#theveryfasttrain

Thought experiments are a marvellously economical and powerful tool that takes imagination and logic into realms where experiments are difficult. I can find no account of Einstein actually performing real physical experiments, but he certainly did use thought experiments to advantage.

Here is a thought experiment I have conjured up to try to convey what happens to concepts of length, time and speed in the weird and wonderful world of Special Relativity. Out of respect for tradition I have used the example of a very, very fast train.

For tidiness, round off the speed of light to be 300 million meters per second. Now imagine a very long straight train platform 300 million meters long. At the west end is a stationmaster called P, equipped with an accurate clock and a camera. At the east end, one light second away, is an observer called Q and he has a clock synchronised with the stationmaster’s clock, and he also has a camera. If P looks at Q (and vice versa) they can each see that other’s clock is showing a time one second less that theirs, which means that the two clocks are well synchronised.

Alongside the platform there is an infinitely long straight railway line. Hurtling towards the station from the west there is a very long and very fast train, travelling at half the speed of light. At the front of the train there is a driver equipped with a clock and camera. At the back there is a guard who also has a clock and camera. The train, when measured at rest, is 300 million meters long, the same length as the platform. The guard and driver have synchronised their clocks in their reference frame so that each sees the other’s clock showing one second less than theirs.

Now, the driver is very punctual and he passes the stationmaster exactly when the stationmaster’s clock reads zero seconds. And through careful planning, the train driver’s clock also reads zero seconds. This is Event 1.

The stationmaster sees the train whizzing past at 150 million meters per second, as measured by the platform and the platform clocks. So observer Q sees the driver’s end of the train flash by at local time of 2 seconds precisely. This is Event 2.  Q takes a picture showing his clock, the driver and the driver’s clock, all in the same photo frame.  

The question is – what is the time showing on the driver’s clock at Event 2? Well, we have two rules (Postulates 2 and 3 in an earlier blog) that we have to follow:

the driver and the guard have to measure the speed of light in their reference frame to be 300 million meters per second, and the stationmaster and observer Q have to measure the speed of light in their reference frame to be 300 million meters per second,

both pairs of observers have to agree that they are passing each other at the same relative speed (again measured locally), which in this example is half the speed of light.

Now, suppose at Event 1 the driver sends a pulse of light back down the train to a mirror being held up by the guard and observes that pulse when it comes back again.  In his reference frame the light has travelled twice the length of the train and so it will be seen again precisely 2 seconds later on his clock.  But where is the end of the station and Observer Q when this happens?

The driver saw Observer Q standing at the far end of the station whizz past a fraction of a second earlier. In fact, when his (the driver’s) clock showed 1.732 seconds.  Something must have happened to the station platform. It must have shrunk!

Or maybe the driver’s instruments are wrong and the train is not travelling at half the speed of light? How can he tell? If external references like the length of the platform have started to get all rubbery and unreliable, how can he check his speed?

The best option for the driver is to use the length of the train as his benchmark ruler and to time how long it takes for the stationmaster to go from being level with the front of the train to being level with the back end of the train. The first event here has already been called Event 1 and we will call the second part Event 1a. Assume that the driver prearranged with the guard that the guard will photograph this Event 1a. When the driver receives this photograph of Event 1a he can see that the guard’s clock was showing 2 seconds.  (This follows from Einstein Postulate (#2) that classical relativity must hold true, irrespective of the relative velocity). So this confirms The train driver’s belief that his train was travelling at half the speed of light.  

But then the driver notices something odd in the background of the guard’s photograph. The stationmaster’s clock is showing not 2 seconds, but 1.732 seconds. The poor stationmaster not only has a short station, he also has a slow clock!

And just to recheck the length of the platform the driver later has a discussion with the guard. The driver says to the guard “I reached the end of the platform (Event 2) at 1.732 seconds on our clocks, where were you at that moment?” The guard replied “I was always one light second behind you, remember. Furthermore, my return of your light pulse travelled the length of our 300 million meter long train and reached you at exactly that Event 2. But to be honest, I did not see anything of the station platform next to me at that time. I had not reached the start of the platform yet, and I did not see the stationmaster until a fraction of a second later. In fact, it was not until 2 seconds on my clock that I was abreast of the stationmaster. Where were you at 2 seconds?”  

The driver replies “I was well past the end of the platform at that time. So, using our train length as the benchmark, we can safely conclude the platform is shorter from our perspective than it is from their perspective.”  

Meanwhile, back on the platform observer Q is developing his photograph of Event 2. It shows his clock at 2 seconds, the driver of the train and the train driver’s clock. But that’s odd. The train drivers clock says 1.732 seconds.  It must be running slow.  

“How fast was that train actually going”, asked the stationmaster while having a beer with observer Q after work. Q replied “Well, you clocked the driver entering the platform at 0 seconds and I clocked him leaving it at 2 seconds, so it was half the speed of light alright. But how long did it take for that train to pass you by?” The stationmaster replied “The driver passed me at zero seconds on my clock and guard passed by me at 1.732 seconds. I thought that that was a bit odd.” 

“I noticed that too, ” said Observer Q.  “The only way I can explain it is if that train was not as long as it was advertised to be. In fact, I can prove it.” 

“How so?” asked the station master. 

“Well,” says Q, “the best way to measure the length of a moving object is to chalk mark the front and the back of it against a reliable ruler, but of course this has to be done at exactly the same time. I rigged up a clock at every million meter interval along the whole platform, and they were all equipped with a camera set to go off exactly at 2 seconds.”

“Which camera recorded the end of the train?” asked the stationmaster.  

“As I suspected, it was the camera number positioned 173 million meters along from your end of the platform, i.e. from the start of the platform,” said Q, “confirming that that the train was only 173 million meters long”.

“Ha!” said the stationmaster. “I had arranged those cameras to also go off at exactly the moment that the guard passed by me. And here is the photo that shows the driver at that place and moment, which I have labeled Event 1b. It was taken by the camera 173 million meters up the platform. So I can also confirm that that was a short train.”

“Can you see the clock reading and also the driver’s clock at event 1b, and what do they show?” asked Q.

“I can.  Our station clock number 173 shows a time of 1.153 seconds and the driver’s clock shows a time of 1 second,” said the stationmaster. It looks like the driver’s clock is running a bit slow. What does the driver’s clock show in your photo of Event 2?”

“I’ve already told you,” said Q. “My clock said 2 seconds and the drivers clock said 1.732 seconds, and here is the photograph to prove it. So I agree with you that the driver’s clock was running slow.”

Reality or Illusion?

It takes a bit of mental adjustment to come to terms with the logic of Special Relativity. We can talk about clocks and trains and light signals all we like, but somehow it still seems weird. It seems to defy logic that A is slower/shorter than B but B is slower/shorter than A. Are relativistic effects real or are they an illusion?  

The very fast train thought experiment above anticipated the question. The photographs taken at Events 1b and 2 showed the train’s clocks to be running slow compared to the platform clocks, and the photograph taken by the guard at Event 1a showed the platform’s clocks running slow compared to the train’s clocks. So yes, the time dilation effect is real. The pictures prove it. It may be counter intuitive but there it is.

Note that if the clocks and cameras had a mirror on them, then the mirror images in the photographs would have shown each observer’s own clock to be exactly what the observers themselves see it to be directly – i.e. ahead of the other clock. There is no logical violation on that score. And all the platform observers saw all the train clocks running slow, while all the train based observers saw all the station clocks running slow, so there is consistency on that score as well.

It all comes down to “how do you measure a time interval” and “how do you measure a distance interval.”

To properly measure a time interval you have to use your own clock, or a set of clocks that has been properly synchronised in your reference frame. You cannot rely on moving clocks and you certainly cannot assume that what you “see” is correct.

And to properly measure lengths you need to use a ruler in your own reference frame. If you want to measure the length of a moving object you need to make sure that you mark both ends simultaneously in your own reference frame. If you do all this, and you follow the rules for proper measurements, then you logically arrive at the effects described above.

Special Relativity obliges you to abandon your inherent, naive notions of time and length and speed. All you can rely on is your local clocks and rulers, plus the constancy of the speed of light, plus the rule that physical outcomes are not a changed by arbitrary choices of inertial reference frames, plus the rule that if A observes B to moving at relative speed v, then B must observe A moving at the same relative speed v in the opposite direction.