The Heretics Penitence: A Sermon on Special Relativity

Some things are hard to understand because they are complicated, some things are hard to understand because you can’t believe that you have understood them correctly. Special Relativity is in the second category.

The Heretics have no beef with Special Relativity: its math is simple and pure; its predictions are fully and repeatedly tested and confirmed. However, there are problems with almost everything said about Relativity, statements like ‘You can’t go faster than the speed of light’ that are almost entirely untrue. There is nothing in Relativity that prevents you getting into a rocket and going between places light-years apart in a time as short a time as straightforward physical constraints permit – getting enough fuel, physically tolerating the g-force of acceleration, etc.

But still people have problems with time dilation and the ‘twins paradox’, problems of believing they have understood them. To address these the Heretics have tried to find the simplest and most down-to-earth explanations of both and put them into a sermon in the hope that at least a few more people may understand the simplicity of Relativity correctly.

The Sermon on Special Relativity

A light-year is a measure of distance, the distance light travels in a year: that is 9,500,000,000,000,000 metres. The light from near the centre of our galaxy, the Milky Way, takes about 26,000 years to get to Earth, so the centre of our galaxy is about 26,000 light-years away. Imagine you have a spaceship that accelerates fast enough to make you feel the same weight as gravity on earth does, that is, 1g, which is equal to accelerating at 35 km. per hour (kmph) every second or 9.8ms^-2. Imagine, too, that the spaceship has enough fuel to accelerate like this for years. Set off for the centre of the galaxy at this rate and you will get there in just over 11 years. You have covered a distance of 26,000 light-years in just over 11 years, as measured by the clocks you have on board. There is no magic wall or speed limit that stops you from doing this – after all, you are still travelling at a speed of zero relative to yourself and your rocket. You could cover the distance sooner if your rate of acceleration was faster, but that would be less comfortable.

In this scenario, would you have gone faster than the speed of light[1]? Strictly speaking, the answer is no; you cannot get ‘close to the speed of light’, ‘go fast’ or even go at any speed at all. All speeds are speeds relative to you and the things immediately around you. This is known as your ‘frame of reference’ and is, by definition, stationary. What confuses us is that on Earth, we have a different convention, which is that that everyone measures their speed by how fast they are going relative to the surface of the earth. On Earth, we also have the convention of calling moving away from the surface of the earth ‘going up’ and moving towards or below the surface ‘going down’. But in space, there is no ‘up’ or ‘down’, nor is there a special place everyone agrees to measure speed from. Stars, planets, and galaxies all move at different speeds relative to you, so relative to them, you are moving at lots of different speeds, but none of them is special, none of them is ‘your speed’. So, to keep things simple – actually, to make any sense at all – we use own point of view, our ‘frame of reference’, as stationary and describe all speeds as they are relative to us. Anyway, this is the only practical way that we can actually measure speeds.

If you need to work out when you will get somewhere, you can calculate how long you have been accelerating since you left. This also tells you how fast you are travelling away from your start point – or to be pedantic, which is a good idea here – how fast your start point is now moving away from your frame of reference. For example, if you have been accelerating at 1 g, after one minute you will calculate ‘your speed’ as 588 m.sec^-1 and, after a year, you will calculate ‘your speed’ as 309,262,514 m.sec^-1. This will tell you accurately when you will arrive at your destination (by your clocks), even though the calculation also tells you that you are going faster than the speed of light, 300,000,000 m.sec^-1. With a powerful enough rocket and enough fuel, there is nothing to stop you from getting anywhere you want, as soon as you want, so long as you can take the acceleration stress. After all, you are still stationary, relative to your rocket, so there can’t be any weird effects.

Your colleagues back on Earth see your trip to the centre of the galaxy differently. As you get further away from Earth, the flashes of light you send back to them once a day arrive after ever-longer periods. This is because the light of each flash must travel a greater distance to get back to them. To them, it looks like you and your clock are getting slower and slower as your spaceship accelerates away. Soon there will be weeks and, after a while, years and then decades between the arrival of each flash and, if they could see your clock, it would be getting slower and slower. Relative to them, our rocket never gets to the speed of light (the speed of the Field) but, because they can see that your time is going slower on the rocket, relative to them, they can see how you might think you are going faster than light relative to them (if you could see them, they would look like they had been slowed down too, everything is symmetrical so far). When you send them news that you have arrived at the centre of the galaxy and are setting off back home, they only get the message just over 52,000 years after you left Earth. From their point of view, you have taken over 26,000 earth-years to get there, and the message of your arrival takes 26,000 years to travel back. From your point of view, you have taken 11 years to get to the centre of the galaxy or, if you have slowed down to stop there before you turn around, around 20 years to get there.

After you arrive at the centre of the galaxy and avoid the giant black hole there, you turn around to go back home and get back just over 40 years after you left (measured by you, 20 years each way, allowing for slowing down). But, on Earth, you arrive back just over 52,000 years after you left – and only about a year after your message saying you had arrived at the centre of the galaxy was received on Earth. That’s all there is to it – it takes you 40 years to send a flash of light every day, going to and from the centre of the galaxy, and it takes just over 52,000 years for Earth to receive them.

Now many people would say ‘but this is all about appearances, what it looks like, not what it actually is’. But we can only measure time by counting events, ticks of a clock, vibrations of quartz or changes in the state of a caesium ion. No extra magic. If the rocket clocks have counted 40 years of events and the earth’s clocks have counted 52,000 years of events and other planets and other rockets have measured other periods of time between your departure and arrival back on Earth, where does the ‘absolute time’ come from? How is it measured? Who is measuring it? How much time has passed during these events: 40 years or 52,000 years, or somewhere in between? We know that all events that we can count show the effects of two different passages of time. There is no ‘absolute up’, no ‘absolute down’, no ‘absolute speed’ and no ‘absolute time’. Up, down, speed and time are relative to the observer, they are measured by each observer and differ between different observers and each different measurement is just as ‘valid’ as the rest. Difficult to really ‘feel’ these ideas that are so far from our earth-bound experience, but just because weightlessness is outside our experience does not mean that it does not happen in circumstances we have not experienced.

Returning to our voyage, you would expect the light flashes from your ship to be seen on Earth closer and closer together on the way back – and they are indeed seen coming quickly one after the other. On your journey back, all your flashes, 20 years at the rate of one flash each day, 7,300 flashes, would arrive back on Earth starting just about a year before you get home, 52,000 years after you left Earth. If they could see you in your ship, your movements over this one year would look speeded up by about 11 times. But overall, the effect is so small as to be irrelevant – 20 years speeded up coming back, against 52,000 years slowed down going out. Meanwhile, if Earth had sent you one flash a day for the 52,000 and a little bit years you were away, you would be flying back into a storm of flashes from Earth – 19 million flashes, only 7,300 of them received on the way out (the same as the number of flashes you sent them), the rest coming in a rush as you head back towards Earth. If you could see the people on Earth as you come back, they too would be speeded up, just as you are to them.

It is agonisingly difficult to keep the frames of reference clear in your head for the journey back and you may find yourself saying things like ‘But if all 7,300 flashes I send to earth come back in just over a year and the journey started 26,000 light-years away, that implies….’ Just remember that on the journey back all the 7,300 flashes of light you sent on your journey back will arrive on Earth before you do. Just.

Why does your clock, viewed from earth, not speed up on the way back as much as it slows down on the way out? The reason for this difference between one way and the other lies not in the theory of relativity, but in the misbehaviour of simple numbers. Ordinary digits occasionally go completely off-piste and provide very counter-intuitive answers, especially where speeds are concerned and it is here, in simple arithmetic and not in fancy physics, that the counter-intuitive result arises. Let’s look at the homely example that shows the same problem of numbers going one way resulting in a totally different timescale to the numbers going the other.

Take an old Mississippi paddle steamer, the Southern Belle, that plies the 10 km between Starting City, Louisiana, and the upriver city of Destination, Arkansas. Its engines can move it through the water at 10 km/h and, often, in the summer, when the river is still and effectively ceases to flow, the Southern Belle does both the journey there and the journey back in one hour each. In the autumn, however, the river returns to flowing at 5 km/h. This reduces the upriver speed of the Southern Belle, seen from the riverbank to 5 km/h, engine speed minus the speed of the current, so the journey to Destination takes two hours. The return journey downriver fails to compensate, because, going at a combined speed of 15 km/h (ship speed plus current), only 20 minutes are saved on the 10 km journey back to Starting City. So, although the current opposing travel upstream is exactly the same as the current aiding speed downstream, the effect is not symmetrical, and the overall journey there and back takes 40 minutes longer.

And it gets worse in winter, much worse. The winter current rises to 9.9 km/h and the Southern Belle is in trouble. It now takes 100 hours to get upstream to Destination, while the downstream return trip is only 10 minutes less than it was in the autumn. Although the current has less than doubled its speed, the return journey from Starting City to Destination and back is nearly 42 times longer than it was in the autumn. An apparently symmetrical change – the current that slows the boat down going upstream speeds it up by the same amount going downstream – creates a hugely asymmetrical outcome. Also, small numbers, like a 10 km/h drive speed and a 9.9 km/h current, can start to generate much larger numbers, such as ‘100 hours’ and ‘42 times longer’. If the current were 9.99999 km/h it would take one million hours to do the trip upstream. Perhaps you can see why the time dilation effect is small when the speed of an object is not near the speed of light, relative to an external observer, but becomes very large as it approaches that speed relative to the observer.

You may have noticed that there is little reference to the fact that the speed of light is constant in this story of travelling to the centre of the galaxy. Many people think this is a weird fact or claim, and that it is quite counter-intuitive. After all, if you throw a stone forward out of a speeding train, it will go a lot faster than if the train is stationary, so why not light? How is it that two lights shone out of a speeding train, one backwards and one forwards, will be seen by someone outside to be emitting light at the same speed? If you have mirrors down the line that reflect both lights back to your position exactly halfway, the light from both will both arrive at the same time.

There are two ways to understand this. The first is to realise that the speed of any wave is not connected to the speed of the source. You hear a car coming towards you at the same moment as another car, the same distance away, moving away from you. Otherwise, if they crashed (head-on), you would hear the crash twice, the first from the car coming towards you, the second from the one going away and this doesn’t happen. Waves from a moving object change in wavelength, not speed. This is the famous Doppler effect, the ‘wee-yow’ sound as a racing car passes you. As something noisy comes towards you, the sound waves are shortened because each peak and each trough is emitted a little closer to you and shorter wavelengths have a higher pitch. As the noisy thing moves past you and away from you, the peaks and troughs of the sound waves are further apart when they are emitted. This is heard as a lower pitch: the wavelength changes but not the wave speed. The speed is governed by the medium through which it is moving. The speed of sound, for example, varies in different mediums, generally the denser the medium, the faster the speed of the wave through it; it is faster in water than in air and faster still through rock.

As we discussed earlier, the second reason why light always travels at the same speed is because you are always stationary relative to light. Are you reading this in a building? If so, you see yourself as stationary. Which you are, relative to the surface of the earth. But relative to the centre of the earth, you are going around at quite some speed, how fast exactly depends on how far north or south you are. At the equator, you are going round at 460 m.sec^-1. Then add in the earth’s speed, relative to the sun, orbiting at about 30,000 m.sec^-1 or 108,000 km/h. The solar system is going around the galaxy at around 220,000 m.sec^-1 and the galaxy seems to be closing in on an unknown object called the Great Attractor at about 1,000,000 m.sec^-1. Relative to distant galaxies, you are travelling at more than 90% of the speed of light. As you can see, there is no such thing as speed, only speed relative to something specific, something with a location. In Relativity, we are stationary, so the speed of light, relative to us, never changes[2].

There is another problem with the time dilation effect predicted by relativity that has tormented many: surely the position of the space traveller in a fast rocket and those staying on Earth is essentially symmetrical? And if this is the case, how can their experiences be so different?

The classic way that this question is posed is called the twins paradox. The theory of relativity predicts that, if one twin was to leave Earth on a spaceship, travel away at close to the speed of light relative to the other twin, and then return to Earth, the travelling twin would have aged less than the twin that stayed on earth – ignoring any effects from gravity that we will discuss elsewhere. The apparent paradox here is not that time goes slower, but that the two twins’ experiences seem to be symmetrical: each twin has separated from the other and then returned to meet again. The fact that one was on Earth and the other was in a spaceship is irrelevant: they separate, then they re-join.

You should remember how reasonable and obvious it seems now that the two twins do the same thing, because the problem with relativity is our intuition going wrong, not with the ideas under Einstein’s zany hairdo. When we look carefully, we find that the twins do not do the same thing relative to each other.

We need to go through the experience of each twin from their own point of view. To make the situation more familiar, we will move the twins to a simpler situation and put each of them into one of two identical, slow-moving ships, far out to sea: SS Earth and SS Spaceship.

As we start, SS Earth and SS Spaceship are right next to each other but moving past each other at a speed of 5 km/h – that is, each sees the other ship moving at 5 km/h relative to their own ship, which each sees (of course) as stationary. They continue to separate at this speed for one hour. Then SS Spaceship turns around and returns to re-join SS Earth, getting there after one further hour. A simple, apparently symmetrical situation – they move apart from each other and then back together again – that exactly replicates the twin paradox, but at low speeds.

But let’s look at each twin’s experience. The twin on SS Earth sees himself as stationary throughout. He sees his twin moving away from him at 5 km/h for an hour and then returning back to him at 5 km/h for an hour. The twin on SS Spaceship also sees himself as stationary and his twin travelling away from him at 5 km/h for an hour. So far, so symmetrical. But, after an hour, the SS Spaceship twin starts his ship’s engine and moves toward his twin. As SS Earth will be 5 km away from him when he starts his engines and is still moving away from him at 5 km/h, so to catch up SS Earth in one hour, SS Spaceship needs to go at 10 km/h (calculated by how much he uses his engines – of course, he is still stationary relative to himself). All this time the twin on SS Earth has done nothing and not moved as far as he is concerned.

So, the experience of the twins on the two ships is not at all the same. One has not moved, the other has moved 10 km at 10 km/h. What seemed symmetrical when we first looked at it from our imaginary vantage point in outer space is not at all symmetrical to the twins involved: to repeat, one has not moved at all, the other has moved, has started his engine and used fuel. Obviously, in the case of the twins on ships, the time dilation effect is completely negligible, but with the rocket twins, the speeds involved can start behaving like the Southern Belle in peak flood season.

If you recall our trip to the centre of the galaxy, everything was symmetrical up to the point where you turned the rocket around and headed home: you had sent 7,300 flashes of light and had received 7,300 flashes of light from Earth. The trouble only started when you turned around and accelerated back, back into the stream of nearly 19 million more flashes of light from Earth finally catching up with you. Once you turned around, the situation, was no longer symmetrical. If you had stopped and stayed at the centre of the universe, things would still be symmetrical, your time passing at the same rate as their time, receiving one flash every day, sending one flash every day, except you would both think that the other was 26,000 years behind them.

To repeat the earlier and essential point, ‘real time’ is altered and goes at ‘different speeds’. Real time is what is measured on your clock and, however sophisticated your clock is, the only way it works is by counting something. It used to be counting pendulum swings in a grandfather clock, then it became counting escapement clicks in your mechanical watch, then counting vibrations of quartz in your electronic watch, and then the official one second became a count of 9,192,631,770 cycles of the radiation produced by the transition between two levels of the caesium 133 atom. Time is always – and can only be – measured by counting events. That is all we have done: count flashes. It would be the same if we counted heartbeats or clockwork ticks. Time is no more than a word for the passing of a number of ticks. The mistake many make is to elevate the process of counting ticks to a grand, abstract absolute, the Goddess Time, and then say this grand but non-existent thing has an eternal quality that overrides counting how often our clock ticks, our heartbeats, or our rocket-born light flashes. On our trip to the centre of the galaxy and back, far more clicks, beats and flashes happened on Earth (19 million flashes, compared to 8,000 sent from our rocket), with 52,000 years on their clocks compared to our long, but easily-within-a-lifetime trip.

Hopefully, these examples make the predictions of relativity more intuitively understandable and less of a paradox. This is pretty much all there is to Einstein’s ‘Special Relativity’; lights being flashed as things move past each other and counting the flashes. The weird effects, such as time dilation, follow from these simple thoughts. The math is uncomplicated and pure. These ideas also explain why the science fiction concept of ‘warp speed’, faster-then-light travel, is misguided. As we saw from our initial journey to the centre of the galaxy, if you have the power and the fuel, there is no problem with travelling between places in less time than light does. For ‘warp speed’ to work as it is normally presented in Star Trek, it would have to be a time machine that slowed down time for everyone else in the universe, in order to match their perception of your time slowing.

There is another problem we need to resolve, though. If I think I am going a million kilometres an hour away from earth, calculated by how much and how long I have been accelerating, but the folk on Earth think I am going much slower from the rate my flashes are coming back to them, how hard would I hit a planet I crashed into that was stationary relative to Earth? As hard as the million km/hour I have calculated I am going (relative to them) or more gently, at the much slower speed they think I am going relative to them? The answer is simple but a bit odd: from my frame of reference, I will hit it at a million km/h. On earth, they see me hit it at the slower speed but hit it just as hard because, to them, I have got much heavier. We can observe this effect all the time in accelerators where we can get fundamental things like electrons up to speeds close to the speed of light relative to us. First, if we speed up short-lived things – mesons, say – they last much longer then they would if they were stationary relative to us, showing that their time is dilated relative to us. Then, if we whack them into other things, they wallop them with a force that shows that they are much heavier than the same type of particle going slowly.

We should not be surprised by this if we have taken on board the heresy that things like electrons are ball-waves. If one is coming towards you at high speed, its wavelength will look much shorter to you because it moves closer to you between ‘pulses’ of its wave[3]. This is simply the Doppler effect. Shorter wavelengths mean higher energy, so when they hit something the ‘spring’ is much tighter and the reaction of the hit object is to move much faster, as though it had been hit by something more massive.

Effectively, much of the energy we put into accelerating the thing (relative to us, ‘rtu’) turns, not just into speed (rtu), but into mass (rtu). Mass has been created from energy. You can then play with the equations to ask how much energy there is in a non-moving object and the answer comes out that the energy content of something with mass is equal to its mass times the speed of light squared. It simply fell out when some of Einstein’s equations were manipulated and he originally regarded it as just a funny bit of math of no significance. However, in time, it led to the atom bomb.

You can put this into the Heretic’s picture of the proton, neutron and electron being ball-waves. The question is ‘how much energy would be released if the reciprocating ball-wave stopped bouncing back in and simply became an F-wave, propagating outwards indefinitely’. What must happen to have this result is that the force that makes the ball-wave stop and reverse direction to move back in has to cease – and this can happen, when a positron and electron meet; their charges (hFv and lFv) cancelling each other out, so there is nothing to make the wave come back in. The energy of the new F-wave is the speed of the Field (the speed of light) across its 2-D spherical wavefront, the ‘speed of light squared’. But this is based on how powerful the original wave inside the original ball-wave(s) was – and the more powerful and short that wave it the more massive we measure the ball-wave to be. It is the measured mass allowed to go outwards as a longitudinal wave with a 2-D wavefront at the speed of the field.

Here ends the Sermon on Relativity, not with a whimper but with a bang.

[1] In this chapter we will use the term ‘speed of light’ throughout, since it has such a history in this area. It is, of course, exactly the same as the ‘speed of the Field ‘ used in earlier chapters.

[2] From the Heresies it should also be noted that light, like sound, is a longitudinal wave and these always go at the speed of the medium they are in. However here we are keeping things nice and tidy and need not get involved in those issues.

[3] But, of course, the pulse cannot be shortened so much that two beats become one or overlap and the speed of the waves inside the ball-wave is the speed of light…