and down. – Kris Walker Sep 18 '17 at 23:41
  • Note: outside geostationary orbit. – Martin Bakardjiev Sep 18 '17 at 23:49
  • Ah. Well, if the Earth is rotating/moving below you, then the train is also moving relative to you, so you would see the light take a longer path. – Kris Walker Sep 18 '17 at 23:53
  • The second drawing is not very good. There should be Mirror0 next to Clock90 and Clock1 next to Mirror1 and Mirro2 next to Clock2. The artist just wanted to chow where the clock and mirror are at three different times and made it confusing. – C. Towne Springer Sep 19 '17 at 00:58
  • The second drawing is not very good. There should be Mirror at t0 next to Clock t0 and Clock t1 next to Mirror t1 and Mirror t2 next to Clock t2. The artist just wanted to show where the clock and mirror are at three different times and made it confusing. I also think the train can not go fast enough for the angle of the light, as seen from the station, to be more than pi/4. ( Poorly done example. Is this from Tyson?) – C. Towne Springer Sep 19 '17 at 01:02
  • @MartinBakardjiev I know it's being picky but even in the first picture it may not be a straight line because the earth is moving 30,000 km/s. – Bill Alsept Sep 19 '17 at 01:57
  • If you are using the equation speed = distance/time it does not make sense to take the point of view of an observer standing still on the ground (reference1) for the distance traveled by light but when you measure the time to use the "clock" on a moving train for that measurement (reference2). – Martin Bakardjiev Sep 19 '17 at 10:25
  • @KrisWalker If the clock we are talking about is the light clock then it measures the wrong distance in the proposed experiment. The light does not travel a longer distance from the point of view of the clock, therefore the time measured by the clock does not run at a different rate. – Martin Bakardjiev Sep 19 '17 at 10:49
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    Possible duplicate of I am confused about the idea that the speed of light is independent of the speed of the source of light – sammy gerbil Sep 19 '17 at 12:07
  • So, if you just consider the ping-pong game. From the perspective of a person standing next to the train, at some moment the ball is next to them; a minute later it's a mile down the track. What was its path? –  Sep 20 '17 at 11:20

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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.

Maury Markowitz
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  • I would like to thank all of you for your efforts in explaining this. I no longer have any seconds thoughts on whether the light traveled a longer distance with reference to the stationary observer. – Martin Bakardjiev Sep 20 '17 at 20:38
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Okay, since the comments aren't doing it, I'm gonna have a go at explaining time dilation in this answer.

When Einstein introduced the special theory of relativity in his 1905 paper "On the Electrodynamics of Moving Bodies," he outlined two postulates. The first, that the laws of electrodynamics and optics are valid for all frames of reference for which the equations of mechanics hold good, and the second, that the speed of light is constant for all observers, regardless of motion. This is important, it means that regardless of your speed relative to the light, you always observe it travelling at $c$ (299,792,458 m/s) (well, until you get into General Relativity, where warped space can make light appear to slow down, but that's beside the point).

What you need to realize is that there is no absolute frame of reference. When you are in an inertial reference frame (no acceleration), there is no way of telling if you're moving or if the things around you are moving. In fact, from your point of view, you're stationary.

Now, let's get into what the question is about. Imagine two mirrors spaced at a distance $L$ apart (as pictured). Now imagine a light beam is projected from the bottom mirror at time $t_0$ in the direction of the top mirror; this beam is then reflected off the top mirror at time $t_1$ before returning to the bottom mirror at time $t_2$. The time between $t_0$ and $t_2$ can be expressed as

$$\Delta t=\frac{2L}{c}$$

Now imagine you see both mirrors start moving uniformly to the right. For someone moving with the mirrors, they will be stationary in their reference frame, so the situation will play out exactly as described above. However, for you, the light takes a longer path. After leaving the first mirror at time $t_0$, the light traverses the new distance $D$ and hits the top mirror at time $t_1^\prime$ before reflecting back into the bottom mirror at time $t_2^\prime$. The time between $t_0$ and $t_2^\prime$ can be expressed as

$$\Delta t^\prime=\frac{2D}{c}=\frac{2\sqrt{L^2+(\frac{1}{2}v\Delta t^\prime)^2}}{c}$$

The path of the light beam as seen from someone moving with the mirrors (left) and the path of the light beam as seen from someone stationary relative to the moving mirrors (right).

Now, you may be thinking, "wouldn't this be the same as if you bounced a ball off the ceiling of a moving train?" The answer is no. That ball has the velocity of the train added on to it and so to a stationary observer it moves with a higher velocity than what the person bouncing the ball sees. That is not the case with light. Remember that second postulate I mentioned? What that means is that both the observer moving with the mirrors and the observer stationary with regards to the moving mirrors record the light moving at the same speed. Think about that. From the point of view of the stationary observer the light traces out a longer path and takes longer to hit the top mirror: there is more time between "ticks".

This is what Einstein realized. Moving clocks run slower from the perspective of a stationary observer.

Also, you know all that stuff I said about no absolute inertial reference frame? That means that the observer moving with the mirrors also sees the other observer's clock run slow; that's because to them, they're stationary and the other observer is moving.

I hope that cleared things up.

Kris Walker
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  • Thank you for all your efforts in writing this, it must have taken you some time but you have overdone it because I already know and understand this hypothesis and I argue there is a flaw in it. Also it does not address my point. The distance in the right direction (1/2 v delta t prime) is traveled by the train and has nothing to do with light. Light travels only L. Movement in the right direction of a light particle will occur even if light was hanging still in the air (assume it has infinitely small speed) because the source of the light is carried by the train with it's speed. – Martin Bakardjiev Sep 19 '17 at 20:24
  • This kind of adding the movement upwards to the mirror and movement in the direction of the train is wrong for the purpose of timekeeping. If you design such a clock and place it on a train you will consider only the distance traveled relative to the train. The center point is that the clock measures the time if we agree that the unit of time is the time for which it bounces between the two mirrors. It would bounce with the same rate and frequency between the two mirrors regardless of whether the mirrors are moving or not. – Martin Bakardjiev Sep 19 '17 at 20:33
  • Offtopic: all reasoning for time dilation is based on clocks which work by observing the movement of something. But nothing need to move in order for time to pass. It is just that this is the only way we know how to account for time. In this theory we argue that time "slows down" because something seems to be traveling a longer distance. This could also be questioned. – Martin Bakardjiev Sep 19 '17 at 20:38
  • To focus the attention on the real problem that made me post this I want to remove light from the discussion. Imagine a clock that is not based on light with mirrors but based on a mechanical object that has a mass and is moving between two plates/boards. We know the speed that objects has and this speed is caused by a force (since the object has a mass). It may not be that fast but it does not matter because we know the speed, the distance and from there we can keep account of time. The capability (and speed) of the object to travel between the plates has nothing to do with the moving train. – Martin Bakardjiev Sep 19 '17 at 20:58
  • @MartinBakardjiev you can't focus on one particle. Light consists of trillions of photons in random directions. – Bill Alsept Sep 19 '17 at 21:46
  • @MartinBakardjiev In the post, I addressed several of the things you have just said. For the observer moving with the mirrors, the light bounces off the mirrors, therefore the same happens for the other observer, but it happens slower due to the increased distance. In the paragraph below the image I specify that it isn't like a mechanical object because the speed of light is constant for all observers, thus the time between reflections the observer moving with the mirrors observes is less than what the other observer sees. – Kris Walker Sep 19 '17 at 22:32
  • Martin, I suggest you work it out for a ping-pong ball and a table being played perpendicular to the train's motion. – C. Towne Springer Sep 20 '17 at 04:36
  • @C.TowneSpringer that is exactly how I work it out.
    1. A static ball placed on the train will travel in the direction of the train. It's the train speed that's relevant, not the ball speed or capability to travel. Same for light. The capability to travel right is of the train and is related to the train speed. The capability to travel perpendicular to the train's motion is due to the light or tennis ball capability to travel and therefore related to the speed of light or speed of ball. These two need not get mixed. Also, light does not need to be involved the clock can be mechanical.
     – Martin Bakardjiev Sep 20 '17 at 10:09
  • If it were as you suggest, you add the distance traveled to the right, to the distance traveled by the light, why stop there? Add the speed of the earth, the sun, and the galaxy. It's just wrong.

    Why would the travel to the right be attributed to light and dependent on speed of light, it simply doesn't.

     – Martin Bakardjiev Sep 20 '17 at 10:13
  • The whole point (which I mentioned several times in my answer) is that light moves at a constant speed regardless of motion. The light's speed is in no way effected by the speed of the train. The person moving with the mirrors sees the light move at $c$ and the person stationary relative to the moving mirrors also sees the light travel at $c$, but across a longer distance. – Kris Walker Sep 20 '17 at 10:16
  • What would an experiment with a mechanical clock (not optical) look like then to demonstrate time dilation? – Martin Bakardjiev Sep 20 '17 at 10:23
  • It wouldn't demonstrate time dilation (though it would still occur, obviously). If instead of light, you bounce a ball off the ceiling of a moving train, that ball already has the velocity of the train and so for a person inside also moving with the train, the ball would go straight up and down.But for a person watching the train go by, the ball would retain its horizontal velocity as it is thrown upwards and so will travel in a diagonal path. This doesn't demonstrate time dilation though, it's just addition of velocity (which doesn't happen with light). – Kris Walker Sep 20 '17 at 10:30
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You decide to describe what is happening in this experiment using the frame of reference attached to Earth.

At some moment the first event happened: "very short flash of light was emitted". You know exactly where this event occured: at some point $A$ ("clock t0" on your picture).

Later on the second event happened: "the light reached the mirror and reflected".

And still later on the third event happened: "the reflected light was detected".

What's the coordinates of the third event? Where has it occured? At the moment of first event the detector was at point $A$. Definitely the third event happened after the first event, and as the detector is installed on a moving train, it was on some other point $C$ when the light was detected.

Where the second event happened? It must be somewhere along the upper line of your picture (because the mirror is moving along this line). It happened at some point $B$.

So, the description of the experiment in this frame of reference is: "At the moment $t0$ light was in point $A$, later on at the moment $t1$ it was at point $B$, and still later on at point $t2$ it was at point $C$. You can calculate distances between these points. Then suppose that speed of light is $c$ and calculate $t1-t0$ and so on...

lesnik
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