Non-fiction about one aspect of Special Relativity
The other day somebody sent me a recording of the play, Insignificance, by Terry Johnson, made into a radio production. In it, at one point, a character – who happens to resemble Marilyn Monroe – explains – to a character resembling Albert Einstein – parts both of Einstein’s theory of Special Relativity and General Relativity. In particular, she refers to something known as The Twins Paradox and adds that Special Relativity is inadequate to explain how it works, and that the General Theory is required.
I will not digress to comment here on a play where characters resembling Marilyn Monroe and Albert Einstein discuss Relativity (not to mention that characters resembling Joe DiMaggio and Senator MacCarthy show up) nor ponder why the play is called Insignificance. However, I will state here and now quite categorically that you do not need General Relativity to explain The Twins Paradox and that the Special Theory is perfectly adequate. Indeed, using the General Theory would probably mess up your answer while the Special Theory gives you the right answer. What’s more, I’ll show what that answer is and prove it is right!
And no maths.
(Just a tiny bit of arithmetic, and you can even skip that.)
So what’s all the fuss about? Well, it gives an opportunity to look into what Relativity really means and hopefully make it a bit more easy to understand for more people. I’ll show you don’t have to be a maths genius to understand it too.
Why Two Theories?
Just to explain, for the moment, why there are two theories of Relativity – or, more accurately, two parts – Relativity is all about things moving. It concerns the speeds of things. The word "velocity" is sometimes used in place of speed (in physics, "velocity" has a more precise meaning.) But, for simplicity, I have used the word "speed" most of the time to help keep things clear and used the words physicists prefer when it matters. For the moment, what you need to know is that Special Relativity deals with constant speed while General Relativity is about changing speed, going faster or slower. This is not just playing with words – if you were moving at constant speed smoothly enough – on a train or a plane, for example – you might feel as if you were standing still. If you were on a train, plane or anything else that was speeding up or slowing down, you would always know that you were moving. This difference is vitally important. There will be more about this later.
I have used the word "mass," as preferred by physicists, in place of "weight," just to keep the physicists happy. But they mean almost the same thing.
Einstein was probably the most famous physicist – indeed the most famous scientist – of the 20th century. His face is instantly recognisable in any picture even today, and, as most people are aware, his work led to the development of atomic energy and the atomic bomb (though he emphatically did not work on the atomic bomb himself, as he was a pacifist.) His work also explains how stars burn (though I won’t be going into that here.)
Not surprisingly, Einstein won a Nobel prize for physics. More surprisingly (and quite unfairly, in my view) he did not win it for either versions of his theory of Relativity, but for a piece of work that led to an area of physics called Quantum Mechanics, which a lot of people have never even heard of (although I talk about some bits of it in other articles on this website.)
(His work could also have saved someone from committing suicide, but alas this tragedy was not averted – the news of yet another discovery by Einstein that proved the existence of atoms did not reach, as far as we know, the ears of Ludwig Boltzmann, also a brilliant physicist who worked on a theory of atoms. Boltzmann suffered from manic-depression, it is believed, and was sometimes ridiculed by other physicists. With no proof that he was in fact right, he killed himself a year after Einstein’s proof might have lifted his mood and saved him.)
So what is all this theory of Relativity about then? I’ll try to keep this brief, but if you’ve already had an introduction to this subject, concerning the speed of light, you may want to skip this altogether and get straight to The Twins Paradox.
Light Speed
It had been known for some time that light had a finite speed, although a very great one. Sir Isaac Newton did some work on light (also discussed elsewhere on this website) but he did not, as far as I am aware, investigate what the speed of light might be. He may have been hampered by the absence of telescopes and very accurate clocks during his lifetime, though Newton was such a clever-clogs he probably could have got round this, if he’d set his mind to it.
By the nineteenth century, all the necessary equipment was available. It was still no mean feat to measure something so fast. A man named Albert Michelson came up with a clever method which I will sketch out here. Imagine light from a lamp focussed to shine off a flat mirror to another mirror some distance away – a distance we have measured with extreme precision. The light reflects back to the first mirror and then back to the experimenter with his face just above the lamp. You can see that a bright lamp and a telescope would be quite handy for doing all this. Now, if the first mirror is in fact mounted on the edge of a wheel that is made to turn after a flash from the lamp has reflected off it, the returning flash will no longer be visible to the experimenter – the mirror will now be turned to the wrong angle.
To get round this (literally) how about covering the rim of the wheel with a series of mirrors all placed at precise angles, and spinning the wheel? If the wheel is going fast enough, by the time the flash of light gets back from the second mirror in the distance, another mirror on the wheel will have moved into just the right position for reflecting the light back to the experimenter – he will see the reflected lamp again!
All you need to do is have enough mirrors on the wheel, and spin the wheel fast enough to make this work. And if you know how fast the wheel is spinning (and, as I said before, the exact distance to the far mirror) you can get an exact measure for the speed of light, because you know how long it takes the wheel to turn one mirror’s-worth and therefore how long it takes the light to cover the distance. Nice and simple (and I’ve not used any maths to explain this, notice.)
For a moment we need to consider what speed is. Well, obviously, it’s distance covered in a certain amount of time. If only things would stay that simple.
Let’s think about two cars colliding – fun, I know, providing no-one gets hurt, but who said this couldn’t be fun? If a car runs into a brick wall at 60 miles an hour it hits the wall with a speed of – d’ur! – 60 miles an hour. If it runs into a car driving towards it at 59 miles an hour, the speed of the collision is 119 miles an hour. This seems obvious – we just add up the two speeds. (This is about as hard as the maths gets in this writing so I hope you are keeping up.)
However, if the second car was driving away at 59 miles an hour, the collision would be just 1 mile an hour. Simple.
(Only – it isn’t. For reasons I’m not going to go into here, combination of velocities is not determined by addition of velocities, but don’t worry about it – addition is good enough for speeds much slower than the speed of light, and we will not be looking at faster collisions.)
Another, less obvious thing is that there is no such thing as the speed of something on its own. The speed has to be relative to something else. For cars, this is usually the ground. The important thing is that you have to have two things moving relative to one another to have a value for speed. It’s more obvious if you talk about rate of change of distance between two things, if a bit clumsy. You simply cannot have rate of change of distance between something and nothing else. (It’s a bit like the sound of one hand clapping… but we won’t go into that!)
Now back to the speed of light. From what I’ve just said, if we are moving towards or away from a lamp flashing at us, we ought to get different answers for the speed of light. Also, there are other ways in which we should be able to alter the answer, by shifting the whole experiment about. The thing is, when Michelson tried doing tricks like this, he didn’t get any variation in the value for the speed of light. He always got exactly the same answer! This should be impossible. But what turns out to be impossible is to get the speed of light ever to change!
Various means were used to attempt to explain this. One by one, however, they didn’t work.
Speed Invariance, and Special Relativity
Meanwhile, along comes Einstein. He didn’t worry about how or why the speed of light was fixed. He simply accepted that it never changes, and that was that. He never liked the name ‘theory of Relativity’ for his theory – he wanted to call it the ‘theory of invariance.’
The thing is, if light never changes its speed, you’ve got to fiddle around with other mathematics in order to get the sums to add up. In fact, the maths is not difficult, it’s just that what the answers mean sounds a bit weird. But here we go.
Imagine you are sat still in a room, wearing a watch. On the mantle-shelf stands a clock. As your watch ticks, so does the clock. Time seems to go at the same rate everywhere.
Now we will need a bit more arithmetic here. To make things easier, let’s imagine that light-speed is 10 centimetres per second. (It’s a lot faster, but using made-up figures makes it easier to see what’s going on and, apart from the values, doesn’t have any effect on reality.)
Suppose the clock now began to slide smoothly across the mantle-shelf towards you at one centimetre per second (never mind what’s making it move.) A flash of light comes from behind the clock, passes it and on to you. In one second the light would go 10 centimetres. But if you measured how far the light had gone past the clock, using the clock’s time-keeping and measure of distance, it would only have gone 9 centimetres. If you used the moving clock to measure the speed of light, you would get only 9 centimetres per second – which is the wrong answer!
It’s easy to get back to the right answer (though you have to ignore how weird the answer seems to be.) All you have to do is have the moving clock run more slowly. By stretching out a second on the moving clock, light again travels 10 centimetres in one clock-second.
A similar correction works for any other speed.
This is the first bit of Special Relativity – time goes more slowly, relative to someone watching, for moving objects. It has to, in order to get the speed of light to be constant.
However, if you were moving with the object, you’d notice no difference.
How about – instead of using the clock’s timing, we used the thickness of the clock to measure the distance light travels in one second? If the clock was itself 10 centimetres thick and it was moving at 1 centimetre per second, in one second the light would appear to move 9 centimetres past the clock in travelling from the back to the front. Again, the wrong answer. Again, it is easily fixed, mathematically. We just say that the clock gets thinner in its direction of travel when it is moving. Once again, we get that light takes one second to move from the back to the front of the clock because the clock, while moving at one centimetre per second, is only 9 centimetres thick, from our point of view, giving a light-speed of 10 clock-centimetres per second.
Again, a similar correction works for any other speed.
This may sound very peculiar and not at all true, but it is, in fact, exactly how the Universe works, and that is that. The only ‘fiddling’ I’ve done is to make the numbers easier to work with.
This is the second bit of Special Relativity – things get shorter, relative to someone watching, when they are moving, in their direction of travel.
However, if you were moving with the object, you’d notice no difference.
Using our imaginary light-speed of 10 centimetres per second, it’s easy to see that, if the clock itself were sliding towards you at 10 centimetres per second, it would have stopped ticking altogether, and it would have no thickness at all! This still gives the right answers for the speed of light, as measured by the moving clock. It also shows that to travel at or faster than the speed of light is impossible for a physical object – time can’t go backwards and you can’t have negative thickness or length.
A third problem arises. When a thing with a certain mass is moving at a certain velocity, it has momentum which is calculated by multiplying its mass and velocity together. If it collides with something else – which may also have its own momentum, it’s a law of the Universe that the total amount of momentum after the crash must be the same as the total momentum before. It doesn’t matter if the things get smashed up and the parts scatter in all directions – do the arithmetic properly and you will see that no momentum has been gained or lost. This is known as The Law of Conservation of Momentum, and it’s been known about and proved to be absolutely true for ages.
The thing is this. If you stuck clocks on all the moving objects involved in the crash, those clocks start running slow, depending on how fast the things are moving. As velocity is distance covered in a given time, you start getting speeds that are too slow and the momentum doesn’t add up properly any more. The Law of Momentum seems to be broken – which it can’t be – so something again has to altered to allow for this. The only thing is left is the mass, so the Universe needs to increase that to make up for the loss of apparent velocity.
This is the third bit of Special Relativity – things get more mass, relative to someone watching, when they are moving compared with when they are still. (We’ll see where they get the extra mass from in a moment – you don’t get anything for nothing!)
However, if you were moving with the object, you’d notice no difference.
In fact, if you had something moving at very nearly the speed of light, its mass would become very nearly infinite. You would therefore need a very nearly infinite amount of energy to push it that little bit faster – and even then this energy would be turned into mass – the fat thing you were pushing would get fatter rather than faster. It was from this that Einstein realised that energy would turn into mass, and could be turned back again, in certain circumstances, giving us atomic power, the atomic bomb, radioactivity, why the core of the Earth is still hot after so many years of trying to cool down, and how stars burn. But we’re not going to go into that.
So, in a nutshell, in Special Relativity, when things are going at constant speed, things get shorter, more massive and their clocks runs more slowly.
Now, in practice, light goes an awful lot faster than this. But it is absolutely true to say that, every time you move, relative to your surroundings, all the things around you get more massive, shorter, and have time going slower. It’s just that at ordinary everyday speeds, you never notice it and can ignore it. (Global Positioning System satellites works to such high precision, their movement and altitude do have to be taken into account, however.)
But, just for a moment, let’s go back to the clock sliding towards you at constant speed. From its point of view, the clock feels as if it isn’t moving, and that you are. So the clock sees you get heavier, thinner and your watch running more slowly. How can they both be running slow, and which one is right? What time will each show after you’ve been moving for a while? This is where The Twins Paradox comes from.
First of all, let’s look at the clock and the watch both looking as if they are going slow from the point of view of the other, while each ‘feels’ as if it is running normally. This is a bit like watching a ship sail over the horizon (possibly through a telescope, for a clearer view.) From someone watching on land, the ship appears to sink into the sea (and, if we could look very carefully, the mast of the ship would appear to tip away from us.) But, from the point of view of someone on the ship, you and the land would appear sink into the sea (and tip slightly backwards) while the ship stayed afloat and its mast vertical. What we have is two different horizons caused by being in two different positions. In a similar way, we have two different ‘time horizons’ for things moving. Neither clock is ‘right’ while the other is ‘wrong.’ If you brought the two clocks to the same speed – that is, stationary relative to each other – the two clocks would show time going at the same rate, much as bringing the ship back to the same place, at shore, shows neither of them sinking and that vertical things point straight up.
The maths of all this, when things are moving at constant speed, is really quite simple and any teenage maths pupil at school should be able to do it (though I have deliberately left as much maths out as I possibly could.) Things get tricky when you start looking at things speeding up and slowing down, accelerating or decelerating. The maths for this probably needs you to be a graduate maths student. Einstein’s Special Theory of Relativity is so called because it deals with the special case of constant speed. When he wanted to work out what was happening generally, when things are changing speed, he had to resort to much more complicated maths. Surprisingly, some of the maths had been done already by a man called Berhard Riemann, about thirty years earlier, but Einstein was such an appalling student (from the point of view of his lecturers) that he skipped the lectures where he would have learned about it so, sadly in some ways and impressively in others, it took him longer to come up with the General Theory as he had to work out the maths for himself. Notice that, in the General Theory, if you have a velocity change of zero, that is a special case – the same as the Special Theory and why the Special Theory is so called.
However, we want to avoid the maths of the General Theory as much as we can, and this is why The Twins Paradox is interesting. Many people think you can only explain The Twins Paradox with the General Theory. But they are wrong!
But what is The Twins Paradox? Well, I’ve hinted at it already, but here it is in full.
The Twins Paradox
Take a pair of twins – who are, naturally, the same age. To distinguish them, we’ll have fraternal twins, a boy and a girl. The girl becomes an astronaut. She gets on a rocket, cruising steadily at a sizeable fraction of the speed of light (which is not forbidden, but would require a pretty powerful rocket) and she flies to the star nearest to Earth, which is Alpha Centauri, 4 light-years away. (A light-year is the distance light travels in a year. It is not a time, it is a distance.) She turns round and comes back to her stay-at-home brother, still on Earth. OK so far?
Looking over the theory of Relativity as explained above, because she’s been a gal on the move, her clock has been running slower than her brother’s on Earth, so she is now younger than her twin brother!
As odd as this is, things get worse. Also according to what I’ve said earlier, the brother has seen her fly away and come back, so it’s from his point of view that her clock looks slow. But it also depends on whose point of view you are looking from. From her point of view, he has moved away (along with the Earth) then come back, so, as she sees it, his clock has run slow and he’s younger.
They can’t both be right. Are they the same age? No – as we shall see. Which one is younger? Well, it depends – in some way something that has happened to one of them is different from something that has happened to the other. The questions are: what? To which? And with what result?
The usual explanation – and the one referred to in the play Insignificance, is that she has had to undergo an acceleration in order to fly away from Earth. She also had to slow down when she got to Alpha Centauri, turn round and speed up again to fly back. She probably had to put the brakes on as well when she got back to Earth so that she could have a chat with her ageing, Earthbound brother. This involves changes of speed, so we have to use General Relativity with all its horribly difficult maths to explain why – as it turns out – she is younger than her twin.
This is true. The astronaut twin ages less. She is younger. So that’s the answer to that one. But it’s nothing to do with General Relativity.
That’s because we can cut out the bit about speeding up and slowing down. It’s not so important that the two people are twins now – any two people will do – we just want to see which one ages less – who has the slower clock, and why.
Suppose that we use stopwatches to time everything and our lady astronaut sets off in the opposite direction from Earth, away from Alpha Centauri, to start with. She turns round, has a good run-up and gets to her steady cruising speed just as she passes her brother on Earth, and both of them start their watches. Onwards she travels. As she gets level with Alpha Centauri, she stops her watch and applies the brakes; back on Earth he stops his watch at the same time. There is a problem with this – she is 4 light years away so can’t actually see her pass Alpha Centauri for another 4 years (and with a very good telescope.) But he knows, for a given cruising speed, how long it should have taken her, from his point of view, so he trusts nothing has gone wrong and stops his watch by dead reckoning. (This might not sound very convincing, but it is not a fiddle – read on.)
Sister now turns her spaceship round, accelerates to steady cruise speed just as she gets level with Alpha Centauri, and, as she does so, she and her brother both start their stopwatches once more. (To do this, they will have had to plan the mission out very carefully and stick to the plan, but as long as they do so, everything will be fine.)
Now, she gets level with Earth and, as she does so, both brother and sister stop their watches a final time. (She can now get on with braking and coming back to Earth for a soft landing.) The important thing is – both watches have only been timing the part of the journey when the rocket was flying at a steady, unchanging, Special Relativity-friendly speed. So what do the watches show?
They show her watch is slow compared with his! She has aged less! How is this possible, when, from the point of view of each of them, it’s the other that has been moving?! Surely, now that we’ve cut out the speeding up and slowing down bit, they have experienced exactly the same things!
Not so. It’s a bit like a magician’s trick. We’ve been looking at the wrong thing about the experiment.
The thing is – she has flown to Alpha Centauri and back, a distance of four light years as seen by the brother on Earth. Earth and Alpha Centauri have not been moving, as far as the brother is concerned. But she has been moving, and quite quickly too. And from her point of view, Earth has moved away and Alpha Centauri has moved towards her. Or, thinking of it another way, the Earth-Alpha Centauri route are two ends of something moving past her, like light going past our old clock on the mantle-shelf.
Anything that moves, shrinks in length of the direction of travel. Remember?
From her point of view, the journey to Alpha Centauri, and back, has been shorter than it looks to the brother on Earth. Because the journey is shorter, as she sees it, her watch hasn’t had enough time to run for as long as her brother’s, nor has she aged as much. The two siblings have not had the same experience and that’s why they’ve not had the same number of birthdays by the moment she gets back. Speeding up and slowing down have nothing to do with it – at least as far as our stopwatches are concerned, because they weren’t running for that part of the experiment.
So that is the answer and explanation to The Twins Paradox. The travelling twin ages less and is no longer as old as her brother, but you only need Special Relativity to explain it. General Relativity can be kept out of it.
That’s about it, really.
The End
(Much of the information for this article was drawn from lecture notes made available on line by Michael Fowler at
http://galileo.phys.virginia.edu/classes/252/ - however, all errors, faults and general confusion are my fault. For much more – puzzling, interesting and accurate information on Relativity and other stuff - see this site.)
Addendum - For all you Doubting Thomases out there
(Gary, this means you!)
Some people have doubted that the difference in ages can be a Special-Relativity-only effect. Here is a worked example using actual maths and figures to demonstrate that the twins will age by different amounts even without considering acceleration and deceleration – that is, constant speed.
The situation.
One twin – the female – is going to fly to Alpha Centauri, a distance of 4 light years, and back, 60% of the speed of light (v = 0.6c) while the male twin stays at home.
During the trip, each will send off a flash of light from a beacon, once a month. In other words, each will have aged one month between emitting each flash.
It is vital to realise that each flash from its sender means that one month has passed for that person. In other words, counting up the flashes sent by either person shows how much they have aged.
As she sets off at 0.6c, she sees flashes arriving from him just once every two months, just as he does from her. This is partly owing to the ever-increasing distance, (an ‘optical’ effect) and time dilation (a relativistic effect.)
Not convinced? And I think I can hear you say - Hang on - after 1 month travelling at .6c, the next flash ought to be after 1 + 0.6 = 1.6 of a month! But you are forgetting that time dilation is slowing her clock. The amount by which it is slowed down is worked out using the following formula: Time observed = Time at rest / square root (1 – v^2 / c^2 ) where v is her speed and c is the speed of light. Her speed is 0.6 so v^2 = .36. So v^2 / c^2 = 0.36. 1 – 0.36 = 0.64. The square root of 0.64 is 0.8. So her time is dilated 1/0.8 which is 1.25. In other words it takes her 1.25 months between signals, owing to time dilation. But in that time, she has travelled 0.6 * 1.25 light-months which is 0.75 light-months. Therefore her next light signal takes 1.25 + 0.75 = 2 months to reach Earth. (And, of course, swapping things around to her point of views, his signals reach her only once every two months as well.)
On the way back, the situation is reversed as far as the change in distance is concerned. Each signal is sent out, as seen by the other, with time dilated to 1.25 months. But in that time, the distance has closed by 0.6 * 1.25 = 0.75 light-months. So the signal arrives at the other end of its journey 1.25 – 0.75 = 0.5 of a month, in other words, two signals a month.
From the Earth, the total journey time (excluding speeding up slowing down and turning around) is 8 light years at 60% the speed of light = 13 years 4 months, or 160 months.
What does she see?
The distance of the trip to Alpha Centauri, where she sees the star rushing towards her, is shrunk by length contraction. The formula for this is the original distance multiplied by the square root of 1-v^2/c^2. This equals 80% of 4 light-years, which is 3.2 light years. (The square root of 1-0.36, which equals square root of .64, which is 0.8.) It therefore takes her 64 months to get there (3.2 light-years/0.6 = 5 years 4 months.) Therefore she sees 32 flashes from her brother.
She turns round and heads back, and immediately sees the frequency of flashes from her brother increase. This is despite time dilation as she is shortening the distance to her brother. She now sees flashes twice a month. It takes her another 64 months to get home, and she sees 128 flashes. By the time she reaches Earth her brother has flashed, and aged, 32 + 128 = 160 months. This is just what we would expect (see above.)
What does he see?
From his point of view, the distance to Alpha Centauri remains 4 light years, so he expects her to take 4 light years / 0.6c = 6 years 8 months = 80 months to get there. But, because Alpha Centauri is 4 light years away, he doesn’t see her turn round for another 4 years = 48 months. (This is a key difference in their experiences – she sees the flashes from her brother change immediately she turns round. He doesn’t see her turn round till 4 years later.) So it is 80 + 48 = 128 months into the mission before he sees her turn round, during which she has flashed 64 times and aged 64 months.
There is now only 160 – 128 = 32 months left before she gets back Earth. Because the distance is closing and despite time dilation, he too sees flashes from her at twice a month, 64 flashes in total.
By the time she arrives back, he has seen 64 + 64 = 128 flashes from his sister who has therefore aged only 128 months to his 160. So his twin sister is now 32 months younger than he.
Note that this difference is regardless of accelerations and is a constant velocity, Special-Relativity-only effect.
Don’t take my word for it, doubt all you want to – but do the maths
(I apologise for breaking my word about there being no maths in this article, but sometimes sums speak louder than words!)
P.S. Notes
Equations of Special Relativity
Length observed = Length at rest * square root (1 –
v^2 /
c^2 )
Time observed = Time at rest / square root (1 –
v^2 /
c^2 )
Mass observed = Mass at rest / square root (1 –
v^2 /
c^2 )
.jpg)
