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Rutherford did say something like this but suggesting a barmaid instead of a six-year-old, while Hilbert suggested the first man on the street. Technically, if we assume the statement correct, then nobody has understood anything about science, ever. But I'd like to have my go at explaining Physics-y stuff as simply as I can, and, perhaps equally importantly, without too much of the "woaaah quantum mechanics is so weird, you just have to accept it like this even though it's completely non-intuitive" crap one often finds in science articles written for the laymen.

Thursday, April 2, 2015

A relatively special post

In the end of the 19-th century, it seemed like Physics was almost complete. Analytical mechanics was practically finalized with the work of Newton, Lagrange, Hamilton, and of course many others, and Maxwell had just written down a complete description of electromagnetism, building upon the work of Gauss, Faraday, Ampere, etc. There appeared to be a few things left to straighten out here and there, a few small outliers that didn't exactly fit the otherwise beautiful theories, but few people expected any big surprises. You won't believe what happened next!
The beginning of the twentieth century saw the development of two theories - both arising from scratching these dimples of remaining inconsistencies - that completely changed our understanding of the world. This is by now a textbook cliché, but it's also the simple truth, so I'm OK with re-iterating it. 

The first of these theories is Quantum Mechanics, about which I've already written (every word is a separate link). The second one is the theory of Relativity, which I'll (start to) explain here. A remark right off the start: there is the Special Theory of Relativity (STR) and the General Theory of Relativity (GTR); the former is much simpler and came first, which might or might not sound strange (it depends on whether you think deduction is a more natural train of thought than induction). This post is about the STR only.

The story of STR - and how Einstein came up with it - follows, in a nutshell, the same pattern that has brought some of the greatest insights in the history of science: someone refusing to take for granted something which is obvious for everybody else.

In the case of Einstein, the statement was, 'obviously, if I see you moving with velocity v1, and I see him moving with velocity v2, then you see him moving with velocity v2 - v1'. This 'obvious truth' has a slightly more complicated formulation than the ones above (it involves math!), but I'm sure that anybody would agree that it does seem pretty obvious if we only restrict our thinking to practically everything we experience in our lives, ever. But Einstein was totally, like, 'Hmmmm..."

Of course, he wasn't just toying with an alternative idea out of an abstract what-if-like curiosity. To abandon such 'truths' (postulates), which are deeply in-grained in our thinking, we usually need a very strong nudge. In the case of the STR, this came from the fact that Maxwell had derived an equation for the speed of light, and it had turned out to be a constant - without assuming any particular reference frame! In other words, you assume you're sitting on Earth, you write Maxwell's equations, and you get that you should see light moving with a velocity c. But if you assume that you're moving with velocity c with respect to the Earth, and you write the equations, you get again that light should be moving with velocity c! Initially, people just thought that this is some strange feature of electromagnetism (light is just electromagnetic waves, by the way). They thus tried to propose various extensions to the 'classical' theory, but as experiments kept proving them wrong, the add-ons themselves became more and more convoluted, even for philistine physicists who wouldn't usually care about aesthetics in equations. Einstein always cared about the aesthetics of Physics, by the way.

Maybe this is the reason why he realized there is an alternative way: instead of adding ever-more-unlikely complications, remove just one assumption, even though it seems completely and obviously true - but who knows. So instead of taking the addition of velocities as a postulate, he took a new one: 'light propagates with a constant velocity regardless of the reference frame' (obviously, you cannot have both of those true at the same time). Then, he set out to derive the new laws describing how to go from one reference frame to the other. This transformed the universality of c from a peculiarity of electromagnetism into a fundamental property of space-time. A clarification: by talking about space-time, I'm not trying to sound all sci-fi-y. Instead, this only refers to the fact that if we want to describe any event happening in the world, we have to say where and when it happens. The property of events of having a 'where' and a 'when' makes them embedded in some measure of space and some measure of time, which, combined, we call space-time. Furthermore, I should define more rigorously what a 'reference frame' is: it is the 'where' and the 'when' of the world - so, the space-time - as seen from some particular standpoint. But a standpoint need not be standing with respect to other standpoints, and in fact the relationship between moving standpoints is what the STR is all about. Simple, huh?

I'd like to highlight what I think are the most important aspects of the theory.
1. Einstein assumed just two postulates: that c is constant in all reference frames, and that the laws of Physics are the same in all reference frames (after having been properly translated). In other words, there is a fixed relationship between what I see and what you see, regardless of whether we're moving with respect to one another.
2. Given those two assumptions, he derived the way in which I can translate what I see to match exactly what you see. The theory of relativity, in its purest form, is nothing but a prescription of how to describe an observation to someone who is moving (fast!) with respect to you, in a way that would agree with the other person's observations.
3. All the crazy effects you might have heard of - like space contraction and time dilation, or the new law of addition of velocities (a modified version of the v2 - v1 above that complies with c - c = c) - as well as the not-so crazy but super-fundamental ones - like E = mc2 - can be derived based on those two postulates (and the other standard postulates of Physics, like energy conservation, and, you know... mathematics). I always found it truly amazing how much one can achieve with so little to start with.

In the remainder of this post, I’ll focus on the effects of length contraction and time dilation. These refer to the fact that different observers might disagree on the distance between two points, or on how much time passes between two events. That is to say, they would disagree as long as they don't use STR to 'translate' their observations in a way that makes them agree. These effects can be derived in many ways; below, I present one illustration of why something fishy has to happen either space or time - or both - if light is to move with the same velocity in different reference frames. Assume that a beam of light travels between two mirrors. If we think about the distance that light covers in a round-trip, there's obviously some difference to be observed depending on whether we are moving with respect to the system (which is equivalent to the system moving with respect to us). 

Most generally, in the first case, light travels some distance 2L in some time t, and in the second case - a distance 2L' in time t'. Our intuition would strongly suggest that t' should be the same as t, and that L' should not be the same as L. But if you think about it, this couldn't possibly be the case! Light has to travel with the same speed in both systems, and speed is just distance over time. So 2L/t has to equal 2L'/t', which implies that either we're wrong about t = t', or we're wrong about L' not equal to L, or we're wrong about both... In that particular case, we're wrong about t = t', but this is a detail. The big picture implied by STR is that both distances and time periods vary from one observer to the other. These are in fact two manifestations of the same, more general effect: space-time itself is different for different observers. 

Now, the illustration above gives a good intuition of why something fishy must happen when we think of light propagation, but the interesting twist is that, since something fishy actually happens to space and time itself, all physical processes pass on different time-scales in different reference frames. This, by extension, also includes chemical and biological processes, and, ultimately, what we perceive as the passing of time in any meaningful definition (also, aging). This has been experimentally verified in various experiments. One of the seminal ones involves the decay of particles called 'muons' (they’re kinda like electrons, but heavier) created by cosmic rays in the top layers of the Earth’s atmosphere.

We can create muons here on Earth, in a lab, where we can measure their lifetime (how long they 'live' before decaying into other, more stable particles), which turns out to be quite short. We also get to detect the muons that come from the cosmic rays, which 'rain down' on us with an enormous speed (close to that of light). The funny thing is, though, that if we multiply their speed and their expected lifetime, we should get the maximum distance that the muons could travel before decaying, and... it turns out that the cosmic ones shouldn't be able to get to us at all! The distance they can travel is smaller than the width of the atmosphere, if we don't take the STR effects into account. Thus, we shouldn't be able to observe the cosmic muons here on Earth - yet we do. Fundamentally, these muons are identical to the ones we make in a lab; the difference comes only from the motion with a 'relativistic' velocity.

This is a great experiment to demonstrate why time dilation and length contraction are two different manifestations of the same effect. The fact that the muons do reach the Earth can be explained in two different ways, depending on whether you want to stay in a frame in which the Earth stands still (which you are probably used to[citation needed]), or in one where the muon stays still (remember that the Physics has to be the same in those two frames, namely, in both of them the muons have to reach the surface). Let's first have a look at the less intuitive frame: the muon is standing still, minding its own business, and the Earth is moving up towards it with a speed close to c.

The life-time of the muon in this reference frame is exactly the one that we measure for muons created on Earth; however, since the planet, together with its atmosphere, is moving towards the particle, the atmosphere appears shorter than its length as measured on Earth. And so, the particle reaches the surface – or actually, in this case, the surface reaches the particle!

We can also look at the case where the Earth is standing still, minding its own business, and the muon is free-falling.

In this case, the length of the atmosphere is the one we are used to, but the lifetime of the muon is dilated, since it is moving fast with respect to us – so it has more time to traverse the atmosphere and reach the surface. And so, as expected, the laws of Physics are preserved.

Many people, when they first learn about the relativity of length and time intervals, are tempted to ask questions like, 'What happens exactly? Does the atmosphere actually get thinner? Does time actually pass more slowly?' The problem with these questions is that the whole point of the theory of relativity is that there is no 'actually' - hence the name of the theory! There is no 'absolute' series of events in the Universe - they are always described relatively to a given reference frame. In some reference frames, some lengths appear shorter; in others, time seems to pass by more slowly; but in the end, everybody agrees that the muons reach the surface of the planet, one way or another. 

P.S. There is some polemic about Einstein's role in the whole matter, and how fair it is for him to get all the credit, which he often does. Here's my opinion: it's true that a lot had already been done - the constant speed of light is within Maxwell's equations, and Lorentz had derived the famous transformations that also imply time dilation and length contraction, but it really was Einstein who realized that these two effects are not some strange, minor aspect of electromagnetism, but instead carry a fundamental implication about the very nature of the world. This is as far as STR is concerned; regarding the GTR, Einstein deserves even more credit. So, no, I don't think he's overrated.

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