The Theory of Almost Everything: The Standard Model, the Unsung Triumph of Modern Physics - Robert Oerter (2006)
Chapter 2. Einstein’s Relativity and Noether’s Theorem
How deep is time? How far down into the life of matter do we have to go before we understand what time is?
—Don DeLillo, Underworld
Take a peek into an alternate universe:
“Good morning, ma’am, Yellow Cab Company.”
“Of course we can take you to the train station. How far from it do you live?”
“Thirty miles? We’ll need to leave at least an hour early. Can’t drive over 30 miles per hour—it’s the law, you know.”
“Well yes, if you bring your laptop you can get some work done on the way. But I wouldn’t worry about that too much if I were you. The trip only takes 10 minutes.”
“No, no! We need to leave your house more than an hour before the train’s departure. Thirty miles at almost 30 miles an hour means we need more than an hour to get there. But your ride in the cab will only take 10 minutes. It’s the Time Effect, you know. The cab travels at 29.6 miles per hour. That’s pretty close to the speed limit.”
“May I ask where you’re going? Washington to Los Angeles! That’s a long trip. If you leave on Sunday you’ll arrive
Thursday. You’ll want your laptop—the train ride is 14 hours.”
“Sure, you’d have a shorter trip if you went by plane. That’s only a one-hour trip. The plane goes faster, nearly 29.998 miles per hour.”
“No, you’ll still arrive on Thursday. The plane only saves you travel time, not ground time. Of course, you pay through the nose for it.”
“Right, then. Train it is. We’ll pick you up on Sunday. Have a good trip!”
What is going on here? The speed of light in this alternate universe is just 30 miles per hour. According to special relativity, nothing can travel faster than the speed of light. (“It’s not just a good idea, it’s the law.”) So, a trip of 30 miles will always take at least an hour, and a trip of 2,500 miles (Washington, DC to Los Angeles) will take at least three and a half days, in “ground time” as the cab dispatcher calls it. However, special relativity also tells us that there is a Time Effect: time runs at different rates depending on your state of motion. The effect gets stronger as you travel closer to the speed limit. The 83-hour (ground time) trip from DC to LA only takes 14 hours for train passengers (traveling at 29.6 miles per hour), and only one hour for plane passengers (at 29.998 miles per hour).
The strange effects of special relativity described in the alternate universe actually occur in our universe. However, these relativistic effects only become significant at speeds close to the speed of light. The reason we don’t hear conversations like the one above is that the speed of light in our universe is 300,000 kilometers per second (186,000 miles per second), instead of 30 miles per hour, so the Time Effect is normally much too small to notice.
Let’s return to our own world. Picture a flight attendant on an airplane traveling with a constant velocity in level flight. A bag of peanuts slips out of her fingers and falls to the floor. Now, intuition might lead us to think that, since the plane is moving forward during the time the peanut bag is falling, the bag will land toward the back of the plane from where the flight attendant is standing. This isn’t what happens, however. The peanut bag lands right at her feet, exactly as if she had been standing on the ground.
In fact, life aboard a plane is remarkably unremarkable: coffee pours the same way it does on the ground, electrical devices function normally, voices sound the same. It is only when air turbulence suddenly lifts or drops the plane that cookies fly off trays and coffee leaps out of cups. As long as the plane is in uniform level motion, no experiment performed on the plane will reveal that it is in motion. Only by looking at an outside reference point (the ground, say) can it be determined that the plane is moving. We can summarize this in the principle of relativity:
All steady motion is relative and cannot be detected without reference to an outside point.
Galileo propounded this principle already in the seventeenth century, when he noticed that if you dropped an object from the mast of a moving ship it landed at the base of the mast, in the same position relative to the ship as it would land if the ship was at rest.
The laws of physics tell us about the locations of objects at a particular time, and those locations must be measured from some reference point. The location of the falling peanut bag, for instance, might be specified as “20 meters from the back wall of the passenger compartment and 1 meter above the floor.” We also need to choose a reference time, some event that tells us when to start the stopwatch. Such a choice of reference points in space and time is called a frame of reference. A frame of reference allows physics to become mathematics: Instead of a vague description—for instance, “the ball falls to the floor” —we can say where the ball is and when it is there. The numbers that specify the where and the when are meaningful because we have defined the reference points with respect to which they are measured.
The plane’s frame of reference is in uniform motion with respect to the ground, as long as the plane is in straight, level flight at constant speed. Now, suppose some law of physics is different when in motion; the law of fluid flow, say, so that coffee pours differently when the plane is in motion. Then, all we would have to do to find out if we are moving would be to test that law by pouring a cup of coffee. If it pours as in a stationary frame, we must be at rest; if it pours as in a moving frame, we must be in motion. We would then have a way of detecting motion without reference to any outside point. Thus, another way to state the principle of relativity is this:
The laws of physics are the same in any uniformly moving frame of reference.
A Relatively Moving Experience
As a high school student in the 1890s, Albert Einstein wondered about relative motion. What would happen, he asked himself, if you traveled fast enough to catch up with a light beam? What would it look like? Was there such a thing as light that stayed in one place?
Ten years later, Einstein had finished a degree in physics and had taken a position as a patent examiner third class in Bern, Switzerland. During his studies, he had learned Maxwell’s equations, which purported to encapsulate everything there was to know about light. What would Maxwell’s equations say about Einstein’s high school conundrum? To his astonishment, Einstein found that it was simply impossible, according to Maxwell’s equations, to move at the same speed as a light beam. This was a shocking discovery. Nothing in Newton’s laws of motion suggested the possibility of an ultimate speed limit. Could the 200-year-old laws of motion be in error?
Other physicists who were aware of the dilemma assumed that it was the recently discovered electromagnetic equations that were incorrect. Einstein, younger and impatient with authority, assumed the opposite. What if Maxwell’s equations were correct and Newton’s laws of motion were wrong? Once the audacious first step was taken, the logical consequences could be derived using high school algebra. In a paper published in 1905 titled “On the electrodynamics of moving bodies,” Einstein laid the foundations of a new dynamics, replacing Newton’s laws of motion with the laws now known as special relativity.
Einstein’s bold step was to add a new postulate to go along with the principle of relativity.
Einstein’s Postulate: The speed of light is the same in all reference frames.
All of the weirdness of the Time Effect stems from this deceptively simple postulate. To see how, let’s return to the airplane. Suppose a second flight attendant standing at the rear of the passenger compartment tosses a new bag of peanuts to the first attendant, who is 20 meters away. Let’s suppose the bag was in the air for one second. From the reference frame of the plane, the speed of the bag was 20 meters per second.
However, things look different from the ground. During the time the peanut bag was in flight, the plane moved forward some distance, say 200 meters.
So, the bag went 200 meters + 20 meters = 220 meters, as viewed from the ground. Since the bag was in flight for one second, we conclude that its speed with respect to the ground was 220 meters per second. In other words, velocities add: The speed of the bag with respect to the ground is the sum of its speed as measured with respect to the plane and the plane’s speed with respect to the ground.
We would expect the same thing to happen if the flight attendant had a flashlight instead of a bag of peanuts. Light travels at 300,000 kilometers per second, so one second after the flashlight is turned on, the forward edge of the beam will be 300,000 kilometers from the attendant. (We have to allow the flashlight beam to pass through the windshield of the airplane, or else assume we have a very long airplane!) Viewed from the ground, though, the beam will have traveled farther, just as the bag of peanuts traveled farther in the previous example.
In the reference frame of the ground, the beam, it seems, has traveled (300,000 kilometers + 200 meters) in one second. The beam moves a larger distance, but in the same amount of time: The speed of the beam is larger in the reference frame of the ground than in the reference frame of the plane. The addition-of-velocities rule clearly contradicts Einstein’s postulate: If Einstein is right, the addition-of velocities rule must be wrong. But we came to the addition-of-velocities rule considering only the everyday properties of distance and time. If the addition-of-velocities rule is wrong, our intuition about those “everyday” properties must be wrong. Our intuition about space and time comes from experiences with relative speeds much below the speed of light. Because the relativistic effects are so small at these speeds, we don’t notice them. When talking about the speed of the flashlight beam, we assumed (incorrectly) that “one second” on the plane is the same as “one second” on the ground. If we lived in the alternative universe where the speed of light is 30 miles per hour, the time effect would be so familiar that we would never make that mistake.
If the speed of light is to be the same in both reference frames, it must be the case that either distance or time is measured differently in the two frames, or, perhaps, both. The answer, Einstein discovered, was “both.” Neither time nor distance is absolute; they both depend on the relative motion of the observer. Space and time are thus inextricably intertwined. If you alter your rate of movement through space, you also alter your movement through time. As Hermann Minkowski put it in one of the first public lectures on special relativity, “Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.”1 The union of the two is what we now call spacetime.
Time runs differently for different observers: This is the explanation of the cab dispatcher’s apparently nonsensical comments in the story that began this chapter. The 30-mile trip to the airport takes only 10 minutes for the person in the cab, but you still need to leave an hour early. When the passenger gets out of the cab, her watch will show that 10 minutes have passed since the cab ride began. For everyone else, however, an hour will have passed. Similarly, the plane trip from Washington, DC, to Los Angeles takes only an hour, but on arrival she will find that three and a half days have passed since she got on the plane.
Even more mind-boggling is that relativity requires that the situation be symmetric. If someone on the ground looked in through the windows of the train as it passed they would see the passengers moving in slow motion, breathing, talking, and eating six times slower than normal. The passengers in the train, on the other hand, look out of the window at people on the ground and see them in motion relative to the train, so those on the ground are seen as moving in slow motion.
Here, we have an apparent paradox: If each reference frame sees the other as slowed down, whose clock will be ahead when the passengers leave the train? The resolution of the paradox comes from the fact that the train must slow down and come to a stop in order for the passengers to disembark and compare watches with those on the ground.
When slowing down, the train is no longer in uniform motion, so the situation is no longer symmetric. It took Einstein 10 more years to extend his theory to cover nonuniform motion. The result was general relativity, a theory that dealt with cases of nonuniform motion, and incorporated gravity as well.
In the real world, the speed of light is not 30 miles per hour but 186,000 miles per second. The Time Effect only becomes large for speeds very close to the speed of light. Even the space shuttle moves at only a tiny fraction of light speed, so we do not notice these effects in everyday life. For elementary particle physicists, who accelerate particles to 99.999995 percent of the speed of light, the effect is enormous. These fast-moving but short-lived particles survive 3,000 times longer when in motion than at rest. If you could travel at that speed in a spaceship for a year, when you returned to earth 3,000 years would have passed. For the particles (and the physicists who study them), there is no if: The effect happens just as Einstein predicted. The Time Effect isn’t science fiction or wild speculation; it is science fact.
The World’s Most Famous Equation
Plant an acorn, and watch as it grows year by year to a tall tree. The tree’s bulk obviously wasn’t contained in the acorn. Where, then, did it come from? Some early scientists thought there must be a vital force associated with living things that created matter from nothing. Later, careful experiments showed that this was not the case. If you kept track of the water added, the weight of the soil, and, especially, the gases absorbed by the tree from the air, the mass of the tree was completely accounted for. The tree grows, not from nothing, but literally from thin air.
By the nineteenth century, the principle of conservation of mass was well established: Mass can neither be created nor destroyed. Technically, mass is a measure of the inertia of an object: how much it resists being pushed. More loosely, we can associate mass with the weight of an object. Think of a sealed box from which nothing can escape. Put anything you like inside the box before sealing it: a chemistry experiment, a potted plant with a battery-powered grow light, a pair of gerbils with enough air, food, and water for a lifetime. The conservation of mass implies that no matter what physical processes or chemical reactions are going on inside the box, no matter what creatures are being born there, growing, or dying, the box will always weigh the same.
Energy, according to nineteenth century physics, is a completely different beast. An object’s energy depends on both its velocity and its mass. A bullet thrown by hand won’t do any damage, but the same bullet projected at high speed from a gun can be lethal. A loaded dump truck that crashes when traveling 60 miles per hour will cause a worse wreck than a compact car traveling the same speed. An electromagnetic wave also carries energy, even though it is not made of particles—it is “pure” energy. In 1847, Hermann von Helmholtz proposed the law of conservation of energy: Energy can neither be created nor destroyed; it can only be converted from one form to another. For instance, sunlight shining into a car carries electromagnetic energy that is absorbed by the car seat and converted to heat energy.
In special relativity, energy and mass are no longer independent concepts. Einstein considered an object that emits electromagnetic waves. From the special relativity postulates, Einstein deduced that the object loses an amount of mass equal to the energy of the wave (E) divided by the speed of light (c) squared, (old mass) - (new mass) = E/c2. He concluded that mass is really another form of energy. If the object could continue radiating energy until all its mass is gone, it would release an amount of electromagnetic energy equal to E = mc2. The speed of light is very large: c = 300,000 kilometers per second, so a tiny amount of mass produces a large amount of energy. A grain of salt, if all its mass could be converted to energy, could power a light bulb for a year.
To put it another way, suppose you had a microwave oven that didn’t just heat the food, it actually created it out of electrical energy. No need to put anything into the oven, just spin the dial to Hamburger, press Start, and out pops a steaming quarter-pounder. Sound enticing? But it would take about three billion kilowatt-hours of electricity, at a cost of about a hundred million dollars. Suddenly McDonald’s is looking pretty good.
Even Einstein had misgivings about overturning the time-honored conservation laws of mass and energy. In a letter to a friend, he wondered “if the dear Lord ... has been leading me around by the nose”2 about mass-energy equivalence. Today, the conversion of mass into energy is a matter of course: Nuclear power plants operate on this principle. Perhaps the most dramatic demonstration occurred when the first atomic bomb was exploded in New Mexico on July 16, 1945, converting a raisin-sized amount of mass into energy.
The equation E = mc2 is valid for an object at rest. For an object moving with speed v, Einstein derived a different formula:
According to this equation, as the object’s speed approaches the speed of light, its energy grows to infinity. An infinite amount of energy is impossible to achieve, therefore nothing having mass can ever reach the speed of light. It’s like a race in which the contestants move halfway to the finish line each time a whistle blows: They never reach the finish because they always have half of the distance remaining. Similarly, each time energy is added to an object the increase in speed is less. When the object is moving at a speed very close to c, it takes an immense amount of energy to increase the speed by even a small amount. Particle physicists spend vast amounts of the taxpayers’ money to build huge machines to nudge the particles a bit closer to the speed of light. As we will see, it is not the tiny increase in speed that interests these researchers, but the great gain in energy that accompanies it.
A massless particle, however, can travel at the speed of light; in fact, it must do so. If such a particle could exist, it would carry energy, but it could never be brought to rest and weighed. For this reason, physicists say such particles have no rest mass. Because they have energy, it isn’t strictly correct to call them massless. A box full of such particles zipping back and forth would weigh (very slightly) more than the same box when empty. These massless particles will reappear in later chapters; keep in mind, though, that we’re using the word “massless” to mean particles that have no rest mass, carry energy, and always move at the speed of light.
The idea that the amount of “stuff” in the universe doesn’t change—that is, that mass is conserved—makes intuitive sense: You can saw up a log into boards, but the total weight of the boards, plus the weight of the splinters, shreds of bark, and sawdust left over from the sawing must be the same as the weight of the original log. Energy is a much more abstract idea. A fast-moving object has more energy than the same object when moving slowly. Only by making careful measurements and combining the measured values in the correct mathematical relationship do we discover that the particular combination we call energy has the same value at all times. Einstein’s discovery of the equivalence of mass and energy reveals that energy is just as fundamental as mass; energy counts as part of the “stuff” of the universe, too. What Helmholtz’s principle of energy conservation had hinted at, special relativity made indisputable. Energy is not just a mathematical tool; it is a fundamental physical entity.
In addition to the equivalence of mass and energy, the space/time connection in special relativity also has deep philosophical consequences. Physical facts have meaning only insofar as they pertain to a particular observer. If Albert and Betty clap nearly simultaneously, one observer may report that Albert clapped first, whereas a second observer, in motion with respect to the first, may report that Betty clapped first. It makes no sense to ask, “Who reallyclapped first?” The question assumes that one viewpoint, one reference frame, is valid or “real” and the other is not. But time is not absolute; it is a property of a particular frame of reference. Both observers’ viewpoints are equally valid. Do not be confused by the term viewpoint into thinking that the difference is merely a matter of opinion. A viewpoint here has the very specific meaning of a frame of reference, a choice of reference points in space and in time from which all measurements are made. We are talking about differences of measurement, not of opinion. Moreover, an observer who understands special relativity can easily change viewpoints, converting all his measurements into the reference frame of the other person. Doing so allows him to understand the other’s conclusion about the order in which the events occurred.
Special relativity taught physicists that only things that can be measured have meaning: There is no way to measure which event really happened first, so the question is meaningless. It was the beginning of a fundamental shift of philosophy in science, from asking questions of what is to asking what can be known. This shift would become even more prominent in the rise of quantum mechanics.
A Radically Conservative Idea
There is a deeper meaning behind the conservation of mass-energy: It is a necessary consequence of a fundamental symmetry of the physical universe. The connection between symmetries of nature and conservation laws was discovered by a young German mathematician, Amalie Emmy Noether, who had been working with the great mathematician David Hilbert on Albert Einstein’s new theory of gravity, the general theory of relativity.
Noether was forced to struggle against the institutional sexism of her time. Hilbert tried to get her a paid position at the University of Göttingen in 1915, but was turned down on the basis of “unmet legal requirements,” a roundabout way of saying “we don’t hire women professors.” At a faculty meeting, Hilbert replied, “I do not see that the sex of the candidate is an argument against her admission as Privat-dozent (Lecturer). After all, we are a university, not a bathing establishment.” Unfortunately, his eloquence was ineffective. After several years as an unpaid lecturer at Göttingen, she was finally given a paid position in 1923 and was allowed to oversee doctoral dissertations. She became one of the founders of a new branch of mathematics known as abstract algebra. This is not the same algebra you learned in school. Ordinary high-school algebra deals with the properties of numbers and the rules for manipulating them. The algebraists noticed that other mathematical objects obey some of the same algebraic rules as ordinary numbers. By abstracting and generalizing the principles of algebra, abstract algebra pulled together results for numbers, vectors, matrices, and functions. Results proved in the general theory automatically applied to any system that obeyed the general rules. Abstract algebra is still a corner-stone of mathematics today.
Because of her Jewish background, Noether was forced to give up her position at Göttingen when the Nazis came into power. She moved to the United States to take a teaching position at Bryn Mawr College, a small women’s college in Pennsylvania. Norbert Weiner wrote in support of her application, “Leaving all questions of sex aside, she is one of the ten or twelve leading mathematicians of the present generation in the entire world and has founded ... the Modern School of Algebraists.” 3 4 Sadly, she died only two years later after a surgical procedure.
To the mathematician, Emmy Noether is the founder of a fundamentally important branch of mathematics and the author of many important theorems. To the physicist, there is one result of hers that is so important it is known among physicists simply as Noether’s theorem. This was a minor part of her post-doctoral work that grew out of her work on general relativity. Noether’s theorem (as I shall call it, too, since this is a book about physics, and bar my door against enraged mathematicians) relates the symmetries of a physical system to the conserved quantities, like energy, that can be found for the system.
We usually think of symmetry in terms of objects like snowflakes.
If someone were to rotate a perfect snowflake by 1/6 of a full circle (or 60°) when you are not looking, you would have no way of knowing it was rotated. The snowflake is said to be invariant under such a rotation. It is also invariant under rotations of 120°, 180°, 240°, 300°, and, of course, 360°.
The snowflake is also invariant under mirror symmetry: It looks just the same when viewed in a mirror. The human body very nearly possesses mirror symmetry; however, if you part your hair on one side rather than in the center you will look different in the mirror than in a photograph. Even if you attempt to part your hair exactly down the center, there are subtle asymmetries that will give the game away. The procedure of replacing something with its mirror image is called the parity operation; if the situation remains unchanged it is said to be invariant under parity.
It is easy to think of objects with other symmetries. The 12-sided dodecagon is invariant under rotations of one twelfth of a circle (30°), or any multiple of 30°. A perfect circle, on the other hand, is invariant under any rotation whatsoever. Because the circle can be continuously rotated, it is said to be invariant under a continuous symmetry. Rotations by discrete amounts, as for the snowflake and dodecagon, are called discrete symmetries. Parity (invariance) is another discrete symmetry.
Symmetry in physical systems carries a different meaning than simple geometrical invariance. Instead of asking whether an experiment looks identical when rotated (geometrical invariance), we ask whether the laws of physics are invariant. In other words, do objects behave in the same way when the system is rotated? A collision between two billiard balls doesn’t have geometrical symmetry; it is easy to tell if the billiard table has been rotated. In physical terms, though, nothing important has changed. If the speed of the incoming balls and the angle between them is the same in both experiments, the speed of the outgoing balls will also be identical in the two experiments, as will the angle between them. It is in this sense that physicists talk of the rotational invariance of a physical system. Rotating the initial setup leads to a rotated, but otherwise identical, outcome. Rotational invariance is a continuous symmetry; the billiard table can be rotated by any amount without affecting the outcome.
Suppose we have a theory that we want to check for rotational invariance. We obviously can’t solve the equations of the theory for every experiment that could conceivably be done, and then check that we get the same answer again when the experiment is rotated. Fortunately, this is not necessary. We can instead check the equations themselves for symmetry. The equations may involve directional quantities, those quantities that are represented by arrows (known as vectors). For instance, the Lorentz force law involves the particle’s velocity, the electric field, and the magnetic field, all of which are vectors. Other quantities, such as the mass of a particle, have no direction associated with them and so are unchanged by a rotation. To check the equations for rotational invariance, we mathematically rotate all of the directional quantities, plug the rotated values back into the equations, and check if the resulting equations have the same form as the original equations. If they do, the theory is rotationally invariant. Maxwell’s equations, the Lorentz force law, the laws of dynamics, indeed, all known laws of physics, are rotationally invariant.
Most physical systems are endowed with two additional continuous symmetries, which we can call the space shift and time shift symmetries. (Physicists call them spatial translation invariance and time translation invariance.)We expect that any experiment can be moved four feet to the left, or moved to New York, Helsinki, or Canberra, without affecting the outcome. Assuming, that is, that any purely local conditions, like the altitude, temperature, or local magnetic field, don’t affect the outcome. A system that can be moved from place to place without affecting the outcome has space shift invariance. Similarly, we expect that it won’t make a difference what time we start an experiment. Starting two hours later, or a week from next Thursday, will only change the timing of the subsequent events, not the ultimate outcome. Again, local effects must be excluded: If you have a date for tonight you’d better not try the experiment of showing up a week from next Thursday! Setting a time for the date creates a local condition: All times are no longer equivalent, so the situation no longer has time shift invariance. The laws of physics admit no privileged time: They have time shift invariance.
Here’s where Noether’s theorem comes in. The theorem declares that there is a conserved quantity associated with every continuous symmetry of a physical system. Previously in this chapter, we discovered that energy is a conserved quantity—it can’t be created or destroyed, only converted from one form to another. Is there a corresponding symmetry? Yes, in fact, it is time shift invariance. Noether’s theorem provides the connection: From the time invariance of the theory, the expression for the conserved quantity can be derived, and it turns out to be just what we call the energy.
The other two continuous symmetries we have encountered, space shift invariance and rotational invariance, naturally correspond to other conserved quantities. The conserved quantity corresponding to space shift invariance is momentum, the inertia of an object due to its forward motion. Conservation of momentum is what tells us that, in a head-on collision between a Mack truck and a VW Beetle, the combined wreckage will be traveling in the direction the Mack truck had been traveling before the collision. The conserved quantity corresponding to rotational invariance is called angular momentum, which is, roughly speaking, the amount of spin of an object. Conservation of angular momentum explains why an ice skater who is in a spin will rotate faster when he pulls his arms in. (It’s fun to test this yourself using a swivel chair. Hold your arms out to the side and start the chair spinning, then pull in your arms. The effect is increased if you hold a heavy book in each hand.)
As an example of space shift invariance, think about a skateboarder in a half-pipe. Viewed from the side, the half-pipe doesn’t have space shift invariance. If the pipe were suddenly shifted to the left, the skateboarder would find herself in midair. A space shift in this direction causes a change in the physics of the situation. Viewed lengthwise, however, the half-pipe does have space shift invariance.
If the pipe were suddenly shifted lengthwise, the skateboarder wouldn’ t notice that anything had changed. As long as she wasn’t near either end of the pipe, she could complete her maneuver as if nothing had happened.
As a result, there is a conservation law for the lengthwise direction, but not for the crosswise direction. A skateboarder riding lengthwise down the pipe moves at constant velocity. Here, velocity is the conserved quantity. Actually, the conserved quantity for space shift symmetry is momentum, which is mass times velocity. In this case, the skateboarder’s mass isn’t changing, so constant momentum implies constant velocity. In the cross-pipe direction, though, momentum and velocity are not constant. The skateboarder speeds up as she descends the pipe and slows again going up the other side. No symmetry means no conservation law.
Noether’s theorem guarantees that whenever a theory is invariant under a continuous symmetry, there will be a conserved quantity. It allows us to go from the seemingly trivial observation that the result of an experiment doesn’t depend on what time of day the experiment begins to the deep fact that there is a quantity, the energy, that remains the same before, during, and after the experiment. Equally important for elementary particle physics is the fact that we can sometimes go the other direction: If we notice that some quantity is conserved in all our experiments, the theory we are looking for may be invariant under some symmetry. Identifying the correct symmetry may lead us to the correct theory. There is as well a beauty to a symmetrical theory, as with a symmetrical face. Although experimental test is the ultimate arbiter, aesthetics can sometimes be a guide in developing new theories. The most beautiful theory is not necessarily the best theory, but it sometimes happens that theories developed for purely mathematical reasons turn out to be useful in describing nature. The search for symmetries has been a fundamental guiding principle of elementary particle physics for the last 50 years, and has led in the end to the Standard Model.
Special relativity dramatically changed physicists’ ideas about space and time, mass and energy. It left intact the concepts of particles and fields as the stuff of which things are made. The actors remained the same; only the stage on which they were acting was changed. Quantum mechanics, the other great conceptual development of early twentieth-century physics, would retain nineteenth-century ideas of space and time but would revolutionize ideas about particles and fields.