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One more chance at Special Relativity

In Danish, by Daniela Milton:
In Polish, by Alice Slaba:

You were good at Physics in school. You were not bad at Maths. But you never understood those things about Einstein's relativity, with those explanations about trains and lightning flashes traveling in all directions. One problem is that lots of authors, journalists and encyclopedia article writers don't understand anything either on the subject. They just repeat what they heard, reproduce the drawings they saw and suggest that if you don't understand, that's because you're idiotic. Let's give it another chance...

The speed of light is 299,792.458 km/s but let's round it to 300,000 km/s inside this text.

What did you see in those articles about Special Relativity? There was that drawing with that rocket that looks shorter because it travels very fast. Then that assertion according to which a clock inside the fast rocket runs slower.

To straighten things out, let's say that you live inside a space train. It is made of 10 wagons and each wagon is 300,000 km long. Between each wagon hangs a clock. Those 11 clocks are synchronized: they show the same hour. This means that if you are in the middle of a wagon and you look with a telescope at the clock in the front of the wagon and at the one in the back of the wagon, they show the exact same hour. (But if you are at the end of the wagon, just below the clock, then the clock in the front of the wagon will look to be 1 second back. That's because the image of that clock took 1 second to travel through the wagon and reach you.) Every passenger inside your train has a synchronized clock.

Whatever speed your train travels, it obviously has the same length for you. If you go walking from the front to the rear with very precise measuring instruments, the length of the train always is exactly 3,000,000 km. You can even spacewalk out of the train, billions of kilometers aside of it, and take a photograph of the whole train. It perfectly shows to have a length of 3,000,000 km. And of course, all the clocks aboard obviously run at a normal rate. The mechanical clocks, the water clocks, the electronic clocks, your hearth beat... all run at one second per second exactly. You feel it... You can compare them...

What says Special Relativity? Suppose your train is loosely hanging high above in Space. If another train, identical to yours, shears along yours at 150,000 km/s, it will be shorter. Instead of being 3,000,000 km long, it will be only 2,597,476 km long. No kidding: if you ask the passengers of your train where the front and the rear of the other train were at exactly 11:34 am, they will confirm the photograph you took while spacewalking: the other train is shorter. Its front was behind your front and its rear was further than your rear. Let's say it once more: at 11:34 am and 00 seconds exactly, a passenger in the middle of your front wagon was seeing the front of the other train right through his window. And a passenger in the middle of your last wagon was seeing the rear of the other train right through his window. On your photograph, each wagon of the other train seems as shrunk lengthwise a little bit. Weird... (But the other train's wagons still have the same height as yours.)

And the clocks inside that train indeed seem to run slower. Let's survey the clock in the front of the other train. When it reached the back of your train, it showed 08:00:00 pm exactly (a person in the back of your train confirms this very tightly). The other train travels at 150,000 km/s, so its front reaches the front of your train after 20 seconds. So then that clock should show 08:00:20 pm. Well, no... a person in the front of your train says that the clock in the front of the other train then showed 08:00:17 pm. It obviously runs slower. Whatever photograph or video you take, will confirm this.

Two things are strange:
Actually this is easily acceptable. Their train was shortened by the high speed but so where the measuring instruments they use to measure its length. If your shirt shrunk in the washing machine and a witch shrunk you too during the night, you will not get the feeling that your shirt has shrunk when you put it on in the morning. The same way, if every clock inside the other train is slowed down exactly the same way, then when comparing them, they will see no difference. They will get no hint towards a slowing down of their clocks. They are slowed down themselves.

Now here come the true and frightening paradoxes:
How in Space can it ever be possible that your train looks shorter to them and altogether their train looks shorter to you? Let's admit their train is shorter... but then yours must look to be longer to them!

How on Earth can it ever be possible that your clocks look slower to them and altogether their clocks look slower to you? Let's admit their clocks run slower... but then your clocks must look to run faster to them!

That's the paradox that Einstein solved. He realized that the clocks inside the other train will not look synchronized to you. Actually, the passengers inside the other train synchronized their clocks perfectly, using the exact same methods as you used to synchronize your clocks in your train. For them, their clocks look perfectly synchronized. But this makes their clocks be unsynchronized from your point of view... To express this with precise numbers: when you see that the clock in front of their train marks 08:00:40, then you see that the clock at the rear of their train marks 08:00:45. And all their clocks in-between show a time in-between. There is nothing magical about this. Simply stand by and observe a passenger in the middle of a wagon inside the other train. He looks at the clock in the front of his wagon. The image of that clock speeds towards him at 300,000 km/s. And he speeds towards the image at 150,000 km/s. So, he will "hit" that image really soon. On the other hand, the image that originates from the back of the wagon sure travels towards him at 300,000 km/s, but he flees that image frontwards at a speed of 150,000 km/s. So it will take a lot more time till that image reaches him. So, if he has to see the same hour on both clocks, the clock at the back of the wagon must be a short time later. That way he will see the same hour on both clocks. He is absolutely sure both clocks are synchronized. He sees that with accurate precision. But you see that they are not. But you understand why he sincerely believes his clocks are synchronized. And in a flash you understand that he must see your clocks as being not synchronized! Though you see them as perfectly synchronized.

The clocks of the other train are unsynchronized from your point of view... So, now you understand why they honestly believed that your train is shorter. They made the measure using their clocks and they got a result that looks false to you... but you understand why they made that "error". From their point of view, it is better to consider things as they obviously seem to be: your train is shorter. And from your point of view, their train is shorter, in all honesty too. And if a passenger of the other train went out spacewalking aside of his train, and took a picture, on that picture, his train appears to be 3,000,000 km long and your train appears to be shorter. (And if he sees you and wants to shake hands with you and he fires his retrorockets to slow down 150,000 km/s to get immobile compared to you, and he turns his head to look at his train, he will now see, just like you, that it is shorter than yours. And that your clocks are perfectly synchronized.)

Einstein's main discovery actually is that light always travels at 300,000 km/s. If you see a passenger in the other train fire a bullet frontwards at 1 km/s, you will see it travel at 150,001 km/s; the speed of the bullet plus the speed of the train. But what if he lights a torch and directs it towards the front of his train? You would think that you see that light travel at 450,000 km/s; which is 300,000 km/s plus the 150,000 km/s of the train. Well, no... you see it travel at 300,000 km/s. But the inhabitants of the train see that light travel at 300,000 km/s too. Are they mad? No... their measuring tools are shorter, their clocks run slower and they are unsynchronized! So, when you observe them measuring the speed of that light, you understand why in all honesty they conclude it travels at 300,000 km/s. And whatever light you produce, in whatever direction, you will see it travel at 300,000 km/s and the passengers in the other train too will always see it travel at exactly 300,000 km/s. And they will understand, while observing you, why you believe your light and their light always travel at exactly 300,000 km/s. (This is for the speed of light in vacuum. Einstein perfectly knew that light travels slower in air, glass, water...)

The magic Special Relativity tells about our universe is that:
Weird... Seems paradoxical... But things are so that in fact there is no paradox at all. Everything fits neatly and mathematically together. This is how our universe is. This has been verified by countless experiments.

Some further consequences are that:

If you'd like formules:

Thanks to Albert Frank for his suggestions.

Eric Brasseur  -  November 1 till 3 2008