Special Relativity

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This page is an experiment. I don't necessarily understand Special Relativity, but I think I've got a pretty good handle on the problems other people have when they try to understand it. As a result, this page is here to try to talk about the concepts and underlying techniques, not necessarily to explore the detail of the maths.
To understand where SR comes from we need to take a few things for granted, and then question everything else. When we talk about measuring something we need to say "How" and reply with excruciatingly detailed processes. So if you say that two things are a metre apart, I'm going to say "How do you know?" If you say two clocks have been synchronised, I'm going to say "How?"

This is really where the whole thing starts. Warning, in places this will sound incredibly patronising. For that I apologise in advance, but I'm not sure which bits I can gloss over, and I need to be careful and explicit.

I'm going to start with a question asked recently by a friend of a friend. I'll paraphrase, but it's close enough.

If we use the conventional c for the speed of light, that means that D is c times 1 second.
Suppose we have two spaceships, A and B, that are a certain distance apart, and I'm looking at them from off to the side. For convenience, suppose we all agree that light takes 1 second to go from A to B. For convenience I'll call the distance between them D.

Now suppose I see them, again distance D apart, and they're travelling along a line, the same line as the one that joins them. Think of them as on rails, travelling along, with B leading the way and A coming up from behind. They're going at the same speed, let's say they cover distance D in 2 seconds, so their speed is D/2.

This is Zeno's paradox (or one of them) but let's not get distracted!
When I see A send a pulse of light to B, I see it travelling at c and catching up with B, but by the time it's been going for a second, and it's got to where B was, B has moved on. B is now D/2 further on, so the pulse has to go further. Half a second later it's got to where B was, but now B has gone another D/4. And so on.

So after 2 seconds B has gone distance D, the total distance the light has travelled is 2D, and the light pulse hits the target B.

We see that the light has travelled further at the usual speed, and has taken 2 seconds.


This is one of the really big assumptions, but it doesn't come from nowhere. The speed of light was measured in the direction of the Earth's orbit, and against the Earth's orbit, and they got the same result. Even more, it's really born of the assumption that the laws of physics should look the same to all observers. Light, when considered as a wave, is an interaction of electric and magnetic fields, and it can only be self-sustaining if it travels at speed c. If I'm moving relative to you, and the laws of physics are the same for us both, we should both see the same field interactions, so we should both measure the same speed for the wave.
We're told that the speed of light is the same for all observers. That should imply that A measures the speed of light as c, and the distance to B as D, and so it should still take 1 second to get from A to B.

But it doesn't. It takes 2 seconds.

Where is the problem.

That's a pretty good question, it's more-or-less the one that bothered Einstein, and in the remainder of this page we'll investigate it.

So here we have A and B travelling down a railway line distance D apart and at speed c/2, when A sends a pulse of light to B. Already there are a few things we should worry about.

So let's start with the first of those. From our point of view, looking at it all from the side, how do we measure the gap between A and B?

See how careful we have to be in specifying the process? Measuring distances, synchronising clocks, communicating, observing, all have to be defined very precisely.
What we can do is create a lot of clocks along the length of the line, all synchronised. We synchonrise the clocks by sending a pulse from the leftmost to the rightmost, then reflecting it back. Each clock notes the time the pulse passes them in each direction, and declares the midpoint of those times to be 0.

Now at a certain time they all look to see if either A or B is alongside, and if so, they raise a flag. Then we measure the distance between the raised flags, and that gives us the distance between A and B. Let's now assume that the distance is D.

In a similar way we can answer the second question - how fast are A and B travelling. We simply ask all the clocks to raise a flag if A is opposite, then 1 second later do the same again. That gives us the distance A has travelled in one second, so we have the speed.

Finally, each clock can observe the event of the pulse leaving A and the pulse leaving B. If it happens immediately by them then they raise a flag on which is written the time by their clock when it happened. If A and B are travelling at c/2, and the distance between them is the ditance light travels in one second, then we'll measure the distance between the flags as 2D, and the time difference as 2 seconds.

So far so good?

OK, so now I'm happy that I see A and B travelling distance D apart, and at speed c/2. Next we'll see how things look from A and B's perspective.


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