Gravity waved! (Q&A)

If you've been online even for a moment in the last few days, you might have heard something about it. Gravitational waves have been detected! The starting point to understand what those are is general relativity, and I am planning to write a post about it one day. But, due to popular demand,^{[citation needed]}  I will go ahead of myself and answer some questions about this exciting observation. 

Note that there are many other sources from where you can learn about gravitational waves in general, and the measurement in particular. Just a few suggestions: the American Physical Society viewpoint (the paper was published in Physical Review Letters, a scientific journal published by the APS), the New York Times take on the story, or simply this cool PhD comics video. I hope that the Q&A below will further quench the thirst for understanding of the scientific enthusiast. 

Q: So is that just another type of a wave?
A: This one is quite unique. Other types of waves, like waves in water, sound waves, or electromagnetic waves (light), are in some ways very different from one another, but they do share an important characteristic. Namely, they can be described as an oscillating modulation of something physical, be it the water particles, air pressure, or electric field intensity. Mathematically, we would write this as some amplitude that depends on position and time - A(x, t) - and the wave nature appears in the fact that there is some periodicity in the time dependence. Gravitational waves, however, are different and kinda hard to imagine properly, because it is the position and time intervals themselves that change! Mathematically, one might write the effect of a gravitational wave as something like x'(x, t), t'(x, t), which is to say that space and time themselves change when such a wave passes through. (Note: this is not the way one would typically express the waves mathematically, but I think it's a good illustration.)

Q: So, like, a wave stretching the fabric of space and time?
A: You hear that a lot these days, but I don't like the 'fabric' part of the statement. It implies that space and time are made out of something. As I've already discussed, Einstein's view was rather that there are objective, absolute events, which appear to us imbued with the qualities of a location and a moment in time. Measuring these qualities defines our subjective space-time, and the laws of relativity tell us how to communicate with observers with different subjective space-times. In any case, whether space-time is physical or a figment of our perception is a big can of philosophical worms. I would thus much rather just say 'a wave stretching space-time' and throw the 'fabric' away to keep the interpretation load to a minimum.

Q: So what did we detect?
A: General relativity explains gravity as curvature of space-time. Our Sun for example curves our space-time in such a way that if we try to follow a 'straight' path, we end up orbiting around it. This is a fairly static effect: mass implies curvature just by being somewhere. Another prediction of the theory, however, is that waves of space-time curvature can be radiated by accelerating massive objects. An extreme example of accelerating masses is a system of two black holes rotating around one another. This is what we detected. As black holes can pack a lot of mass in a tiny volume, the centrifugal acceleration in such a system is tremendous, and so is the amplitude of the emitted gravitational waves - at least as compared to any other emitters. Compared to the sensitivity of our apparata, the amplitude was still barely big enough for us to measure - and we've been trying for decades. In short, we detected two black holes that circled around one another and eventually collided 1.3 billion years ago.

Q: Ok, I know enough astronomy to know that we see something that happened 1.3 billion years ago because it happened 1.3 billion light years away from us. So did we, like, see two black holes about to collide somewhere (1.3 billion light years away), point our gravitational wave detector in that direction, and see the waves coming in? 

A: The first part of the statement is correct: gravitational waves also travel at the speed of light, so indeed the black holes were 1.3 billion light years away. The second part is wrong. For starters, we cannot 'point' our gravitational wave detector in any particular direction: it's a giant device that 'listens' for space-time disturbances coming from anywhere (do see the PhD comics video). More importantly, however, we had no other way to detect this black hole collision: we only know about it because of the LIGO measurement. In fact, even though the first-ever gravitational wave detection is what made the headlines, it is worth noting that this is also the first-ever detection of a black hole binary system, and of two black holes colliding to form one! 

Q: Well that's pretty cool.
A: Yes! A great deal of the excitement comes from the fact that we now have a whole new tool to explore the Universe with. Black holes in particular are very elusive; up until recently, we didn't even know whether they existed. It's easy for us to observe stars, since they emit so much light, but light cannot be emitted from the inside of a black hole. Thus, our observations are always based on indirect methods that typically rely on the extreme space-time bending in the vicinity of black holes. For example, light also gets bent, which results in the so-called gravitational lensing that we can observe and consequently infer the presence of a black hole. Gravitational-wave experiments will now add another dimension to our observations - and hence understanding - of the Universe.

Q: What did the collision look like?
A: To us, it looked like this:

B. P. Abbott et al. (LIGO Scientific Collaboration and the Virgo Collaboration), Phys. Rev. Lett. 116, 061102 (2016)

The top curve shows the signal that was recorded by the LIGO detector. The bottom curve shows our prediction (based on General Relativity) of what the signal from two colliding black holes would look like. The agreement between the two curves confirms the observation (careful statistical analysis was made to substantiate that claim). It is also worth noting that the same signal at the same time was detected by a similar detector located in Louisiana. This is important for us to be sure that it wasn't some fluke due to random noise. 

Q: Isn't that a bit contrived? To me, that doesn't look anything like two black holes colliding...

A: Your brain does similar inferences all the time. Everything that you perceive as 'real' is your brain's interpretation of the signals it receives from your sensors, like your eyes. 'Seeing' something is an electromagnetic signal recorded by your retina not too different from the one in the figure above. It's your brain who is responsible for your subjective feeling of 'really' seeing it. We tend to think that we are really perceiving something when the interpreting process is sub-conscious. What we are doing with gravitational waves is, we are using our brains again, but this time consciously - through Einstein's theory. This is because they just don't have the capacity to record and interpret the gravitational-wave signal on their own. 

Q: Does this change everything we thought we knew about the Universe?
A: On the contrary, it confirms once again the extremely successful theory of General Relativity. Einstein predicted the existence of gravitational waves exactly 100 years ago, and, like a prophet who is actually worth listening to, he was once again correct. Actually, General Relativity works so well when it comes to astronomically-sized objects that I don't think there were many physicists doubting the existence of gravitational waves. For me at least, their existence was obvious, I just wasn't sure if I'd live to see a detection, since they are so elusive. It's pretty cool that I did. And of course it's very important to have the experimental confirmation, as we should never over-trust our theories. All in all, this is one more beautiful testimony to the fact that science just. works. bitches. 

Q: Is this quantum gravity?
A: No, this is classical General Relativity stuff. We actually don't have a complete theory of quantum gravity due to the infamous difficulty of combining gravity with the other forces that are now explained by the Standard Model on the quantum level. A theory of quantum gravity might contain a graviton - a particle that 'carries' the gravitational interaction - but this is for now hypothetical. And since I'm not a big fan of the 'particle' notion, I'd rather think of the graviton in relation to the gravitational waves: the graviton is the quantum of a gravitational wave, i.e. the smallest indivisible amount of energy that can be radiated by such a wave. There is a perfect analogy between this and a beam of light which is composed of photons - indivisible packets of light energy. In the case of a macroscopic number of quanta, or packets, the quantum nature is lost and both light and gravity can be described by the corresponding non-quantum theories (Maxwell's equations and General Relativity, respectively).

Still, it's worth noting that the LIGO experiment also managed to set an upper bound on the mass of a graviton, provided it exists.

Q: Thanks!
A: No prob. 

Let me teal you about the cyansce of color

If you were among the asocial few who actually listened to their Physics teacher, you'll probably know most of what follows. But maybe you were cool instead, and even if not, surely some of the details go beyond what you know. In any case, this is one more post (after this one) on the way of explaining what I did during my PhD. Enjoy.

What gives objects color?

Color, of course, has everything to do with light. Now, light is actually electromagnetic radiation, and as such propagates in waves, and every wave has an associated wavelength. Classically, this is the distance between two consecutive crests. Quantum mechanically, the wavelength also determines the smallest amount of energy that a beam of light can carry - namely, the energy of one light 'quantum', commonly called a photon. The higher the wavelength, the lower the photon energy. I’ll get back to this later on. Importantly, individual wavelengths, or photon energies, are also associated to various colors. Or, rather, the other way round - to every color, a particular wavelength can be associated, but not every wavelength has an associated color. This is simply due to the limitation of our perception - we cannot register all the wavelengths, and the concept of color is only associated to the ones we do. You should be familiar with some version of the following image, which illustrates the full range of possible wavelengths, and what we call the corresponding radiation:

"Electromagnetic-Spectrum" by Victor Blacus. Licensed under CC BY-SA 3.0 via Commons - https://commons.wikimedia.org/wiki/File:Electromagnetic-Spectrum.svg#/media/File:Electromagnetic-Spectrum.svg

As you can see, there is only a small range of wavelengths that we can see, namely the one marked as 'visible light'. Now, a beam of light of a fixed wavelength is called monochromatic light, and indeed it has a fixed color (if it is within the visible range). What is important to understand, however, is that light around us is virtually never of a single wavelength - instead, it has contributions of varying intensity at all possible wavelengths. The intensity of the light as a function of its wavelength is what is called the spectrum. Color is almost the same thing, but it requires in addition a brain to recognize the spectrum and give it the color-label. For color is certainly in the eye of the beholder, and the way two identical spectra are recognized differs among different animals, and even among different people. Here, I won't be discussing the human eye; instead, let's just assume a generic observer that registers equally well light of wavelengths from 400nm to 700nm, see image above (nm stands for nanometer, which is one billionth of a meter). Anything outside of this spectral range is, in our discussion here, irrelevant. In other words, the following two light sources appear to be of the same color to a human, since they have the exact same intensity vs. wavelength dependence in the spectral range that we register.

In short, the color of a beam of light is determined by the contributions of the various colors - or wavelengths - that make it up. In the example above, the intensity is the strongest in the blue region, and so we would probably register both of the spectra as blue-ish. However, in general, all the contributions mix up - proportionally to their intensity - to determine the final color. Much like in an artist’s palette, or in primary school art lessons.

Now, the range of wavelengths that our eyes detect is not at all arbitrary. This is what the spectrum associated to sunlight reaching the Earth looks like:

"Solar spectrum en" by Nick84. Licensed under CC BY-SA 3.0 via Commons - https://commons.wikimedia.org/wiki/File:Solar_spectrum_en.svg#/media/File:Solar_spectrum_en.svg

Our eyes have naturally evolved to detect light in the range in which there is the most of it! If it’s not clear why, I’d have to explain natural selection to you, and that’s well beyond this post. But sunlight brings me to the next point: so far we only discussed color in relation to the spectrum of a beam of light. With this in mind, the color of objects is naturally determined by the light coming from them. However, since most objects around us don’t emit light on their own - at least not in the visible range - color is implicitly also determined by the spectrum of the light illuminating them. In other words, if we were to illuminate any object with monochromatic blue light, it would appear either blue or black, or something in between - because no other hue can mix up in the spectrum. Thus, in a sense, we should call the 'true' color of an object the one we see after illumination with light of equal intensity at all wavelengths, otherwise known as white light. Sunlight is to a good approximation white light, see image above.

Finally, I come to the main point. What determines the color of an object once we fix the observer (an average human) and we fix the illumination (white light)? Both of these factors are important, but they are extrinsic to the object itself. There must also be some intrinsic property that gives an object its hue. Indeed, its physical and/or chemical structure determine its optical features, and more specifically - the absorption, reflection, and transmission properties. These are related to a wide variety of phenomena and can be incredibly complex to study and predict, but here we take them for granted for illustrative purposes. The important point is that there are several possible outcomes for light hitting an object, and the interplay between those determines its color. Namely, light can be transmitted, reflected, scattered, or absorbed, and normally all of those happen simultaneously - to a various extent.

It is in principle possible that the object itself emits some light, which adds to or even dominates its color. However, as mentioned earlier, the color of the vast majority of everyday stuff is determined by the four properties illustrated above, and not by emission. Actually, all objects around us, including ourselves, do emit electromagnetic radiation, but the intensity is peaked in the infra-red, making it invisible to us. The peak of the emission spectrum is determined by the temperature of the object. This is why incandescent light-bulbs, for example, in which the wire becomes much hotter than room temperature, do emit light at wavelengths that we see. But, again, we neglect emission in our discussion, and focus on illumination. Here are several basic examples of colors and how they arise from the interplay of transmission, reflection, scattering, and absorption. Black bodies absorb most of the incident light:

And white things scatter most of the light back.

Here, a distinction should be made between scattering and reflection. When the reflection dominates, the object is still technically white. However, we would be more prone to calling this object a mirror, because we'd be more used to seeing the exact same color of light that hits it. Still, if you think about looking at a mirror from far away when it's in the sunlight, it looks like a bright white object. The difference is that the reflected light is directional, and thus far stronger in the reflection direction than the scattered light, which goes everywhere. Have a look at this Vsauce video for further insight into the color of mirrors.

Finally, when there is mostly transmission, the object appears translucent because we see the light coming from behind rather than the illuminating light. 

To make matters more complex still, color depends on the interplay between these four phenomena not just in general, but at every wavelength within the visible range. Thus, for example, an object could be mostly absorbing at all wavelengths apart from blue light, which gets mostly scattered. Such an object would appear blue.

And, as for white objects above, if the blue light were mostly reflected rather than scattered, the color would be brighter, glossier blue, with some reflections of nearby objects appearing (in blue). Think car paint.

These qualitative consideration give a lot of intuition about color. So, while we are at it, why not answer the age-old mystery: why is the sky blue? In fact, it's not really - or rather, it can take on various colors, blue being just one of them. The first important note is that the sky is actually mostly translucent (assuming no clouds). When you look at the Sun, you see it (right?), and when you look back at Earth from space, you see it. Furthermore, the Sun looks white (technically kinda yellow-ish because the spectrum is not exactly white light), and around the Sun the sky looks white, too. It doesn't look blue, because we're mostly seeing the light coming directly from our star.

You can have a quick look to make sure that's the case, but, needless to say (I hope), please be careful with burning your retinas out.  

Apart from that, the sky is indeed blue when we look away from the Sun, at the light that is scattered by the constituent particles of our atmosphere. Now, it turns out that light of a given wavelength gets scattered stronger the lower the wavelength. This means that light in the blue-violet part of the spectrum gets scattered the most (in the visible range). And that's why we see a blue color when we look away from the Sun.

Of course, that's not the full story. We also all know that the sky appears red at sunset and sunrise. The explanation is pretty much the same, only now sunlight travels a long distance through the atmosphere. Since red is on the long-wavelength part of the spectrum, the red component of the light is the last to scatter out before reaching us. Thus, now, when looking directly at the sun, it appears red-ish, as does the region around it.

On the other hand, if we look away from our star, the sky is still blue-ish, or white-ish, or just getting dark, because most of the light gets scattered away before reaching us.

All this constitutes the well understood, and maybe boring, part of the story. When physiology and psychology kick in, it gets much, much messier, as color perception gets very subjective. Note for example that language shapes the way we perceive colors. Blue is a particularly weird color, and believe it or not there's evidence that you might not know that the sky was blue had you not heard that all your life. But, at least when it comes to Physics, you now know why it should be somewhat blue-ish. Apart from the cases when it's white-ish or red-ish... 

Wibbly-wobbly timey-wimey... stuff

Disclaimer: this post quickly becomes quite technical, but it could give some extra STR intuition to those brave enough to stick with it.

I'll try to explain relativity geometrically. This is actually a very common approach, because space and time themselves are best understood geometrically. As discussed in the 'introduction', these notions are intricately related to our experience: our perceptions tell us that things can be in different places and can happen at different moments, the collection of which we call space-time. Very often, the need arises to quantify the places and moments in order for us to, you know, function as humans and as a society.  In our day-to-day lives, we go about this in an informal way, choosing just the right amount of vagueness that is needed for  particular situation. 

The formal (i.e. boring?), scientific way to describe positions is to use a system of coordinates, and the most commonly used one is Cartesian coordinates, introduced, just like the evil demon, by René Descartes. In two dimensions, these look in this probably familiar way:

What I mean by two dimensions is that the space described by the example above (e.g. the plane of your monitor) is such that any location can be precisely pinpointed by two numbers. These could be the x- and y-positions in the Cartesian frame above, but the concept of dimensionality is much more general. In fact, we are intuitively used to a very complex two-dimensional space, since we are tied by gravity to the surface of the Earth [1]. This surface has an extremely complicated shape, which the x-y plane above cannot capture well, but it is still two-dimensional, in the sense that any location can be uniquely defined by two numbers, as anybody who's used GPS coordinates knows. So 3D beings as we are, we do anyway spend our lives tied down to what is, to a good approximation, a 2D world. 

The Cartesian coordinate system seems very natural. Right angles make sense. Well, I don’t know if it is a priori aesthetic - in fact I think that a priori aesthetic is an oxymoron - but we humans definitely seem to like it a lot. We even like to organize our cities in this way. There's not much difference between saying '42nd and 5th' and saying 'at x = 3.2, y = 2', or something.

Now, to illustrate relativity, we're going to decrease the spatial dimensions to one. That means that we’re only going to keep the x-axis. It also means that anything living in this toy 1D world can only be in front or behind anything else. The reason for this simplification is because we want to use the other axis to represent time:

This is now a graphical representation of space-time, and already hints at the fact that those are kinda like different aspects of the same thing. Note that we always had some intuition that time is very similar to space, or in any case certainly before Einstein or Descartes. This can be seen in our language. Interestingly, there are different ways in which different cultures view time.  What is practically ubiquitous among cultures, however, is the fact that the words we use to speak about time are very similar to the ones we use for space: there is a conceptual metaphor of time as a path through space. The examples are numerous. Plenty of time ahead of us. At 8am. In French one even says, 'le moment où quelque chose s'est passé', which literally means 'the moment where something happened.' 

An important remark is due. The space-time reference frame shown above is neither unique nor absolute. It is in fact observer-dependent, so you should think of everyone as having one of those attached to them.

The Universe itself consists of events which are absolute in the sense I discussed here: all observers agree on them happening, but, to do that, every observer relies on their own reference frame. This is what this looks like.

This way of talking about events by naming both their spatial and temporal position is perhaps a bit awkward, but you’d better get used to it if you want to understand relativity. Space-time should always be thought of as indivisible.

The example above, where two observers at a different place compare their observations of an event is fairly straightforward to understand. It's also possible to compare reference frames at different times. This is slightly less intuitive, but still nothing too complicated.

In both cases, to see whether the observers agree, they simply have to properly add or subtract the distance between them (both in space and in time) from their corresponding observations. This transformation of the reference frames is mathematically known as a translation (technically, not related to language, but somehow this connotation is also appropriate here). All this was well understood already in pre-relativity physics. The main innovation of STR is what happens to the reference frames of observers that are moving with respect to one another. This is what we used to think the transformation looked like:

The two reference frames above correspond to two observers moving with respect to one another, which at t = 0, x = 0 (in both frames) find themselves at the same time and place. This is a graphical representation of what is known as a Galilean transformation. The fact that the time axis tilts while the space axis doesn't is deeply connected to our (wrong) understanding of time as something that is absolute for everybody. Consider this. If we have two events which happen at the same time with respect to observer one, then they also happen at the same time for observer two. 

This shared simultaneity of events means that we can come up with a common time-counting scheme for the two observers, such that t = t' for all events. This would represent the absolute-ness of time. However, if two events happen at the same place for observer one, they don’t happen at the same place for the second guy: space is relative. 

To see what the situation looks like in relativistic physics, let’s first take care of a small technicality. In measuring and plotting x and t, we have to choose some units. We actually have some freedom in that, and the units we choose affect the scale of the axes. Let’s say we take the standard choice of measuring time in seconds. A standard choice for measuring positions is the metre, unless you have the misfortune to come from one of the three countries in the world where it isn't. For the purposes of relativity, however, it is particularly convenient to choose a different unit: the light-second. This is analogous to the light-year (which we would use if we chose to measure time in years instead of seconds) in that it’s defined as the distance that light travels for one second. With this choice of units, the reference frames for observers moving with respect to one another turn out to look like this:

We chose to use the 'proper' measurement units so that everything is nicely symmetrical. Physicists like stuff being symmetrical. The 45-degree line bisecting the reference frames marks the propagation of a light beam, which should be thought of as a series of absolute events. Note that this line bisects both reference frames. As opposed to the Galilean transformation above, in this relativistic (and correct) transformation, the x- and the t-axes of the moving reference frame tilt towards the light-line, at the same angle. Faster speeds mean a larger tilt, and in the limit of the speed of Observer 2 going to the speed of light, the two axes merge:

With this in mind, we can see where all the 'weird' stuff in relativity comes from. For example, simultaneity can no longer be defined in an absolute (i.e. observer-independent) way. Events that happen at the same time for one observer can happen at different times for another one:

Effects like length contraction and time dilation are very strongly related to the fact that simultaneity (and thus time) is relative. Those can also be inferred by looking at the graphs above and thinking a bit about how lengths and time intervals translate from one observer to a second, moving one. This is left as an exercise to the reader. *EVIL LAUGH*

The title of this post comes from a famous quote by the Doctor, namely, 'People assume that time is a strict progression of cause to effect, but actually from a non-linear, non-subjective viewpoint - it's more like a big ball of wibbly-wobbly timey-wimey... stuff.' One thing that Einsten always insisted upon, however, was kinda the opposite: the absolute nature of causality. In other words, he did subscribe to the assumption that the Doctor claims is wrong: the idea that cause always precedes effect. This is what is known as causality, and is usually held as a fundamental principle of nature. An illustration from everyday language is when we say that you cannot eat your cake and have it whole; this is because the cause - eating the cake - results in the effect - you no longer having it whole. Einstein made it clear that although some funky things happen according to STR, nothing as funky as eating a cake and having it whole can ever happen, in any reference frame. Pictorially, this looks like this.

It’s clear that regardless of the exact velocity of the second observer, the world-line of the cake always precedes the event of your eating it. Or does it? Remember how the faster the observer moves, the closer their x and t axes come to the light-line, until they practically merge? Well, if we let an observer move faster than the speed of light, and neglect a small imaginary detail (we can, cause it’s imaginary, right?), what happens is what you should expect to happen - the angle of the tilt of the axes continues to increase, and they switch places! The space-axis comes above the light line, while the time-axis pops up below. The crazy thing is that if you now map the cake, in this new reference frame the event of eating it precedes the series of events of having it whole:

What is perceived as cause becomes an effect, and vice versa: according to this super-luminal observer, people eat their cakes in order to make them! And then I guess they enjoy them in a moment of un-cooking that succeeds both eating the cake and having it whole. This sounds like nonsense, and is one of the arguments against the possibility of anything moving faster than the speed of light. But this is a philosophical argument: should we dismiss something just because it sounds like nonsense? There is another strong argument about why we ourselves could never achieve such speeds: the acceleration would require infinite energy, and then some more. But there is no conclusive argument against the existence of matter that is already moving faster than the speed of light. In a similar fashion to us, such matter would not be allowed to decelerate to below that speed, as that would require more than infinite energy. Still, such matter could in principle exist, and, a few decades ago, the theoretical study of tachyons - i.e. particles moving faster than the speed of light - was very hot.

The enthusiasm has by now somewhat died out, though. Even though we cannot prove them impossible, most physicists don't consider the existence of tachyons very likely. More precisely, the possibility to observe these particles even if they do exist is considered unlikely, which is practically an equivalent statement. That's some food for thought for you: in terms of physics, is there any difference between something not existing, and something existing but not interacting with us?

Einstein would have agreed that tachyons cannot be a part of our world. Basically, interacting with such particles would immediately break causality and result in a ton of paradoxes of the kill-your-grandfather type (and the even more disturbing Futurama version). Again, this is technically not a proof of the impossibility of backwards-in-time travel. But, while strictly speaking nothing can be proven impossible, some things just look mighty improbable. In any case, Einsten considered the cause-and-effect realtionship a fundamental principle of nature, and its breaking - impossible. In fact he was much more concerned with that than with determinism, despite his famous 'God does not play dice' quote. And I must say that once again I find it easy to agree with the great man. I would say that it is, in fact, safe to assume that time is a strict progression of cause to effect, at least until we have seen even the tiniest reason to think otherwise. Which, right now, we haven't.

Bottom line, while Einstein's theory does illustrate that time is wibbly-wobbly (i.e. not absolute), I don't think he would've been much of a fan of the Doctor's dismissal of causality. 

Some stuff Einstein actually said

These blog posts of mine always turn out much longer than I initially expect. What follows was supposed to be just an introduction, but it became so long that, knowing your attention span (or rather, judging about it based on my attention span when it comes to blog posts), I thought it best to just split it off and give it a separate entry kind of status.

I’d like to talk a bit more about space and time (what are those things anyway?), and about absolutes in the theory of relativity. Yes, not everything is relative.

By the way, Einstein himself has written something like a book (it’s more like a collection of lecture notes) about relativity. It’s called The Meaning of Relativity. Yes. By Einstein himself. If you’re up to the challenge, you can try to understand relativity directly through Einstein’s words. But I wouldn’t recommend that to people who haven’t had at least a basic course on the subject - otherwise the book will be quite anticlimactic.

Actually, I think the book is a bit anticlimactic for anyone, since Einstein is such a deity in science that you expect every word of his to bring you noticeably closer to the true meaning of the Universe.

In fact the book is a normal, at times even boring, exposition of relativity, with a good number of abstruse (also due to their slightly archaic notation) equations. However, he does share some non-mathematical insights, and some of those I just cannot put better than Einstein. Let’s start with time. He writes,

The experiences of an individual appear to us arranged in a series of events; in this series the single events which we remember appear to be ordered according to the criterion of earlier and later, which cannot be analysed further.

While he does not specifically talk about space (the book is really concise), it is straightforward to extend the statement in that direction: our experiences appear to us positionally arranged, so that we can for example say that an object is closer to us than another object. Thus, our concepts for both space and time stem from our experience. Obviously, these are then always associated to a particular individual - the one doing the experiencing - and there is a priori no need for any overlap between various individuals. However, as Einstein writes,

By the aid of speech different individuals can, to a certain extent, compare their experiences. In this way it is shown that certain sense perceptions of different individuals correspond to each other, while for other sense perceptions no such correspondence can be established. We are accustomed to regard as real those sense perceptions which are common to different individuals, and which therefore are, in a measure, impersonal. The natural sciences, and in particular, the most fundamental of them, physics, deal with such sense perceptions.

This is a truly outstanding definition of the object of physics. There seems to be some order in the world in the sense that we all see some things happening in a certain way. Why is that? We don’t really have to, if you think about it, but the fact that we do implies some underlying laws that unite our experiences - and that's what physicists try to analyze. 

Of course, Einstein allows for experiences which are not shared by various observers, but these are not physical. For example, you “totally going all the way with Judy last night” won't be of interest to physicists, especially if it was an experience not even shared by Judy. Now, an obvious question springs up. Does this definition of the physical world as 'shared experiences', well, suck, because it's too anthropocentric? And on a related note, is there space and time beyond the human observation of those? Is there a physical world beyond our perception of it? And how could we ever separate one from the other? 

What is important to realize is that this question is a philosophical and not a physical one. And the philosophers have given it a lot of thought - we have Descartes' cogito ergo sum, which to a first approximation suggests that we can be certain of our existence (through the very act of questioning it), but of nothing beyond that. Descartes also has an evil demon that helps him extend that idea. The modern version of the demon is the brain-in-a-vat, while the ultra-modern Wachowski-siblings version is the backbone to the plot of one of my favorite movies. Which makes me wonder if The Matrix would've been even cooler if it were called The Evil Demon. 

Incidentally, I think therefore I am is a little bit like the E = mc²  of philosophy. Many people know it, but not so many have any idea what the point is. For those, The Matrix is a great starting point to dive in the implications of the cogito ergo sum. On the other hand, it’s not a good starting point to learn any physical laws, as they are consistently broken both in and out of the Matrix... (Still a great movie!). Anyway, coming back to physics, I like Einstein’s stance:

The only justification for our concepts and system of concepts is that they serve to represent the complex of our experiences; beyond this they have no legitimacy. I am convinced that the philosophers have had a harmful effect upon the progress of scientific thinking in removing certain fundamental concepts from the domain of empiricism, where they are under our control, to the intangible heights of the a priori. For even if it should appear that the universe of ideas cannot be deduced from experience by logical means, but is, in a sense, a creation of the human mind, without which no science is possible, nevertheless this universe of ideas is just as little independent of the nature of our experiences as clothes are of the form of the human body. This is particularly true of our concepts of time and space, which physicists have been obliged by the facts to bring down from the Olympus of the a priori in order to adjust them and put them in a serviceable condition.

As usual, I find it really easy to agree with the great mind.

The main point of this post can then be stated as a summary of everything above. The world in the framework of the theory of relativity is composed of absolute events which are embedded in an observer-dependent space-time. The absolute events are the 'shared experiences' that Einstein talks about. Those are observer-independent in the sense that everyone agrees that they 'happened'. However, when describing the events, everyone has to inevitably refer to their own space-time reference frame, which is by definition observer-dependent. So, in order for different individuals to compare their experiences by the aid of speech, and find that certain sense perceptions of different individuals correspond to each other, a 'common language' is necessary. This provides the means of a translation from one individual's very own, personal space-time to the next. And as I've mentioned before, the theory of relativity is simply a recipe for how to do the translation.

Notice then that the big conceptual leap in the theory is not that everything is relative - in fact it's important that there are absolutes, the shared experiences. Also note that we knew that some things were relative even before Einstein. In particular, we are quite comfortable with the notion that space is relative: if something is two meters away from me, I wouldn’t generally expect it to be two meters away from you - that would only be true in some very particular cases. The main innovation of relativity was actually the fact that time is also relative. We used to think (and still often do) that if something is two minutes away from my 'now', then it's also two minutes away from your 'now'. However, just as with space, this turns out to be true only in some particular cases, which just happen to fully encompass our everyday experiences, which is why it took us so long to figure that out. Better said, it’s not possible to define our 'nows' in a way that they match under all circumstances (in particular when we move fast with respect to one another). Better still, space and time are inseparable and kinda mixed up and only kinda absolute if thought of as a single unit: space-time. However, they are always relative when separated. 

OK, this is getting a bit too technical. But I’ll try to illustrate all these points even further in the next post.

'Energy' is a great name for a band*

^*Actually not really. It sounds cool but it would be impossible to look up on google, and  anyway I would guess it's already being used. But I don't know cause it's hard to look up on google. 

Where was I? Ah, yes: there's a fascinating relationship between the microchips that allow you to read these words and the colorful patterns on the wings of a butterfly.

The original idea of this post was to explain what I'm working on, at least qualitatively. I thought a good introduction to that could be the wings of a butterfly, but then I realized an even better starting point would be the unlikely relationship that they share with microprocessors, so I thought I'd start off there... and then I finished the post and never got to talking about what I'm working on, and neither did I get to the butterfly. But I will get to both, soon.

Anyhow, hopefully I got you interested. Good. Now I'll go into the boring stuff. Semiconductors seem to be boring. Honestly, even I was bored when I first had to learn the details. Think about it - do you even have an idea what a semiconductor is? My observations are that the majority of scientifically interested people (not physicists, though), who have an idea about quantum mechanics and relativity and superconductivity and dark matter and the Higgs boson, have a hard time defining what a semiconductor is, and, consequently, why they are the backbone of practically any gadget around us.  Occasionally someone would know that a semiconductor is a material with conductivity between that of a conductor and that of an insulator, but a fairly intelligent person can practically read that off their name. Essentially, they're not quantum-black-hole-God-particle-wormhole material, they’re boring stuff... which has meanwhile had more impact on our lives than literally anything else in literally the whole history of humanity. So I'll dare try to lay out the basics here - please bear with me if you start yawning.

Let's start all the way back: throwing a ball. Or a potato. When you throw one (or the other), there is a fixed relationship between its energy and its momentum. The harder you throw it, both of those increase, but the energy increases with the velocity v squared (E = mv2/2), while the momentum scales just with v (p = mv). So, if we were to draw a graph of energy versus momentum, it would look something like that. 

This holds true for any 'classical' (= large enough) moving object, but it turns out that even with quantum mechanics taken into account, the energy-momentum graph for massive elementary particles still looks like the one above! If you're ever asked to name one similarity between an electron and a potato, you're welcome. No, really, if you do get that question and say that energy is proportional to the momentum squared, you'll make an impression.

By the way in line with my previous posts, do notice that this does not pre-suppose any particle nature of the electron - energy and momentum can be defined independently of that. However, in both cases the graph above only holds true when we consider free propagation. If there is interaction with the surrounding world, the graph could change significantly. In most situations of practical interest, this is straightforward to take into account for potatoes and other objects from the classical lot, but elementary particles become much more tricky when interactions are turned on. To make matters worse, electrons love to interact with their environment, because they are charged particles, and there are usually other charged particles around them (including, but not limited to, other electrons). This makes characterizing the way electrons move in a given material an extremely difficult task. On the other hand, it is arguably the most important characterization we could do, since a large number of a material’s physical, chemical, optical, and electrical properties can at least in principle be extracted on this basis. And so, this study, usually referred to as solid-state physics (the term is a bit broader than just the motion of electrons, but that's the gist of it), has produced an astonishing number of PhD theses.

Now, in all crystals, which is to say materials with some structure in them, you have a lattice of ions through which the electrons propagate. This of course modifies the Energy-momentum graph, and it starts looking somewhat like this.

The jump discontinuities occur at exactly the same intervals, and so the graph above is usually drawn in the "folded" version, like this

It is easy (well, for a physicist, at least) to know where to 'fold' the graph - the fact that the crystal lattice is periodic ensures the periodicity of the 'special points' on the momentum axis, and you fold at the first one of those. This often has some deep physical significance, but anyway, you could as well just think of it as a more compact representation of the full plot.

Hope you're still with me here – we're getting to the point where you'll understand the title of the post. The energy intervals within which an energy-momentum graph exists are called 'energy bands', cause, well, they kinda do look like bands when you highlight them:

How much of the energy spectrum is covered by these bands, and in what way, determines a lot of the properties of the material. In fact, perhaps even more importantly – although technically carrying the same information – are the regions outside the energy bands, which are called band gaps:

The significance of those is a fairly bizarre one: electrons of that energy simply cannot exist in the material! That's now quite different from throwing a potato – you can throw it with any speed you like, and it will propagate with the corresponding kinetic energy – there are no forbidden energies. That's obviously not always the case for an electron in a crystal. Now, remember that electrica current is nothing else but motion of electrons. Because of this, the energy bands - and the presence of band gaps - fully determine whether a material is a conductor, a semiconductor, or an isolator. To see this, we need to add the last piece to the story, which is the fact that in every material there are a number of electrons that could move around, if 'pushed' (which in physical terms is done by applying voltage). If they're not pushed, they occupy all the lowest energy 'states' – this is because nature in general is quite lazy, and does everything with the lowest possible amount of energy, unless it has a very good reason to do otherwise. Now, every material can be characterized by a certain number called the Fermi energy, whose significance is that there are electrons with all energies below that level (and none above, at least not at 0 temperature). This together with the 'band structure' of the material determine its electrical properties in the following way. 

In brief, if the Fermi level crosses a band, then arbitrarily small voltage makes the electrons move, i.e. creates current, which the material 'conducts'. If the Fermi level is in a band gap, the applied voltage needs to be large enough so that the difference in energy between the Fermi level and the closest higher-energy band can be overcome. Thus, technically, there is no difference between a semiconductor and an isolator: it is in principle possible for both of them to conduct electricity, but only if enough voltage is applied. What makes a material an insulator is then only the practical consideration that an impossibly high amount of voltage is needed to make its electrons move in a certain direction, while for a semiconductor that value is achievable. 

Now, another important aspect of semiconductors which ultimately renders them useful for CPUs is that they can be doped

The red dots in the image above represent 'impurities' which are artificially introduced to make the semiconductor either more or less conductive (we can do both). I'll skip the rest of the details: the important point is that by making contacts ('junctions') between differently doped semiconductor pieces, you have a lot of non-trivial control on the way current flows through a device. Two examples are diodes (the D in LED) and transistors. The latter, very schematically, work in the following way

Essentially, the transistor is a 'switch': if there is no current at the 'gate', the switch is off, and no current can pass from the source to the drain. The switch can be turned on by current flowing at the gate. And that's all you need to make a logic circuit! No current is your 0, yes current is your 1. You connect the drain of one transistor to the gate of another, and your 0 or 1 influences the next 0 or 1. Depending on how you make the connections, there's no end to the possibilities of the allowed operations that you can hard-wire in a chip. Of course you only go for some basic ones, and then leave it to a programer to play around in arranging those in ever-more-complicated patterns, so that you can all enjoy your cat pictures and your instant connectivity with anyone anywhere around the world and your daily dose of videos of naked people doing dirty stuff and reading this post and God knows what else you're into that's only made possible because of what I explained here.

Hope it wasn't so boring after all.