It's been a year since I've given a public outreach physics and astronomy talk in Second Life. I used to do these things fairly regularly as a part of MICA (the Meta-Institute of Computational Astronomy). However, the MICA project has completed, its island in Second Life has gone online, its Second Life groups have been disbanded, and MICA no longer really exists. (Its website is still up, and should stay up for at least a little while. If I were smart, I'd probably make sure to download and archive elsewhere all of the audio recordings of my own talks....) A write-up of what MICA did and was all about is available at arxiv.org/1301.6808, and was published in the conference proceedings of a SLActions conference on virtual worlds
I've always meant to find other venues for continuing to do popular talks in virtual worlds. Someday, I'd like to escape from Second Life's walled garden and start doing these talks in an OpenSim grid, and even did the first steps for trying to get set up to do them in my own region on OSGrid. However, of course, the audience in Second Life for now is still far bigger.
Fortunately, the Exploratorium, the excellent science museum in San Francisco, has a presence in Second Life. This Saturday (tomorrow, 2013 April 6) at 10AM pacific time (17:00 UT) I'll be giving a talk about the Higgs boson in the Exploratorium region in Second Life. Remember, basic Second Life accounts are free. Drop by if you're interested.
You may have seen announcements that Dark Matter has been "found". I don't believe there's a publicly available scientific paper on this yet, so the original source for this is two press releases from CERN: One from four days ago and one from today.
First, I want to say what is meant by dark matter being "found" here. It's not evidence that previously-uncertain Dark Matter exists. We already know that Dark Matter exists; the Bullet Cluster observations several years ago was unambiguous confirmation that non-baryonic dark matter exists. We don't know what it is from that, but we know it exists. (Here is a podcast I did three years ago about the evidence for the existence of Dark Matter, including the Bullet Cluster.) So what does it mean to say that these new CERN results may have "found" Dark Matter?
Although we know Dark Matter exists, there remain a huge number of mysteries about it. Many of these can be summarized under: what is it? All we really know is that it's not made out of baryons, that is, protons and neutrons. So, it can't be an excess of dim stars or rogue planets (a model that was once considered a real possibility for our Galaxy's dark matter). Thus far, we've observed it because of the effects of its gravity. We've seen it in comparisons of the structure in the Universe to models of structure growth from early-Universe conditions; in the dynamics of galaxy clusters and galaxies; and through gravitational lensing. It would be nice to observe it in other ways.
To "find" Dark Matter, we'd like to do one (or more) of two or three things. Either, we'd like to see the results of decay products in our atmosphere or in space because of interactions of Dark Matter particles out in space. Or, we'd like to have an actual Dark Matter particle interact with a particle detector we have on Earth (analogous to how we see neutrinos from the Sun). Or, we'd like to actually make some of the stuff in a collider like the LHC at CERN in Switzerland, and see its decay products or signature there.
The current announcement from CERN is potentially of the first type. There is a detector, the "Alpha Magnetic Spectrometer" or AMS, on the International Space Station. This spectrometer is measuring electrons and positrons (the antiparticles of electrons) coming from space— that is, cosmic rays that are electrons and positrons. They see too many positrons for what we'd expect. One possible reason for the excess of positrons is that they are the result of very rare Dark Matter annihilations in our Galactic halo. (Although such annihilations, if they are happening, would be rare, there is so much bloody Dark Matter out there that if it's doing this, it would produce enough excess positrons for us to observe.)
What's really been detected is a positron excess, which is interesting all by itself. Whatever it turns out that this positron excess is coming from, it's going to be at least new astronomy, and potentially also new physics. It may not be as sexy and headline-worthy as "WE FOUND TEH DARK MATTER!!!1!!one!", but it will still be interesting, and will tell us something about nature. What's been seen is consistent with it coming from the Dark Matter halo of our galaxy, but other sources can't be ruled out yet. As more data is collected, the investigators running this experiment will be able to test whether the details of what is seen remain consistent with what would be expected from Dark Matter, versus other possible sources.
Holy cow, it's been a long time since I blogged.
The class I'm teaching right now is 3d Computer Modelling and Animation. Perhaps the hardest thing about it is figuring out if the word Modelling has one or two l's in it... it depends on whether you're in the USA or Canada, I think.
For this class, I'm making all of the students do a major project. Some of them are doing some pretty interesting things, and already several of them have figured out things about Blender (the 3D software we're using, a quite powerful free package that you should check out yourself) that I don't know myself. A couple are playing around with motion tracking, in order to add 3D rendered elements into a live action video scene. One is building a game using the Blender game engine. Others are doing various other animations.
I've decided to take on a project myself. For this project, I am going to model a white dwarf in a mutual orbit with a main sequence or red giant star, pulling matter off of it into an accretion disk. During the animation, the white dwarf will go critical, and explode in a supernova, blowing itself way, and blowing off some of the outer layers of the companion star.
So far, I've managed to create the basic progenitor model, and do a little bit of animation of the textures so that the disk is spinning, the star's surface is roiling, and the gas bridge between the star and the disk looks a little like it's streaming. Here's a rendered frame from what I've done so far:
I'll certainly post the full animation once I've completed it. Next, I'm going to have to start worrying about how to deal with the supernova. Eventually, I'll set the whole thing to music.
This being election day, you're seeing a lot of people telling you to get out and vote, saying that if you think your vote doesn't count you're abdicating democracy, reminding you how many people who died so that you could vote, etc.
I have to admit I find these exhortations both facile and manipulative. If we're talking about the presidential election, sure, you can make an argument that "every vote counts". That argument is not really practical, however, because of the electoral college. Every vote for president counts only in a small number of battleground states. I used to live in California; everybody knows that California is going to go to Obama. The Obama and Romney campaigns certainly know it; how much time and effort did they spend trying to sway California voters? Now, strictly speaking, if everybody who would vote for Obama figured it didn't matter and as a result didn't vote, then, yes, Romney could pull out a surprise win. But, while that's a theoretical possibility, let's be realistic here about how likely that is. It's not going to happen. As a result, if you tell somebody in California that their vote for the president really matters, you come across looking either naive, or manipulative.
So should you vote anyway? Yes. Two reasons.
First, to stay in practice. The USA's current system of elections is horribly corrupt. Jimmy Carter has spent a lot of time overseeing elections in other countries, and he says that about our elections. Also, check out rootstrikers.org, a website related to Lawrence Lessig's book Republic Lost. It's easy to become cynical, to realize that everybody running for any office is dancing to the tune of large campaign donors, and to give up and not bother voting. However, you must vote, both because there are differences between candidates, and because you need to stay in practice, and we need to keep voting as "a thing" that we do in the USA, in hopes that we do manage to fix the corrupt system.
The second reason is: there are elections other than president. If you live in California, no, it doesn't matter who you vote for for president; Obama's going to get your electoral college votes, whether you like it or not. However, there are congressional districts whose representatives are not a foregone conclusion. And, in many states, the Senate seats may well not be a foregone conclusion. Congress matters. You need to vote there. Additionally, there are going to be state and local elections that matter. You need to vote there. With all of these other things, there isn't an electoral college making your votes irrelevant; in these other races, every vote does matter.
So, yes, get out and vote. But, please, let's stop pretending that every individual vote for president matters, because that's simply not the reality of the situation.
There's a fair amount of bruhaha in the popular press what with the announcement of the discovery of what is probably the Higgs Boson at CERN last summer. In describing why the Higgs Boson is so important, you will read that the Higgs Boson "gives other particles mass". This is the so-called Higgs Mechanism, and is an esoteric thing arising from the mathematics of Quantum Field Theory (QFT) that's very difficult to understand at a popular level. You can find a number of analogies out there, such as the description of the famous actor walking through the crowd at a Hollywood Party, and acquiring inertia (i.e., mass, i.e., resistance to being accelerated) as a result. (You can find that description here at the Exploratorium's site, along with a brief mention of the W and Z bosons, a description of whose properties is what led to the Standard Model of Particle Physics predicting that the Higgs Boson must exist.)
I have to admit, this analogy has always left me cold. Part of the reason is that my familiarity with QFT (remember, that acronym stands for Quantum Field Theory) is quite shaky, and I didn't really know how the Higgs mechanism worked. This analogy didn't give me, as a physicist, any insight into the actual Higgs mechanism. It's still on my list to become much more conversant with QFT. After all, QFT and GR (General Relativity) are the two most fundamental theories of reality that we've got, and while most physicists don't actually work with them on a daily basis (they're either working with more complicated and practical things like solids or fluids, or, as with this year's Nobel Prize Winners, they're working on fundamental physics in a manner that is adequately, and more easily, described with the non-relativistic version of quantum Mmechanics).
In this post, I'm going to try to describe, at a popular level, the view of reality that QFT gives us, and then, using only one equation (and it's one you've seen before, many times), try to tell you how it is that the existence of the Higgs Boson leads to other particles having mass. here are the sections below:
- Fields in Physics
- The Magnetic Field
- The Photon, and the Electromagnetic Field
- Fields and Particles in General, and an Analogy to a Lake
- Energy in QFT
- Massive Fields
- The Higgs Field and Mass Terms in QFT
- Pithy Summary
Fields in Physics
One of the problems in physics (or nearly any other technical area of study, including, unfortunately, law) is that there is a lot more new vocabulary than there are new words. That is, there are words with everyday definitions that are different from the technical definition in physics. One of the most common ones you hear is "theory", which is used in everyday parlance to mean "speculation", but is something very different in science. In physics, we talk about "fields", and it often sounds very scary and esoteric. However, the basic definition of a field is actually fairly straightforward:
field, n.: something that has a value everywhere in space
That's it. I'll give you a few examples of fields below, including everyday things you can understand, physics things you've heard of before, and esoteric physics things that might surprise you. Here's a physics example: a temperature field. Look around the room you're in. Imagine putting spatial coordinates on that room. That is, in one corner, draw three axes. Label the two on the ground the x and y axes, and label the one sticking up the z axis. Now, at any point in your room, in principle you could measure the temperature, with a thermometer or some other device. If, say, you've got a hot plate (or, equivalently, a computer) running in your room, just above it the temperature will be higher than it is somewhere else in the room. That is, the temperature has a different value everywhere in the room. You can just talk about the temperature at one point, but if you talk about all of the values of the temperature everywhere in the room, you could call that the "Temperature Field", and even give it mathematical notation, T(x,y,z), and then deal with it conceptually and mathematically as if it were one thing— albeit one thing that's got different values at different places.
The Magnetic Field
One physics field you've almost certainly heard mentioned is the magnetic field. Consider, for example, a bar magnet. There's a magnetic field around it, and you may have seen it visualized similar to the following:
A magnetic field is a little different from a temperature field. A temperature field's value is just a single number, with temperature units. A magnetic field has both a strength and a direction. The arrows in the picture above tell you the direction of the field; the strength is weaker the farther away you get from the magnet.
This magnetic field can interact with other things. You could, for example, hold the magnet near a nail on a table, and pull the magnet; if you're careful, the nail will be dragged along behind the magnet. (If you're not careful, the nail will jump and stick to the magnet, or you'll get the magnet too far away and the field won't be strong enough to overcome friction from the table.) There are two jargon terms I want to introduce here: interaction and coupling. The magnetic field is clearly interacting with the nail. How strong the pull of the field on the nail is depends on how strong the iron in the nail couples to the magnetic field.
Indeed, there's another coupling going on here that I want to draw your attention to. There's also iron (or some other sort of atom) inside the bar magnet, which is itself coupling to the magnetic field. Wait! Did you notice what I just did there? The word "the" in "coupling to the magnetic field" is very important. Previously, I'd been saying "a" magnetic field, as if the magnetic field from the Earth and the magnetic field from the bar magnet were two different things, so that each one is just a magnetic field. However, really, each point in space has a magnetic field strength, whatever it was that gave rise to it. So, really, there is just one magnetic field, which is everywhere in space. If you go to a very empty part of space, the value of the field might be zero everywhere nearby, but you can still describe the field there, just by saying what the value (even if zero) is everywhere in space. In this way of looking at it, the iron atoms in the bar magnet have properties that couple them to the magnetic field in such a way that the strength of the magnetic field around the bar magnet takes on certain specific values.
The Photon, and the Electromagnetic Field
Two new concepts here. The first new concept is the electromagnetic field. Just as there is a non-zero magnetic field around a bar magnet, there is a non-zero electric field around a charged object, such as an electron, or a Van de Graaff machine that you've charged up.
Way back in the 19th century, physicists figured out that the electric and magnetic fields are in fact two aspects of the same thing, which today we call the electromagnetic field. If you've ever built an electromagnet, say, by wrapping a wire around a nail many times and connecting the wire to a battery, you've been playing with the unification of the electric and the magnetic fields (albeit mediated by the moving charges in the wire). The different voltage on the two terminals of the battery create an electric field, which pushes around the charges in the wire, and the current (which is the moving charges in the wire) gives rise to a magnetic field.
In fact, you don't need charges there at all. If you have a varying electric field, it automatically gives rise to a magnetic field, and vice versa. The classical physics of this is described by Maxwell's Equations. Indeed, Maxwell's Equations show that you can get waves in the electromagnetic field, and that these waves will freely propagate through space. We call such a wave an electromagnetic wave, although you probably more often call it by its more common name, light. Light that you see, the "heat" that you feel radiating off of a burner (which isn't really heat in the physics sense), the ultraviolet radiation that gives you sunburns, the microwaves produced by a cellphone or a microwave oven, and radio waves are all the same physical phenomenon: electromagnetic waves. Another couple of vocabulary words here: these electromagnetic waves represent a disturbance, or an excitation of the electromagnetic field. If there's a light wave passing by, the electromagnetic field doesn't just have a zero (or even a constant non-zero) value, but it's got some wiggly business going on.
Classical physics allows a disturbance of any size in the electromagnetic field. In particular, the energy in an electromagnetic wave can be arbitrary small. Have an electromagnetic wave with a certain amount of energy? Divide all the strengths of the electric and magnetic fields, and the energy in that wave goes down by a factor of four, but everything still solves the equation. This is what happens in classical field theory.
Here's the next new concept. Quantum field theory (QFT), on the other hand, only admits quantized excitations of the field. With the electromagnetic field, a disturbance of a given frequency (which we would see as a given color, if it were a frequency in the range that our eyes can detect) can't have any old energy. Rather, it must come in quantized steps of energy. There's a minimum energy that you can have in a light wave of a given color, and the total energy you have must be an integer multiple of that minimum energy. When you have an excitation of the electromagnetic field that has this minimum energy, we call that excitation a photon.
You've probably heard the photon described as "the particle of light", and this is an accurate description. However, when we say "particle" in quantum field theory, what we really mean is a disturbance of a field. So, a better way to describe a photon is to say that it is a disturbance of the electromagnetic field, or an excitation of the electromagnetic field. The field's natural state, or vacuum state, is no electric or magnetic field anywhere. If there is any light propagating, it must come the form of these quantized excitations of the field, that correspond to some multiple of the energy that one photon represents.
Fields and Particles in General, and an Analogy to a Lake
So, great. We've got the electromagnetic field. It may be a newish concept, although almost certainly you've heard the word "electromagnetic" before; but, at the very least, you've heard about magnetic fields. And, we've got the idea that light is a wave propagating through this field, and that we might describe that as a "disturbance" or "excitation" of the field. And, we have the concept of the photon, as the minimal allowed disturbance (at a given frequency) of the electromagnetic field. We also will sometimes refer to this minimal disturbance as a particle, and thus call the photon the particle of light.
Let me give you another example of a field, that has a disturbance in it: the surface of the water on a lake.
Here, the field is a height field. Rather than having a value everywhere in three-dimensional space, it have a value everywhere in two-dimensional space. That is, you could draw x and y axes on the surface of the lake to represent a coordinate system to figure out where you are on the lake. For every value of (x,y)— that is, everywhere on the lake— the water level has a height above its average height. I chose this image because the field value is zero most places on the lake. You can see that it's very still water, and the height field of the lake is undisturbed.
However, over on the left of the lake, you can see a localized disturbance. There are ripples propagating through the height field represented by the surface of the lake. In QFT terms, we'd say that this excitation of the field would represent the presence of a fair number of "lakeon particles" in that general area of the lake, just as a disturbed electromagnetic field represents the presence of photons. With a nod to Heisenberg's Uncertainty Principle, you can't figure out exactly where the lakeon particles are. Indeed, you could view the height field of the lake as being some sort of abstract thingy that you could use to figure out a probability density for there being a lakeon present. Where there's some wiggly business going on, as is the case on the left side of the lake, there's a non-zero probability that you'd find a lakeon at that spot were you hypothetically able to pinpoint the lakeon.
Here's the fun thing: in QFT, everything is described by fields. The photon is the particle of electromagnetism, but rather than being what you'd think of as a particle (a little spec), in QFT it's really a disturbance of the electromagnetic field, some wiggly business that propagates around like wave. Well, everything else is the same way. You might be used to thinking of particles like electrons as little specs the same way you might talk about particles of dust. That's not how our most fundamental of theory describes them. Rather, there is an electron field, a field that is a weird and abstract thing that's much harder to visualize than the height field of a lake or even the electromagnetic field. However, it is in fact a field in the physics sense of the word, in that it has a value everywhere in space. If you disturb this field, and get a wave moving through it, that excitation of the field is what an electron is. Or, to be more precise, the minimal allowed disturbance of that field would be an electron; a larger disturbance would represent a larger number of electrons.
Energy in QFT
Physicists are obsessed with energy. Isn't everybody? It turns out, though, that when people talk about energy in popular parlance, they are talking about something vaguer, that incorporates aspects of both energy and entropy (and maybe perhaps your force of will and current capacity for focused cognitive activity). But that's neither here nor there.
In physics, identifying the energy in a system is often a very useful thing to do, because energy is conserved. Saying that "energy is conserved" is very different from the popular parlance version of "energy conservation", which is about keeping energy in a useful form. In physics, energy is neither created nor destroyed, so there's no need to try to conserve energy; it just happens, always. That fact, plus a whole lot of math, allows us to make all kinds of predictions about how physical systems will behave.
In QFT terms, there are a few ways that energy can arise. There is the self-energy of a field. If you've got an electric field in space, there's an energy density associated with that field. (I say "energy density" rather than just "energy", because the field isn't all at once place, but is distributed through space. So, within a given volume, there will be a certain amount of energy; energy within a volume divided by that volume gives you energy per volume, or energy density.) There is also the energy of interaction between fields. So, because a charged particle like an electron can interact with the electromagnetic field, there is an energy associated with the interaction of the electron field and the electromagnetic field. How much energy depends on the values of the two fields— that is, how probable it is that there are one or more electrons or photons at various points in space— and the coupling strength between those two fields. Another kind of particle, the neutrino, has no electric charge; it does not interact with the electromagnetic field at all, and so we would say that it does not couple with the field, or equivalently that the coupling strength the neutrino field and the electromagnetic field is zero.
For the electromagnetic field, that's basically all you have to worry about: the self-energy of the field, and the energy that comes with coupling from other fields.
Many of the fields in QFT, on the other hand, are massive fields. These are fields like the electron field, the quark field (quarks being the particles that make up protons and neutron), and several other of the fields we know about. A massive field is a field such that if you have an excitation of that field— that is, a particle— that particle has mass associated with it. The mass of the electron is not zero, nor are the masses of quarks zero. There are a couple of massless fields in QFT, such as the electromagnetic field— photons have zero mass.
It turns out that mass is just another form of energy. If you have something with mass m, it has energy E just as a result of its mass; the amount of mass-energy E that you've got when you have something of mass m is given by the most famous equation in all of physics:
E = mc2
In this equation c is the speed of light. The equation is just the conversion factor between mass and energy; it tells you how much energy there is associated with a particle of mass m. Among other things, this means that it's possible to create mass "out of nothing", although you're not really creating it out of nothing, you're just converting other forms of energy into mass energy. If two photons with the enough energy come together and interact in just the right way, it's possible that they'll disappear and create two particles, a positron and an electron, the positron being a antimatter particle that's sort of like the opposite of an electron. The mass of the positron is exactly the same as the mass of the electron.
Notice that before, you had two photons, and zero mass; after you have mass. If you've taken a chemistry class, you may have learned about the law of "conservation of mass". This law is in fact not strictly correct. For chemical reactions, the amount of mass that gets converted to energy and back is typically about a billionth the amount of mass present, so it's correct to very good approximation. But, when we're talking particle physics, it's not true at all. You can convert mass energy to and from other forms of energy.
This means that when you're writing down the energy expressions in QFT, you have to include not only the self-interaction of fields, and the coupling of fields to other fields, but also the mass energy associated with the particles of that field. Unfortunately, the way that you add this mass energy in QFT is rather ad-hoc. The coupling of the fields together come in a fairly elegant way (although the actual coupling strengths are arbitrary, and as of right now we have to take them as "just the way nature is" rather than determining them from fundamental principles). However, the mass terms show up in an ugly and tacked-on way.
The Higgs Field and Mass Terms in QFT
And so, finally, we come to the Higgs field. Now, if you've been paying attention, by introducing the "Higgs field", I'm saying that there's a new kind of particle, which we'd call the "Higgs particle". In fact, you hear it called the "Higgs boson", because physicists categorize things (for reasons that aren't important here) as either fermions or bosons. Electrons, for instance, are fermions, while photons are bosons. The Higgs field is predicted by the standard model of particle physics in a fairly esoteric way. Suffice to say that the part of the standard model of particle physics that predicts the Higgs field also predicts other things that had previously been measured in physics experiments. That is, we have a theory that predicts various things, and some of the predictions of that theory had been validated. So, we had reason to take this theory seriously. The theory also predicted that there would be this Higgs field, and that it would be a massive field. In other words, there was a prediction for a new field, and excitations of that new field would show up as a massive particle. The mass of the particle is still tiny tiny tiny compared to everyday masses, but it's huge compared to the masses of the other fundamental particles we know about. As such, it took accelerators that were able to accelerate other particles to very high energies before there was enough energy to create this new massive particle.
The Higgs field, however, has a key difference from the other fields. Above, I talked about the analogy to the surface of a lake. The "vacuum state", that is, the natural, undisturbed, zero-energy state of the field was a field with a zero value everywhere— a level lake everywhere at its average height. Fields don't necessarily have to be this way. For instance, it would be entirely possible to have an electric field that is doesn't have any waves moving through it, but that is constant everywhere in space. The electric field inside some kinds of capacitors is very much like this. This electric field would have energy associated with it though (which is why capacitors work!), and so we wouldn't call it the vacuum state of the field. The vacuum state of the electromagnetic field is in fact a field value of zero.
The Higgs field is different. It's vacuum state is in fact not a field value of zero. This has consequences. One consequence is that when you figure out the interaction of the Higgs field with other fields, you get an additional energy term in the equations of QFT describing the energy of everything. The neat thing is that that extra energy term resulting from the non-zero vacuum value looks exactly like the mass-energy term of the other field. Consider, for example, the electron field. Where we used to have an ad-hoc mass term, we now just have another elegant field coupling term with the Higgs field, but that term looks, mathematically, just like the mass term. The energy we would have called the mass energy of the electron is in fact something that arises with the interaction of excitations of the electron field with the vacuum state of the Higgs field.
Notice that this doesn't mean that the Higgs boson gives particles their mass. In the "star walking through a crowed in a party in Hollywood" analogy, you might be tempted to think that all the people in the room represent Higgs bosons. They do not. The sea of people, as it were, together represent the vacuum state of the Higgs field. Even though there aren't any actual Higgs bosons tooling around, the interactions of other fields with the field of which the Higgs boson is an excitation is what gives rise to the mass energy terms in the QFT equations.
So what about photons and other massless particles? They don't couple to the Higgs field; they ignore it, and so no mass-like terms show up in the equations for them. The different masses of all the other particles arises because of the different coupling strengths between those particles and the Higgs field.
The mass energy of particles in quantum field theory is in fact the result of interaction of the fields associated with these particles as they couple with the non-zero vacuum state of the Higgs field.
My wife graduated from from Oberlin college in 1992, and as such she gets the Oberlin alumni magazine. The summer 2012 issue includes a one-page article entitled "The Entirety of Relativity", which I find to be a very unfortunate presentation of Relativity. (As a pedantic point, it's only talking about Special Relativity (SR), and doesn't address General Relativity (GR) at all, but that really is a pedantic point. When a physicist says "Relativity", she likely means GR (especially given that SR is a subset of GR, so nothing is lost), but when presented publicly we often use "Relativity" as a shorthand for SR.)
The basic problem with the article is that it presents the theory as if its nature were the way that SR has been taught to students for a long time. The article starts with three things that are correct as far as they go: moving clocks run slow, a moving rod is short, and moving clocks aren't synchronized. Where the article loses me, however, is on point number 4, "That's All There Is To It.".The brief text after this says that the first three points are the basis of relativity, and the rest of the article claims that all of SR is a consequence of these three points. This is at the very least a perverse way of describing the theory.
A lot of texts at both the high school and college level present Relativity by first presenting these three points. You're given formulae for each of these consequences; parts of them resemble each other, but they're each presented as if they were a fundamental formula that couldn't be derived from anything else, for you to memorize (or, in a more modern way of thinking about it, look up) and use. However, this is a back-assward way of presenting SR, and I would argue that stating that the rest of SR is a consequence of these three observations is not just back-assward, but in fact wrong.
In fact, these three points are themselves consequences of the theory of Relativity. The formulae for them can be derived from more fundamental considerations. They're no more fundamental than all of the various kinematic formulae you memorize or look up (such as d=½a
Special Relativity itself starts with just two very simple postulates— "simple" in the sense of "not complex", not in the sense of "easy to understand". Those postulates are:
- The laws of physics are the same for every freely-falling observer
- The speed of light is one of those laws of physics; every freely-falling observer will measure the speed of light in a vacuum to be 2.998×108 meters per second.
Everything else in SR— including moving clocks running slowly, separated moving clocks not being synchronized, and moving rods being short, as well as other things (such as the Doppler shift, focusing of light emitted by a moving object in the direction of motion, an apparent rotation of a moving object) are consequences of these two postulates.
I should note that both of these postulates do require more explanation to be truly precise. For the first postulate, you have to carefully define "freely-falling observer". You get it basically right if there are no net external forces other than gravity acting on that observer. (However, if you allow gravity to be around, things can get a little subtly complicated. It doesn't ruin the postulates, but you have to be careful in treating the consequences.) For the second postulate, in fact it's not the speed of light that's absolute, it's the speed of any object that both carries energy and is massless. Light just happens to be the thing that we think about the most that works like this, and thus we call the cosmic speed limit "the speed of light", even though we really ought to call it "the speed of spacetime" (at least in the context of Relativity).
One of the most interesting consequences of these two postulates it that you have to change the way you think about time. Most of us live our lives with a Galilean/Newtonian view of time: it's an absolute, that advances at the same rate and is the same for everybody. However, you can't maintain that idea and have the speed of the same bit of light be measured at the same rate by everybody regardless of how they're moving. Galileo and Newton would say that the latter is wrong; Einstein's postulate, from which all of Relativity springs, was that in fact it's this speed of massless objects that is absolute, and as such we just have to give up on the idea of absolute time. Some of the consequences of this are that separated moving clocks aren't synchronized and moving clocks run slow... as well as other things.
I'm fond of the way that Thomas Moore's Six Ideas That Shaped Physics presents Special Relativity. (This is the book series that I currently use when teaching introductory calculus-based physics.) His Book R of the series is written for college-level physics who have had Calculus (and indeed have had some Calculus-based Newtonian physics). It presents SR not in the old-fashioned and unfortunate pedagogical way that the Oberlin article does— by starting with the consequences such as time dilation and with their formulae, and only later getting to the fundamental structure of spacetime implied by Einsteins postulates— but rather by starting with the fundamental structure of spacetime implied by Einstein's postulates, and then developing the consequences out of that
Yes, it's easier to just learn the formulae and do calculations about time dilation and so forth, and presents fewer difficult abstract conceptual challenges to students coming across this for the first time. However, if you learn it this way, you're given a warped perspective of what the theory of Special Relativity really is. My beef with this Oberlin alumni article is that it presents Relativity as if the theory itself is based and structured in the way that it has often been taught.
When Andrew Hacker asks "Is Algebra Necessary?", why doesn't he just ask "Is High School Necessary?"
Yes, I admit, the editorial at the New York Time entitled "Is Algebra Necessary?" pushes my buttons. Hacker makes some valid and relevant points, and I'll get back to that. However, the core of his argument is the ultimate in anti-intellectualism. What's worse, it's the kind of anti-intellectualism that you get from intellectuals, the sort of thing that sprouts from those on the math-ignorant side of the "two cultures" identified by C. P. Snow.
Andrew Hacker's argument against making algebra necessary for high school and college students is essentially: Math Is Hard. Having to do it gets in the way of people who could be amazing at other things, because they will drop out of high school because Math Is Hard. So, rather than stop them from achieving all that they might achieve, we should just remove algebra from the high school curriculum. He points out that failing math is one of the main reasons students leave school. Now, I might think that this is a reason to look at our educational culture, at how math is taught, at the fact that it is somehow deemed acceptable and indeed normal to find basic math impenetrable. But, if you're on the other side of the two cultures, evidently this means that we as a society should just give up on the general teaching of basic algebra. Evidently, it's OK that the elites who understand the simplest things about science become that much more separated from the general educated public, and that the generally educated public know that much less about them.
There's one particular part of the argument I want to highlight:
Nor is it clear that the math we learn in the classroom has any relation to the quantitative reasoning we need on the job. John P. Smith III, an educational psychologist at Michigan State University who has studied math education, has found that “mathematical reasoning in workplaces differs markedly from the algorithms taught in school.” Even in jobs that rely on so-called STEM credentials — science, technology, engineering, math — considerable training occurs after hiring, including the kinds of computations that will be required.
So, because algebra isn't what's needed in jobs, we shouldn't be teaching it. This is absolutely the wrong way to think about a lot of education.
If you accept that argument, we need to reevaluate the entire high school curriculum, and the entire core curriculum of all colleges and universities. I think most people would agree that you need to be able to read and write in order to function in today's society. Do you really need to be able to interpret themes in literature, however? Honestly, is anything that you do in high school or college English classes really necessary in the workplace, any more than algebra is? The kind of reading and writing that most people need is something that students should already know by the time they're out of middle school. Likewise, history, biology, all the rest: everything that they study in high school is not going to be necessary for their jobs. And, really, if the purpose of high school and college is to train people to function like good little Betas and Gammas within our economic system, why is Andrew Hacker singling out algebra for attack? If we're going to dumb down the curriculum because we don't like that right now some people aren't mastering it, why don't we just dumb it down all the way?
The simple fact is that a college or university education is not job training. In recent decades, it's become conflated with job training, at least in North America, and this is too bad. A liberal arts education is all about expanding your mind, all about being able to think. It's not about gaining skills that you are then going to use in a job. Too many of us professors tend to not have any clue what somebody is supposed to do to earn a living after a liberal arts education other than go to graduate school (so that your liberal arts education is "training" for what you do next). That's because that was our own life trajectory, and it's what we know. Liberal arts education is to make people into good citizens, not into good workers. They are to acquaint you with the intellectual achievements of humankind. That is why we read the Iliad, why we watch a performance of Hamlet, why we learn about the history of ancient Greece, and, yes, why we study algebra. Because we want people to be educated so that they understand the intellectual achievements that have made our society what it is today, and that will drive our society in the future. We're training people to be members of civilization, not employees.
I will say that Hacker makes some good points. There are other kinds of quantitative reasoning, which too many of those coming into college and too many in our society completely don't grasp, that people should learn. A better understanding of basic statistics may at this point be more important to the citizen of a democracy than an understanding of algebra. So, yes, I would agree that we could and perhaps should de-emphasize algebra in favor of making time for statistical awareness, and perhaps in filling in the basic number sense that students failed to get out of elementary school. However, to me this is a bit of a red herring. Yes, we should always be evaluating what the subject matter of mathematical high school education is. But, right now, the problems are bigger than that. That so many people through high school without basic quantitative reasoning skills is not a reason to throw out algebra. We do, however, have to figure out why it is somewhere around fifth grade that individuals and society both get the "Math Is Hard" meme so firmly embedded. Why it becomes normal not to "get" math and indeed a little weird to actually understand and like those classes. Why it becomes OK to not like and not try at math and just do what's necessary to get by without actually learning anything. I strongly believe that there are serious problems with a lot of the math education that's done at the later elementary, middle school, and high school level. But that's not a reason to give up. We might as well point at various studies of how little so many people know about the state of the world to say that teaching geography and international history just isn't worth doing any more.
Perhaps the problem, or part of the problem, is that we have conflated vocational and liberal arts education. Anybody who is interested in a liberal arts education does not deserve a degree if they are completely ignorant of algebra, and any society that values liberal arts education cannot neglect algebra. However, perhaps not everybody needs such a liberal education. If we have the problem right now of too many people failing out, it may be that we're pushing them through the wrong kind of education. This does not mean that a liberal arts education needs to jettison those parts of it that are hard for people on the wrong side of C. P. Snow's divide!
Algebra is fundamental to nearly all of "higher math". Even if you want to do more than the most basic of things with statistics, you need to know some algebra. To give up on that would be right on par with the giving up on the teaching of history as anything other than memorizing the occasional date, and to give up on the teaching of English literature as anything other than being able to read a short document for simple surface content and to put together a simple declarative sentence. If you want people to be educated beyond elementary school and beyond "job training", then algebra is one of the intellectual foundations of our civilization that simply cannot be neglected.
GUILDENSTERN: ...Four: a spectacular vindication of the principle that each individual coin spun individually is as likely to come down heads as tails and therefore should cause no surprise each individual time it does.
—"Rosencrantz & Guildenstern Are Dead" by Tom Stoppard
There has been a lot of bru-ha-ha over the last few days about the much anticipated discovery of what looks to be the Higgs Boson at CERN. Among many other things that you have probably read is the statement that the confidence that the signal is real is 99.9999%. You might be wondering, why so many 9's? That is, they had a signal a while back that was already 99% or thereabouts certain. If I had 99% confidence in winning the lottery I would go out right now and spend $1000 on lottery tickets. Why was a 99% confidence limit not good enough to indicate discovery? Indeed, the announced discovery, with 99.9999%, is at the statistical confidence level that is considered the minimum for a particle physicist to announce a discovery. Why do they have to be so damn confident?
Rather than talking about the energy spectra of interaction cross sections, let's talking about flipping coins. At the opening of Tom Stoppard's play Rosencrantz & Guildenstern Are Dead, the two courtiers are flipping coins (and have been doing so for some time). They are approaching a streak of 100 flips of heads in a row. Rosencrantz (who wins a coin each time it comes up heads) is not concerned about this, but Guildenstern is so disturbed by the seeming violation of the laws of probability that he philosophizes at length about what it is that's going on. (The real thing that's going on is that he's a character in a play, not a real person.) Let's keep it more modest, though.
Suppose I were to walk up to you with a quarter, and flip it six times in a row. If the quarter is normal, and if I'm not cheating, the probability that all six flips of the quarter will come down heads is about 1.5%. In other words, if I do flip six heads in a row, you can be 98.5% sure that it was not due to random chance, that I must have been cheating somehow. (Ask me to show you this sometime.) You're not 100% confident, because there is a small chance that six heads will come up in a row just randomly, but it is a very small chance... and so you would be well within your rights to think that something was probably up. It may not be good enough to convict somebody in a courtroom, but it's certainly good enough to bet on.
Suppose instead, however, that 30 people come up to you, and each one of them flips six coins in a row. The probability that at least one of those people will flip six heads in a row is 38%. So, while it won't happen every time this crowd of coin-flippers accosts you, you shouldn't be particularly surprised that somebody flipped six heads in a row if a whole bunch of people tried it. Even though it's extremely unlikely that any given coin flipper will flip the coin six times, the probably that somebody somewhere will is entirely reasonable. Lightning has to strike somewhere. (See Randall Munroe's much more concise take on this, and on overreactions to it.)
This same principle applies to particle physics. The particle physicists looking for the Higgs Boson were not sure at exactly what energy the particle would show up. Here's one of the plots from the CMS collaboration:
The signature of the Higgs Boson is the extra bump of events at an energy of 125 GeV. There are lots and lots of events at all energies in the plot; there's a little something extra there, which indicates that something is going on there, and that something is probably the production of a short-lived Higgs boson. But they didn't know before they found it to look right at 125 GeV; it could have been at other energies, too. If all they were after was finding something that was "a little extra" at 95% confidence, they could have found it lots of places; indeed, there's a data point hanging out there at a bit over 135 GeV that is that far away from the background. But since there's 30 data points in the plot, I'm not the least bit surprised to see that. Randomly, you'd expect to see at least one of those more than 3/4 of the time somebody showed you a plot like this with 30 data points, even if there are no new particles.
The physicists in these collaborations were doing the equivalent of looking at a whole bunch of people flipping coins, and trying to find somebody who was flipping more heads than tails. If you look at 30 people who flip 6 coins and you find one person who has flipped 6 heads in a row, you have no right to declare that you've found a person who is cheating at flipping coins; the chances of that happening randomly are too high. Similarly, if you look at a whole bunch of different energies, and you see a single place where more is going on to 99% confidence than you'd expect from random fluctuations, you don't have much confidence that you've really found anything... because if you look at enough different energies, you will eventually find the unlikely random fluctuation. This is why for a particle physicist to be confident that she really has discovered something, she needs six nines in her confidence.
As for why the Higgs field (the "same thing" as the Higgs Boson... it's complicated) gives particles mass... that I really don't understand.