String Theory – part 2: Special Relativity – the Picture of Space and Time

Solving the First Conflict

110 years ago, in June 1905, Albert Einstein, a young person from a Swiss bureau, sent a paper titled “Zur Elektrodynamik bewegter Körper” to the German journal Annalen der Physik in which he provided a solution to the speed of light paradox that had puzzled him 10 years earlier. This was his famous Special theory of Relativity in which he threw down a gage to our very understanding of space and time that had persisted since the times of Newton.

According to the Newton’s laws, if you were to move with the speed of light, you would see light being stationary. However, in the middle of the XIX century Scottish physicist James Clerk Maxwell combined previously separate notions of electricity and magnetism into a unified model of the electromagnetic field. This field represents electric and magnetic flux lines. If you spread iron dust around a magnet you will see that they form a particular pattern because they follow the magnetic field’s flux lines. Taking off a woolen sweater you could hear a crackle which demonstrates the presence of an electric field. But apart from everything else, Maxwell’s model suggested that electromagnetic waves moved always with a constant speed which is equal to the speed of light irrespective of observer’s motion. Based on this, Maxwell concluded that light represents an electromagnetic wave. Moreover, Maxwell’s theory demanded that any electromagnetic wave never stops and never slows down in a vacuum. Thus, such a thing as stationary light just did not exist. Here we’ve come to the contradiction between Newton’s mechanics and Maxwell’s electromagnetic field model which was mentioned as the first conflict in physics in the previous article.

In this article we are going to explore the basics of Special Relativity which are provided in Brian Greene’s book “The Elegant Universe” where you can find a more detailed explanation.

Distance and Time are Affected by Motion

Our intuition plays an important role in our lives, allowing us to solve various problems through inductive or deductive reasoning and leading us to great insights from time to time. However, a scientist, seeking for the hidden patterns of Nature, should not always rely on his intuition since these patterns may sometimes be highly counter-intuitive.

Let us start with a simple example of an intuitive understanding of motion. Imagine a great sunny day in your city. You are walking down the street, approaching a traffic light which is signalling red and you are waiting for a signal to go. You see some buildings on the other side of the street and these buildings seem stationary, i.e. not moving relative to you. But for a driver in a car, which is passing by down the street, they are moving (i.e. they change their position relative to our driver). On the other hand, the control panel in the car is stationary for the driver, but is moving from your perspective. This is a completely intuitive picture for us, so we do not usually take notice of it.

But in his Special theory of Relativity Einstein took this picture into account and showed that such an unnoticeable, rudimentary thing represents a fundamental principle of physics. According to Special Relativity, two observers moving relative to each other will not agree on which object moves and which does not and, what might seem completely unexpected, they won’t agree on what distance they measure either. Likewise, they will measure different rates at which time passes. And this has nothing to do with the accuracy of their clocks but represents a fundamental property of time itself.

Let us take an example. Suppose your friend Jim has bought a new sport car and asks you to help him in his experiment. He is going to drive along a track, whose length is 1 km, with a constant speed of, say, 200 km/h, and you need to measure the time period in which he passes the whole track. Let’s say Jim also wants to measure the time and he takes a clock completely identical to yours, so both clocks are measuring how fast Jim’s car would finish the track. Before Einstein developed Special Relativity, our intuition would have told us that both clocks will measure the exact same period of time. But all of a sudden, this is not what actually happens. Instead, your clock will measure 18 seconds, whereas Jim’s clock will show 17.99999999999969 seconds – a tiny fraction of a second less (the numbers are taken from Brian Greene’s book). Similarly, if Jim asks you to measure the length of his car while it is in motion, your tape measure will show the car being 4.999999999999912 m long, whereas Jim would measure the length of 5 m.

In both situations the difference is extraordinarily tiny such that there are no clocks or tape measures of such precision at the moment, but we have numerous other confirmations of these effects, so they do occur with no doubt at all. Should we care about such a tiny difference? Well, in fact we should since the strength of these effects is dependent upon an object’s speed. The higher the speed, the stronger the effect for an observer. For example, suppose Jim’s car moved 935 million km/h (it is roughly 90% the speed of light) instead of the previously given speed. In this case you would measure the length of his car being 2.5 meters which is 2 times shorter than it is from Jim’s perspective! Likewise, the time measured by your clock would have been 2 times longer than that by Jim’s clock. Those effects are completely unfamiliar for us just because we never see objects moving at so high speeds, but if we were, these effects would have been just as ordinary as those we’ve discussed at the start of this section.

The Principle of Relativity

Special Relativity is based on two main concepts: the principle of relativity and the speed of light. Let us start with the first one. The principle of relativity had been known since the time of Galileo but it was first formalized by Einstein at the start of the XX century. It says that when you are talking about a quantity of some physical parameters, such as speed, distance or time, you should necessarily specify the observer measuring these quantities and his reference frame.

Let’s say you have bought a car and are now able to perform some experiments with Jim. Suppose you’ve happened to find yourself in empty space with no planets, stars or galaxies around. You are now in complete weightlessness and your car does not move in any direction. Then you notice a faint light far away. Gradually, the light appears brighter and eventually you can see your friend Jim’s car floating next you and flicking out of sight again. Exactly the same picture would appear for Jim. He would say that his car was completely stationary, then he noticed a faint light far away which has gradually increased in brightness and eventually Jim recognized your car floating next him and disappearing in the darkness.

These stories depict the exact same situation from two different points of view. Both you and Jim counted himself (with his car) stationary and the other one moving. Nobody has any reason to count himself right and the other one wrong – you both are right. This situation vividly demonstrates the principle of relativity in action. It says that the very motion is relative and the statement “Jim’s car is moving with a speed 200 km/h” is devoid of meaning until we specify in relation to whom it is moving. In other words, there is no absolute motion. Motion is relative.

What’s important here is that both you and Jim were moving with a constant speed in the aforementioned example. If you were to press the gas pedal, you would certainly notice your motion. This feeling is internal and you may have noticed it when a car, in which you are sitting, starts moving. When the car accelerates your back certainly feels the seat. Likewise, when travelling in a subway you feel when it starts turning to either side. Thus a more correct form of our statement would be “free motion with a constant speed is relative but accelerated motion is not”.

Here we’ve considered a situation in empty space, but the same reasoning can be applied for ordinary situations on the Earth. Suppose you fell asleep in a train with no windows, the train which is moving down a straight railway with a constant speed. When you open your eyes you will have no idea whether the train is moving or sitting stationary. For if the train is not accelerating you cannot distinguish between being stationary and moving with a constant speed. And there is no experiment that would allow you to determine whether or not the train is moving. This is again due to the principle of relativity. As we have seen, free motion is relative and it acquires physical sense only in comparison with some other objects.

Einstein realized that not only can you not determine whether or not you are moving, but the very laws of physics are completely invariant for an object moving with a constant speed. If you and Jim were to try to perform various experiments in order to determine who is actually moving, you would get the exact same results, hence would not be able to differentiate between your reference frames.

The Speed of Light

The second main concept of Special Relativity is the speed of light. You might wonder why such an ordinary thing as speed demands a thorough consideration. As we shall see, it affects our very notions of distance and time dramatically.

Suppose you are snowboarding in the mountains with an average speed of 60 km/h. All of a sudden, you notice a snow avalanche approaching you with a speed of 100 km/h. You are standing on your snowboard and speeding up downwards at 70 km/h trying to elude a situation in which there will be big trouble. Taking into account that the speed of the avalanche is 100 km/h and your speed is 70 km/h we can easily calculate with what speed the avalanche is approaching you. It is now 30 km/h but if you were staying still, it would be 100 km/h, so you would definitely have much lesser chance to evade the avalanche. Similarly, if you were, for some reason, running in the opposite direction, i.e. towards the avalanche, with a speed of 10 km/h, the resulting speed of your convergence with the avalanche would be 110 km/h.

This is all well and good, but what would happen in a situation with light? Suppose you are living in the XXV century where people have managed to develop cars that can travel with a speed 200,000 km/sec (2/3 the speed of light). You would like to test Einstein’s Special Relativity and are looking for an experiment where your car reaches the maximum speed and your colleague shines a beam of light which travels parallel to your track. According to our previous example, we would expect you to see the beam of light receding from you at 100,000 km/sec. And if you are on the other side of the track and start moving towards the beam, you would expect to see it converging with a speed of 500,000 km/sec. Interestingly, this is not what actually happens. Either way you would see the speed of light being exactly 300,000 km/sec. Whatever speed you are travelling at and whatever direction, the speed of light in a vacuum is always constant irrespective of your reference frame.

You have probably learned the speed formula at school. It is v = d/t, where v stands for speed, d – for the distance travelled and t – for time. So these three quantities are intimately related to one another, and if the speed of light is constant, this leads to an unprecedented shift of our understanding of the other two. Let us see how time and distance are affected by motion in a bit more detail.

Influence on Time

Let us start with an example taking light clocks into consideration. Such a clock is basically two parallel mirrors and a photon bouncing between them. If our mirrors are placed 15 cm apart, then it would take one billionth of a second for a photon to perform one cycle, i.e. two successive touches of either the bottom or the top mirror. So the photon makes a billion cycles a second.

We can use such a clock for measuring the time with which some continuing event occurs. For example, let’s take the above situation where we should have measured the time with which our friend Jim’s car passes through a track. Previously we measured the time being pretty darn close to 18 seconds. With our light clocks that would correspond to 18 billion cycles of the photon. Albeit these clocks can be used in such a situation, they are apparently highly impractical. Instead, we shall use them to show that the passage of time actually depends on motion.

For this we shall place our stationary light clock at some point and make another one move with a constant speed nearby. Suppose the photons on both clocks touch the bottom mirror simultaneously. Now the photon on the stationary clock still measures one billionth of a second for each cycle, but the photon on the moving clock has to travel a longer path to reach the top mirror and, likewise, to reach the bottom mirror then. This situation is illustrated on the picture below.

Then, the constancy of the speed of light tells us that both photons are travelling with the same speed, but since the photon in the moving clock has to pass a longer path, it performs one cycle in a longer period of time. This tells us that time for the clock in motion does inevitably slow down relative to stationary clocks! And this has nothing to do with the way we measure time, this is an inherent property of time itself. You can get a great illustration of this effect in this video by Brian Greene.

What I’d like to emphasize is that the strength of this effect – which is called time dilation – is intrinsically dependent on the magnitude of speed. The faster the speed – the larger the time dilation effect. This can be easily seen in the above example. The faster the second light clock is moving, the longer the path for the second photon, hence it would take longer to make a cycle. But in our everyday life we are only familiar with speeds which are very low in comparison with the speed of light. Thus, such effects are completely unfamiliar for us even though they permanently occur. If our moving clock goes with a normal speed, say, 100 km/h, then the time dilation it would experience is so tiny that the effect remains completely unnoticeable for us. On the other hand, if it were to travel with a speed, say, 3/4 the speed of light, then its time passage would correspond to 2/3 of that measured by a stationary clock. This is a noticeable difference!

Let us take another example. Suppose there is a train capable of reaching a very high speed, a significant portion of the speed of light. There are two passengers in the train who are sitting there, playing tabletop games, drinking tea and doing whatever they want. As we have seen, they are feeling stationary, and their reference frame (the train) is also stationary, because they cannot distinguish between being stationary and moving with a constant speed. They see the countryside rushing by, and from their perspective everything outside the train is moving at high speed. Then the train enters a tunnel where a trainspotter watches the train passing by and measures its speed.  As we have seen, the flow of time in the train reference frame is slowing down from the trainspotter’s point of view. And what’s particularly significant for us here is that not only is the passage of time in our passengers’ clocks affected, but the passengers’ motion, speaking, ageing and any other process are also slowing down! That is to say, our trainspotter would see them just as in slow motion.

We shall come back to this example later on, but now let me show you yet another example which became one of the first proofs showing that time dilation actually occurs. It considers particles called muons – heavier cousins of electrons – that are created when high energy cosmic rays strike Earth’s atmosphere from space. Since these rays were directed towards the Earth, created particles have to continue their motion in that direction due to the conservation of momentum. They indeed do that, and our muon detectors on the surface find plenty of them that are getting through. But the reason why this was a bit of a surprise is because those muons don’t live very long, they have an extremely short decay time. According to the calculations based on Special Relativity, even if they were travelling at the speed of light, there would not be enough time for them to reach the surface, most of them should have decayed before they do. The resolution to this turns out to be time dilation. When these particles move extremely fast, close to the speed of light, time in their reference frame is highly dilated from our point of view. So every process occurring with those particles, including their decay rate, becomes tens or even hundreds times longer than it takes for a stationary muon. And with time dilation they do have enough time to reach the surface.

Influence on Distance

Let us now see how motion affects distance. For this we shall firstly take our example with muons. Previously we considered the situation from our point of view. But we can ask what would happen from the muon’s perspective. From its point of view it suddenly comes into existence in the Earth’s atmosphere. Again, from its perspective it is stationary and the Earth is rushing up to smack into it. But as soon as it is stationary, it has a normal decay rate, so again, there would not be enough time for the Earth to meet it, and it will decay into an electron and a neutrino before. So there has to be another effect which allows it to reach the Earth. This effect is known as Lorentz contraction, which says that not only is time being affected by motion, but distance is getting affected by motion as well. When something is moving at high speed, along its line of sight distance is getting contracted. From our muon’s point of view, instead of travelling a hundred kilometers to reach the surface it has to travel only a few hundred meters or so. And even though its decay time is now 2 millionths of a second, when travelling close to the speed of light there is enough time for it to reach the surface. This is how Special Relativity explains the detection of those particles. What we need to remember is that this contraction occurs only along the direction of your movement. If our muon travelled towards the highest skyscraper on the Earth it would see it being of normal width and normal length, but suddenly a few centimeters high.

Bearing this in mind, let’s return to the example involving a train and a tunnel. Recall that we had a trainspotter staying in the tunnel and passengers sitting in the train. What we need to assume now is that the train is of exact same length as the tunnel.

Now, because the train is moving relative to the trainspotter it is Lorentz contracted, which means that it is shorter than it would be in a stationary position. So even though its length is equal to the length of the tunnel, it now completely disappears in the tunnel for a moment.

On the other hand, from the observer’s sitting in the train perspective, the tunnel is moving relative to them, so it is now shorter than the train, which means that the train always shows itself from one end of the tunnel or from both.

In this case you could say that it is just a matter of perspective and the question as to whether or not the train disappeared in the tunnel is just relative.

But there is another famous thought experiment which considers the following situation. Suppose someone has installed two large guillotines on both ends of the tunnel. When the train enters the tunnel our trainspotter pulls a lever which makes these guillotines come down and then go back up. Here from the trainspotter’s point of view it is absolutely fine, because the train was shorter than the tunnel, so he could do this without doing it any damage. But from the passenger’s perspective the tunnel is shorter than the train so either the front end of the train or the back end of the train gets chopped off, or both. But this is not something you can say that it is just a matter of perspective, either the train gets smashed or it does not. This is something everyone has to agree on!

This is known as the relativity paradox and its solution lies in one of the fundamental principles of the theories of relativity, which is called the relativity of simultaneity. It tells us that two events happening simultaneously in one reference frame are not necessarily simultaneous in another one. In our example this means that both the trainspotter and the observer in the train do agree that the train is fine, but the reason for them is different. As we have seen before, the trainspotter will say that the train does not get smashed simply because it is shorter than the tunnel, whereas the passenger will see something quite different. From his perspective when the train enters the tunnel, before its front side passes through the far end of the tunnel, the front guillotine comes down and then goes back up, and after it passes through (the back end of the train passes through the back end of the tunnel) the back guillotine comes down. Eventually, the train passes through the tunnel unharmed.

This of course might sound as some kind of a trick allowing us to solve the paradox, but numerous indirect evidence confirm that this is the way the Universe actually works. The whole concept of simultaneity just goes away in Special Relativity, and two observers moving with different speeds won’t agree on which events are simultaneous and which are not.

Motion through Space-Time

What I want to introduce you to at the end of this article is a very helpful picture which shows us the significance of the combined notion of space and time. From the very early moments in our lives we are familiar with 3 spatial dimensions which can be thought of as width, height and depth. We are also familiar with time which seems to be flowing in one direction, from past to future, throughout our lives. What Einstein showed, based on earlier works of German mathematician Hermann Minkowski, that both space and time are intrinsically bound together to form a unified entity which we now call space-time.

Suppose your friend Jim is called to test a new model of a car. This car always speeds up extremely fast to the speed of 200 km/h and keeps this speed until its engine is switched off. For this example we can safely ignore the time needed for acceleration and can assume that the car keeps the constant speed while it is moving. Jim was invited to a track whose length is exactly 20 km and it is absolutely straight, with no turns. We know that we can calculate the time dividing the distance travelled by the speed. In this case we see that Jim should finish his track in one tenth of an hour – or 6 minutes. Jim has performed 10 test runs today but he was pretty tired for the last three of them, so when he looked at the results he has seen that the first 7 trials took him exactly 6 minutes each, as was expected. But in the last three trials his results have been worse – 6:30, 6:45, and even 7 minutes. To get an idea what has happened we need to take into account that the track is quite broad, being, for example, a kilometer wide (a highly impractical track).

For the first seven test runs Jim was riding straight forward along the whole track, that is to say, he used one dimension for his movement. But when he got tired he could not concentrate enough to do the same, so the movement of the car started diverging from the straight line. When this happens, the car is moving in 2 dimensions, which means that a part of its speed is being used for covering additional distance.

(The above image is taken directly from Brian Greene’s book)

You might be wondering what it has to do with Special Relativity. What Einstein realized in his theory is that everything actually moves with the speed of light! But this motion represents the overall motion in 4 dimensions. For example, if something is stationary and does not move in any direction, all of its motion propagates through the time dimension. But when it starts moving, its motion spreads out to a space dimension, so it is now moving in 2 dimensions. And with the constant speed, with which everything is travelling, this means that a part of this speed is used for covering a spatial dimension, hence it is moving slower in time! I familiarized myself with this description while reading the aforementioned Brian Greene’s book and it was absolutely fascinating for me how intuitively clear this idea happens to be. Everything we have been concerned with in this article can be expressed in this idea.

What’s also important here is that motion in the time dimension also diverts the motion through space. If our muon from the above example, for some reason, significantly slows down before reaching the Earth, then its decay time becomes extremely short again and it can’t manage to make it through the atmosphere all the way down to the surface. We had one extreme situation where we were talking about stationary objects that do not move through spatial dimensions at all, but we have another extreme case where something is moving only through spatial dimensions. This extreme example is of course photons of light. Since they are moving with the speed of light, all of their motion diverts to spatial dimensions, hence they do not experience time at all. The photons of CMB (Cosmic Microwave Background radiation) are of exact same age as they were emitted. They do not age at all.

Next time we shall consider another breakthrough in physics made by Albert Einstein, General theory of Relativity.

Thank you.