1.000000000000000858

A BOLT OF RELATIVITY


Among all the usual football-related paraphernalia there was something different under the Christmas tree that year. It was a dictionary, one of those classic Collins ones that could serve as a barricade should the need ever arise. I’m not sure why my mum and dad thought fit to buy their ten-year-old son a dictionary when, at that stage, I had shown relatively little interest in words. In those days, I had two passions in life: Liverpool Football Club and maths. If my parents thought this present would broaden my horizons, they were sorely mistaken. I considered my new toy and decided I could at least use it to look up massive numbers. First I searched for a billion, then a trillion, and it wasn’t long before I discovered a ‘quadrillion’. This game went on until I happened upon the truly magnificent ‘centillion’. Six hundred zeroes! That was in old English, of course, before we embraced the short-scale number system. Nowadays a centillion has a less inspiring 303 zeroes, just as a billion has nine rather than twelve.

But this was as far as it went. My dictionary didn’t contain a googolplex or Graham’s Number or even TREE(3). I would have loved them back then, these leviathans. Fantastic numbers like these can take you to the brink of our understanding, to the edge of physics, and reveal fundamental truths about the nature of our reality. But our journey begins with another big number, one that was also absent from my Collins dictionary: 1.000000000000000858.

I imagine you’re disappointed. I’ve promised you a ride with numerical leviathans, but this number doesn’t seem to be very big at all. Even the Pirahã people of the Amazon rainforest can name something bigger, and their number system includes only hoí (one), hói (two) and báagiso (many). To make matters worse, it’s not even a very pretty or elegant number like pi or root 2. In every conceivable sense, this number appears to be remarkably unremarkable.

This is all true until we start to think about the nature of space and time and the extremes of our human interactions with them. I chose this particular number because it’s a world record for its size, revealing the limit of our physical ability to meddle with the properties of time. On 16 August 2009 Jamaican sprinter Usain Bolt managed to slow his clock by a factor of 1.000000000000000858. No human has ever slowed time to such an extent, at least not without mechanical assistance. You may remember this event differently, as the moment when the 100-metre world record was shattered at the athletics world championships in Berlin. Watching in the stadium that day were Wellesley and Jennifer Bolt, whose son hit a top speed of 27.8mph (12.42m/s) between the 60- and 80-metre mark of the race. For each second experienced by their son in those moments, Wellesley and Jennifer would experience a little more: 1.000000000000000858 seconds, to be precise.

To understand how Bolt was able to slow time, we need to accelerate him up to the speed of light. We need to ask what would happen if he were able to catch up with it. You can call this a ‘thought experiment’ if you like, but don’t forget that Bolt managed to break three world records at the Beijing Olympics, fuelled by a diet of chicken nuggets. Imagine what he could have achieved if he ate properly.

To have any hope of catching light, we must assume that it travels at a finite speed. That is already far from obvious. When I told my daughter that the light from her book did not reach her eye in an instant she was immediately very sceptical and insisted on conducting an experiment to find out if it was really true. I typically get a nosebleed whenever I stray too close to experimental physics, but my daughter seems to have acquired more of a practical skill set. She set things up as follows: turn the bedroom light off, then turn it on again and count how long it takes for the light to reach you. This is exactly the same sort of experiment carried out by Galileo and his assistant using covered lanterns four hundred years ago. Like my daughter, he concluded that the speed of light ‘if not instantaneous … is extraordinarily rapid’. Rapid, but finite.

By the mid-nineteenth century physicists such as the wonderfully named Frenchman Hippolyte Fizeau were beginning to home in on a reasonably accurate – and finite – value for the speed of light. However, to properly understand what it would mean to catch up with light, we need to first focus on the remarkable work of the Scottish physicist James Clerk Maxwell. It will also illustrate the beautiful synergy that exists between maths and physics.

By the time Maxwell was considering the behaviour of electricity and magnetism there were already hints that they could be two different sides of the same coin. For example, Michael Faraday, one of England’s most influential scientists, despite his lack of formal education, had previously discovered the law of induction, showing that a changing magnetic field produced an electric current. The French physicist André-Marie Ampère had also established a connection between the two phenomena. Maxwell took these ideas and the corresponding equations and tried to make them mathematically rigorous. But he noticed an inconsistency – Ampère’s law, in particular, defied the rules of calculus whenever there was a flux of electric current. Maxwell drew analogies with the equations that governed the flow of water and proposed an improvement on what Ampère and Faraday had to offer. Through mathematical reason, he found the missing pieces of the electromagnetic jigsaw and a picture emerged of unprecedented elegance and beauty. It is this strategy, pioneered by Maxwell, that pushes the frontiers of physics in the twenty-first century.

Having established his mathematically consistent theory, unifying electricity and magnetism, Maxwell noticed something magical. His new equations admitted a wave solution, an electromagnetic wave, where the electric field rises and falls in one direction and the magnetic field rises and falls in the other. To understand what Maxwell found, imagine two sea snakes coming straight for you on a scuba dive. They are travelling along a single line in the water, the ‘electric’ snake slithering up and down, the ‘magnetic’ snake slithering left and right, and to make matters worse, they are charging towards you at 310,740,000m/s. The last bit of the analogy might be the most terrifying, but it is also the most remarkable part of Maxwell’s discovery. You see, 310,740,000m/s really was the speed that Maxwell calculated for his electromagnetic wave – it just popped out of his equations like a mathematical jack-in-the-box. Curiously enough, that figure was also very close to the estimates for the speed of light that had been measured by Fizeau and others. Remember: as far as anyone was aware at the time, electricity and magnetism had nothing to do with light, and here they were, apparently consisting of waves travelling at the same speed. Modern measurements of the speed of light through a vacuum place its value at 299,792,458m/s, but the parameters of Maxwell’s equations are also known to a greater accuracy and the miraculous coincidence survives. Because of this coincidence, Maxwell realized that light and electromagnetism had to be one and the same thing: an astonishing connection between two apparently separate properties of the physical world revealed by mathematical reason.

It gets better. Maxwell’s waves didn’t just include light. Depending on their frequency of oscillation or, in other words, the rate at which the sea snakes slither from side to side, the wave solutions described radio waves, X-rays and gamma rays, and although the frequencies were different, the speed at which they moved was always the same. It was the German physicist Heinrich Hertz who actually measured radio waves, in 1887. When he was quizzed about the implications of his discovery, Hertz humbly replied, ‘It is of no use whatsoever. This is just an experiment that proves Maestro Maxwell was right.’ Of course, whenever we tune a radio station to the desired frequency, we are reminded of the real impact of Hertz’s discovery. But even if he underplayed his own importance, Hertz was right to describe Maxwell as a maestro. He was, after all, conductor of the most elegant mathematical symphony in the history of physics.

Before Albert Einstein revolutionized our understanding of space and time, it had been widely assumed that waves of light require a medium through which to propagate, much in the way that waves on the ocean need to propagate through a body of water. The imagined medium for light was known as the luminiferous aether. Let’s assume, for a moment, that the aether is real. If Usain Bolt were to catch up with light, he would have to travel through the aether at 299,792,458m/s. If he did get up to speed, then once he is running alongside the light ray, what would he actually see? The light would no longer be moving away from him so it would just appear as an electromagnetic wave oscillating up and down and left and right but not actually going anywhere. (Imagine the sea snakes slithering to and fro but ultimately staying in the same place in the ocean.) But there is no obvious way to adapt Maxwell’s laws to allow for this sort of wave, which suggests that the laws of physics would have to be radically different for the supercharged version of the Jamaican sprinter.

This is unsettling. When Einstein drew the same conclusions, he knew that something had to be wrong with this idea of catching up with light. Maxwell’s theory was much too elegant to abandon just because somebody happened to be moving quickly. Einstein also needed to find a way of taking into account the strange results of an experiment carried out in Cleveland, Ohio, in the spring of 1887. Two Americans, Albert Michelson and Edward Morley, had been trying to find the speed of the Earth through the aether using some clever arrangement of mirrors, but the answer kept coming out as zero. If correct, this would have meant that the Earth, unlike almost all of the other planets in the solar system and beyond, just so happened to be running right alongside this space-filling aether, at exactly the same speed and in exactly the same direction. As we will come to appreciate later in this book, coincidences like that don’t tend to happen without good reason. The simple truth is that there is no aether – and that Maestro Maxwell is always right.

Einstein proposed that Maxwell’s laws, or indeed any other physical laws, would never change, no matter how quickly you move. If you were locked away in a windowless cabin on a ship, there would be no experiment you could do to detect your absolute velocity because there is no such thing as absolute velocity. Acceleration is a different story, and we’ll come to that, but as long as the captain of the ship set sail at constant velocity relative to the sea, be it at 10 knots, 20 knots or close to the speed of light, you and your fellow experimenters in the cabin would be blissfully unaware. As for Usain Bolt, we now know that his chase would be futile. He would never catch the light ray because Maxwell’s laws can never change. No matter how fast he ran, he would always see the light as if it were moving away from him at 299,792,458m/s.

This is all very counterintuitive. If a cheetah runs across the plain at 70mph and Bolt chases after it at 30mph, then everyday logic would suggest that the cheetah will extend its lead on Bolt by 40 miles every hour, simply because its relative speed is calculated as 70mph – 30mph = 40mph. But when we are talking about a ray of light travelling at 299,792,458m/s across the plain, it doesn’t matter how fast Bolt runs, the ray of light will still move relative to Bolt at 299,792,458m/s. Light will always travel at 299,792,458 m/s,1 relative to the African plain, relative to Usain Bolt, relative to a herd of panicking impala. It really doesn’t matter. We can sum it up in a single tweet:

The speed of light is the speed of light.

Einstein would have liked this. He always said that his ideas should have been described as ‘the Theory of Invariance’, focusing on their most important features: the invariance of the speed of light and the invariance of the laws of physics. It was another German physicist, Alfred Bucherer, who coined the phrase ‘the Theory of Relativity’, ironically while criticizing Einstein’s work. We call it the special theory of relativity in order to emphasize the fact that all of the above applies only to motion that is uniform, in other words, with no acceleration. For accelerated motion, like a Formula One driver hitting the gas or a rocket being fired into space, we need something more general and more profound – Einstein’s general theory of relativity. We’ll get to that in detail in the next section, when we plunge to the bottom of the Mariana Trench.

For now, let’s stick with Einstein’s special theory. In our example, Bolt, the cheetah, the impala and the ray of light are all assumed to be moving with constant velocity relative to one another. Those velocities may differ, but they don’t change with time, and the most important thing is that, despite those differences, everyone sees the light ray speeding away at 299,792,458m/s. As we have already seen, this universal perception of the speed of light certainly contradicts our everyday understanding of relative velocities, in which one velocity is subtracted from another. But this is only because you aren’t exactly used to travelling around at speeds close to the speed of light. If you were, you would look at relative velocities very differently.

The problem is time.

You see, all along you have been assuming that there is a big clock in the sky that tells us all what time it is. You might not think you are assuming this, but you are, especially when you start subtracting relative velocities using what you believe to be common sense. I’m sorry to disappoint you, but this absolute clock is a fantasy. It doesn’t exist. All that ever matters is the clock on your wristwatch, or on my wristwatch, or the clock ticking along on a Boeing 747 as it flies across the Atlantic. Each and every one of us has our own clock, our own time, and these clocks don’t necessarily agree, especially if someone is hurtling around close to the speed of light.

Let’s suppose I jump aboard a Boeing 747. Taking off from Manchester, by the time it reaches the British coast at Liverpool, the aircraft is cruising along at several hundred miles per hour. I decide to bounce a ball a couple of metres across the floor of the cabin, to the slight irritation of the other passengers. My sister, Susie (who happens to live in Liverpool), is on the beach as the plane flies over and, from her perspective, the ball moves considerably further, some two hundred metres or more. At first glance, this doesn’t seem to require any major revision of our everyday concept of time. After all, the ball just gets a piggyback from the fast-moving aircraft – of course she sees it move further. But now let’s play a similar game with light. I switch on a light on the floor of the cabin, shining a ray vertically upwards, perpendicular to the direction of travel of the aeroplane. In a very short time, I see the light climb up to the cabin ceiling. If Susie were able to see inside, she would see the light travel along a diagonal, rising from floor to ceiling but also moving horizontally with the aircraft.

Trajectory of light ray as seen by Susie on the beach.

Her diagonal distance is longer than the vertical distance I measured. That means that she saw the light travel further than I did and yet she saw it travelling at the same speed. That can mean only one thing: for Susie, the light took longer to complete its journey; from her perspective, the world inside the aircraft must be ticking along in slow motion. This effect is known as time dilation.

The amount by which time is slowed depends on the relative speed, of me with respect to my sister, of Usain Bolt with respect to his parents in Berlin. The closer you are to the speed of light, the more you slow down time. When Bolt was running in Berlin, he hit a top speed of 12.42m/s, and time was slowed by a factor of 1.000000000000000858.2 That’s the record for human relativity.

There is another consequence of slowing down time – you age more slowly. For Usain Bolt, it turns out he aged about 10 femtoseconds less than everyone else in the stadium during the race in Berlin. A femtosecond doesn’t seem like much – it’s only a millionth of a billionth of a second – but still, he aged less, so when he came to rest he had leapt into the future, albeit very slightly. If you aren’t much of a runner, you can take advantage of some mechanical assistance to slow down time and, chances are, you will do even better. Russian cosmonaut Gennady Padalka spent 878 days, 11 hours and 31 minutes in space aboard both the Mir Space Station and the International Space Station, orbiting the Earth at speeds of around 17,500mph. Over the course of these missions, he managed to leap forward a record 22 milliseconds in time compared to his family at home on Earth.*

But you don’t have to be a cosmonaut to time-travel in this way. A cabbie driving through the city for forty hours a week for forty years will be a few tenths of a microsecond younger than he would have been had he just stayed put. If you aren’t impressed by microseconds and milliseconds, consider what could happen to any bacteria hitching a ride aboard the Starshot mission to Alpha Centauri. Starshot is the brainchild of billionaire venture capitalist Yuri Milner, who plans to develop a light sail capable of travelling to our nearest star system at one fifth of the speed of light. Alpha Centauri is around 4.37 light years away, so we would have to wait more than twenty years on Earth for it to complete its journey. For the light sail and its bacterial stowaway, however, time would slow down to such an extent that the journey would take less than nine years.

At this point, you may have spotted something suspicious. Travelling at one fifth of the speed of light for nine years, the intrepid bacterium will cover less than two light years – which is less than half the distance to Alpha Centauri. It’s the same with Usain Bolt. I told you that he ran for 10 femtoseconds less than you might have thought, which suggests he didn’t actually run as far. And it’s true – he didn’t. From Bolt’s perspective, the track was moving relative to him at 12.42m/s and so it must have shrunk by around 86 femtometres, which is the width of around fifty protons. You could even argue that he didn’t quite finish the race. For the bacterium, the space between Earth and Alpha Centauri was moving very quickly and as a result it shrank to less than half its original length. This shrinking of space, or of the racetrack in Berlin, is known as length contraction. So you see, running will not only make you age less, it can also help you look thinner. If you ran close to the speed of light, anyone watching would see you flatten out like a pancake, thanks to the shrinking of the space you occupy.

There is something else you should be worried about. I just said that the track was moving relative to Usain Bolt at 12.42m/s. That means that his parents were also moving, relative to their son, at exactly the same speed. But given everything we have established so far, this means that Bolt would have seen his parents’ clocks slow down, which is very weird, because I already told you that they also saw his clock slow down. In fact, this is exactly what happens: Wellesley and Jennifer see their son in slow motion (!), and Bolt sees them in slow motion. But here’s the really troubling part: I also said that Bolt managed to finish the race 10 femtoseconds younger than he would have been had he stood still. Couldn’t we flip things around and look at it from Bolt’s perspective? Time is ticking more slowly for his parents, so couldn’t it be they who age less? It seems we have a paradox. This is known as the twin paradox, because of the narrative usually used to explain it, but unfortunately Usain Bolt doesn’t have a twin. No matter. The truth is that it is Bolt who ages less, who stays that little bit younger. But why him and not his parents?

In order to answer this question, we have to consider the role of acceleration. Remember, everything we have discussed so far applies to uniform motion when there is no acceleration. In those moments where Bolt is running at a constant 12.42m/s, he and his parents are what we would call inertial. This is just some fancy jargon that says they aren’t accelerating – they don’t feel any additional force speeding them up or slowing them down. Whenever this is the case, the laws of special relativity apply and so Bolt will see his parents in slow motion, and vice versa. However, Bolt doesn’t run at a constant speed for the entire race: he accelerates from zero up to his top speed before slowing down again at the end. In those periods when he is accelerating or decelerating he is not inertial, in contrast to his parents. Accelerated motion is a very different beast. For example, locked away in a cabin of a ship, you would certainly be able to tell if the ship was accelerating because you would feel the force acting on your body. Too large an acceleration could even kill you. Bolt was never at risk of death, but his acceleration and deceleration were enough to break the equivalence between him and his parents. This asymmetry takes care of the paradox – a more detailed analysis, carefully factoring in Bolt’s accelerated motion, reveals that of all the protagonists it was indeed Bolt who aged that little bit less.

It is important to realize that this isn’t just some fun with equations. These are real effects that have been measured. Fast-moving atomic clocks have been seen to tick more slowly than their stationary counterparts, ‘ageing less’, just as Usain Bolt did in Berlin. Further evidence comes from a microscopic particle called the muon and its apparent stay of execution. The muon is very much like the electrons you find orbiting the nucleus of an atom, but it’s about two hundred times heavier and it doesn’t live anywhere near as long. After about two millionths of a second it decays into an electron and some little neutral particles called neutrinos. There is an experiment at Brookhaven National Laboratory in New York in which muons are accelerated around a 44-metre ring at 99.94 per cent of the speed of light. Given their short life span, you would expect the muons to complete only 15 laps; somehow, though, they make it around 438 times. It’s not that they live any longer – if you were travelling alongside one at the same speed, you would still see it decay after two millionths of a second – but then you would also see the circumference of the ring shrink to 1⁄29 of its original size. The muon gets around 438 times because it has less distance to travel, thanks to length contraction.

Length contraction and time dilation help us understand why nothing – not even Usain Bolt – can travel faster than light. As he gets closer and closer to light speed, Bolt’s time appears to slow to a standstill and the distances he encounters shrink to nothing. How can time slow down any more? How can distances shrink to any less? There is simply nowhere to go. The speed of light now presents itself as a barrier and the only reasonable conclusion is that no one can go any faster.

As he accelerates towards the speed of light, Bolt takes on more and more calories to try and accelerate faster and faster. The speed of light looms large as a barrier not to be crossed and so eventually his speed begins to plateau and his acceleration slows down. The closer he gets to the speed of light, the harder it becomes. His resistance to acceleration or, in other words, his inertia, just gets larger and larger. That is the problem with trying to accelerate up to the speed of light: inertia blows up to infinity.

But where is this inertia coming from? Well, the only thing that Bolt is bringing into the system is energy, and so that energy must be the source of Bolt’s extra inertia. Energy never goes away, it just changes how it looks, moving from one form to another. So, inertia must be a form of energy, and this must still be true even when Bolt is resting. The cool thing is that for a resting Bolt, we know exactly what his inertia is: it’s just his mass, because the heavier he is, the harder he is to move. Mass and energy become one and the same or, as Einstein put it3: E = mc2. The terrifying thing about this formula is quite how much energy (E) you can get from mass (m), thanks to the enormous value for the speed of light (c). A resting Usain Bolt weighs around 95 kilograms, and if you were to convert all of that mass into energy it would be the equivalent of 2 billion tons of TNT. That is more than a hundred thousand times the energy released by the Hiroshima bomb.


Copyright © 2022 by Antonio Padilla