Special Relativity for Children

Scruffy Rat was very scruffy. His whiskers were bent, his hair stood up, his coat was rumpled and he smoked a pipe.

One day Scruffy Rat decided he needed a bath. He went to the river to wash; but the water was cold, he was in the shadow of a tree and the sun was shining on the other side, so he decided to cross the river where it would be warmer. He looked around for a floating log and sure enough he found one, very conveniently drifting by.

Scruffy Rat jumped onto the log and ran across it at speed c, but when he'd gone the length of the log, x', he was still too far from the opposite bank. He ran back across the log, again at speed c, again a distance x', but now he couldn't reach the tree where he's started because the log had drifted away at speed v. "Oh well", thought Scruffy Rat, "I'll just have to swim!", and he did, struggling up onto the river bank bedraggled and scruffier than ever.

Along came Toad and Mole and asked Scruffy Rat why he'd been swimming with his clothes on, but Scruffy Rat was embarrassed by his own foolishness in running out on the log and would not admit his mistake.

"Don't change the subject!" roared Scruffy Rat, desperately changing the subject. "If I run across that log at speed c and a distance x' then I'm travelling at speed c+v from the river bank and the time it takes is x'/(c+v); running back across the log takes the same time, x'/(c-v), so there!"

"No", reasoned Toad, "your speed c is measured with respect to the air above the log, so you ran out at c-v and back at c+v".

"No matter", said Mole, "he calculated the time wrong, x'/(c+v) can't equal x'/(c-v); besides, he's wet and it's not raining, he must have been swimming!", but Toad and Scruffy Rat were too busy arguing theoretical physics to listen.

"Look, said Scruffy Rat to Toad, "I'll prove it on paper, even doing it your way", and this is what he wrote:

If we place x'=x-vt, it is clear that a point at rest in the system log must have a system of values x', y, z, independent of time. (What this means in ordinary English, boys and girls, is that the length of the log doesn't change from day to day.)

We first define t as a function of x', y, z, and t. To do this we have to express in equations that t is nothing else than the summary of the data of clocks at rest in system log, which have been synchronized according to the rule given in § 1 (which was "we establish by definition that the "time" required by a rat to travel from A to B equals the "time" it requires to travel from B to A", boys and girls).

From the origin of system log let a rat be emitted at the time t₀ along the X-axis to x', and at the time t₁ be reflected thence to the origin of the co-ordinates (back along the log but not to the tree on the bank; Scruffy Rat got wet, remember, boys and girls), arriving there at the time t₂; we then must have ½(t₀+t₂) = t₁, or, by inserting the arguments of the function t and applying the principle of the constancy of the velocity of rats in the river bank system:

Hence, if x' be a very short log (infinitesimally small),

$\begin{displaymath}\frac{1}{2}\left(\frac{1}{c-v}+\frac{1}{c+v}\right)\frac{\par... ...au}{\partial x'}+\frac{1}{c-v}\frac{\partial\tau}{\partial t}, \end{displaymath}$

(Which has now gotten rid of those awkward coordinates that were put there to confuse you so that you didn't read it too carefully.)

$\begin{displaymath}\frac{\partial\tau}{\partial x'}+\frac{v}{c^2-v^2}\frac{\partial\tau}{\partial t}=0. \end{displaymath}$

Since $\tau$ is a linear function (because Scruffy Rat says so, but it isn't linear at all), it follows from these equations that

$\begin{displaymath}\tau=a\left(t-\frac{v}{c^2-v^2}x'\right) \end{displaymath}$

where a is a function $\phi(v)$ at present unknown, and where for brevity it is assumed that at the origin of log, $\tau =0$ , when t=0.

With the help of this result we easily determine the quantities $\xi$ , $\eta$ , $\zeta$ by expressing in equations that rats (as required by the principle of the constancy of the velocity of rats, in combination with the principle of relativity) are also propagated with velocity c when measured in the moving river system. For a rat emitted at the time $\tau =0$ in the direction of the increasing $\xi$

$\begin{displaymath}\xi=c\tau\ {\rm or}\ \xi=ac\left(t-\frac{v}{c^2-v^2}x'\right). \end{displaymath}$

But the rat moves relatively to the initial point of log, when measured in the river bank system, with the velocity c-v, so that

$\begin{displaymath}\frac{x'}{c-v}=t. \end{displaymath}$

(Which is where Scruffy Rat divides the fixed length of the log, x-vt, by [the distance travelled by the rat from the river bank to the end of the moving log also known as x, divided by the time it takes the rat to reach the end of the log, also known as t, so that c-v = x/t]. Confused, boys and girls? That's what Scruffy Rat wants you to be. Think it through. )

(x-vt) / (x/t) = t ?

If we insert this value of t in the equation for $\xi$ , we obtain

$\begin{displaymath}\xi=a\frac{c^2}{c^2-v^2}x'. \end{displaymath}$

In an analogous manner we find, by considering rats moving along the two other axes, that

$\begin{displaymath}\eta=c\tau=ac\left(t-\frac{v}{c^2-v^2}x'\right) \end{displaymath}$

when

$\begin{displaymath}\frac{y}{\sqrt{c^2-v^2}}=t,\ x'=0. \end{displaymath}$

Thus

$\begin{displaymath}\eta=a\frac{c}{\sqrt{c^2-v^2}}y\ {\rm and}\ \zeta=a\frac{c}{\sqrt{c^2-v^2}}z. \end{displaymath}$

Substituting for x' its value, we obtain

$\begin{eqnarray*} \tau & = & \phi(v)\beta(t-vx/c^2), \ \xi & = & \phi(v)\beta(x-vt), \ \eta & = & \phi(v)y, \ \zeta & = & \phi(v)z, \ \end{eqnarray*}$

where

$\begin{displaymath}\beta = \frac{1}{\sqrt{1-v^2/c^2}}, \end{displaymath}$

Scruffy Rat then went on to boringly prove $\phi(v)$ = 1, leaving the well-known cuckoo malformations that all good little children learnt the way they learnt their times tables, i.e. by rote, ignoring what went before because it was too hard for them:

$\begin{eqnarray*}\tau & = & \beta(t-vx/c^2), \ \xi & = & \beta(x - vt), \ \eta & = & y, \ \zeta & = & z, \ \end{eqnarray*}$

where

$\begin{displaymath}\beta = \frac{1}{\sqrt{1-v^2/c^2}}, \end{displaymath}$

which has since been renamed g (gamma) .

"Very impressive", said Toad, "I understand it completely now", not understanding a word of it but wanting to appear knowledgeable and academic.

"So if a twin rat stayed under the tree and didn't get wet when it rained", said Toad, ignoring Mole, "he'd be older and dryer than the twin rat on the log. Isn't that a paradox?"

"Exactly!", said Scruffy Rat, puffing himself up with pride and relighting his smelly wet pipe, "but the rain was accelerated sideways by the wind! Willows, y'know - can't trust 'em".

"Of course, that's the Lorentz contraction", said Scruffy Rat, forgetting he'd said he had stayed under the willow tree.

"But he's wet and it's not raining!", screamed Mole at the top of his voice, "that's his woolly brain that is shrinking!"; but nobody was listening.

You see, boys and girls, Scruffy Rat had to swim back from position 8 at speed c, and that takes time. t() is not a linear function as Scruffy Rat claimed, for

So everyone except Mole agreed the velocity of rats is c in all inertial frames of reference; then they all went home for tea and argued happily ever after.

Once Scruffy Rat made the mistake it went all the way through to cloud cuckoo land, producing the well-known twin paradoxes and pole-in-the-barn paradoxes.

What was Scruffy Rat hiding by suggesting
the speed of rats along logs from A to B is c-v,
the speed of rats along logs back the other way from B to A is c+v,
the "time" each way is the same?

Would you like to grow up to be wet theoretical physicists who can't do mathematics but like to pretend that they can, boys and girls?