Understanding the Theory of Relativity

The theory of relativity is much easier to understand than you might think. The only problem is that the historic development of the theory wasn't exactly straightforward. At the end of the 19th century and beginning of the 20th century there were two different theories in physics that contradicted each other: Maxwell's theory of electromagnetism on one hand and Newtonian mechanics on the other.

What Lorentz, Poincare and finally Einstein did was to change Newtonian mechanics in such a way that the electromagnetic forces would fit inside it. The resulting new theory of mechanics is the special theory of relativity. After obtaining this theory, which suited electromagnetism perfectly, physicists were disconcerted to find that gravitation was left on the outside. Newtonian mechanics had no problem incorporating the gravitational force (Newton himself solved that problem), but special theory of relativity couldn't do it. Eventually, after more than 10 years, Einstein and Hilbert delivered the general theory of relativity which solved the problem of gravitation. In the process, they ended up with a theory of gravitation that proved to be even more precise than Newtonian gravitation.

This history is not exactly straightforward! And the problem with presenting the special theory of relativity using a historic approach is that you need to present electromagnetism (which is also the theory that explained light). However, once the special theory of relativity was in place, it delivered various mechanical predictions which were empirically tested. So, I'll use these experiments because they are easier to understand and thus I'll present you the theory in a totally unhistorical manner.

The entire theory of special relativity can be deduced out of two things. These two things seem at first to contradict each other, and for understanding the theory of relativity you only have to understand how is it that in fact they don't.

1. The principle of relativity (the relativity of space):

This principle says that everything happens in the exact same way in two frames of references that move in a straight line at constant speed relative to each other. For example when you are in a train that moves in a straight line with 300 km/h (relative to the ground) everything seems normal to you - it's exactly as if you were on the ground. In other words, you cannot feel speed, you can only feel acceleration (the change of velocity). The principle of relativity says that the reason why you cannot feel speed is not that you have some sort of biological inability (in the same way as you, for instance, cannot see radio-waves), but because this is an intrinsic, objective aspect of the physical reality: frames of reference moving uniformly relative to one another are entirely equivalent.

For understanding why this principle is true you may wonder what it means that something stands still. Imagine for example that the entire universe was empty and only a single ball existed in it. Is this ball moving? In what sense could you say, "five minutes ago it stood still, but now it is moving"? You couldn't say something like this. It is moving relative to what?! If it's the only thing that exists, the idea of motion doesn't even have meaning. Now suppose you have two balls in this otherwise empty universe. You still couldn't say "one ball stands still, the other one is moving" because in order for one ball (random picked) to move relative to the other one, the distance between them would have to change in time. But how could you realize whether this distance changes? You would still have nothing (no other distance) to compare it with. Now suppose you have three balls. Now we're getting somewhere! We have three distances that we can compare with each other. The concept of motion can finally have meaning and motion can finally exist. But, as it should be obvious, there is a matter of choice which balls you happen to declare "standing still". In other words, the idea of "standing still" is subjective, not objective. And no matter how many more balls you would add, even an entire universe of particles of all sorts, this idea wouldn't become any more objective.

This is what the principle of relativity says. That it is a matter of choice whether you dub something as "standing still" or as "moving with constant speed in a straight line relative to another object that stands still". This idea of relativity of space was discovered by Galileo and it is a fundamental part of Newtonian mechanics (as well as the theory of relativity).

2. The relativity of time.

Above I talked more about space, and assumed that we have some way of measuring time. But how do we measure time? The best way is to use various particles that have an intrinsic rate of spontaneous disintegration. These particles' rates of disintegration are unaffected by anything, temperature, pressure etc. You can artificially disintegrate them by smashing other particles into them, but otherwise, they have a very constant rate of spontaneous disintegration. You can easily verify that this rate of disintegration is indeed incredibly constant: all you have to do is take two identical such "clocks", start them simultaneously, and, as "time goes on", watch whether any difference appears between them. None appears! Among such particles are the radioactive atoms or other various subatomic particles called mesons.

The classic experiment regarding the relativity of time involved mesons, but the experiment was also conducted with atomic clocks. Mesons can be produced in the lab by smashing other particles. After producing such mesons physicists could measure the rate of their spontaneous disintegration (they spontaneously change into other particles after a while).

Another way such mesons come into existence is when cosmic rays (such as high energy light coming from the Sun or protons from outer space) hit the upper layers of Earth's atmosphere. The mesons generated this was fly downwards toward Earth's surface. The only difference between the mesons produced by the cosmic rays and the mesons produced in the lab is that the former move at huge speeds while the latter are almost standing still (relative to a human experimenter I mean).

The experiment:

Physicists got on the top of a mountain and measured the amount of mesons there at high altitude. Then, they went to the bottom of the mountain and measured the number of mesons there. Some mesons disintegrate on their way from the top of the mountain to the bottom of the mountain. But, remember, the physicists already knew at what rate the mesons disintegrated in the lab. Knowing how many of them are at the top of the mountain, how many of them go "puf" in each unit of time (the disintegration rate) and how fast they fly towards the bottom of the mountain you can compute how many mesons should be at the bottom of the mountain. But the physicists could also measure the actual number of mesons reaching the bottom of the mountain. The result is that the actual number of mesons at the bottom of the mountain (determined experimentally) is far greater than the calculated number.

So what happened?

Think about an alternative experiment: You have a sandwich. If you leave the sandwich on a table, it has a certain rate at which it decomposes. It becomes uneatable after a day or so. But if you place it in the fridge it doesn't rot even for a week or more. So, what happens? Does time flow more slowly inside the fridge? Do Eskimos live longer than Africans as a result of the colder northern climate?

Well, in case of the sandwich, the explanation is easy: temperature influences the rate of sandwich disintegration. Its disintegration ultimately involves a bunch of chemical reactions which are more likely to happen if the entire system (the sandwich) is supplied with more energy - i.e. if the temperature is higher. Thus, it's no miracle that a warm sandwich disintegrates faster than a cold one.

But what about the mesons? Their situation seems analogous to that of the sandwich. It seems that their speed influences their rate of disintegration. A bunch of mesons standing almost still (relative to the experimenter) disintegrate much faster than a similar bunch of mesons moving at huge speeds (relative to the experimenter).

Hm... But this contradicts the principle of relativity! According to that principle (and we have already seen why this principle is true) the speed of something cannot influence what actually happens to it. Speed is subjective, remember?

So, the issue cannot be what actually happens to the mesons due to their speed, because nothing happens to them due to their speed. The issue must be what happens with the experimenter's perception of the mesons. The meson's intrinsic, objective rate of disintegration cannot be affected by its speed relative to the experimenter. The only thing that can be affected is the experimenter's perception of this rate.

Where am I getting at? There are two different times involved here: On one hand we have the moving mesons which have their intrinsic clock, and on the other hand we have us looking at the moving mesons and having our own (equally precise) clock. The huge discovery here is that these two clocks do not show the same time.

And it's all a matter of perception: We are seeing the fast moving mesons' clock lagging behind our clock. But from the mesons' perspective, they are the ones standing still and we are the ones moving at incredible speed towards them: so, they see our clock lagging behind theirs. We are seeing the cosmic ray mesons as "living longer" than out lab mesons. But the cosmic mesons "see" our lab mesons as "living longer" than them.

This is the relativity of time and it's the fundamental cause of the weirdness of relativity theory. According to Newtonian theory, and to our everyday intuitions, both clocks should show the same time, none of them should lag behind. But this is not what actually happens in nature. (Although at the speeds we are moving this de-synchronization of clocks is, fortunately, very hard to observe - but it does exist even at our speeds and it has been measured.)

How can one make sense of this nonsense?

The idea that saves the day is the idea of space-time. The point is that in the same way as we are seeing the moving meson at a certain position, we are also seeing it at a certain time. But what time should we consider? As we're seeing it or as it's seeing itself?

The correct solution is that the meson's position in space-time is the position in space and the time as it is measured by us. This seems logical because we are measuring the meson's position in space-time relative to ourselves - so we should use our time (the way we see it) and not the meson's time. But the use of intuition is already pretty uncertain. So we need some experiment to tell us that in order to represent the meson's position in space-time we should indeed choose our time and not the meson's time.

The experiment is simple (at least conceptually). The idea is that two things collide with each other when the distance between them becomes zero. So we simply go on and see: When do particles collide - when the distance in space-(our)-time is zero, or when the distance in space-(one-of-their)-time is zero? The answer is that they collide when the space-(our)-time is zero. Thus, the correct choice is (x, y, z, t) and not (x, y, z, s).

How real is this space-time? Is it just a mathematician's toy? It is worth mentioning that collisions are as real as you can get and, generally speaking, collisions don't happen when the distance in space at a certain moment of time is zero (as Newton would have it), but when the distance in space-time is zero. It is possible to have two particles situated at two different positions in space and each at two different moments of time that nevertheless collide with each other because the space-time distance between them is zero. When does this collision happen? If you think in terms of the time you associated to one particle it happens at one moment in time, if you think in terms of the time you associated with the other particle it happens at a different moment in time! In other words, you see the same collision twice in two different locations in space (but only once in particles' proper time and at a single location in space-time). Thus, the experiments with colliding particles demonstrate that space-time is very real and that the Newtonian separation between three-dimensional space and time is in fact an illusion.

This illusion is the result of the fact that we are used to things moving slowly relative to us and in these cases the distinction between moving clocks is much too small (given the precision with which we perceive time). Physicists first stumbled upon relativity when they studied light - which moves a very high speeds and the relativistic effects cannot be ignored.

Another way to understand why the meson's space-time involves the time measured by our clock is to understand that the "time" in the space-time is not really time in the sense of the thing that measures change - it is simply a way to identify where the particle is relative to our orientation devices. The true time (or "proper" time as it is called by physicists) is the meson's time s - that time really describes the intrinsic change experienced by the meson until its spontaneous decay.

In Newtonian physics we understand motion as the change of the spatial position of a particle in time - how from time to time the particle moves from one place to another. In relativity we understand motion as the change of the space-time position in proper time. This is weird, but if you think about it, it shouldn't be very mysterious. The change from Newtonian mechanics to special relativity isn't really that big. It just incorporates the fact that the two clocks moving at constant speed one relative to the other are not perceived by the observer as both showing the exact same time - there's always a difference between them (as the experiments with mesons prove).

Einstein tried to justify why it is that there's always this difference between the perception of the two clocks - for instance he tried to link this fact to the idea that information cannot travel faster that a certain speed. But there are all sorts of problems with the attempts to derive the relativity of time from something else (such as the speed limit). For example in his attempt, Einstein understood "speed" in the Newtonian sense (because in the relativistic sense it simply doesn't make sense to speak of a limit speed). In the end I think that the fact is more important (and after all more general) that these speculations about how information (or other people say energy) is supposed to travel. You can of course wonder about why does this difference between the perception of clocks exists, and it may seem to be a pretty important question, but I think that insofar your guess is as good as anybody else's (including Einstein).

Common misunderstandings

The most common misunderstandings of relativity are what we could call "the rotting sandwich analogies": ideas that the speed influences the stuff that moves. The very essence of the theory of relativity is that speed cannot influence what actually happens to an object because speed is subjective; it depends on the chosen frame of reference.

For example it is sometimes claimed that something that moves gets slightly compressed ("length contraction") or that its time flows slower ("time dilatation") or that is mass gets larger. These ideas are just plain wrong. Length contraction and time dilatation have everything to do with somebody's perception of the moving thing and nothing to do with what actually happens to the thing. The thing that moves in a straight line at constant speed doesn't actually contract, it's clock experiences exactly nothing and its mass doesn't change at all.

Moreover, there is no such thing as an actual speed limit. If you were to get on a spaceship and start accelerating, your speedometer won't hit some impassable limit (the "speed of light"). Your speed as you measure it will get higher and higher toward infinity. But the interesting thing is this: the Newtonians on Earth looking at you would indeed see your speed as going asymptotically toward some "speed limit"! This happens because the Newtonians measure your speed as x/t, while you measure your speed as x/s. You are correct, they are wrong - the correct speed is the one computed using the proper time and not the time coordinate (it doesn't make much sense to divide one coordinate to another coordinate).

The idea that mass increases as you speed up is a mathematical trick that some people find useful because it allows them to retain the Newtonian concept of velocity. However, the whole point is that the Newtonian concept of velocity is wrong. The funny part is that when you start talking about an increasing mass you are forced to say that Newton's second law, F = ma, is wrong, but in fact this law is correct even in the theory of relativity (provided that you understand velocity properly)! This law is very important because it tells you that in order to predict the motion of an object you only need to know its position and velocity at one moment of (proper) time. This holds in the theory of relativity as well as in Newtonian mechanics and messing up with F = ma doesn't help understanding one bit; it just obscures a very important fact about mechanics.

Finally, the theory of relativity is often presented as having a lot to do with light. However, the theory of relativity actually describes the motion of everything (including light) and light doesn't play any fundamental part in the story. Light is just one of the things that moves and that fits well into the theory. The history of the theory of relativity has a lot to do with light, but what the theory itself tells us about nature isn't linked to light in any special way.