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=mc²?

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, x² + y² = z², 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 v² over c² 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=mc².

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. 

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Quotations from Why does e=mc²? (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:

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