Lorentz and Einstein

When I was at York University earlier this week I took a break from computer science to remind myself of some of the basics of (special) relativity and was struck, while reading the opening few pages of Rindler’s Essential Relativity: Special, General, and Cosmological (which would appear to be the set text at York), just how simple the basic maths of one of the core concepts of relativity is.This is the idea that observers in one inertial frame see objects moving at high speed foreshortened (and time dilated). This insight is not Einstein’s, but Lorentz‘s (though Lorentz did not see the importance of time dilation and it was Einstein who understood the fundamental nature of the result and so built a whole new dynamics on top of it).

Though using a contemporary laser, this Michel...
Though using a contemporary laser, this Michelson interferometer is the same in principle as those used in the original experiment. (Photo credit: Wikipedia)

The maths of this are such that any good GCSE student should get it – and it might suit teaching physics better if that was emphasised rather than the supposedly counter-intuitive nature of the relativity principle it leads on to – because I am sure that puts people off.

I am going to use the example used in the book – someone swimming up and down and back and across a river – to explain it. If you are happy with Pythagoras’s theorem then you will have no problem with this.

But first a little historical background – James Clerk Maxwell formulated the theory of the electromagnetic field and from that came the idea that light travelled as a wave. But in what medium? After all, you drop a stone in the water and you see waves, but they are waves of water.

It was speculated that the equivalent of water was ‘the luminiferous ether’ – an all pervading medium through which the waves of light undulated.

But as the Earth moved through the universe – solar rotation being the biggest factor – that should mean we would see light move faster when it was travelling in the same direction as the Earth’s motion (just as you can swim faster down stream with the current pushing you along). Except that there was no sign of this.

A famous and sophisticated experiment – the Michelson-Morley experiment – and many others – sought to measure the difference in the speed of light when it travelled in the direction of, and orthogonal to, the Earth’s motion. None was seen.

To see the size of the effect that was expected – imagine that you are a swimmer who travels at speed V in a river with a current that moves with speed v . To swim downstream to a fixed point at a distance P away would take \frac{P}{V+v} seconds, while to swim back to our starting point would take  \frac{P}{V-v} (as now the current is against us) – the total time then is \frac{P}{V+v} + \frac{P}{V-v} = \frac{2PV}{V^2-v^2}.

Now let us assume the river is wide, of width P and we decide to swim straight across and back. Here too we have to fight the current, as it tends to make us drift downstream – so we have a classic right angled triangle of forces, with the effective distance we swim being the hypotenuse of the triangle.

So to get across takes \frac{P}{\sqrt{V^2-v^2}} and across and back takes \frac{2P}{\sqrt{V^2-v^2}} .

Plainly these two times (downstream and across) are not the same – but the Michelson-Morley experiment suggests that for light, they are. So Lorentz suggested that objects travelling in the direction of motion were contracted – substituting c for V and L for the length of an object in the direction of travel and W for the length of the object when orthogonal to the direction of travel we get (remember the light is our swimmer here):

\frac{2W}{\sqrt{c^2 -v^2}} = \frac{2Lc}{c^2-v^2}

So \frac{W}{\sqrt{c^2-v^2}} = \frac{Lc}{c^2-v^2}

And \frac{L}{W} = \frac{c^2-v^2}{c\sqrt{c^2-v^2}} = \frac{\sqrt{c^2-v^2}}{c}

Or, as it is more conventionally written \sqrt{1-\frac{v^2}{c^2}} (square both sides, divide by c^2 – then take the square root).

In other words as an object approaches the speed of light then an observer at rest will see it contract. For jet travelling at 700 km/h, the contraction is somewhat less than of the order of one part in a trillion, nothing you would notice! (In fact it is so small I cannot get the calculator on my computer to give me a useful answer).