The title is a bit of a holiday joke. Of course there's nothing easy about understanding the theories of special and general relativity first developed by Einstein, and now expanded to include multiple theories about gravitation and how the universe really works. Since relativity is very far away indeed from how we understand the perceptible three-dimensional space we occupy, and which we've evolved to be able to navigate, it's a concept at which we're all literally beginners.

Brian Cox and Jeff Forshaw, however, are well beyond where most of us are at on the subject, and fortunately they've written a book about it for the rest of us: *Why does e=mc ^{2}*?

Cox is a particle physicist and works on ATLAS, one of the four main projects going on at CERN's Large Hadron Collider outside Geneva, Switzerland. Yes, it's been in the news lately, but more on that later. Forshaw is a theoretical physicist and one of the youngest people on the planet ever to understand this stuff. In their book, the two scientists set out to do something elegantly, beautifully simple (well, simple for them anyway): derive Einstein's famous equation about the relationship between matter and energy starting with nothing more complicated than Pythagoras's 2500 year-old theorem about triangles, x^{2} + y^{2} = z^{2}, where x and y are the lengths of approximate sides of a triangle and z is the opposite longest side, and ending up with a description of the work being done right now at CERN. Along the way, they attempt to explain, in as simple a fashion as possible, the theory of special relativity, and, in what is more of a postscript, give some outline to general relativity as well.

**Ah, Mr. Faraday...**

The story starts with experiments by Michael Faraday (1791-1867) into electromagnetism--how currents can be made to travel along a conducting material in the presence of a magnet, and how magnetism is effected by electricity--and the powerful equations of James Clerk Maxwell (1831-1879) which helped to explain the processes involved.

Maxwell's work in turn provided predictions about the velocity of light-- about 299,792,458 meters per second--and once the obscuring idea of "ether" was dispelled, scientists came to the very strange conclusion that the speed of light will be the same everywhere, for everyone, regardless of the source of the light or our (the observer's) position in relation to it.

This was a disturbing conclusion to reach, as Cox and Forshaw point out, because previous work by some very convincing minds, including Galileo, had already made it clear that the problem with measuring the distance traveled by objects in space is that there literally is no fixed place on which to stand and from which an absolute result can be obtained. This effectively ruled out the idea of absolute space, while retaining the idea of absolute time (for the time being).

The rest of the picture had to wait to be filled in for Einstein's discovery that time, too, was relative. (As an aside, Einstein was born the same year that Maxwell died.) Einstein did this with, amongst other proofs, the famous thought experiment involving an observer standing on a train platform watching a train whizz past, with another person as a passenger on the train measuring time with a "light clock" . . .

**The Light Clock Experiment**

The "light clock" is a device made up of two mirrors between which a beam of light bounces back and forth, each circuit equalling one "tick" of the clock. If the mirrors are 1 meter apart, then the light has to travel 2 meters (at the speed of light, 299K+/meters per second) in order to measure a single "tick," or approximately 150 million "ticks" in a heartbeat.

But, to let Cox and Forshaw explain, since the train is moving "the starting point of the light beam's journey is not in the same place as its end point according to the person on the platform, because the clock has moved during the tick." That is, if the clock continues to tick at the same rate "the light must travel a little bit faster," which is what happens in Newton's world-in-a-box type physics (which works just fine on a rough scale in three-dimensional perceptible space). But according to Einstein's example, following Maxwell, the speed of light is the same for everyone.

Therefore, from the perspective of a caroler singing away on the platform the clock must take longer to complete one tick than it does for the caroler sitting beside their light clock crooning away on the train. Time *slows down* for the caroler on the train *relative to* the caroler on the platform. In this case, it's a very negligible difference, since we'll assume the train is traveling at very low velocity compared to the speed of light. For instance, if the train is going 300 kilometers per hour, traveling on it for 100 years would extend your life one-tenth of a millisecond relative to the person on the platform. But if you could go very very fast, then you could draw out your Ho-ho-ho's almost forever.

This tells us that not only measuring distance in space, but time as well, is relative. Using nothing more complicated than Pythagoras's theorem about triangles, and a simple equation (distance = speed x time, or time = distance / speed), we can derive an expression to tell us by how much the light clock on the train will run slow as measured by the caroler on the platform, a quantity known as gamma: 1 to the square root of 1 less v^{2} over c^{2} where "v" is the speed of the object measured by a known quantity and "c" is the speed of light. If you look at this closely you'll see that as long as the speed is small compared to the speed of light, gamma will remain close to 1; when the speed reaches a significant fraction of the speed of light, gamma starts to deviate from 1, and all sorts of interesting things start to happen.

In fact as the quantity for the velocity of an object becomes appreciably closer to the speed of light, the effect becomes more extreme, until it seems that one could extend the object's lifespan almost indefinitely. This is what is predicted to happen at the event horizon of a black hole where the matter being sucked in is speeded up to very near the speed of light and time appears to stop. That is until the elongated, squeezed matter reaches the singularity, a point of extreme density, and gets sucked down into it.

This is a little much to take in, but it's been proven many times by, amongst other methods, accelerating muons (a heavy type of electron) in accelerators and observing that as they increase to about 20% the speed of light their lifespans are extended many times. Instead of making about 60 circuits before decaying and breaking down, they make about 400. In this way you can watch while time slows down for them, just like the lucky caroler on the speeding train.

**Minkowski Had a Plan**

So all this not-having-any-firm-place-on-which-to-stand business is beginning to add up. But we're only part of the way to Einstein's beautiful equation E=mc^{2}.

If space and time really act this way, then, as Einstein's contemporary Hermann Minkowski put it: "From henceforth, space by itself, and time by itself, have vanished into the merest shadows and only a kind of blend of the two exists in its own right." In other words, we need the concept of spacetime if we want to continue.

But this presents us with some pretty onerous problems if what we want to reach are invariant quantities and rules about how the universe behaves. So far all we have that we know we can rely on are the speed of light and gamma. And it looks as if measuring distances in spacetime will require abandoning regular old Euclidean geometry. If we stick with it, we end up with some truly ridiculous results, including messing up causality, that is, instead of Event A causing Event B, we end up with a universe where B can happen before A. We have to switch to using hyperbolic space or Minkowski spacetime.

This also means that we're limited to a "cosmic speed limit", which we'll call *c*, and which no caroler and nothing else in the universe can exceed. It's part of the structure of the universe, of the way things act. This means that it's an invariant speed. It's beginning to look a lot like the speed of light, but we haven't proven that yet.

Actually it turns out that whether *c* is the speed of light or not is not all that important. Light just happens to use up all its speed in spacetime with its motion. This is because, as far as we know, photons (bits of light) are massless, and anything that's massless has to travel at the "cosmic speed limit" in order for Minkowski spacetime to work and to keep B from happening before A. If we discover at some point in the future that photons __do__ have a tiny mass, that won't mean that Minkowski spacetime, and Einstein's theories, fall apart, but rather that *c *is still a constant, and still the "cosmic speed limit," just not the speed of light. Either way the quantity *c* will remain the speed of massless particles.

And what we're implying when we talk about speed really is energy. Matter and energy, therefore, must have a deep relationship. Where matter is packed very closely together, like near the center of an atom, the nucleus, where the strong nuclear force is acting, tremendous amounts of energy must lie hidden.

This energy can be freed by bombarding the closely packed particles with other particles and breaking the strong nuclear force so that some matter is destroyed and energy is released. Or it can be done by getting certain particles, especially protons, close enough together so that their repelling force is overcome and they "fall" toward each other and produce other particles and a burst of energy. The former process is known as fission, the latter as fusion. Fusion occurs at very high temperatures, which is an increase in kinetic energy--acceleration. Besides accelerators, fusion happens at the cores of stars like the sun. Every second the sun converts around 600 million tons of hydrogen into helium through fusion. One type of particle produced as an effect of the fusion of hydrogen atoms near the sun's core are neutrinos. An enormous bombardment of neutrinos is constantly striking the earth, about 100 billion per second per square centimeter. A rather famous experiment near Hida, Japan (the Super-Kamiokande) involving a large buried tank of pure water surrounded by photomultiplier tubes, extremely sensitive detection devices, bears this out. The neutrinos pass through almost everything, but show up as flashes of light when some of them hit electrons in the water.

By now it should be obvious--as it was for Einstein by working much of this out mathematically--that mass, or the amount of stuff in what we call matter, and energy, are different expressions for the same thing in the universe, and that the speed of light, or the "cosmic speed limit," is the exchange rate in the transformation of one into the other. Therefore the equation, Energy = mass x the speed of light (or speed of massless particles)^{2}. This completes the picture for the theory of special relativity.

But this leaves us with a very thorny question: Where the heck does mass come from to begin with? That is, why is anything attracted to anything else? Or repelled from anything else for that matter? Why are there *things* in the universe, and not just light and tiny particles whizzing around?

**Return All to the Void...***

Cox and Forshaw point out that as modern physics has gotten more complex, and our understanding of the universe, especially of the interactions of particles, has expanded, our ability to observe experimental results to back up or bar hypotheses has been reduced. What once required simple materials and a laboratory, now requires complicated and expensive equipment. Faraday conducted most of his investigations into electromagnetism using materials anyone could find. The design wasn't too esoteric either.

At the Large Hadron Collider (LHC) tests are being done to try and find an elusive and very important particle predicted by theory to be a determining force in how all the most basic pieces of the universe operate. The LHC is the biggest and most complicated scientific experiment ever undertaken, however, it's still based on something rather simple.

During the 1950s and 60s particle physicists and mathematicians worked out the Standard Model, the most mathematically streamlined expression of all the possible interactions between the elementary particles in the universe, including protons, different types of electrons, neutrinos, and some ghostly things known as the *W* and *Z*. Add in gluons, the weak and strong forces, and electromagnetism, and you have a working model of the universe--minus gravity.

One of the things predicted by the Standard Model--which works pretty well as far we know from experiments done at previous colliders, including the LEP, the antecedent to the LHC on the same site--is a particle known as the Higgs boson. At first physicists weren't even sure that it was a particle. Perhaps it was a universal equivalent, they thought, a shadow of how all the fundamental particles and forces interacted. But a recent detection at the LHC seems to suggest that they've closed in on the Higgs, and that it's located, or rather that it occurs, exactly where the Standard Model predicts it to be.

Hold on though. It's not a done deal yet. The data obtained so far only squeezes the window for the allowed mass range of the Higgs. Within the still allowable mass range there are some fluctuations. ATLAS and CMS, the two projects at the LHC looking for the Higgs, will continue to work within the allowed mass range and should eventually report a detection. Still it's good to keep in mind that the Standard Model expresses everything in probabilities (in keeping with quantum theory) and uses field equations, with "gauge" phase values, to narrow things down to a limited number of possible locations for each type of particle and precisely when, during the phases of each of these fields, we might find one. This expression defines, from the point of view of the Standard Model, what a particle *is*.

If we find the Higgs it might mean that string theory, one of the candidates for a unified theory--a theory that could be used to determine how gravity fits into the picture--has outlasted its rivals. String theory, the M-theory, predicts that there should be partner particles for all the most basic particles included in the Standard Model, and that at some point we should look for these, too. Other theorists would argue that if the Higgs turns out to be exactly where the Standard Model puts it (as the case appears to be shaping up) Einstein's theory of general relativity, along with the processes described by the Standard Model and confirmed by experiment, should be enough. And that the so called partner particles might be the shadows, the equivalents for the interaction of the Higgs with everything else. And then there are other candidates for a unified theory . . .

What they all have in common is that they agree on the principles set forward by Einstein's theory of special relativity. None of them deny the existence of spacetime or its special (that is, its local and invariant) effects. If you look far enough back in time at the conditions in a very young universe, things were different--even the most fundamental forces acted differently. Nevertheless, we're all subject to the laws of an old, complicated, and rapidly cooling universe. (It's only a few degrees above absolute zero.) And we live in this universe because . . . well, because this is where we are. The universe is the way it is, according to physics, because it can't be any other way. Even theoretical models that predict alternate universes--stacked up beside each other like infinitely thin slices of pie--have to provide for certain invariant conditions like causality and relativity and their significance in our understanding of spacetime. There's no way out of that.

Probably.

______________

Quotations from *Why does e=mc ^{2}*? (

*And Why Should We Care*?) by Brian Cox and Jeff Forshaw, Da Capo Press, Cambridge, MA, 2009.

*from Hung Ying-ming, *The Roots of Wisdom*, translated by William Scott Wilson.

Brian Cox explaining what goes on at the Large Hadron Collider:

**

## Comments

what facinates me these days in the world of Cosmology and Quantum Physicis is the return (ironically) of the Cyclic Universe supported by recent observations regarding a special direction in the universe combined with the Hubble/Schwartzchild radius and the University of Indiana kid's Eculidean math that all points towards the Universe BEING a black hole that self-regenerates during endless big bangs and big sucks.

Temporally this would seem to indicate that "all this has happened before and will happen again."

Which would technically make the Cylons from Battlestar Galactica right...

Which proves to me that the Universe has a sense of humor, or at the very least supports the mounting evidence along side everything else.

Thanks for the refresher!

i also like the fact that cox & forshaw go to great pains to point out that most of what we consider to be matter is empty space, that the universe is, for all intents and purposes, a void, a positively charged one but still a void. the recent nobel winning work that showed the universe is accelerating and will probably go on forever is rather a downer though...bleaker and bleaker, darker and darker, and colder and colder....

ah well.

p.s. the mayan thing is funny. he's really grand when he wants to be, easy to forget he was a rock star once. have you seen the video where he berates a student who can't understand gravity waves? student, whining: "but it's really hard..." cox: "tough."

I always get lost when it comes to vertices and vectors and the conservation of momentum. It seems like something is missing. Wait, so we're still in space-time, but these things still apply? Geez, I'm lost.

Have a good "God particle" holiday.

rate

( you bastard ~

Iwas just about to write this. )It's all the holiday reading I'll need, & thank you.

Rated

stu pot - Saw the video you mention, I liked his comment, "Are you getting weird with me now?" And I love how he turned the Mayan thing into a function--statistically more of them will be eliminated, and the rate of advance will increase...

themanhattankid - Conservation of momentum still applies in spacetime because you can construct a vector in spacetime where the length in the space direction represents a conserved quantity for velocities small compared with the speed of light, and the length in the time direction must be a conserved quantity equated with energy! Therefore the relevant term here is the conservation of energy, and the conversion of mass to energy is the logical next step. It's structured into everything. And I'm not too enamored of the "God particle" buzz phrase. Seems silly, doesn't it?

Dr William Lee - I'm not too sure we're ready to announce the winner of the unified theory race yet. Some people are jumping the gun. See my response to Davey Marx below...

Kim Gamble - A good read is worth a thousand Higgs detections. Well, not really, but it sounds nice on a card. Thanks and have a happy holiday.

skinnydave - Einstein's layman's version is pretty tough, I agree. He wanted to include some of the original math, and yes, I think he was writing for a pretty educated audience--not exactly a pop aesthetic. Cox and Forshaw found a nice balance and provided a long missing piece in the culture. I waded through Hawking and Ellis's book at some point, but it's a blur. I want to read about the theorizing of the Higgs next: Guralnik, Hagen, Tom Kibble and Peter Higgs' own work. But I need to find a more general text. There's one by Natekar that looks promising.

http://www.newscientist.com/blogs/bigwideworld/2011/12/in-the-thick-of-it-at-the-large-hadron-collider.html

And no on getting younger. Sorry. Here's why the past is protected from the future:

http://www.youtube.com/watch?v=LfCv9GLwvYE&feature=relmfu

I think Kane and other string theorists are getting ahead of themselves (hah!) when they predict the Higgs from the current findings at LHC. All that can be said for sure is that they closed the mass range down to where the Standard Model says it should be, about 127 GeV. It's there, somewhere. We just need to repeat, and analyze, repeat, and analyze...Ugh, tedious I know, but that's real science!

But yeah, Kane is way out ahead of the data when he says we should be looking for the partnered particles already. They will anyway at LHC, but we'll probably need the upgrade (at least) to see them.

I'm not a big fan of the GR-only+Standard Model route. It seems you run into things that are insoluble without string theory. The behavior of twinned or partnered particles around a black hole's even horizon, for instance, which allowed Hawking to explain why there's radiation being emitted. I mean, it shouldn't be!

And I want a collider of my own. A big one. :)

rated.

(insert universe here)

Rated.

mrvoulezvous - Thank you. Although I don't think it'll fit. Unilaterally. Even in a tranformable quantum states sort of way. I mean, what would Bob and Alice think? Yeah, well, that's question....

"They argue, on the basis of recently published studies by two top U.S. physicists, that the neutrinos pumped down from CERN, near Geneva,

should have lost most of their energy if they had travelled at even a tiny fraction faster than light."Neutrinos should have lost most of their energy if they had travelled at even a tiny fraction faster than light.Why?

(The original paper is here, with no less than 35 authors!)

As far as I can tell, their calculations are based on the expected loss of energy from Cherenkov radiation, which typically occurs when a

chargedparticle passes through a dielectric medium at a speed greater than the phase velocity of light in that medium. So obviously the phase velocity of an out-and-out super-luminal particle will be greater than the phase velocity of light in any material, and presto! Cherenkov radiation!So the energy spectrum of the supposedly super-luminal neutrinos would have been distorted by generating Cherenkov radiation, and since it wasn't, the world of physics can happily conclude that the neutrinos did not travel faster than the speed of light, and the physics that physicists have been doing for the last 100 years isn't junk.

Hurrah!

But it's also worth observing that this happy conclusion is based on a very complicated prediction about a phenomenon (super-luminal neutrinos) which has never been observed, and based on physics which this particular observation would have essentially discredited, so the putative "refutation" of the original finding doesn't really amount to much more than an exceedingly unscientific and even anti-scientific pronouncement...

"It doesn't fit our theory, so it didn't happen."

And meanwhile you keep claiming that positive photon-mass wouldn't be a big deal, and maybe you're right if you don't mind losing absolute conservation of charge along with gauge invariance for QE, but what the heck! Standard physics would still cover levers and pulleys!

But instead of going back and forth about papers you haven't read, instead I will make you a lovely gift, in the form of a truly outstanding exposition of "The Concept of Mass," by Lev Okun in an old edition of Physics Today, not available online unless you're plugged into the physics web.

Message me your email and I'll send you the pdf, and it's isn't exactly easy, but you'll enjoy it.

Of course that is all beside the point, because as it turns out the Standard Model is INCREDIBLY accurate in its predictions. If not, then how do you explain the fact that all the particles have been found already at EXACTLY where the Standard Model predicted them to be, right up to the tau electron, the last one to be located at the LEP? Magic? Or perhaps a conspiracy being perpetuated by tens of thousands of physicists and mathematicians all over the world? No, it's probably because it's correct. Now they're narrowing in on the Higgs--and, once again, it's in exactly the mass range predicted, heavy on the side of 115 to 127 GeV. I still don't understand your complaint...

And I brought up photons because that's what this is really all about. Ever since Einstein said that the speed of massless particles must be the greatest speed in the universe, the race has been on to figure out what a massless particle is. The best candidate is still a photon, and I don't see how any of the work you mention disproves that. And, once again, conservation remains invariant regardless of whether photons turn out to have a very very tiny mass.

http://lhc.web.cern.ch/lhc/

There's no new run planned until after the holiday break.

skinnydave - Bob and Alice come from communications theory, where there are two participants in a process and they have to relay some information between themselves. There is a response, and counter-response, and so on, and all the time there is a process that continues. The placeholders are usually employed in expressions of how information is lost, degraded, and in how encryption systems work. In physics they're used to explain what happens when more than one observer's results are factored in to quantum descriptions. For instance, in orthogonal matrices, which are used to underpin complex field equations, one finds that certain observations yield better results, and that there are limitations to what one can and cannot say about one's results. In other words, it depends on what you're looking for, and how. Ultimately this is another way of expressing indeterminacy. And again, I think there's some confusion in some of the commenters' remarks. The point of relativity theory, and quantum mechanics, is not to GET RID of indeterminacy. Einstein accepted its necessity, eventually, and it's very much part of GR, where there is no attempt to factor it out. Both sides of the problem, and both approaches, macroverse theory and the microverse of quantum states, are ways of looking at a universe where indeterminacy plays a significant part. But, once again, this is science, so the perspective is not morbid. There are a numbe of ways of expressing indeterminacy, and just because it's part of the universe doesn't mean you can't say anything for certain. In fact we can say a GREAT DEAL for certain, right down to being able to predict the tiny, tiny mass ranges where the elementary particles of the universe should be found. And sure enough, there they are...

http://press.web.cern.ch/press/PressReleases/Releases2012/PR17.12E.html

The complete analysis of data will be published at the end of July. Now, partner particles anyone.....?

And for God's sake, let's stop calling it the "God particle." That's stupid.