I've just given a couple of lectures on special relativity to a class of first years. It's clear that grasping the key ideas is going to take some time. The results are so far removed from everyday experience that there is a certain air of bewilderment in the classroom. Here's an example of what I mean.
Suppose two students are travelling on skateboards, both at 10 km/h, but heading towards each other. In the frame of reference of one, what is the velocity of the other?
The answer is simple: Add up the speeds – one sees the other coming at a speed of 20 km/h.
Now make the skateboards a little quicker. To be precise, make them both travel at 0.8 times the speed of light. Now what does one of the skateboarders experience?
Our immediate reaction might be to say 0.8 + 0.8 = 1.6 -they see the other approaching at 1.6 times the speed of light. But that would be wrong. At high velocities, it just doesn't work that way. The correct answer would be (from the Lorentz addition formulae) 0.98 times the speed of light. That is hard to grasp. There are at least two reasons. First, we never experience skateboards going at 0.8 times the speed of light, so the question is not physically meaningful. Secondly, it is so far removed from our physical experience it just doesn't make any intuitive sense.
One can do the correct relativistic calculation on our first example – two skateboarders each heading at 10 km/h (or 9.26 billionths of the speed of light). This time we end up with 19.999 999 999 999 998 3 km/h. It is little wonder that calling it 20 km/h is an approximation that works for us! Putting it in context – if we travelled at this speed for an hour (thereby covering 19.999 999 999 999 998 3 km), we would be about 2 picometres short of 20 km. That's something much less that the size of an atom (but rather larger than the size of a nucleus). Little wonder we get away with calling it 20 km without any trouble.
A consequence of special relativity is that time and space are 'relative' – meaning that different observers will disagree on the time between two events, and the distance between two events. This is measurable – put an atomic clock on an aircraft and one on the ground, and fly the plane around for a few hours. On landing, the two clocks will be different, showing that time has been experienced (very slightly) differently.
There is, however, one very readily measurable consequence of relativity – one for which we are all familiar. That is magnetism. Magnetic fields and electric fields are part of the same entitiy. Just as observers will disagree on the time and distance between two events, so two different observers will disagree on the strengths of magnetic and electric fields in a system. A magnetic field becomes an electric field to a different observer. The reason we experience magnetic fields at all is down to the extreme neutrality of matter – the number of electrons and protons in a sample of material being incredibly well balanced. I didn't try to explain that one – but it makes a nice bit of analysis for third-years.