A Roller Coaster Ride through Relativity

Preface
This book is written for anyone who has studied a bit of Mathematics and has a working knowledge of basic algebra. It is ideally suited to A level or High School students of Physics and/or Maths and the intention is to explain the theories of Special and General Relativity as far as is possible using concepts and language appropriate to that level of knowledge.

Contents

 * 1) Introduction
 * 2) *The initial climb up
 * 3) *The River Race
 * 4) Time Dilation
 * 5) *The 1g rocket problem
 * 6) *At the top of the second hill
 * 7) Length Contraction
 * 8) *The long straight
 * 9) Relative Velocity
 * 10) *The big loop
 * 11) Simultaneity
 * 12) *The switchback
 * 13) Adding Speeds Together
 * 14) *The spiral
 * 15) The Doppler Shift
 * 16) *The third hill
 * 17) Mass Inflation
 * 18) *The second roller coaster
 * 19) The Bending of Light
 * 20) *Down we go again
 * 21) Gravitational Time Dilation
 * 22) *Oblivion
 * 23) Black Holes
 * 24) *The last encounter
 * 25) Appendices
 * 26) *Appendix A: The 1g rocket problem
 * 27) *Appendix B: Length Contraction
 * 28) *Appendix C: Time on a distant star
 * 29) *Appendix D: The addition of speeds
 * 30) *Appendix E: Travelling 'faster than light'
 * 31) *Appendix F: Relativistic Kinetic Energy
 * 32) *Appendix G: The relation between energy and momentum
 * 33) *Appendix H: The Gravitational Potential at the surface of a star

Introduction
Books on Relativity generally fall into two categories. There are, of course, many undergraduate level text books which take the reader the whole way from simple concepts such as time dilation right through to Minkowsky diagrams and tensor theory. There are also a great number of books which try to explain Relativity without using a single equation (or perhaps just one!). Regrettably, many of these are overly simplified and some are, frankly, wrong. As Einstein himself said, 'you should explain your theories as simply as possible - but not more so.' For the mathematically literate student and general reader, neither of these approaches fit the bill. This little book is an attempt to fill the gap by explaining Einstein's theories using plenty of simple algebra and numerical examples but without introducing any mathematical techniques beyond some very simple calculus (which can, of course, be ignored if necessary).

So how much is it possible to achieve using just A level ideas? The answer is quite a lot. The fundamental theorem of special relativity (time dilation) requires nothing more than Pythagoras' theorem and the remaining theorems require as much mental agility as algebraic competence. All the proofs given have been chosen carefully and the proofs of the 1g rocket problem and the addition of velocities are new in the sense that I have not seen them proved this way elsewhere. The approach to the famous equation E = mc2 is also slightly novel and avoids the complications of quantum theory which is a very unsatisfactory feature of any proof which relies on the behaviour of a photon in a box. At each stage in the argument I have been careful to show how the relativistic expressions reduce to Newtonian ones when v is much less than c.

When it comes to General Relativity, one can really only get as far as Einstein's 1911 paper in which he describes the bending of light and gravitational time dilation. The extra effects due to the distortion of space and time revealed in the full theory which appeared in 1917 is outside the scope of this book. Nevertheless, it is possible to calculate some of the predicted properties of those extraordinary objects known as black holes and even to start discussing the numerical properties of the universe as a whole.

There is a story about Sir Arthur Eddington who wrote many books explaining and popularising Einstein's theories. A journalist once remarked to him that he was one of only three people in the world who really understood the General Theory of Relativity. Eddington was silent. When asked why he did not say anything he replied "I was just wondering who the third person might be."

In the century that has passed since Einstein's first publication the scientific world has gradually come to an acceptance that the General Theory of Relativity – at least on a large scale – is the way the world works and its predictions have now been verified countless times to an amazingly high degree of accuracy. But as to the number of people who really understand it, you can probably still list their names on a single sheet of paper – and you won't find my name on the list! On the other hand, the basic ideas of Relativity and some of its bizarre consequences including the existence of Black Holes have become common knowledge and anybody with an inquiring mind will want to know something about how these claims are justified. If you find the sheer number of equations and formulae in this book a bit daunting, it is well to remember that, as Galileo said, that the Book of Nature is written in the language of Mathematics and if we are to understand the world we live in, we must accept that fact and continue to practice our facility with that language even when we have left our formal mathematical education behind. Besides – equations have an important property which, like other theorems which occur in this book, is so important I shall print it in a fancy box:

A mathematical proof is like the Title Deed to a house. It is very important that the Deed exists and that it is kept somewhere safe. When you first buy a house, you might be curious enough to glance through it to see what it says but you are unlikely to understand much of the legal language it is written in; nevertheless, you have employed a good solicitor to make sure that it is in order and you can, at least, admire the fancy paper on which it is written.

I urge you to regard the equations and proofs in this book in the same way. Have a glance through them; try to get to grips with a few of them but don't think you have to understand every equation – just sit back and admire them. Do take the trouble to get your calculator out and verify some of the figures though; otherwise you may find them difficult to believe. Then when you have finished the book you can put it in a safe place in your bedroom and go to sleep in the comfortable knowledge that even if you still can't really believe that clocks in motion go slow and that pennies bend when they accelerate, the proofs are quietly sitting there on your shelf so it must be true after all.

The initial climb up
Let me take you on a roller coaster ride through Relativity. You will see many strange sights along the ride and hear many strange stories - many of which you will find hard to believe. But at the end of the ride, you will be able to look at the world about you and the stars above with a new and altogether deeper understanding. Have you got your ticket? Then let's get going. ..

Together we climb into the train and sit down. There is a clanging of bells and with a tremendous jerk, we are on our way.

There's not much to hang on to, you say.

No, you're right there. In front of us there is absolutely nothing except what looks like a single gear stick with a black knob on the end. On the knob there is some writing engraved in white which says simply: 'The Fundamental Principle of Special Relativity'.

That's all there is, I am afraid, and you have to hang on to it like glue. If ever you let go of this Principle, you are lost. The Principle itself is seemingly innocuous, almost self evident, and yet Einstein showed that it leads to an amazing series of almost incredible consequences. While the roller coaster climbs to the top of the first hill, let me tell you what this wonderful Principle is.

Is that it? you ask. I thought that was obvious. Well, yes, it is. After all, when you pour a cup of tea from a tea pot into a cup, it doesn‘t matter if you, the cup and the teapot are all hurtling down a (straight) railway line at a (constant) speed of 125 mph in a railway carriage. Nor does it matter that the whole train and indeed the whole Earth is hurtling round the sun at an (almost constant) speed of 30 km s-1, nor does it matter that the whole solar system is hurtling round the galaxy even faster than that! The laws of physics which govern the way the tea falls are just the same. Nor would you expect calculators to give different answers or musical instruments to make different sounds just because they were moving. Surely all the laws of physics are the same whether you are moving or not.

We can restate our Principle in an equivalent form like this:

Sounds very plausible, doesn't it?

Sounds pretty obvious to me.

I agree. But there is, perhaps, one way in which you just might be able to tell if you were moving or not. What if you were to measure the speed of light travelling in different directions? Suppose that you discovered that the speed of light measured in one direction was greater than the speed of light travelling in the opposite direction? What would you infer then? Surely it would be reasonable to suppose that your laboratory was in fact moving through space and in one direction the speed of light and the speed of the lab were adding together while in the other direction the two speeds were subtracting.

It is actually very difficult to do this experiment because you have to measure the speed of light very accurately but eventually, as we shall see later, a suitable experiment was performed by two physicists called Michelson and Morley, but the results were disappointing. The speed of light seemed to be constant in all directions.

While most scientists tried to explain this result away by means of various subterfuges, Albert Einstein merely accepted it as a necessary consequence of the Fundamental Principle namely:

(It is worth noting that Einstein was not aware of the results of the Michelson-Morley experiment when he first worked on the theory of Special Relativity) We have now cranked our way to the top of the first hill. Let's pause a minute and take a last look around at our cosily familiar world. Far below you can see your twin brother thumbing through what looks like a travel brochure. Later we will discover that he is planning a short holiday on Alpha Centauri. Away to your right you can see some workmen doing some maintenance on a large clock which towers over the rifle range. Studying the behaviour of the bullets will tell us something remarkable about how speeds add together. The shrieks of the children riding the Ghost Train catch your attention and you watch for a moment as the long train rushes into a tunnel. You notice the rear of the train vanish into the tunnel at the precise instant that the engine begins to emerge from the other end. A detailed consideration of this circumstance will turn everything we thought we knew about time on its head. Down below you notice some children playing shove-halfpenny on a cracked old table and two boys tossing a couple of footballs. In the distance you can see a fast flowing river. Wait a minute - a race is about to begin...

The River Race
Two friends, Albert and Beatrice if you like, have agreed to a rowing race on a fast flowing river. Albert is going to row 100m directly across the river and back again. He knows that in order to proceed at right angles to the flow he will have to aim upriver a bit and this will slow him down on both halves on the race, but he doesn't think he will be slowed down too much. Beatrice is planning to row to point B 100 m upstream and back again. She knows that it will be hard work rowing up stream but she reckons that she will be assisted just as much on the way home as she is hindered on the way out and that she will win the race overall.

Who do you think will win the race?



You watch. As expected, Albert makes steady progress across the river and easily beats Beatrice, who is pulling hard upstream, to the first turn around point. But when Beatrice eventually reaches her turn-around the excitement and the cheering begins to mount because the current is rapidly carrying her back to the start while poor Albert is still struggling across the river. Nevertheless, Albert has gained too much of a head start and in spite of the assistance of the current, Beatrice cannot make up the lost ground and loses the race.

Why is this? Let’s work it out using some typical figures.

If we assume that the river is flowing at 1 ms−1 and that both rowers row at 2 ms−1 through the water, it is easy to use Pythagoras' theorem to see that Albert will move across the river at a resultant speed of √3 ms−1 and that he will complete the race in 115 s.

Beatrice, on the other hand takes 100 s to get to the turning point (travelling at an effective speed of 2 - 1 = 1 ms−1) and in spite of the assistance of the stream, she cannot make it back in time to win the race because even at a speed of 2 + 1 = 3 ms−1, the return journey takes her 33 s.

The difference in journey times is due to the different geometries of the two paths and of course it depends on the speed of the river. It is worth pointing out that if the river was flowing at 2 ms−1, the same speed that Albert and Beatrice can row, neither rower would complete the race at all but it turns out that at speeds less than 2 ms−1, Albert always wins.

What is the significance of this pretty little story? Well in 1887 a famous attempt to measure the speed of the Earth through the supposed æther was made by Michelson and Morley using a device called an interferometer. The apparatus looked (schematically) something like this:



Light from a monochromatic light source is split by a half silvered mirror into two beams which travel out to two distant mirrors A and B (just like the two rowers Albert and Beatrice). When they return, the same half silvered mirror recombines the two beams into one again. If the two arms are of exactly equal lengths (and if the Earth is stationary) the two light beams will take exactly the same time to get to the mirror and back and will therefore arrive back exactly in phase and will interfere constructively (i.e. they will produce a strong interference fringe). If the Earth is moving, however, according to the æther theory of light, the beam which is travelling parallel to the direction of the Earth's motion through space will be delayed with respect to the other beam and the fringe pattern will shift, perhaps showing destructive interference instead of constructive interference. In order to eliminate the effect due to the inevitable slight difference in arm lengths, the whole apparatus was designed to be rotated slowly through 90° at a time.

Now the Earth moves round the Sun at about 30 km s−1 - that is about 0.01% of the speed of light. This makes the calculations a bit difficult because the important figures occur in the 10th decimal place and your calculator may not be accurate enough - but the result nonetheless is that Michelson and Morley expected to see a fringe shift of about ± half a wavelength. Not much, but easily detectable all the same.

Well, what did they see? They looked for a whole year and found precisely - nothing. No fringe shift at all. Whichever way you looked at it the experiment was either a triumphant failure or a dismal success - for it seemed to indicate that Earth wasn't moving at all!

Various theories were put forward to explain the result. It seemed incredible that the Earth was the only thing in the Universe with zero speed, so perhaps the Earth 'dragged' the æther around with it, or perhaps the arms of the apparatus changed in length according to how they were moving.

None of these are correct. The truth of the matter is, of course, that the result of the experiment does not need explaining! It is simply a fundamental fact about the universe we live in. The speed of light in a vacuum is a fundamental constant and will always be the same even when measured by observers in relative motion.

I don't have a problem with that.

You will.

Why?

The problems all arise when you consider measurement made of the speed of light by two different observers who are in relative motion. The first thing you will have to accept is:

The roller coaster gives a violent jerk and suddenly you are accelerating rapidly down the long descent. 'Hang on to that Principle' I shout as your arms flail around wildly trying to find something to grab hold of. As we gather speed I glance at the clock tower and note with satisfaction that the hands of the clock above the rifle range are not moving quite as fast as they were...

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