Stranger Than We Can Imagine (4 page)

Could we instead define our position with the very centre of the earth? This isn’t fixed either, because the earth is moving around the sun at over 100,000 km/h. Or perhaps we can define the sun as our fixed point? The sun is moving at 220 km/s around the centre of the Milky Way galaxy. The Milky Way, in turn is moving at 552 km/s, relative to the rest of the universe.

What of the universe itself? As a last-ditch and somewhat extreme attempt to locate a fixed point, could we not declare the centre of the universe to be our omphalos? The answer, once more, is no. There is no ‘centre of the universe’, as we will see later, but for now we can also reject the idea for being ridiculously impractical.

So how can we say anything definite about our position, or that of the cup of tea? There may not be a real ‘fixed point’ which we can use, but we are still free to project our own frames of reference wherever we like. We can create a reference frame centred on ourselves, for example, which allows us to say that the tea is moving relative to us. Or we can create one centred on the tea, which would mean that we were moving relative to the cup. What we can’t do is say that one of these frames of reference is correct or more valid than the other. To say that the tea moved past us would be to declare our innate, tea-ist prejudice.

There is an apt example of how one frame of reference is no more valid than another in Einstein’s 1917 book,
Relativity
. In the original German-language edition, he used Potsdamer Platz in Berlin as the frame of reference in one example. When the book was translated into English, this was changed to Trafalgar Square in London. By
the time the book was out of copyright and made available online as an eBook, this had been changed to Times Square in New York because, in the opinion of the editor, ‘this is the most well known/identifiable location to English speakers in the present day.’ What is important about the reference point, in other words, is that it has been defined as the reference point. Practically, it could be anywhere.

The first step towards understanding relativity, then, is to accept this: a statement of position is only meaningful when it has been defined along with its frame of reference. We can choose whatever frame of reference we like, but we can’t say that it has more validity than any other.

With that in mind, let us return to Zurich in 1914.

Einstein gets on a steam train in Zurich and travels to Berlin. He is leaving his wife Marić and their two surviving children in order to begin a new life with his cousin, who will later become his second wife. Imagine that the train travels in a straight line at a constant speed of 100 km/h, and that at one point he stands, holds a sausage at head height, and drops it on the floor.

This raises two questions: how far does the sausage fall, and why is he leaving his wife? Of these two questions Einstein would have found the first one to be the most interesting, so this is what we will focus on.

Let us say he holds the sausage up to a height of five feet above the train floor and drops it. It lands, as you would expect, near to his scuffed shoes, directly below his raised hand. We can say that it has fallen five feet exactly. As we have just seen, such a statement only makes sense when the frame of reference is defined. Here we take Einstein’s frame of reference, that of the inside of the train carriage, and we can say that relative to that, the sausage fell five feet.

What other frames of reference could we use? Imagine there is a mouse on the railway track, and that the train rumbles safely over the mouse as Einstein drops his sausage. How far would the sausage fall if we use this mouse as a point of reference?

The sausage still starts in Einstein’s hand and lands by his feet. But, as far as the mouse is concerned, Einstein and the sausage are also moving over him during the sausage’s fall. During the period between Einstein letting go of the sausage and it hitting the floor, it will have moved a certain distance along the track. The position of his feet when the sausage lands is further down the track than the position of the hand at the moment it was dropped. The sausage has still fallen five feet downwards, relative to the mouse, but it has also travelled a certain distance in the direction the train is travelling in. If you were to measure the distance taken by the sausage between the hand and the floor, relative to the mouse, its path would be at an angle rather than pointing straight down, and that means it would have travelled further than five feet.

This is, instinctively, something of a shock. The distance that the sausage moves changes when it is measured from different frames of reference. The sausage travelled further from the mouse’s perspective than it did from Einstein’s. And, as we have seen, we cannot say that one frame of reference is more valid than any other. If this is the case, how can we make any definitive statements about distance? All we can do is say that the sausage fell a certain distance relative to a particular frame of reference, and that distance can be different when measured from other frames of reference.

This is only the beginning of our troubles. How long did the sausage take to fall? As you can appreciate, a sausage that falls more than five feet will take longer than one that just falls five feet. This leaves us with the slightly disturbing conclusion that dropping the sausage took less time from Einstein’s point of view than it did for the mouse.

Just as we live with the constant fixed point of the ground beneath us, we believe that there is a constant, universal time ticking away in the background. Imagine the bustle of commuters crossing Westminster Bridge in London, with the Houses of Parliament and the clock face of Big Ben up above them. The clock is suspended above the suited people below, ticking away with absolute regularity, unaffected by the lives going on beneath it. This is similar to how
we intuitively feel time must work. It is beyond us, and unaffected by what we do. But Einstein realised that this was not the case. Time, like space, differs according to circumstances.

All this seems to leave us in a tricky situation. Measurements of time and space differ depending on which frame of reference we use, but there is no ‘correct’ or ‘absolute’ frame of reference that we can rely on. What is observed is dependent, in part, on the observer. At first glance, it appears that this leaves us in a desperate situation, one where every measurement is relative and cannot be said to be definitive or ‘true’.

In order to escape from this hole, Einstein reached for mathematics.

According to well-established physics, light (and all other forms of electromagnetic radiation) must always move at a particular speed when it travels through a vacuum. This speed, nearly 300,000,000 metres per second, is known to mathematicians as
‘c’
and to non-mathematicians as ‘the speed of light’. How, though, can this be the case when, as we have seen, measurements differ depending on the frame of reference?

In particular, there is the law of addition of velocities to consider. Consider a scene in a James Bond movie where James Bond is shot at by the henchman of an evil villain. We don’t need to worry whether Bond will be killed, as henchmen are notoriously bad shots. Instead, let us worry about how fast that bullet will be travelling when it sails harmlessly over his head. Imagine, for the sake of argument, the speed of the bullet from the gun was 1,000 mph. If the henchman was driving towards Bond on a snowmobile when he fired, and if the snowmobile was travelling at 80 mph, then the velocity of the bullet would be these speeds added together, or 1,080 mph. If Bond was skiing away at 20 mph at the time, then this would also need to be factored in, and the bullet would then have a velocity relative to Bond of 1,060 mph.

Now back to Einstein on the steam train, who has swapped his sausage for a torch which he shines along the length of the dining
carriage. From his point of view, the photons emitted from the torch travel at the speed of light (strictly speaking the train would need to be in a vacuum for them to reach this speed, but we’ll ignore such details so that he doesn’t suffocate). Yet for a static observer who was not on the train, such as the mouse from earlier or a badger underneath a nearby tree, the photons would appear to travel at the speed of light plus the speed of the train, which, clearly, is a different speed to the speed of light. Here we have what appears to be a fundamental contradiction in the laws of physics, between the law of addition of velocities and the rule that electromagnetic waves must always travel at the speed of light.

Something is not right here. In an effort to resolve this difficulty we might ask if the law of addition of velocities is in some way flawed, or if the speed of light is as certain as claimed. Einstein looked at these two laws, decided that they were both fine, and came to a startling conclusion. The speed of light, nearly 300,000,000 metres per second, was not the problem. It was the ‘metres’ and ‘seconds’ that were the problem. Einstein realised that when an object travelled at speed, space got shorter and time moved slower.

Einstein backed up this bold insight by diving into the world of mathematics. The main tool that he used was a technique called a Lorentz transformation, which was a method that allowed him to convert between measurements taken from different frames of reference. By mathematically factoring out those different reference frames, Einstein was able to talk objectively about time and space and demonstrate exactly how they were affected by motion.

Just to complicate matters further, it is not just motion that shrinks time and space. Gravity has a similar effect, as Einstein discovered in his General Theory of Relativity ten years later. Someone living in a ground-floor flat will age slower than their neighbour living on the first floor because the gravity is fractionally stronger closer to the ground. The effect is tiny, of course. The difference would be less than a millionth of a second over a lifespan of eighty years. Yet it is a real effect nonetheless, and it has been measured
in the real world. If you get two identical, highly accurate clocks and put one on an aeroplane while keeping the other still, the clock that has flown at speed will show that less time has passed than that measured by the static clock. The satellites that your car’s satellite navigation system rely on are only accurate because they factor in the effect of the earth’s gravity and their speed when they calculate positions. It is Einstein’s maths, not our common-sense concept of three-dimensional space, which accurately describes the universe we live in.

How can non-mathematicians understand Einstein’s mathematical world, which he called
space-time
? We are trapped in the reference frames that we use to understand our regular world, and we are unable to escape to his higher mathematical perspective where their contradictions melt away. Our best hope is to look downwards at a more constrained perspective that we can understand, and use that as an analogy for imagining space-time.

Imagine a flat, two-dimensional world where there is length and breadth but no height. The Victorian teacher Edwin Abbott Abbott wrote a wonderful novella about such a place, which he called
Flatland
. Even if you are not familiar with this book, you can picture such a world easily by holding a piece of paper in front of you and imagining that things lived in it.

If this piece of paper were a world populated by little flat beings, as in Abbott’s story, they would not be aware of you holding the paper. They could not comprehend our three-dimensional world, having no concept of ‘up’. If you were to bend and flex the paper they would not notice, for they have no understanding of the dimension in which these changes are taking place. It would all seem reassuringly flat to them.

Now imagine that you roll the paper into a tube. Our little flat friends will still not realise anything has happened. But they will be surprised when they discover that, if they walk in one direction for long enough, they no longer reach the end of the world but instead arrive back where they started. If their two-dimensional world is
shaped like a tube or a globe, like the skin of a football, how could these people explain those bewildering journeys that do not end? It took mankind long enough to accept that we live on a round planet even though we possessed footballs and had the advantage of understanding the concept of globes, yet these flat critters don’t even have the
idea
of globes to give them a clue. They will need to wait until there comes among them a flat equivalent of Einstein, who would use strange arcane mathematics to argue that their flat world must exist in a higher-dimensional universe, where some three-dimensional swine was bending the flat world for their own unknowable reasons. The other flat critters would find all this bewildering, but given time they will discover that their measurements, experiments and regular long walks fit Flat Einstein’s predictions. They would then be confronted with the realisation that there is a higher dimension after all, regardless of how ludicrous this might seem or how impossible it is to imagine.

We are in a similar position to these flat creatures. We have measurements and data that can only be explained by the mathematics of space-time, yet space-time remains incomprehensible to the majority of us. This is not helped by the glee with which scientists describe the stranger aspects of relativity instead of explaining what it is and how it relates to the world we know. Most people will have heard the example of how, should a distant observer see you fall into a black hole, you would appear to take an infinite amount of time to fall even though you yourself thought you fell quickly. Physicists love that sort of stuff. Befuddlement thrills them, but not everyone benefits from being befuddled.

It is true that space-time is a deeply weird place from a human perspective, where time behaves like any other dimension and concepts such as ‘future’ and ‘past’ do not apply as we normally understand them. But the beauty of space-time is that, once understood, it removes strangeness, not creates it. All sorts of anomalous measurements, such as the orbit of Mercury or the way light bends around massive stars, lose their mystery and contradictions. The incident where the cup of tea may or may not pass you in deep
space becomes perfectly clear and uncontroversial. Nothing is at rest, unless it is defined as being so.