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This is a guide to Einstein's theory of Special Relativity aimed at the non scientist. No prior knowledge is assumed, but I will need to assume that you are familiar with basic arithmetic, and that you know something, even if only a very little, about algebra.

## Introduction

Please do not be put off if maths frightens you - I hope to show you that maths is nothing to be feared and what have you got to lose? If you know that algebra uses letters to represent values then that should be enough to get us going. If you are more proficient then so much the better. If you know Pythagoras' theorem about right angled triangles then you already know the most difficult maths in the whole guide.

Special relativity was Einstein's new way of looking at the universe. Specifically it is his way to explain light and movement but not gravity and, therefore, not real objects moving around the earth. That comes in the second part of relativity theory - the General theory. Here I am concerned only with the Special theory (SR) which is, despite the name, much easier to get to grips with than the General theory (GR).

Firstly a bit of context will help to put us in the right frame of mind. Einstein wrote SR in 1905. Why then? Well, from the middle of the 19th century a real puzzle had emerged in science that had just about everyone scratching heads. To understand this we need to go back further still, to Newton and Galileo.

## Galilean fandango

Many people are surprised to learn that the first theory of relativity belongs to Galileo, not Einstein, and is, not surprisingly, called Galilean relativity. What Galileo realised was that speed, specifically either being stationary or moving at a constant rate, is a matter of opinion, not a fixed value. If you take a person below decks in a ship, would they know if the ship was moving? They are allowed to perform any test they like (apart from peeking) - could they tell if they were moving or not? The answer is no, not if their motion was at a constant velocity. So, thought Galileo, who is to say the ship IS moving? Why can we not say that the people in the harbour are moving and the ship is stationary? Finding no satisfactory reason, Galileo developed the idea a bit further. In fact**nobody** can say that they are moving and everyone else is stationary, or moving differently. It is equally valid to say that a train is stationary and the station is moving.

Galilean Relativity

Just because we associate the station with 'solid ground' is not a good reason - after all, that same 'solid ground' is revolving, and moving in at least three different ways as we orbit the sun. It isn't any more 'solid and fixed' than is the train. This is what we mean by*Galilean relativity*. In essence it says that allmovement (at constant rate - acceleration doesn't count) is relative - it depends on the person moving and what they measure that movement against - we need an observer and a subject and the motion is measured by one relative to the other - hence 'relativity'. This is true for all non-accelerating motion. There is a word for non-accelerating which I'll start to use now - mainly because it is easier than keep tying 'non accelerating'. Such motion is called **Inertial**. So whenever we have objects moving in inertial relationships, we can decide to measure that movement however best suits us, using whatever reference point is easiest for our purpose. This worked fine, and it is assumed in Newton's later work - in which we get not only the theory of gravity, but equations of motion for everything - in physics speak we call this '**classical mechanics**'.

## Newton waves

Basically Newton came up with three laws of motion that can be used to model any movement with surprising accuracy. They produce approximate answers because they sometimes don't account for all the forces (friction, wind, uneven surfaces) perfectly - but the answers are very close to what we observe and Newton's laws are still generally used for everything from calculating the ballistics trajectories of missiles and artillery, to planning space voyages far into the solar system.

So, constant motion, or no motion, was known to be all a point of view when it came to measuring things like distance and speed. There is no point in the universe which we can say is the reference point for measuring all motion - we simply compare the*relative* motion of two or more objects to get any answers we need. This was all fine and dandy until science hit a problem.

By the Victorian age, the best minds in science were digging into the problems of light and magnetism and making startling progress. It was known that light behaved like a wave. This was easy to demonstrate. Waves are reflected when they hit a solid object. The angle a wave hits a flat surface is the angle it will leave the surface - this is always the case (*and can be used with non waves, for example, in snooker/pool*). Also, waves change direction when they move through different materials (refraction). You may have seen this effect when light enters water - the path of the light is apparently bent. If you reach for something under water whilst you are still on the surface, you find it is not where your eyes tell you. If you push a straight rod into water, you will see it apparently bend into a new angle.

This can be seen nicely illustrated in this photograph of two glasses of water with straws in them. Since light was known to be a wave, and since scientists knew that waves always move 'through' a medium, the question was, what does light move through?

We know that water waves move through liquid, and sound is waves moving through air. Take away the liquid or the air and there are no waves. When I was at school there was an experiment to show this the teacher would put an alarm clock into a bell jar, and then use a vacuum pump to take out the air. As the air is pumped out the sound goes down in volume until, when nearly all the air is removed, the bell can no longer be heard and is, effectively, silent, though it can still be seen to be ringing away.

Different types of wave move differently through media. If we consider transverse waves, the medium particles move perpendicular to the wave. So if the wave is moving horizontally, the particles of medium through which it moves will be moving up and down.

Here is an example of a**transverse wave**: notice the particles are moving up and down whilst the wave itself is moving left to right. Notice also that the particles do not move overall - they oscillate around a point and stay where they are.

Most people think that water waves and ripples are transverse waves. They are not. Waves in water are a combination of transverse and longitudinal waves.

The best known pure transverse wave would be earthquake S-waves (they usually come after the initial quake. Here is a diagram of a water wave. Notice that individual particles move both up and down and side to side - the examples in yellow illustrate this nicely.

Notice also that, like with the transverse wave, the individual particles oscillate around a fixed point and don't actually change position over time. If you observe closely you will notice a distinguishing characteristic of this type of wave, compared to similar 'Rayleigh waves' - the particles always move in a clockwise direction

Here is a** longitudinal wave**. In this type of wave the particles move in the same direction as the wave itself. Notice that, once again, the particles oscillate around a fixed position. In this type of wave the actual wave-front can move faster than the individual particles. A distinctive pattern of compression and rarefaction is generated. The best example of this type of wave is the normal sound waves moving through air, water, or other medium. The areas of high and low pressure generated cause our ear drums to vibrate at the same frequency and we perceive that as sound. At the 'recording end' the simplest microphones do the same - they have a membrane which is vibrated by the air pressure. This vibration is converted into an electrical signal which is then processed, or stored as needed.

Here is a wave which starts from a single point and moves outwards in all directions. This is called a monopole wave and this is the sort of characteristic I would expect to see in a sound wave coming from a point source.

So the question for light and magnetism is - which kind of movement do they use and what medium do they use to move through?

Pretty much everyone was sure that there must actually BE a medium - I don't know of any scientists of the time who proposed anything other. Since nobody knew anything about the medium of 'empty' space, which light certainly moves through, it was assumed that space must have some type of material in it, maybe particles so tiny they were undetectable. This substance was given the name '**aether**'. Most people assumed it would be something like thin air, although some of the more advanced thinkers - eg Michael Faraday - thought it might be more like the sort of invisible forces we see around magnets if we scatter some iron filings.Newton had considered that the aether would be made up of tiny particles and this is the model he proposed in his 'Optics'.

The model works reasonably well in explaining reflection, but it falls down badly when it is applied to refraction and it failed completely on the issue of diffraction.

Towards the end of the 18th century, two scientists - Thomas Young and Augustin-Jean Fresnel - proposed that the aether was a medium allowing the passage of waves - specifically transverse waves(Newton had considered waves to be only longitudinal and had never considered this possibility). The theory was promising - it could explain a behaviour which had, until this time, been unexplained in all previous models - Birefringence. This is the phenomenon where the propagating medium has a refractive index which depends to some extent on the polarisation of the light being transmitted. The diagram shows this phenomenon. Incoming light in the parallel (*s*) polarization sees a different effective index of refraction than light in the perpendicular (*p*) polarization, and is thus refracted at a different angle. This property is used in the construction of polarising filters for various uses - including CD and DVD reading.

Towards the end of the 19th century two scientists decided to try and measure the aether. We know that waves moving through a medium will move differently depending on how the medium itself is moving. Swimming against a tide is obviously harder that swimming with the tide, so one would expect to see light move faster when it was travelling the same direction as the aether. The two American scientists - Michelson and Morley - decided to build an experiment which would allow them to see if light travelled at different speeds across the same distance. Crucially the entire experiment could be rotated so that it could be run at any angle, thus allowing the movement of the aether to be detected. The experiment is shown in an application below which will let you re-run the experiment here.

A beam of light is split at right angles by a prism, one part of the original carries straight on, while the other travels an equal distance at a right angle. If there is any aether movement the two should arrive back at slightly different times. Michelson and Morley ran the experiment many times and they could not detect any difference. No matter how they rotated the equipment, the two light pulses always arrived at exactly the same time. This is the most famous '*null result*' in science. Nobody expected that they would fail to detect an aether but the conclusion was unavoidable - there IS NO aether.

The experiment has been repeated many times and more sophisticated versions have been tested, but each time we see a null result (ie no results allowing the testing of the thing the experiment is designed to detect). The conclusion is clear and as certain as it is possible to be - there is not aether and light can travel without a medium to go through. This is all happening in the time that Einstein is working on his theory.

## Maxwell

Even more importantly, a few years before the Michelson-Morley experiment a Scottish Mathematician James Clerk Maxwell, had worked out a set of fundamental formulas which described how light and magnetism behaved - the three equations are simply known as the 'Maxwell equations'. At first nobody realised one very important and very puzzling consequence of these equations. When they are applied properly, they show that light (and magnetism) always moves through free space at a fixed speed - what we now call**c** - the speed of light. The puzzling thing was that there is no term in the equations to represent the movement of the observer. The bizarre thing about this is that if two people both observed a light pulse, and they were moving relative to each other, the equations say that they would both see the light pulse move at the same speed. This contradicts everything we know about how things move. Consider this simple example from our everyday world:

## Enter Einstein

This result is extremely peculiar - it just isn't how things behave in our experience. Various explanations were tried - all of them trying to explain why the light couldn't really be moving at the same speed for all observers - obviously there must be some subtle thing that had been missed which would return the system back to normal sensible operation and not this bizarre world where speed and distance misbehaved. In Germany, however, a clerk working at his local Patents office had seen Maxwell's results and decided to accept them without any attempt to modify or reinterpret them. This was Einstein's master-stroke. He didn't reject the Maxwell equations because they produced results which went against experience and common-sense, instead he decided to follow through with his work and see just where the strangeness led him.

We now have the conditions for Einstein's theory to emerge from the Patent Clerk. In fact, we can almost write it for ourselves now.

The theory itself consists of two 'axioms' - statements which are put forward as true and from which the rest of the theory follows. These starting assumptions are:

* 1) As Galileo showed, the laws of physics works the same for all observers who are moving inertially with respect to each other.*

*2) The speed of light is constant, to all observers, regardless of the size and direction of their motion.*

From these two starting points the rest of the theory flows - indeed it cannot help but do so. Given this starting point any competent scientist must arrive at the same conclusions that Einstein did.

So the central paradox is this - how can people moving in different directions all agree on the same speed for something moving relative to them all? It makes no sense to us because we know that is not how things behave - or we THINK we know. In fact it makes perfect sense and everything moves in this way. To see how, we need to move to the most technical part of this guide. Please do not run away or be nervous at this point - I will try to make this understandable for you and if you give it a go I think you will surprise yourself - there is nothing to lose, if you DON'T manage to understand the next part it will be a shame but you can still go on to understand the rest and, hopefully, the next bit will become obvious to you at some future point.

## Thinking time

OK - What I want to do now is what Einstein did. He was famous for his thought experiments. He would imagine himself moving along with a beam of light and see if he could imagine what it would be like and how it might work in reality. We are going to do such an experiment now. The things I will use are :

A fast spaceship for our intrepid experimenter Jill. Two light clocks (more about this in a moment) so that Jill and the observer (Harry) can keep very accurate time. Simple mathematics - the most complex part of this is when I use Pythagoras theorem so please don't be spooked, and do try to follow.

Jill is our spaceship pilot. Harry is our observer. Both have a light clock. This is simple a pair of polished surfaces a fixed distance apart. Light can bounce between the two surfaces and this gives us a basic clock, because distance divided by the speed (c) gives us time. t=d/c

Light Clock

The actual distance d is not important, and we don't need to put the real value for c in at this stage - it is very large and cumbersome, so we will stay with the simple letters/algebra.

Now, Jill is going to fly her spaceship as fast as she can and we will watch what the observer (Harry) sees when she does this.

Imagine that Harry is looking straight ahead as Jill flies across his view. Imagine he can see her light clock (hey, this IS a thought experiment, after all).

What Harry sees is Jill's light clock moving as the light pulses up and down, as follows:

As you can see, the light pulse, from Harrys point of view, is going along a diagonal, not a straight line. This means it is travelling further from Harry's point of view (POV) than it is in Jill's POV. Remember also that the pulse cannot speed up or slow down - light travels at the same speed for all observers - both Harry and Jill see the pulse move at the same speed.

How much difference there is will obviously depend how fast Jill is going. The faster she goes, the shallower the diagonal line and the greater the difference as the light beam has further and further to travel.

I hope you have followed to this point. If you are struggling then please go over this again until you are confident to proceed to the next part. At this point I want to start using a new word - frame (or frame of reference). This is just another word meaning 'Point of View'. A frame is a location where everything has the same point of view - ie everything is stationary with respect to everything else in the frame. So in this example Harry is in one frame, and for our purposes that frame includes everything around him that is not moving. Jill is is another frame and that frame includes everything in the spaceship, and the ship itself.

** Question** - this is how Harry sees Jill's light clock. How does she see Harry's light clock?

Now you will remember that if Relativity is correct, one of the conditions is that light moves at the same rate for all observers in inertial frames. Jill is not accelerating, so Jill and Harry are in inertial frames, meaning that light goes at the same speed for both. But both see the other's clock ticking at a different rate (since light is the same speed, and since the path of the light is clearly longer, it follows, and it MUST follow, that the clock they observe on the other person is ticking at a SLOWER rate than their own clock, when it is observed from their own frame. This is a symmetrical observation - both see the other clock slower than their own. Now this immediately smacks up against common-sense, experience and much of what we think we know. It is almost an insult to our common sense to suggest that two people can both see the other clock running slow. How can that possibly be? At this point I will have to ask for your trust temporarily (anyone who asks for it permanently is selling faith, not science :-). You WILL understand this apparent paradox, but slowly slowly....we have to let each part bed-down before jumping into the apparent paradoxes and problems.

This is crucial and it is important that you understand this before moving on. If necessary stop at this point and let it roll around in your subconscious as you get on with something else. Then come back to it and make sure you follow the logic of what I have described. You don't need to be able to repeat details, but you do need to understand how this business with the clocks is important. You should be able to see, logically, even if you cannot accept it, that the clocks are ticking at different rates - slower and that this is not just some effect on the actual clock - it applies to everything in that frame. Think about it - let's say that Jill takes 5 ticks to blink, 100 clicks to walk 10 metres, 1000 clicks to eat her lunch. To her, the clock is still ticking as it ever did, no difference. To Harry her clock is ticking more slowly and therefore SHE is slower - she takes longer to blink, longer to walk 10 metres and longer to eat her lunch. EVERYTHING is slowed by the same rate as the clock. Please do not move on until you can see WHY this is, even if you cannot see HOW it is.

## Some Sums

OK, let's consider what is going on in more detail now. We have seen that the clocks seem to tick at a different rate for different frames, so we need to know how to convert 1 tick of Harry's clock into a tick of Jill's clock. To do that we need to have an expression which contains both terms. I will use the term**t**_{h} to mean a tick of Harry's clock as seen in Harry's frame. I will use **t**_{j} for a tick of Jill's clock as seen in Harry's frame. We will work entirely in Harry's frame to avoid any confusion.

Here comes the maths. I am not going to dumb it down, so many of you might not completely follow me all the way - don't worry, that is not needed. Follow as far as you can and try to get a 'feeling' for what I'm doing even if you cannot understand the specific operations. Remember the key rules for algebra - what you do to one side of the equals sign, you must do to the other side.

Harry and jill both see their own light-clock ticking normally. The pulse travels distance d at speed c. They both see each other's clock ticking differently - the pulse now travels on a diagonal which is longer, and since light moves at the same rate, the time taken for the clock to 'tick' must be longer as well.

Here is a simple diagram showing what happens.

d is the normal distance for one tick of the local clock. h is the distance each observes for one tick of the other's 'remote' clock.

We can calculate h using Pythagoras' theorem.

*The sum of the squares on the two sides equals the square on the hypotenuse.*

First we need a right angled triangle. Easy - we split the triangle in half by drawing a line from the top vertex to the bottom.

We will call the bottom side** x** for the moment. The other side is **d**, and **h** is the hypotenuse. So we have a sum that looks like this:

\(h^2=x^2 + d^2\) ...(1)

So far so good. Now we can say more about the distance x. It is the distance travelled by the spaceship in one tick of the observer's (Harry's) clock. Think about it - Harry sees the ship move distance x in the time taken for his own clock to do one normal tick. So the distance the ship travels in that one tick of Harry's clock is equal to the velocity of the ship multiplied by the value of Harry's tick - x is the same as velocity of spaceship multiplied by**t**_{h}. So we can put this into 1 and we get

\(h^2= (vt_h)^2 + d^2\) ; ...(2)

At this point we can use Einstein's second assumption - light moves at the same speed for everyone. How long is h then? It is the time (measured by Harry's ticks - \(t_h \) that light travels in one tick of Harry's clock

\(t_h\) so that is \(ct_h=h.\)

If we put this into 2 we get \((ct_h)^2= (vt_h)^2+d^2 \) .

We now need to introduce the t_{j} term into our equation so that we can relate t_{h} and t_{j} to find the relationship.

Now, we can write d as the time that Jill sees one tick of her clock - \(t_j\) and it is therefore\(t_jc\) So we now have\((ct_h)^2=(vt_h)^2+(t_jc)^2\)

Divide both sides by \(c^2\) to give \(t_h^2=t_j^2+\frac{vt_h^2}{c^2} \)Now square root both sides to arrive at the final expression.

\(t_h=t_j \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\)

And there we have it - an expression which gives us t_{j} in terms of t_{h} (or vica versa if needed). What is this telling us. Well Jill's time is running slower than Harry's own time from his frame. Her clock ticks more slowly but that is just a measure of her time (Jill would see everything quite normally, just as Harry does in his own frame, and she would see Harry slowed down, just as he sees her slowed down).

The amount of slowing is given by the term \(\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\)

That is what we must multiply Harry's clock ticks by to give Jill's - it is the conversion factor.

I would not expect most readers to follow the maths all the way, but if you understood the first few lines then well done - you are better at maths than you thought - seriously. If you didn't follow any of it then don't despair - check out the learning zone courses on maths. Any of you who followed me all the way through - your maths is at least level 3 - A level standard, so if this surprises you then I suggest you have a talent you might want to explore.

This conversion factor is always the same and it is more properly known as the Lorentz factor or the Lorentz transformation. It is what we multiply time in one frame by to get time in another non-inertial frame. Looking at it, clearly is is 1 divided by 1 minus something. If that 'something is small then we get almost 1/1 which is no change at all. Only when the 'something' gets larger - 0.1 or more, will we start to notice anything. So let's now examine this 'something'. In English it is the speed of ship squared divided by the speed of light squared. The speed of light is A LOT. It is around 300000000 metres per second (approximately 186,000 miles per SECOND). In order for this 'something' to have a significant value, the speed of the spaceship has got to be something reasonably close to the speed of light - say 1/10th or more. Needless to say we have nothing that goes anywhere near that speed. This explains why our intuition about how speed works is wrong, but we don't see it. It is only when things are moving at massive speeds that this effect is noticeable. At the sort of speeds we are used to it doesn't even register on our most sensitive clocks - even our fastest vehicle - the space shuttle - didn't get over twenty thousand miles per hour - that is an insignificant speed compared to c and if we slot that into the formula we find it gives a value for the Lorentz transformation of : 1.00000000098.

So to put that into figures we can understand, if we went full speed in a shuttle (I know, none left, so imagine..) and we kept the pedal to the metal for the next 50 years before returning to earth, there would be a slight difference between ship time and earth time. That difference would be about \(\frac{1}{10}\) th of a second. If things move really quickly, however, the transformation value gets big quickly. As the speed approaches c the transformation value rockets upwards. Let's examine a few values to get a feel for it.

## Going deeper

There is a general principle in science which is often phrased as '**extraordinary claims require extraordinary evidence'.**

I don't actually like that wording because it sounds as if there is a different KIND of information or evidence, when that is not the point. The point is that any extraordinary claim will be looking to overturn current theory - and the more extraordinary it is, the more of existing theory it probably goes up against. Now that is fine and dandy and long may it continue, but the person or group making the claim need to be clear that existing theory is there because it has proven itself, so your new theory better be on solid ground or it won't even get noticed by the serious geeks in white coats, who you NEED to notice, because those people are the ones who might decide to splash some research grant money checking out your claim.

Well, as claims go, Special Relativity is a pretty big one. It basically requires that tried and tested Newtonian Mechanics be abandoned as the best we can do and admitted to be wrong. This is where a lot of misunderstanding - some deliberate and some genuine - gets written and circulated. Creationists and other religious fundamentalists like to claim that science is always been shown to be wrong. Well, yes and no. Science must always ADMIT of being wrong - any scientist who really things they have the ultimate model, beyond improvement, is either not a real scientist or a real scientist who needs wise council and, perhaps, a long break in quietude and tranquility, away from stress and strain, and, oh yes, away from anything sharp or pointed.

The measurements that were made, showing acceleration due to gravity and fixing a value for g at around 9.81 - those measurements don't suddenly become inaccurate or badly reported, let alone deliberately fiddled. They were correct then and they remain correct. Any new theory must account for them at least as well as the old theory, and must do stuff that the old theory could not do. Nearly all progress of a major kind, over the last century at least, has been enlargement rather than replacement. By that I mean that new theory comes along and it doesn't simply add a couple of decimal points to the certainty, it fundamentally changes the picture so that where we previously saw distinct phenomena, we now see one phenomenon being expressed differently. Einstein's claim was just this sort of change. He didn't try to find parts of Newtonian physics that could be tweaked and improved. He chucked it in the mental dustbin and said 'we are looking at the whole problem from the wrong viewpoint, and if we check out THIS viewpoint we can see that Newton was just a description of a little bit of the whole picture. What we need is to first glimpse that whole picture and then start writing.

Newton produced mathematical models which corresponded really well to observation. That is a reasonable description of a scientist's job and nobody need fear for Newton's reputation at the top of the top table - that is, I think, secure. But in fact Newton had no clue how any of it worked. He might as well have said that the attractive force he called gravity results from a particularly big demon exerting it's will to attract the mass. What Universal Gravitation gives us is an equation which we can use to predict and model systems, but which is not at all explanatory, simply declaring that mass attracts, and bigger mass attracts more. and that the attraction drops off as a square of distance. There is nothing inherently wrong with that. It may be that our brains just cannot cope with some aspects of 'reality' and they will remain beyond us, forcing us to adopt mechanistic calculations to work out what happens, but never seeing under the bonnet of the maths - never understanding what the maths is describing - if anything :-)

Einstein's relativity is a different animal. It starts with imagination and controlled play. Einstein let's himself drift into 'daydreams' in which he explores ideas by putting himself into the model. He imagines what it might be like to ride alongside a photon of light. He thinks about BEING the photon and imagining what his world-experience would then be. These are his famous thought experiments - one of which we have just covered in a small but interesting amount of detail. Newton relies on forces to pull and push things around. Einstein's instinct is to let things go on doing what they do, without some ill explained cause smacking them around and forcing them to do what they don't normally do. In Special Relativity Einstein starts a process which culminates with General Relativity - he does away with the Newtonian notion of forces. I do not want to pursue this too far in this guide, but it will feature heavily in the guide to General Relativity when I get time to seriously work on it.

For the moment it is enough to deal with the way the theory was received and the way it has been tested. The paper was called 'On the ElectroDynamics of Moving Bodies' and was published in the*Annalen der Physik* scientific journal - a respectable German journal. In fact this was just one of the four papers that Einstein published that year. Commentators have said that in that single 'annus mirabilis' Einstein published 4 papers of such importance that they could well be considered the work of an entire career in physics and still seem like a great achievement. In fact these papers were the writings of an amateur with no University position, let alone tenure.

This paper was received, if not rapturously, at least I would say 'warmly'. Einstein earned a very influential ally at the start- Max Planck read the paper and began to evangelise at the scientific gatherings he attended. Naturally a man of Planck's stature was taken quite seriously when he judged the paper a work of brilliance. Einstein also had a stroke of luck, because two years after he published, Hermann Minkowski published a paper in which he took some of the basic ideas and refined them mathematically to present a much improved version (in that it was more rigorous and more elegant - Einstein was never a mathematical physicist and always struggled with that facet of his theories. In fact he frequently sought (and was given) the help of much better mathematicians in the physics community - Minkowski was probably the first to unwittingly help him.

We have seen that clocks (and therefore relative time) respond to increasing velocity. In fact Minkowski refined SR by introducing his 'Minkowski space' which deals with normal (we call it Euclidean, after the Greek geometer) Euclidean 3 space - space which can be uniquely specified using 3 numbers - coordinates. Minkowski adds time to 3-space to get Minkowski-space (*note it is NOT called 4D space or 4 space, because that is taken to refer to 4 dimensional space where each dimension is a physical coordinate, NOT a time coordinate*).

Minkowski space - normally called spacetime - uses 4 numbers to specify every possible point as a unique event in 3-space - a spacetime coordinate. Not only is the '*where'* contained in the coordinate, it also contains the '*when*'. Diagrams drawn using spacetime are called, boringly enough, spacetime diagrams, or Minkowski diagrams. The system to transfer one set of coordinates in Minkowski space to another is precisely the system we derived previously using pythagoras - the Lorentz factor.

## Other changes

We have seen, briefly that time dilates, by which we mean that an observer will see the clock of a person moving at a steady velocity with respect to her slow down. The person moving, on the other hand, has every right to consider himself stationary and the observer moving - and they see the observer's clock slow down in exactly the same manner. There appears to be a contradiction here. If the observer sees the velocity of the ship as v, clearly in the observer's frame of reference, the ship will cover a distance vt in time t (as measured by the observer). The person on the ship would have a clock ticking at a different rate, so if they multiply velocity by time, they get a different distance covered. Surely the ship must cover an agreed distance. It can't cover two different distances in one time interval can it?

Well, yes it can and it does.

The explanation for the apparent paradox is that distances, in the moving frame are contracted with regard to the stationary frame. This only applies to distances parallel to the movement itself. The amount of contraction is specified by the same Lorentz factor that we know applies to time.

So in our example, Harry would regard the distance travelled by Jill as one value, Jill would see that distance as a much reduced one, depending on her velocity. In both cases we can convert between what Harry sees and what Jill sees by using the Lorentz factor.

I'll deal with evidence for Special Relativity in a later section. To finish this section I want to go on from where we have arrived at and show how it leads to the most famous equation in science. The mathematics here is non-trivial - if you don't know what calculus is then it probably won't make a lot of sense, but don't worry about it - I'm including this only for the more experienced reader who may be getting bored.

**Derivation of e=mc**^{2}

The starting point is the relationship of force and distance. Energy is the integral of force with respect to distance. We can express Kinetic Energy as:

\[K=\int_{0}^{s}Fds\]

Using Newton's 2nd law :

\[F=\frac{d(mv)}{dt}\]

It follows by substitution that Kinetic Energy is given by:

\[K=\int_{0}^{s}\frac{d(mv)}{dt}ds = \int_{0}^{mv}vd(mv)=\int_{0}^{v}vd\bigg[\frac{m_0v}{\sqrt{1-\frac{v^2}{c^2}}}\bigg]\]

Integrate by parts:

\[\int xdy=xy-\int ydx\]

Yields:

\[K=\frac{m_0v^2}{\sqrt{1-\frac{v^2}{c^2}}}-m_0\int_{0}^{v}\frac{vdv}{\sqrt{1-\frac{v^2}{c^2}}}\]

\[=\frac{m_0v^2}{\sqrt{1-\frac{v^2}{c^2}}}+\bigg[m_0c^2\sqrt{1-\frac{v^2}{c^2}}\bigg]_{0}^{v}\]

\[=\frac{m_0c^2}{\sqrt{1-\frac{v^2}{c^2}}}-m_0c^2\]

\[=mc^2-m_0c^2\]

The result shows that the kinetic energy of a body is equal to the increase in its mass as a consequence of its relative motion multiplied by c^{2}. This can be rearranged to show:

\[mc^2=m_0c^2+K\]

If the kinetic energy is decreased so that K=0 the body will be stationary, but will still possess energy m_{0}c^{2}. In other words the body contains energy E_{0} when stationary relative to its frame and will have mass m_{0}. This is called the rest mass. This is shown as:

\[E=E_0+K\]

where

\[E_0=m_0c^2\]

This, then, completes the derivation of E = mc^{2} for a body at rest. For a moving body its total energy is given by:

\[E=mc^2=\frac{m_0c^2}{\sqrt{1-\frac{v^2}{c^2}}}\]

QED

I want to summarise the effects that we have discussed up to now.

**Time Dilation**. Moving frames (F_{1}) will be observed from relatively stationary frames (F_{0}) to be slower. That is, any clock in such frames would run more slowly than a clock in the observer's frame - an effect known as time dilation. The effect is real and affects everything within the moving frame. The amount of time dilation is determined by applying the Lorentz Factor - which, in turn, depends on the relative velocity of the moving frame. The faster the frame is moving, the greater the amount of time dilation. As with all relativistic effects, it only becomes significant when the velocity of the moving frame is a reasonable fraction of the speed of light.

**Length Contraction**. Observers in a moving frame see distances parallel to their movement to be contracted (shortened). The amount of contraction, compared to a measurement of the distance in a stationary frame, is given by applying the Lorentz Factor.

**Mass increase**. A moving object has energy because of the movement. An object at rest has a mass known as the rest mass. When it is moving it has the same rest mass, plus the kinetic energy of it's motion. As the velocity approaches c, the energy rises in accord with the Lorentz factor.

\[T_0=T_1\lambda\]

\[L_0=\frac{L_1}{\lambda}\]

\[M_0=M_1\lambda\]

where \(\lambda\) = Lorentz Factor = \( \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\)

It should be noted that as velocity approaches c (speed of light) the Lorentz factor tends to infinity. If it were possible to travel AT c then the Lorentz factor would be infinite as can readily be seen from the arithmetic:

\[1-\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\]

Obviously as v approaches c the term v^{2}/c^{2} approaches 1 which means we have the sum:

\[\displaystyle{\lim_{n \to 0}f(n)=\frac{1}{n}}\]

\[f(n)\rightarrow\infty\]

This tells us that c is not attainable for anything massive, since it would require infinite energy to raise the mass as velocity approached c. For this reason we say that nothing massive (ie with mass) can travel at c, and all non-massive particles - photons, gravitons, ONLY travel at c.*

** Photons and the theoretical graviton cannot travel less than c because the speed of light is the same for all observers, thus it is impossible to construct a frame in which the speed of light=0 - it must always be c. If light were 'stopped' then it would have no rest mass. It would cease to exist. Occasionally one hears that the speed of light has been reduced, in experiment, to some very low level. It is important to realise that this is referring to situations in which a photon is emitted, then absorbed by some material before being re-emitted after some time. It is that delay with apparently slows the speed down - the individual photons still travel at velocity c.*

## Muon Decay

*(All the figures used here will be rounded to two decimal places for simplicity - the potential errors introduced by this are insignificant in the context of the experiment)*

The muon was discovered in 1936 at CalTech by Anderson and Neddermeyer. The muon is a fundamental particle, similar to an electron but much more massive. It's mass results in the muon being highly unstable - it has a half-life of 2.2 \(\mu \)s.The muon is formed by high-energy events - specifically in high-energy collisions. They are produced by particle accelerators like the LHC, but also when cosmic rays strike matter. This happens continuously in the upper atmosphere when cosmic rays from the sun strike gas particles.

It has long been known that large quantities of muons strike the earth's surface, but, since the muons are mostly produced several kilometres above us, and since their short half-life means that the vast majority will decay before they get anywhere near the surface, this was something of a puzzle.

The answer is, as you might guess, something to do with relativity. When the muons are generated by collision they might head in any direction. I proportion will head straight down and they will be travelling FAST. Laboratory experiments show that the muon travels at around 0.98c (that is, 98% of light speed). At those numbers then time dilation is going to be a big factor. This section looks at this issue in some detail and shows how muon decay provides excellent support for relativity theory.

A relatively simple test has been repeated many times. The experiment measures the number of muons detected at the surface of the earth. We know, from laboratory experiments, that a muon is a radioactive particle which decays with a half-life of 2.2 Î¼s into an electron/positron. We also know that it travels at around 0.98 c. The time dilation factor at that speed, according to Einstein's theory, should be 5.

To conduct the experiment we need two muon detectors. We use them simultaneously, one at a height of about 10km (*we know that this is where muons are produced in the atmosphere by the collision of cosmic radiation with atmospheric molecules*) the other on the ground. Since we know the half-life and the distance to be travelled and the velocity, we can easily calculate the number of muons we would expect to see arrive at the ground.

We can perform this calculation on the assumption that the theory is correct (ie we factor-in time dilation of 5, making the half-life five times longer), and again on the assumption that it is not correct (leave the half-life un-dilated). By comparing actual measurements with the two predictions we have a good test of the theory.

So here in the universe with no relativity we would expect to see about 8 in every million muons make it as far as the surface of the earth. For those who are unfamiliar with the mathematics being used, it is a case of simply slotting values into standard formulae. For example, the half-life of a particle is a measure of how long it takes for half of the existing particles to decay into something else.

## Some History of the Muon Experiment

The historical experiment upon which the model muon experiment is based was performed by Rossi and Hall in 1941. They measured the flux of muons at a location on Mt Washington in New Hampshire at about 2000 m altitude and also at the base of the mountain. They found the ratio of the muon flux was 1.4, whereas the ratio should have been about 22 even if the muons were travelling at the speed of light, using the muon half-life of 1.56 microseconds. When the time dilation relationship was applied, the result could be explained if the muons were travelling at 0.994 c. In an experiment at CERN by Bailey et al., muons of velocity 0.9994c were found to have a lifetime 29.3 times the laboratory lifetime.

These calculated results are consistent with

The final table allows you to simulate the experiment yourself from the Earth frame. It will do the calculations for you - simply enter the height to start from. You can change the other parameters if you want to play around with it (encouraged). This simulation will function to a higher level of accuracy than I did in the above examples. I stuck to two decimal places, but the simulation runs with six to nine decimal places. If you try the same figures as used above (and I would encourage you to do so), you will get a different answer, which is simply due to the increased accuracy.

## Other Evidence

Other Support for Relativity

Relativity has been more thoroughly tested than any other theory in science, with the probable exception of quantum theory.

Many of the tests are extremely complicated, or involve deep physics which I have no time (and probably not enough talent) to go into in this paper. I will confine myself, therefore, to examples which any reasonably intelligent adult should be able to grasp, without subject-specific knowledge.

One of the difficulties here is that Special Relativity is only part of the overall theory. It describes a universe without gravity. As soon as gravity becomes significant then we can no longer use Special Relativity - for a quick way of understanding why this is, think of the two conditions for SR - one of them is non-inertial frames. Gravity is an accelerative force - it naturally forces things out of non-inertial frames and into accelerating frames. Objects accelerate under gravity, they do not stay in constant motion. When I am looking at examples, therefore, of where relativity has made predictions which have been tested and found correct, it is very difficult to find examples with only the effects of special relativity. Since we live in a world which has gravity at the centre (in at least two senses of the word) we are never in a scenario in which Special Relativity can operate. If we could transport ourselves into deepest inter-cluster space where even the massive gravity of the galaxies begins to fade to zero, then we could devise some wonderful tests. Unfortunately we cannot currently get there - and I seriously doubt we will ever be able to do so.

We start by looking at what special relativity is and, therefore, what it posits that we can test.

## Calculator

**Relativity Calculator**

Type your value into the box provided it will usually be velocity - either in miles per second, kilometres or fraction of speed of light (c). Then click to indicate which one above correspond to. [The first 2 buttons are self-explanatory. The third button is clicked when the number you entered in the input box is in terms of the speed of light. The value should be 0 and 1 (where 1 is c).]

The 4th button (the factor of change button) is clicked when you wish to do the calculation in reverse - you input the Lorentz factor you require and the calculator works out what velocity that requires. So, for example, if you wanted a Lorentzian factor of 6 you would enter 6 as input and click this box.

I have added an extra field (value) to the calculator to allow the number of decimal places to be specified. This is particularly useful if you are trying to find a precise Lorentz factor - sometimes the velocity has to be specified to many decimal places.

**Calculator**

**A velocity of :**

- Introduction
- Galilean fandango
- Newton waves
- Maxwell
- Enter Einstein
- Thinking time
- Some Sums
- Going deeper
- Other changes
- Muon Decay
- Other Evidence
- Calculator

Special relativity was Einstein's new way of looking at the universe. Specifically it is his way to explain light and movement but not gravity and, therefore, not real objects moving around the earth. That comes in the second part of relativity theory - the General theory. Here I am concerned only with the Special theory (SR) which is, despite the name, much easier to get to grips with than the General theory (GR).

Firstly a bit of context will help to put us in the right frame of mind. Einstein wrote SR in 1905. Why then? Well, from the middle of the 19th century a real puzzle had emerged in science that had just about everyone scratching heads. To understand this we need to go back further still, to Newton and Galileo.

Many people are surprised to learn that the first theory of relativity belongs to Galileo, not Einstein, and is, not surprisingly, called Galilean relativity. What Galileo realised was that speed, specifically either being stationary or moving at a constant rate, is a matter of opinion, not a fixed value. If you take a person below decks in a ship, would they know if the ship was moving? They are allowed to perform any test they like (apart from peeking) - could they tell if they were moving or not? The answer is no, not if their motion was at a constant velocity. So, thought Galileo, who is to say the ship IS moving? Why can we not say that the people in the harbour are moving and the ship is stationary? Finding no satisfactory reason, Galileo developed the idea a bit further. In fact

Galilean Relativity

Just because we associate the station with 'solid ground' is not a good reason - after all, that same 'solid ground' is revolving, and moving in at least three different ways as we orbit the sun. It isn't any more 'solid and fixed' than is the train. This is what we mean by

Basically Newton came up with three laws of motion that can be used to model any movement with surprising accuracy. They produce approximate answers because they sometimes don't account for all the forces (friction, wind, uneven surfaces) perfectly - but the answers are very close to what we observe and Newton's laws are still generally used for everything from calculating the ballistics trajectories of missiles and artillery, to planning space voyages far into the solar system.

So, constant motion, or no motion, was known to be all a point of view when it came to measuring things like distance and speed. There is no point in the universe which we can say is the reference point for measuring all motion - we simply compare the

By the Victorian age, the best minds in science were digging into the problems of light and magnetism and making startling progress. It was known that light behaved like a wave. This was easy to demonstrate. Waves are reflected when they hit a solid object. The angle a wave hits a flat surface is the angle it will leave the surface - this is always the case (

This can be seen nicely illustrated in this photograph of two glasses of water with straws in them. Since light was known to be a wave, and since scientists knew that waves always move 'through' a medium, the question was, what does light move through?

We know that water waves move through liquid, and sound is waves moving through air. Take away the liquid or the air and there are no waves. When I was at school there was an experiment to show this the teacher would put an alarm clock into a bell jar, and then use a vacuum pump to take out the air. As the air is pumped out the sound goes down in volume until, when nearly all the air is removed, the bell can no longer be heard and is, effectively, silent, though it can still be seen to be ringing away.

Different types of wave move differently through media. If we consider transverse waves, the medium particles move perpendicular to the wave. So if the wave is moving horizontally, the particles of medium through which it moves will be moving up and down.

Here is an example of a

Most people think that water waves and ripples are transverse waves. They are not. Waves in water are a combination of transverse and longitudinal waves.

The best known pure transverse wave would be earthquake S-waves (they usually come after the initial quake. Here is a diagram of a water wave. Notice that individual particles move both up and down and side to side - the examples in yellow illustrate this nicely.

Notice also that, like with the transverse wave, the individual particles oscillate around a fixed point and don't actually change position over time. If you observe closely you will notice a distinguishing characteristic of this type of wave, compared to similar 'Rayleigh waves' - the particles always move in a clockwise direction

Here is a

Here is a wave which starts from a single point and moves outwards in all directions. This is called a monopole wave and this is the sort of characteristic I would expect to see in a sound wave coming from a point source.

So the question for light and magnetism is - which kind of movement do they use and what medium do they use to move through?

Pretty much everyone was sure that there must actually BE a medium - I don't know of any scientists of the time who proposed anything other. Since nobody knew anything about the medium of 'empty' space, which light certainly moves through, it was assumed that space must have some type of material in it, maybe particles so tiny they were undetectable. This substance was given the name '

The model works reasonably well in explaining reflection, but it falls down badly when it is applied to refraction and it failed completely on the issue of diffraction.

Towards the end of the 18th century, two scientists - Thomas Young and Augustin-Jean Fresnel - proposed that the aether was a medium allowing the passage of waves - specifically transverse waves(Newton had considered waves to be only longitudinal and had never considered this possibility). The theory was promising - it could explain a behaviour which had, until this time, been unexplained in all previous models - Birefringence. This is the phenomenon where the propagating medium has a refractive index which depends to some extent on the polarisation of the light being transmitted. The diagram shows this phenomenon. Incoming light in the parallel (

Towards the end of the 19th century two scientists decided to try and measure the aether. We know that waves moving through a medium will move differently depending on how the medium itself is moving. Swimming against a tide is obviously harder that swimming with the tide, so one would expect to see light move faster when it was travelling the same direction as the aether. The two American scientists - Michelson and Morley - decided to build an experiment which would allow them to see if light travelled at different speeds across the same distance. Crucially the entire experiment could be rotated so that it could be run at any angle, thus allowing the movement of the aether to be detected. The experiment is shown in an application below which will let you re-run the experiment here.

A beam of light is split at right angles by a prism, one part of the original carries straight on, while the other travels an equal distance at a right angle. If there is any aether movement the two should arrive back at slightly different times. Michelson and Morley ran the experiment many times and they could not detect any difference. No matter how they rotated the equipment, the two light pulses always arrived at exactly the same time. This is the most famous '

The experiment has been repeated many times and more sophisticated versions have been tested, but each time we see a null result (ie no results allowing the testing of the thing the experiment is designed to detect). The conclusion is clear and as certain as it is possible to be - there is not aether and light can travel without a medium to go through. This is all happening in the time that Einstein is working on his theory.

Even more importantly, a few years before the Michelson-Morley experiment a Scottish Mathematician James Clerk Maxwell, had worked out a set of fundamental formulas which described how light and magnetism behaved - the three equations are simply known as the 'Maxwell equations'. At first nobody realised one very important and very puzzling consequence of these equations. When they are applied properly, they show that light (and magnetism) always moves through free space at a fixed speed - what we now call

------> 40mph | |

A train is moving towards a station at a certain speed - we will say it is doing 40 mph. Atop the train is a man hitting tennis balls towards the station. He is hitting the balls at 20 mph. The question is, what would the observer at the station observe about the speed of the tennis balls. How fast would they be going relative to him/her? Most of us know instinctively that the answer is 60mph, and indeed that is the answer. But now we change just one thing. Instead of hitting a tennis ball, the man shines a pulse of light towards the station. The light leaves his torch at the speed of light - c. So what speed should the observer see the pulse approach ? Obviously c+40mph yes? NO. The observer sees the pulse approach as speed c, the same speed that the man sees the light move away from him. Now, how can this be - both are measuring the speed of the same object, both are moving relative to each other, yet they both record the same speed for the light. They try it again and again, with the light shining at different angles, and even away from the station. In EVERY case, both the observer and the man measure the light moving away or towards them at the same speed - c. | |

----> 40mph |

This result is extremely peculiar - it just isn't how things behave in our experience. Various explanations were tried - all of them trying to explain why the light couldn't really be moving at the same speed for all observers - obviously there must be some subtle thing that had been missed which would return the system back to normal sensible operation and not this bizarre world where speed and distance misbehaved. In Germany, however, a clerk working at his local Patents office had seen Maxwell's results and decided to accept them without any attempt to modify or reinterpret them. This was Einstein's master-stroke. He didn't reject the Maxwell equations because they produced results which went against experience and common-sense, instead he decided to follow through with his work and see just where the strangeness led him.

We now have the conditions for Einstein's theory to emerge from the Patent Clerk. In fact, we can almost write it for ourselves now.

The theory itself consists of two 'axioms' - statements which are put forward as true and from which the rest of the theory follows. These starting assumptions are:

From these two starting points the rest of the theory flows - indeed it cannot help but do so. Given this starting point any competent scientist must arrive at the same conclusions that Einstein did.

So the central paradox is this - how can people moving in different directions all agree on the same speed for something moving relative to them all? It makes no sense to us because we know that is not how things behave - or we THINK we know. In fact it makes perfect sense and everything moves in this way. To see how, we need to move to the most technical part of this guide. Please do not run away or be nervous at this point - I will try to make this understandable for you and if you give it a go I think you will surprise yourself - there is nothing to lose, if you DON'T manage to understand the next part it will be a shame but you can still go on to understand the rest and, hopefully, the next bit will become obvious to you at some future point.

OK - What I want to do now is what Einstein did. He was famous for his thought experiments. He would imagine himself moving along with a beam of light and see if he could imagine what it would be like and how it might work in reality. We are going to do such an experiment now. The things I will use are :

Jill is our spaceship pilot. Harry is our observer. Both have a light clock. This is simple a pair of polished surfaces a fixed distance apart. Light can bounce between the two surfaces and this gives us a basic clock, because distance divided by the speed (c) gives us time. t=d/c

Light Clock

The actual distance d is not important, and we don't need to put the real value for c in at this stage - it is very large and cumbersome, so we will stay with the simple letters/algebra.

Now, Jill is going to fly her spaceship as fast as she can and we will watch what the observer (Harry) sees when she does this.

What Harry sees is Jill's light clock moving as the light pulses up and down, as follows:

As you can see, the light pulse, from Harrys point of view, is going along a diagonal, not a straight line. This means it is travelling further from Harry's point of view (POV) than it is in Jill's POV. Remember also that the pulse cannot speed up or slow down - light travels at the same speed for all observers - both Harry and Jill see the pulse move at the same speed.

How much difference there is will obviously depend how fast Jill is going. The faster she goes, the shallower the diagonal line and the greater the difference as the light beam has further and further to travel.

I hope you have followed to this point. If you are struggling then please go over this again until you are confident to proceed to the next part. At this point I want to start using a new word - frame (or frame of reference). This is just another word meaning 'Point of View'. A frame is a location where everything has the same point of view - ie everything is stationary with respect to everything else in the frame. So in this example Harry is in one frame, and for our purposes that frame includes everything around him that is not moving. Jill is is another frame and that frame includes everything in the spaceship, and the ship itself.

Answer - exactly the same. Think about Jill looking over at Harry - to her it is Harry that is moving.

Now you will remember that if Relativity is correct, one of the conditions is that light moves at the same rate for all observers in inertial frames. Jill is not accelerating, so Jill and Harry are in inertial frames, meaning that light goes at the same speed for both. But both see the other's clock ticking at a different rate (since light is the same speed, and since the path of the light is clearly longer, it follows, and it MUST follow, that the clock they observe on the other person is ticking at a SLOWER rate than their own clock, when it is observed from their own frame. This is a symmetrical observation - both see the other clock slower than their own. Now this immediately smacks up against common-sense, experience and much of what we think we know. It is almost an insult to our common sense to suggest that two people can both see the other clock running slow. How can that possibly be? At this point I will have to ask for your trust temporarily (anyone who asks for it permanently is selling faith, not science :-). You WILL understand this apparent paradox, but slowly slowly....we have to let each part bed-down before jumping into the apparent paradoxes and problems.

This is crucial and it is important that you understand this before moving on. If necessary stop at this point and let it roll around in your subconscious as you get on with something else. Then come back to it and make sure you follow the logic of what I have described. You don't need to be able to repeat details, but you do need to understand how this business with the clocks is important. You should be able to see, logically, even if you cannot accept it, that the clocks are ticking at different rates - slower and that this is not just some effect on the actual clock - it applies to everything in that frame. Think about it - let's say that Jill takes 5 ticks to blink, 100 clicks to walk 10 metres, 1000 clicks to eat her lunch. To her, the clock is still ticking as it ever did, no difference. To Harry her clock is ticking more slowly and therefore SHE is slower - she takes longer to blink, longer to walk 10 metres and longer to eat her lunch. EVERYTHING is slowed by the same rate as the clock. Please do not move on until you can see WHY this is, even if you cannot see HOW it is.

OK, let's consider what is going on in more detail now. We have seen that the clocks seem to tick at a different rate for different frames, so we need to know how to convert 1 tick of Harry's clock into a tick of Jill's clock. To do that we need to have an expression which contains both terms. I will use the term

Here comes the maths. I am not going to dumb it down, so many of you might not completely follow me all the way - don't worry, that is not needed. Follow as far as you can and try to get a 'feeling' for what I'm doing even if you cannot understand the specific operations. Remember the key rules for algebra - what you do to one side of the equals sign, you must do to the other side.

Harry and jill both see their own light-clock ticking normally. The pulse travels distance d at speed c. They both see each other's clock ticking differently - the pulse now travels on a diagonal which is longer, and since light moves at the same rate, the time taken for the clock to 'tick' must be longer as well.

Here is a simple diagram showing what happens.

d is the normal distance for one tick of the local clock. h is the distance each observes for one tick of the other's 'remote' clock.

We can calculate h using Pythagoras' theorem.

First we need a right angled triangle. Easy - we split the triangle in half by drawing a line from the top vertex to the bottom.

We will call the bottom side

So far so good. Now we can say more about the distance x. It is the distance travelled by the spaceship in one tick of the observer's (Harry's) clock. Think about it - Harry sees the ship move distance x in the time taken for his own clock to do one normal tick. So the distance the ship travels in that one tick of Harry's clock is equal to the velocity of the ship multiplied by the value of Harry's tick - x is the same as velocity of spaceship multiplied by

\(h^2= (vt_h)^2 + d^2\) ; ...(2)

At this point we can use Einstein's second assumption - light moves at the same speed for everyone. How long is h then? It is the time (measured by Harry's ticks - \(t_h \) that light travels in one tick of Harry's clock

\(t_h\) so that is \(ct_h=h.\)

If we put this into 2 we get \((ct_h)^2= (vt_h)^2+d^2 \) .

We now need to introduce the t

Now, we can write d as the time that Jill sees one tick of her clock - \(t_j\) and it is therefore\(t_jc\) So we now have\((ct_h)^2=(vt_h)^2+(t_jc)^2\)

Divide both sides by \(c^2\) to give \(t_h^2=t_j^2+\frac{vt_h^2}{c^2} \)Now square root both sides to arrive at the final expression.

\(t_h=t_j \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\)

And there we have it - an expression which gives us t

The amount of slowing is given by the term \(\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\)

That is what we must multiply Harry's clock ticks by to give Jill's - it is the conversion factor.

I would not expect most readers to follow the maths all the way, but if you understood the first few lines then well done - you are better at maths than you thought - seriously. If you didn't follow any of it then don't despair - check out the learning zone courses on maths. Any of you who followed me all the way through - your maths is at least level 3 - A level standard, so if this surprises you then I suggest you have a talent you might want to explore.

This conversion factor is always the same and it is more properly known as the Lorentz factor or the Lorentz transformation. It is what we multiply time in one frame by to get time in another non-inertial frame. Looking at it, clearly is is 1 divided by 1 minus something. If that 'something is small then we get almost 1/1 which is no change at all. Only when the 'something' gets larger - 0.1 or more, will we start to notice anything. So let's now examine this 'something'. In English it is the speed of ship squared divided by the speed of light squared. The speed of light is A LOT. It is around 300000000 metres per second (approximately 186,000 miles per SECOND). In order for this 'something' to have a significant value, the speed of the spaceship has got to be something reasonably close to the speed of light - say 1/10th or more. Needless to say we have nothing that goes anywhere near that speed. This explains why our intuition about how speed works is wrong, but we don't see it. It is only when things are moving at massive speeds that this effect is noticeable. At the sort of speeds we are used to it doesn't even register on our most sensitive clocks - even our fastest vehicle - the space shuttle - didn't get over twenty thousand miles per hour - that is an insignificant speed compared to c and if we slot that into the formula we find it gives a value for the Lorentz transformation of : 1.00000000098.

So to put that into figures we can understand, if we went full speed in a shuttle (I know, none left, so imagine..) and we kept the pedal to the metal for the next 50 years before returning to earth, there would be a slight difference between ship time and earth time. That difference would be about \(\frac{1}{10}\) th of a second. If things move really quickly, however, the transformation value gets big quickly. As the speed approaches c the transformation value rockets upwards. Let's examine a few values to get a feel for it.

Speed (fraction of c) | Lorentz Transformation value |

0.5 | 1.15 |

0.8 | 1.666 |

0.9 | 2.3 |

0.99 | 7.1 |

0.999 | 22 |

There is a general principle in science which is often phrased as '

I don't actually like that wording because it sounds as if there is a different KIND of information or evidence, when that is not the point. The point is that any extraordinary claim will be looking to overturn current theory - and the more extraordinary it is, the more of existing theory it probably goes up against. Now that is fine and dandy and long may it continue, but the person or group making the claim need to be clear that existing theory is there because it has proven itself, so your new theory better be on solid ground or it won't even get noticed by the serious geeks in white coats, who you NEED to notice, because those people are the ones who might decide to splash some research grant money checking out your claim.

Well, as claims go, Special Relativity is a pretty big one. It basically requires that tried and tested Newtonian Mechanics be abandoned as the best we can do and admitted to be wrong. This is where a lot of misunderstanding - some deliberate and some genuine - gets written and circulated. Creationists and other religious fundamentalists like to claim that science is always been shown to be wrong. Well, yes and no. Science must always ADMIT of being wrong - any scientist who really things they have the ultimate model, beyond improvement, is either not a real scientist or a real scientist who needs wise council and, perhaps, a long break in quietude and tranquility, away from stress and strain, and, oh yes, away from anything sharp or pointed.

The measurements that were made, showing acceleration due to gravity and fixing a value for g at around 9.81 - those measurements don't suddenly become inaccurate or badly reported, let alone deliberately fiddled. They were correct then and they remain correct. Any new theory must account for them at least as well as the old theory, and must do stuff that the old theory could not do. Nearly all progress of a major kind, over the last century at least, has been enlargement rather than replacement. By that I mean that new theory comes along and it doesn't simply add a couple of decimal points to the certainty, it fundamentally changes the picture so that where we previously saw distinct phenomena, we now see one phenomenon being expressed differently. Einstein's claim was just this sort of change. He didn't try to find parts of Newtonian physics that could be tweaked and improved. He chucked it in the mental dustbin and said 'we are looking at the whole problem from the wrong viewpoint, and if we check out THIS viewpoint we can see that Newton was just a description of a little bit of the whole picture. What we need is to first glimpse that whole picture and then start writing.

Newton produced mathematical models which corresponded really well to observation. That is a reasonable description of a scientist's job and nobody need fear for Newton's reputation at the top of the top table - that is, I think, secure. But in fact Newton had no clue how any of it worked. He might as well have said that the attractive force he called gravity results from a particularly big demon exerting it's will to attract the mass. What Universal Gravitation gives us is an equation which we can use to predict and model systems, but which is not at all explanatory, simply declaring that mass attracts, and bigger mass attracts more. and that the attraction drops off as a square of distance. There is nothing inherently wrong with that. It may be that our brains just cannot cope with some aspects of 'reality' and they will remain beyond us, forcing us to adopt mechanistic calculations to work out what happens, but never seeing under the bonnet of the maths - never understanding what the maths is describing - if anything :-)

Einstein's relativity is a different animal. It starts with imagination and controlled play. Einstein let's himself drift into 'daydreams' in which he explores ideas by putting himself into the model. He imagines what it might be like to ride alongside a photon of light. He thinks about BEING the photon and imagining what his world-experience would then be. These are his famous thought experiments - one of which we have just covered in a small but interesting amount of detail. Newton relies on forces to pull and push things around. Einstein's instinct is to let things go on doing what they do, without some ill explained cause smacking them around and forcing them to do what they don't normally do. In Special Relativity Einstein starts a process which culminates with General Relativity - he does away with the Newtonian notion of forces. I do not want to pursue this too far in this guide, but it will feature heavily in the guide to General Relativity when I get time to seriously work on it.

For the moment it is enough to deal with the way the theory was received and the way it has been tested. The paper was called 'On the ElectroDynamics of Moving Bodies' and was published in the

This paper was received, if not rapturously, at least I would say 'warmly'. Einstein earned a very influential ally at the start- Max Planck read the paper and began to evangelise at the scientific gatherings he attended. Naturally a man of Planck's stature was taken quite seriously when he judged the paper a work of brilliance. Einstein also had a stroke of luck, because two years after he published, Hermann Minkowski published a paper in which he took some of the basic ideas and refined them mathematically to present a much improved version (in that it was more rigorous and more elegant - Einstein was never a mathematical physicist and always struggled with that facet of his theories. In fact he frequently sought (and was given) the help of much better mathematicians in the physics community - Minkowski was probably the first to unwittingly help him.

We have seen that clocks (and therefore relative time) respond to increasing velocity. In fact Minkowski refined SR by introducing his 'Minkowski space' which deals with normal (we call it Euclidean, after the Greek geometer) Euclidean 3 space - space which can be uniquely specified using 3 numbers - coordinates. Minkowski adds time to 3-space to get Minkowski-space (

Minkowski space - normally called spacetime - uses 4 numbers to specify every possible point as a unique event in 3-space - a spacetime coordinate. Not only is the '

We have seen, briefly that time dilates, by which we mean that an observer will see the clock of a person moving at a steady velocity with respect to her slow down. The person moving, on the other hand, has every right to consider himself stationary and the observer moving - and they see the observer's clock slow down in exactly the same manner. There appears to be a contradiction here. If the observer sees the velocity of the ship as v, clearly in the observer's frame of reference, the ship will cover a distance vt in time t (as measured by the observer). The person on the ship would have a clock ticking at a different rate, so if they multiply velocity by time, they get a different distance covered. Surely the ship must cover an agreed distance. It can't cover two different distances in one time interval can it?

Well, yes it can and it does.

The explanation for the apparent paradox is that distances, in the moving frame are contracted with regard to the stationary frame. This only applies to distances parallel to the movement itself. The amount of contraction is specified by the same Lorentz factor that we know applies to time.

So in our example, Harry would regard the distance travelled by Jill as one value, Jill would see that distance as a much reduced one, depending on her velocity. In both cases we can convert between what Harry sees and what Jill sees by using the Lorentz factor.

I'll deal with evidence for Special Relativity in a later section. To finish this section I want to go on from where we have arrived at and show how it leads to the most famous equation in science. The mathematics here is non-trivial - if you don't know what calculus is then it probably won't make a lot of sense, but don't worry about it - I'm including this only for the more experienced reader who may be getting bored.

The starting point is the relationship of force and distance. Energy is the integral of force with respect to distance. We can express Kinetic Energy as:

\[K=\int_{0}^{s}Fds\]

Using Newton's 2nd law :

\[F=\frac{d(mv)}{dt}\]

It follows by substitution that Kinetic Energy is given by:

\[K=\int_{0}^{s}\frac{d(mv)}{dt}ds = \int_{0}^{mv}vd(mv)=\int_{0}^{v}vd\bigg[\frac{m_0v}{\sqrt{1-\frac{v^2}{c^2}}}\bigg]\]

Integrate by parts:

\[\int xdy=xy-\int ydx\]

Yields:

\[K=\frac{m_0v^2}{\sqrt{1-\frac{v^2}{c^2}}}-m_0\int_{0}^{v}\frac{vdv}{\sqrt{1-\frac{v^2}{c^2}}}\]

\[=\frac{m_0v^2}{\sqrt{1-\frac{v^2}{c^2}}}+\bigg[m_0c^2\sqrt{1-\frac{v^2}{c^2}}\bigg]_{0}^{v}\]

\[=\frac{m_0c^2}{\sqrt{1-\frac{v^2}{c^2}}}-m_0c^2\]

\[=mc^2-m_0c^2\]

The result shows that the kinetic energy of a body is equal to the increase in its mass as a consequence of its relative motion multiplied by c

\[mc^2=m_0c^2+K\]

If the kinetic energy is decreased so that K=0 the body will be stationary, but will still possess energy m

\[E=E_0+K\]

where

\[E_0=m_0c^2\]

This, then, completes the derivation of E = mc

\[E=mc^2=\frac{m_0c^2}{\sqrt{1-\frac{v^2}{c^2}}}\]

I want to summarise the effects that we have discussed up to now.

\[T_0=T_1\lambda\]

\[L_0=\frac{L_1}{\lambda}\]

\[M_0=M_1\lambda\]

where \(\lambda\) = Lorentz Factor = \( \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\)

It should be noted that as velocity approaches c (speed of light) the Lorentz factor tends to infinity. If it were possible to travel AT c then the Lorentz factor would be infinite as can readily be seen from the arithmetic:

\[1-\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\]

Obviously as v approaches c the term v

\[\displaystyle{\lim_{n \to 0}f(n)=\frac{1}{n}}\]

\[f(n)\rightarrow\infty\]

This tells us that c is not attainable for anything massive, since it would require infinite energy to raise the mass as velocity approached c. For this reason we say that nothing massive (ie with mass) can travel at c, and all non-massive particles - photons, gravitons, ONLY travel at c.*

The muon was discovered in 1936 at CalTech by Anderson and Neddermeyer. The muon is a fundamental particle, similar to an electron but much more massive. It's mass results in the muon being highly unstable - it has a half-life of 2.2 \(\mu \)s.The muon is formed by high-energy events - specifically in high-energy collisions. They are produced by particle accelerators like the LHC, but also when cosmic rays strike matter. This happens continuously in the upper atmosphere when cosmic rays from the sun strike gas particles.

It has long been known that large quantities of muons strike the earth's surface, but, since the muons are mostly produced several kilometres above us, and since their short half-life means that the vast majority will decay before they get anywhere near the surface, this was something of a puzzle.

The answer is, as you might guess, something to do with relativity. When the muons are generated by collision they might head in any direction. I proportion will head straight down and they will be travelling FAST. Laboratory experiments show that the muon travels at around 0.98c (that is, 98% of light speed). At those numbers then time dilation is going to be a big factor. This section looks at this issue in some detail and shows how muon decay provides excellent support for relativity theory.

A relatively simple test has been repeated many times. The experiment measures the number of muons detected at the surface of the earth. We know, from laboratory experiments, that a muon is a radioactive particle which decays with a half-life of 2.2 Î¼s into an electron/positron. We also know that it travels at around 0.98 c. The time dilation factor at that speed, according to Einstein's theory, should be 5.

To conduct the experiment we need two muon detectors. We use them simultaneously, one at a height of about 10km (

We can perform this calculation on the assumption that the theory is correct (ie we factor-in time dilation of 5, making the half-life five times longer), and again on the assumption that it is not correct (leave the half-life un-dilated). By comparing actual measurements with the two predictions we have a good test of the theory.

So here in the universe with no relativity we would expect to see about 8 in every million muons make it as far as the surface of the earth. For those who are unfamiliar with the mathematics being used, it is a case of simply slotting values into standard formulae. For example, the half-life of a particle is a measure of how long it takes for half of the existing particles to decay into something else.

Muon Experiment## Non-Relativistic | For the maths geeks:The decay is descrbed by the following formula: \(\frac{dn}{dt}=-\lambda N\) (where N is the number of atoms remaining over time t. \(\lambda \) is a constant known as the 'decay constant' - which is different for each element but is constant FOR that element).This is a simple differential equation which solves to give: \(N=N_0e^{\lambda t}\) which simply tells us that the number of atoms left is the number originally present multiplied by e to the power of the decay constant multiplied by the time elapsed. Now, we know that the half-life described the time taken for half the atoms to decay. \(N=\frac{N_0}{2}\) when \(T=t_{\frac{1}{2}}\) If we substitute into our previous equation above, we get:\(\frac{N_0}{2} = N_0e^{-\lambda T_\frac{1}{2}}\) If we take the natural log of both sides, we get: \(ln\frac{1}{2}=-\lambda T_\frac{1}{2}\) \(\therefore ln2=+\lambda T_\frac{1}{2}\) \(\therefore T_\frac{1}{2} = \frac{0.693}{\lambda}\) Don't worry if you can't follow any of this - I have included this as a treat for the more able at maths, to show how we arrive at the formula most often used when dealing with half-lives. The practical upshot of this number chucking is that we expect to see 8 out of every million muons arrive at the surface, if relativity is wrong and the universe is Newtonian. The survival rate can be quickly calculated using \(2^-N_hl)\) or two to the power of the number of half-lives. Here, the journey lasts approximately 17 half-lives, so the sum is approx \(2^-17\) which is, in technical speak...bugger all. |

Muon Experiment## Relativistic, Earth-Frame Observer | However, if Einstein is correct then from our Frame on earth, we would expect to see the 'clock' of the muon running slowly, which is another way of saying we expect to observe time dilation. We need the Lorentz factor to convert our earth time to the muons slower time. Recall that the Lorentz is calculated using : \(\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\) For the muon that gives: \(\frac{1}{\sqrt{1-\frac{0.98^2}{1^2}}} = \frac{1}{\sqrt{0.004}}=\frac{1}{0.2}=5 \)So the muon's local time is five times slower than our local time. When we feed this into the same equations as before, the time to get to earth is still 3.4x10 ^{-6 }seconds, but that now represents only 3.1 half-lives, not 17, as before. We therefore get a survival rate of \(2^{-3.1}=0.116\) which, for a starting population of 1 million, means 116,000 surviving muons at the surface. |

Muon Experiment## Relativistic, Muon-Frame Observer | For the sake of completeness we will now look at the calculation from the frame of the muon. The muon's clock, recall, is running normally from that frame (it would see OUR clocks running slowly). What the muon sees, as an effect of relativistic physics, is easiest to deal with as length contraction. Essentially distances are changed by the Lorentz factor. That is 5 times shorter in this case. So instead of 10 kilometres, from the frame of the muon that is two kilometres. Everything else remains the same, so the calculation is the same, but using 2km rather than 10km. As one might expect, the answer this gives is the same as the answer for the time dilation calculation previously. The different methods for the frames is summarised in the next table for comparison. |

Muon Experiment## Comparison of Reference Frames |

The historical experiment upon which the model muon experiment is based was performed by Rossi and Hall in 1941. They measured the flux of muons at a location on Mt Washington in New Hampshire at about 2000 m altitude and also at the base of the mountain. They found the ratio of the muon flux was 1.4, whereas the ratio should have been about 22 even if the muons were travelling at the speed of light, using the muon half-life of 1.56 microseconds. When the time dilation relationship was applied, the result could be explained if the muons were travelling at 0.994 c. In an experiment at CERN by Bailey et al., muons of velocity 0.9994c were found to have a lifetime 29.3 times the laboratory lifetime.

These calculated results are consistent with

The final table allows you to simulate the experiment yourself from the Earth frame. It will do the calculations for you - simply enter the height to start from. You can change the other parameters if you want to play around with it (encouraged). This simulation will function to a higher level of accuracy than I did in the above examples. I stuck to two decimal places, but the simulation runs with six to nine decimal places. If you try the same figures as used above (and I would encourage you to do so), you will get a different answer, which is simply due to the increased accuracy.

## Muon Experiment## Vary Parameters | |

The calculation will be considered from the Earth frame of reference. The length is then unaffected since it is in the Earth frame. The halflife is in the muon frame, so must be considered to be time dilated in the Earth frame. You may substitute values for the height and the muon speed in the calculation below. | |

Other Support for Relativity

Relativity has been more thoroughly tested than any other theory in science, with the probable exception of quantum theory.

Many of the tests are extremely complicated, or involve deep physics which I have no time (and probably not enough talent) to go into in this paper. I will confine myself, therefore, to examples which any reasonably intelligent adult should be able to grasp, without subject-specific knowledge.

One of the difficulties here is that Special Relativity is only part of the overall theory. It describes a universe without gravity. As soon as gravity becomes significant then we can no longer use Special Relativity - for a quick way of understanding why this is, think of the two conditions for SR - one of them is non-inertial frames. Gravity is an accelerative force - it naturally forces things out of non-inertial frames and into accelerating frames. Objects accelerate under gravity, they do not stay in constant motion. When I am looking at examples, therefore, of where relativity has made predictions which have been tested and found correct, it is very difficult to find examples with only the effects of special relativity. Since we live in a world which has gravity at the centre (in at least two senses of the word) we are never in a scenario in which Special Relativity can operate. If we could transport ourselves into deepest inter-cluster space where even the massive gravity of the galaxies begins to fade to zero, then we could devise some wonderful tests. Unfortunately we cannot currently get there - and I seriously doubt we will ever be able to do so.

We start by looking at what special relativity is and, therefore, what it posits that we can test.

Principle of relativity | Constancy of the speed of light | Time dilation |
---|---|---|

Any uniformly moving observer in an inertial frame cannot determine his "absolute" state of motion by a co-moving experimental arrangement. | In all inertial frames the measured speed of light is equal in all directions (isotropy), independent of the speed of the source, and cannot be reached by massive bodies. | The rate of a clock C (= any periodic process) traveling between two synchronized clocks A and B at rest in an inertial frame is retarded with respect to the two clocks. |

Also other relativistic effects such as length contraction, Doppler effect, aberration and the experimental predictions of relativistic theories such as the Standard Model can be measured. |

The 4th button (the factor of change button) is clicked when you wish to do the calculation in reverse - you input the Lorentz factor you require and the calculator works out what velocity that requires. So, for example, if you wanted a Lorentzian factor of 6 you would enter 6 as input and click this box.

I have added an extra field (value) to the calculator to allow the number of decimal places to be specified. This is particularly useful if you are trying to find a precise Lorentz factor - sometimes the velocity has to be specified to many decimal places.

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