Einstein's Relativity - Physics

Physics 115: Future Physics, Fall 2005

Einstein’s Relativity by Daniel Baumann and Paul Steinhardt

Einstein’s Relativity

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Contents 1 Time and Space prior to Einstein

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2 Special RelativityHow Einstein Transformed our Understanding of Time and Space 2.1

Einstein’s Dreams∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.2

Mr. Tompkins in Wonderland∗ . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.3

Two Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.3.1

Principle of Relativity . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.3.2

Constancy of the Speed of Light . . . . . . . . . . . . . . . . . . . . .

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2.3.3

Just Two Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Time Dilation – Moving Clocks go Slow . . . . . . . . . . . . . . . . . . . . .

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2.4.1

A Light Clock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.4.2

Muon Decay in Cosmic Rays . . . . . . . . . . . . . . . . . . . . . . .

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2.4

2.5

Space Contraction

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Muons Reanalyzed . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.6

Take a deep breath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.7

”Not everything is relative!” . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.8

E = mc2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.8.1

Relative Energy and Momentum . . . . . . . . . . . . . . . . . . . . .

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2.8.2

Rest Mass – another Relativistic Invariant . . . . . . . . . . . . . . .

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2.8.3

The Meaning of E = mc2 – Unifying Mass and Energy . . . . . . . .

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The Absolute Speed Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.10 Simultaneity is Relative . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.5.1

2.9

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Time and Space prior to Einstein

”What then is time? If no one asks of me, I know; if I wish to explain to him who asks, I know not.” St. Augustine ”Time is what prevents everything from happening at once.” John Wheeler ”Absolute space, in its own nature, without relation to anything external, remains always similar and immovable; [...] Absolute, true and mathematical time, of itself, and from its own nature, flows equably without relation to anything external.” Isaac Newton

Prior to Einstein all physical theories were formulated in the passive arena of absolute and universal space and time. Newtonian mechanics describes how bodies change their position in space with time. Even quantum mechanics involves calculating probabilities that vary in space and time. Space and time were believed to form a fixed, unchanging background on which physical events unfolded. Our understanding of space and time has changed dramatically with the advent of relativity.


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Special Relativity How Einstein Transformed our Understanding of Time and Space

2.1

Einstein’s Dreams∗

PROLOGUE1 In some distant arcade, a clock tower calls out six times and then stops. The young man slumps at his desk. He has come to the office at dawn, after another upheaval. His hair is uncombed and his trousers are too big. In his hand he holds twenty crumpled pages, his new theory of time, which he will mail today to the German journal of physics. Tiny sounds from the city drift through the room. A milk bottle clinks on a stone. An awning is cranked in a shop on Marktgasse. A vegetable cart moves slowly through a street. A man and woman talk in hushed tones in an apartment nearby. In the dim light that seeps through the room, the desks appear shadowy and soft, like large sleeping animals. Except for the young man’s desk, which is cluttered with half-opened books, the twelve oak desks are all neatly covered with documents, left from the previous day. Upon arriving in two hours, each clerk will know precisely where to begin. But at this moment, in this dim light, the documents on the desks are no more visible than the clock in the corner or the secretary’s stool near the door. All that can be seen at this moment are the shadowy shapes of the desks and the hunched form of the young man. Ten minutes past six, by the invisible clock on the wall. Minute by minute, new objects gain form. Here, a brass wastebasket appears. There a calendar on a wall. Here, a family photograph, a box of paper clips, an inkwell, a pen. There, a typewriter, a jacket folded on a chair. In time, the ubiquitous bookshelves emerge from the night mist that hangs on the walls. The bookshelves hold notebooks of patents. One patent concerns a new drilling gear with teeth curved in a pattern to minimize friction. Another proposes an electrical transformer that holds constant voltage when the power supply varies. Another describes a typewriter with a low-velocity typebar that eliminates noise. It is a room full of practical ideas. Outside, the tops of the Alps start to glow from the sun. It is late June. A boatman on the Aare unties his small skiff and pushes off, letting the current take him along Aarstrasse to Berberngasse, where he will deliver his summer apples and berries. The baker arrives at his store on Marktgasse, fires his coal oven, begins mixing flour and yeast. Two lovers embrace on the Nydegg Bridge, gaze withfully onto the river below. A man stands on his balcony on Schifflaube, studies the pink sky. A woman who can’t sleep walks slowly down Kramgasse, peering into each dark arcade, reading the posters in halflight. In the long, narrow office on Speichergasse, the room full of practical ideas, the young patent clerk still sprawls in his chair, head down on his desk. For the past several month, since the middle of April, he has dreamed many dreams about time. His dreams have taken hold of his research. His dreams have worn him out, exhausted him so that he sometimes cannot tell whether he is awake or asleep. But the dreaming is finished. Out of many possible natures of time, imagined in as many 1

Excerpt from Alan Lightman’s beautiful novel ”Einstein’s Dreams”.


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nights, one seems compelling. Not that the others are impossible. The others might exist in other worlds. The young man shifts in his chair, waiting for the typist to come, and softly hums from Beethoven’s Moonlight Sonata.

[...] 29 May 1905 A man or a woman suddenly thrust into this world would have to dodge houses and buildings. For all is in motion. Houses and apartments, mounted on wheels, go careening through Bahnhofplatz and race through the narrows of Marktgasse, their occupants shouting from second-floor windows. The Post Bureau doesn’t remain on Postgasse, but flies through the city on rails, like a train. Nor does the Bundeshaus sit quietly on Bundesgasse. Everywhere the air whines and roars with the sound of motors and locomotion. When a person comes out of his front door at sunrise, he hits the ground running, catches up with his office building, hurries up and down flights of stairs, works at a desk propelled in circles, gallops home at the end of the day. No one sits under a tree with a book, no one gazes at the ripples on a pond, no one lies in thick grass in the country. No one is still. Why such a fixation on speed? Because in this world time passes more slowly for people in motion. Thus everyone travels at high velocity, to gain time. The speed effect was not noticed until the invention of the internal combustion engine and the beginnings of rapid transportation. On 8 September 1889, Mr. Randolph Whig of Surrey took his mother-in-law to London at high speed in his new motor car. To his delight, he arrived in half the expected time, a conversation having scarcely begun, and decided to look into the phenomenon. After his researches were published, no one went slowly again. Since time is money, financial considerations alone dictate that each brokerage house, each manufacturing plant, each grocer’s shop constantly travel as rapidly as possible, to achieve advantage over their competitors. Such buildings are fitted with giant engines of propulsion and are never at rest. Their motors and crankshafts roar far more loudly than the equipment and people inside them. Likewise, houses are sold not just on their size and design, but also on speed. For the faster a house travels, the more slowly the clocks tick inside and the more time available to its occupants. Depending on the speed, a person in a fast house could gain several minutes on his neighbors in a single day. This obsession with speed carries through the night, when valuable time could be lost, or gained, while asleep. At night, the streets are ablaze with lights, so that passing houses might avoid collisions, which are always fatal. At night, people dream of speed, of youth, of opportunity. In this world of great speed, one fact has been only slowly appreciated. By logical tautology, the motional effect is all relative. Because when two people pass on the street,each perceives the other in motion, just as a man in a train perceives the tree to fly by his window. Consequently, when two people pass on the street, each sees the other’s time flow more slowly. Each sees the other gaining time. This reciprocity is maddening. More maddening still, the faster one travels past a neighbor, the fast the neighbor appears to be traveling.


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Frustrated and despondent, some people have stopped looking out their windows. With the shades drawn, they never know how fast they are moving, how fast their neighbors and competitors are moving. They rise in the morning, take baths, eat plated bread and ham, work at their desks, listen to music, talk to their children, lead lives of satisfaction. Some argue that only the giant clock tower on Kramgasse keeps the true time, that it alone is at rest. Others point out that even the giant clock is in motion when viewed from the river Aare, or from a cloud.

2.2

Mr. Tompkins in Wonderland∗

We strongly encourage you to read chapters 1 - 4 of George Gamow’s classic ”Mr. Tompkins in Wonderland”.

2.3

Two Principles

”The special theory of relativity, alone among the areas of modern physics, can in large part be honestly explained to someone with no formal background in physics and none in mathematics beyond a little algebra and geometry. This is quite remarkable. One can popularize the quantum theory at the price of gross oversimplification and distortion, ending up with a rather uneasy compromise between what the facts dictate and what it is possible to convey in ordinary language. In relativity, on the contrary, a straightforward and rigorous development of the subject can be completely simple. Nevertheless, special relativity is one of the hardest of subjects for a beginner to grasp, for its very simplicity emphasizes the distressing fact that its basic notions that almost everyone fully grasps and believes, even though they are wrong. As a result, teaching relativity is rather like conducting psychotherapy. It is not enough simply to state what is going on, for there is an enormous amount of resistance to be broken down.” David Mermin, Space and Time in Special Relativity

In this section we will start to develop the formal logic of special relativity. Starting from only two basic postulates we follow Einstein to derive a dramatic revision of our common notions about space and time. 2.3.1

Principle of Relativity

The special theory of relativity rests on two experimental facts. One is a familiar part of our everyday experience, while the other was only revealed by a series of precise measurements. Galileo stated the principle of relativity in the ”Dialogue on the Great World Systems.” It occurs in his refutation of an argument of Aristotle’s that the Earth stands still. According to Aristotle, the Earth cannot be moving because a ball thrown straight up eventually falls to Earth at the point it was thrown from. If the Earth were moving, it would move while


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the ball was aloft, so that as it landed, some new part of the Earth would be beneath the ball. Galileo pointed out that this reasoning is wrong because if the Earth were moving to one side, the ball, originally on the Earth, would be moving in the same way. When thrown into the air it would, in addition, have a new vertical motion, but at the same time it would continue in its original sideways motion. Thus although the Earth would indeed move to the side as the ball went up and down, the ball would move to the side by the same amount and come down on the same part of the Earth from which it was thrown. The conclusion is that nothing can be learned from Aristotle’s experiment. Whether the Earth moves slowly, rapidly, or not at all, the ball will still land in the same place it was vertically tossed from. This observation is an instance of a general principle: One cannot tell by any experiment whether one is at rest or moving uniformly. To illustrate this consider the following situation: Imagine a spaceship with no windows moving at uniform speed. A variety of biological and physical experiments are performed, none of which would give any indication whatsoever of the velocity of the spaceship as long as the ship’s motion was perfectly smooth and uniform. One encounter the same principle in moving trains or airplanes, provided that the motion is truly uniform and there is no jostling or bouncing. Flying smoothly at 500 miles per hour, one observes that coffee poured from a pot falls quietly into the cup below and that dropped objects fall directly down to the floor. One is aware of motion only when looking out the window at clouds or at the ground. Nothing within the moving plane behaves differently from the way it would were the plane at rest or, for that matter, moving with any speed other than the one it has. Alternatively, nothing done within a plane that is either at rest or moving uniformly gives any clue as to whether the plane is at rest or moving uniformly or as to the particular speed with which it moves. Let me give a last example that you might have experienced yourself (I certainly have many times). You are in a plane at rest on the runway waiting for the tower to release your flight. Your plane has a significant delay, so you fall asleep. When you wake up after a while you are for a moment confused whether you are still on the runway or in the air. This is the principle of relativity. This is certainly not obvious, nor the kind of thing one could deduce by sheer logic. It is an experimental fact, which has been repeatedly confirmed. One can easily imagine that it might not be true. One could suppose, for instance, that a chemist might concoct a liquid that boiled at 150 degrees in his laboratory but that when place on a train moving past his lab boiled at 145 degrees when the train moved at 20 mph, at 140 degrees at 40 mph, at 135 degrees at 60 mph, etc. There is nothing inconceivable in this. However, no such substance has ever been found. The principle of relativity has never been violated. It seems impossible to distinguish between states of rest and states of uniform motion.2 In fact, this leads to the 2

A more formal statement of the principle of relativity is the following:


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following corrolary which gives the principle of relativity its name: Motion is a relative concept. Anybody moving uniformly with respect to somebody at rest is entitled to consider himself to be at rest and the other person to be moving uniformly. This denies that there is any absolute meaning to the notion of being at rest and asserts that when we say that something is at rest, we must specify relative to which of all the innumerable, equally good, uniformly moving objects the thing is at rest. The distinction between rest and uniform motion is arbitrary. This becomes obvious when we consider two spaceship in outer space far from any reference points. Spaceship A could claim to be at rest and perceive spaceship B to be moving relative to it. but the reciprocal perspective, where B claims to be at rest and see A moving is of course equally valid. 2.3.2

Constancy of the Speed of Light

The second principle of the special theory of relativity was sparked by a serious of ingenious experiments on the nature of light. It took Einstein’s genius to realize that certain deeply puzzling results arising from experiments on light had an explanation that had to do not so much with the nature of light as with the nature of space and time. The speed of light is enormous but finite. In vacuum its value is c = 3.0 × 108 m · s−1 .

(1)

We should be more precise however. As we just discussed speed is a relative concept. Any value for the speed of an object can only be given relative to a reference object. If I run after a car the relative speed at which the car recedes decreases. This leads one to suspect that when one says the speed of light is c, one must also say with respect to what kind of observer it moves with c. You might be tempted to say that it must be an observer who is at rest, but the principle of relativity tells us that any observing moving uniformly with respect to an observer at rest can also be regarded as being at rest. Here comes the shocker: There is overwhelming experimental evidence for the following The laws of physics have the same form in all inertial reference frames. Here, the laws of physics refer to the accumulated results of all experiments and inertial reference frames are reference frames with uniform relative motion. The specification of inertial reference frames or uniform motion is crucial. If a ship moves steadily on a perfectly calm sea, shut up in a cabin one is completely unaware of the motion. If, however, waves bounce the ship up and down as it moves, one is immediately aware of the nonuniform motion. Similarly, things behave as they do at rest in a train that progresses smoothly at constant speed in a straight line, but if the train veers off to the left, objects hanging from the ceiling will swing to the right, tea will slosh in cups, and standing passengers will have to brace themselves to preserve their balance, none of which phenomena could happen in a stationary or uniformly moving train. Only states of motion in a straight line with constant speed are indistinguishable from states at rest.


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extraordinary result: The speed of light has the same value c = 3.0 × 108 m/s with respect to any observer in uniform motion. Let me describe how this revolution came about. People used to think that light moved with a speed c with respect to a definite something which they called the ether. This seemed reasonable at the time since light was considered a wave and all other waves traveled through a medium, the classic example being sound waves traveling through air. If this were so, prerelativistic reasoning led them to expect that if the Earth moved through the ether with speed v, the speed of light moving past the Earth in the same direction as the Earth’s motion through the ether would be c − v when measured from the Earth, the speed of light moving past the Earth in the opposite direction from the Earth’s motion through the ether would be c + v when measured from the Earth, and in general the speed of light with respect to the Earth in any arbitrary direction would depend on the angle between that direction and the direction of the Earth’s motion through the ether. The famous Michelson-Morley experiment was an attempt to measure this directional dependence of the speed of light with respect to the Earth and thus to determine the speed of the Earth with respect to the ether. They found no effect! The speed of light with respect to the Earth has the same value c whatever the direction of motion of the light. Thus if the ether does exist, it must be managing in a most mysterious way to escape our effort to detect it. As Einstein showed, the way out of this dilemma is to deny the existence of the ether and face courageously the fact that light moves with a speed c with respect to any other observer. Einstein however wasn’t just relying on these experimental facts to announce the principle of the constancy of the speed of light, but took significant intellectual inspiration from Maxwell’s theory of electricity and magnetism. Maxwell was a remarkable English physicist who almost single-handedly derived the mathematical framework for describing all electric and magnetic phenomena. In particular he showed that light may be view as a combination of oscillating electric and magnetic fields propagating through space. He found a formula that describes how electromagnetic waves (light) propagate through the vacuum: ~ 1 ∂2E = 0. (2) c2 ∂t2 Looking at Maxwell’s expression for the propagation of light in vacuum Einstein realized the following important point: The speed of the electromagnetic wave is simply c, independent of the motion of either the light source or the receiver. Maxwell’s theory was totally consistent with Michelson-Morley’s null result. ~− ∇2 E

Realizing that Newton’s and Maxwell’s views were inconsistent, Einstein proposed the following bold resolution: If the speed of light is constant, then the clocks and rulers used to measure velocity must not be constant! Space and time have to change.

Einstein’s Relativity 2.3.3

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Just Two Principles

The special theory of relativity is based on Galileo’s principle of relativity: One cannot tell by any experiment whether one is at rest or moving uniformly. Motion is a relative concept. Anybody moving uniformly with respect to somebody at rest is entitled to consider himself to be at rest and the other person to be moving uniformly. and the constancy of the speed of light: The speed of light has the same value c = 3.0 × 108 m/s with respect to any observer in uniform motion. Accepting just these two principles as our starting points we will use pure logic to come to revolutionary conclusions about the true nature of space and time. These insights will ultimately transform our understanding of gravity and shape the modern view of the Universe.

2.4

Time Dilation – Moving Clocks go Slow

Einstein was the grand master of thought experiments. Let us follow one of his most famous thought experiments to see how the postulates of relativity challenge our most basic ideas about the nature of time. Suppose you are on the Dinky train going from Princeton to Princeton Junction. You are equipped with a powerful telescope to watch a large clock at Princeton station and your friends are waving goodbye, as you move away. Since this is a thought experiment we are allowed to assume that the Dinky moves at the speed of light. Suppose you leave at exactly midday. What would you see? The clock would appear to stand still because the light emitted by the clock would be traveling away from it at the same speed as the Dinky, and light emitted at later times could not catch up with you. Thus, you would always see the clock’s hands stand at 12 o’clock. Indeed all other happenings next to the clock (like your friends waving) would also be seen by you exactly as they were at midday, because the light you receive from Princeton station at all later times is the light that left then. If one could move as fast as light, time would appear to stand still! 2.4.1

A Light Clock

To analyze further how time behaves according to special relativity, we must carefully consider how it is measured by a clock. In general, a clock is a complex mechanism that is difficult to analyze. In order not to get confused by these irrelevant details we consider the


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conceptually simplest clock, a ’light clock’. A light clock is constructed by means of a light source that emits signals which travel a distance d and are then reflected back to the source (Fig. 1a). The time interval between emission and return of the signals to the mirror defines the ’ticks’ of such a clock; they occur a time 2t apart. The ratio of the distance traveled by the light between ticks and the time interval between ticks is equal to the speed of light 2d d d = ⇒ t= . (3) 2t t c Now imagine putting the light clock onto the Dinky speeding away from the platform at Princeton at a speed v = 0.866 c. Alice on the Dinky will still measure the time tA = d/c (Fig. 1a) independent of the state of motion of the Dinky because she considers both herself and the clock to be at rest. Relativity effects only come into play when an observer and a clock (or a ruler) are moving relative to one another. Her friend Bob is on the platform. c=

Figure 1: Light clock on the Dinky. a) Alice’s view. b) Bob’s view. How will the light clock look to him? Since the Dinky and the light clock on board are moving while the light signal travels between the mirrors, Bob who is at rest (relative to the platform) will see the light go on a diagonal path (Fig. 1b) that is longer than the distance between the mirrors! However, both Alice and Bob agree that light travels at speed c (as demanded by Maxwell’s laws of electromagnetism, Einstein, and relativity). The only way to reconcile this with the fact that the light has to travel more distance is if Bob measures a longer time between ticks 2tB . We can even quantify this. The distance traveled by the light according to Bob can be calculated from the diagram using Pythagoras’ theorem l2 = c2 t2B = d2 + v 2 t2B . Substituting d = ctA and solving the equation for tB one finds tA ≡ γtA , tB = q 2 1 − vc2

(4)

(5)


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where the second equality introduces a new symbol γ (the Greek symbol ’gamma’). The factor γ is just shorthand for 1 γ=q . (6) 2 1 − vc2 For the case v = 0.866 c we find γ = 2, so that tB = 2tA ; Bob sees time stretched by a factor of 2 compared to Alice, due to the large speed of the Dinky. The time stretching effect is called time dilation. The factor γ appears over and over in relativity problems, as you will see. It measures how big relativity effects are. If relativity effects are tiny, γ is near one. If relativity effects are important, γ is much greater than one. You should develop the following reflex when you read almost any relativity problem: look to see how fast two observers are moving relative to one another and immediately compute γ because you know you are likely to need it right away to do any problem. Figure 2 is a plot showing how the gamma factor depends on the velocity v.

Figure 2: The dependence of the Gamma factor (γ) as a function of velocity. Note how γ becomes large as v approaches the speed of light. This means that the relativity effects (like the stretching of time) become big for speeds close to the speed of light. It is instructive to look at two extreme limits of the formula for the gamma factor: For very small speeds, γ is very close to 1, so relativity and time dilation are unimportant. On the other hand, as the speed v approaches the speed of light, the gamma factor grows larger and larger, approaching infinity. (The formula for γ does not make sense for speeds greater than the speed of light.) Is time dilation ever important in every day life? Consider that, for most of us, the fastest we ever move (compared to someone standing on the ground) is when we take a trip in an airplane. Its maximum speed is v = 1000 km/h ∼ 300 m/s = 10−6 × c .

(7)


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The corresponding gamma factor is γ=q

1 1 − (10−6 )2

≈ 1.000000000005 .

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This is very, very close to one, meaning that relativity and time dilation are tiny effects. Amazingly enough the small time dilation effect has been measured using highly precise atomic clocks aboard trans-continental flights! 2.4.2

Muon Decay in Cosmic Rays

Cosmic rays are particles from outer space that arrive at the Earth at extremely high relative speeds v (often v/c ≈ 0.99). Their origin and the source of their great energies, is still something of a mystery. At a height of about 20 km above sea level they collide with atoms in the Earth’s atmosphere, and among the particles resulting from these collisions are particles called muons (µ). These also move very rapidly towards the ground (their mean speed being nearly the same as that of the incoming cosmic rays), but they are unstable, decaying rapidly to less massive particles (electrons and neutrinos). One can measure this decay rate in the laboratory; the mean lifetime of a muon at rest is trest ≈ 2.2 × 10−6 s .

(9)

The mean flight time through the Earth’s atmosphere, from where they are created, to sea level is 20 km T ≈ ≈ 6.7 × 10−5 s ≈ 30 trest . (10) 5 0.99 × 3.0 × 10 km/s Given that the time of flight is many times the muon half-life, we expect only a tiny fraction of all the muons created in the upper atmosphere to make it to sea level. Yet, this is not what is observed. In fact, about 100% of all muons make it to the ground. How can this be? The essential point is that we used the half-life of a muon at rest and forgot to take into account the fact that it is moving fast (99% the speed of light!). We need to check if relativity and time dilation effects are important. For this, follow your instinct and calculate γ: 1 1 γ=q ≈ 7.1. (11) =√ 1 − 0.992 1 − v 2 /c2 Sure enough, γ is significantly greater than one, so we cannot ignore relativity. Hence, the half-life as viewed by those of us standing on the Earth is stretched by a factor 7.1. With the extra time, the muons have enough time to reach sea level before they decay. Hence, the fact that many muons are observed at sea level is proof that time is stretched, just as Einstein predicted. We can also demonstrate the same effect by making muons in a high energy particle accelerator on Earth, such as the four-mile long accelerator at Fermilab outside of Chicago. We can directly compare the lifetime of a stationary muon with the lifetime of a moving muon and show that the moving muon lives longer by precisely the factor γ.


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Figure 3: Length Contraction. Bob, who is at rest relative to the rail tracks, sees Alice moving at speed v = 0.866 c (γ = 2) and measures LB = 5 km for the distance from Princeton to Princeton Junction. Alice considers herself at rest and believes the surroundings are moving at speed −v. To her the length of the (moving) rail tracks between Princeton and Princeton Junction is contracted, LA = LγB = 2.5 km.

2.5

Space Contraction

The speed of an object is the distance it travels divided by the time it takes. Relativity says that the speed of light is the same for all observers, but that the time that transpires is different for different observers. The only way this is possible is if distances are also different as viewed by different observers. In fact, having established that times stretches it is now easy to show that distances contract. Consider once more the Dinky racing between Princeton and Princeton Junction at speed v = 260, 000 km/s = 0.866 c (γ = 2). Alice is on the Dinky, Bob is on the platform. The distance between Princeton and Princeton Junction is LB = 5 km as measured by Bob. Alice has the reciprocal viewpoint: The Earth and the track are moving with speed −v relative to the train. Using the fact that distance equals speed times time, we know that for Bob tB = LB /v, where LB denotes the distance from Princeton to Princeton Junction as measured by Bob. Similarly, for Alice, we have tA = LA /v, where LA denotes the distance from Princeton to


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Princeton Junction as measured by Alice.. Finally, we have that Bob and Alice measure different times due to time dilation: tB = γtA . Putting this together we have tB =

LA LB = γtA = γ . v v

(12)

We can compare the second expression to the fourth, cancel the factors of v, and obtain LA =

1 LB . γ

(13)

This says that Alice and Bob disagree about the distance between the train stations. If Bob measures the distance between two pointsrthat are stationary from his point of view, then 2

Alice measures a distance that is 1/γ = 1 − vc times smaller. The distance measurements of Alice and Bob are related via equation (13). Since γ > 1, we have 1/γ < 1 or LA < LB . Moving objects (in this case the rail tracks between Princeton and Princeton Junction) appear shortened, hence the expression ”length contraction”3 . 2.5.1

Muons Reanalyzed

Let us revisit the muon problem of section 2.4.2. This time we take the viewpoint of an observer moving along with the muon. Now there is no time dilation effect for the muon decay, since the muon and the observer are at rest with respect to one another. So, the muon lifetime is trest ≈ 2.2 × 10−6 s , (14) the same as if we produce it at rest in the laboratory. So does the observer traveling with the muon see that it reaches sea level or not? To answer that question, we must first note that, according to our new observer moving with the muon, the surface of the Earth is fast approaching. The approach speed is the same speed v as the observer standing on the Earth measured for the muon (v/c ≈ 0.99). So the new observer finds that the thickness of the atmosphere - between where the muon is created and the surface of the Earth - is measured to be contracted by a factor of 1/γ. The second observer concludes that the atmosphere is short enough so that the muons can reach the Earths surface without decaying. This discussion shows that we have two equivalent perspectives for interpreting the muon data. Muon decay experiences time dilation in the frame of the Earth or the width of atmosphere appears contracted in the frame of the muon. This is an example of a general 3

Length contraction only applies to the measurement of distances along the direction of motion. Distances perpendicular to the direction of motion are uncontracted and both observers agree on them. Notice that in Figure 3 I drew the Dinky contracted along the direction of motion from Bob’s point-of-view. This is because the Dinky is moving relative to Bob and he therefore sees its length contracted. Notice also, however, that I didn’t change the height and width of train between Bob’s and Alice’s views of the Dinky. This is because I was careful to take account of the fact that lengths perpendicular to the direction of motion are not affected by relativistic contraction.


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characteristic: Time dilation for one observer turns into an issue of length contraction for a different observer. Arguments concerning time and space are related by the laws of relativity.

2.6

Take a deep breath

Since this is the essence of (special) relativity, let us summarize our results so far: • The time between two events in a moving reference frame is stretched: An observer in relative motion measures a longer time as compared to an observer at rest. (”Moving clocks go slow”) t = γtrest . (15) Notice, however, the important fact, that by the principle of relativity every observer in uniform motion can consider himself at rest and declare his time to be trest . There is a perfect symmetry between the interpretation of observations of two reference frames in relative motion. This is what relativity is all about. • The distance between two points (or the length of an object) along the direction of motion in a moving reference frame is contracted: An observer in relative motion measures a shorter distance as compared to an observer at rest. 1 L = Lrest . (16) γ It is worth pausing for a moment to contemplate what these ideas imply for your common sense notions of fixed and unchanging time and space. Crazy!

2.7

”Not everything is relative!”

Although the name relativity suggests that everything depends on the observer and nothing is absolute, there are in fact quantities that are the same for all observers. For these quantities, every observer measures the same value. Such quantities are called invariants 4 . In this section, we will give a brief discussion of this important but often misunderstood concept. Recall the Dinky experiment of the previous section. Alice is on the Dinky which is moving at close to the speed of light between Princeton and Princeton Junction. In Bob’s frame the Dinky takes a time tB to cover the distance LB between the two train stations. Since the train moves at speed v, Bob finds that LB = vtB . Alice sees things differently. She is at rest relative to the Dinky, so the Dinky doesn’t change position relative to her at all: both she and the Dinky remain at the same place in her coordinates. Instead, it is the countryside (and Bob) that are moving. She sees first the Princeton station and then 4

In fact, Einstein often expressed that he considered the concept of these invariants the central result of his theory of relativity and that he regretted not naming relativity, ’invariance theory’.


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Princeton Junction station outside her window and she measures the time interval between those two events to be tA . Now, we have already pointed out that tA and tB are different. Also, Alice does not measure the distance between stations to be LB . But let’s compute the following strangelooking quantity: (c × time)2 − (distance)2 , (17) where the time here is the interval between when the train is at Princeton Station and when it is at Princeton Junction, and the distance is how far each sees the train move. For Bob, we get (ctB )2 − L2B = (ctB )2 − (vtB )2 ! v2 2 2 = c 1 − 2 tB c = c

2

tB γ

!2

.

Notice that, even though γ appears here, we have not yet used relativity because we are only discussing Bob and what Bob measures. All we have done is use the fact that LB = vtB (distance equals speed times time) and done some algebra. Now let’s consider Alice. From her vantage point, the distance traveled by the train is zero. So, the quantity (c × time) 2 − (distance)2 is just equal to: !2 t B , (18) c2 t2A = c2 γ where we here we have used relativity (namely, the time dilation results from before) to relate Alice’s time tA to Bob’s time tB . But, then, compare the final expressions we obtained for Bob and Alice: they are identical! Even though Alice and Bob disagree about time and space, they agree completely about (c × time)2 − (distance)2 . If we had repeated the calculation using a third observer, Carol, traveling relative to both Bob and Alice, we would have found the same thing. She would have measured a different time and space interval, but the same value for this combination of time and space. This invariant combination is called the spacetime interval. So, what Einstein meant in his quote above is that he could just as well have called his idea the ”invariance theory” because it says that certain quantities, like the space-time interval are not relative. Instead, he chose to emphasize the fact that other, more familiar quantities, like time and distance, are relative.


2.8 2.8.1

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E = mc2 Relative Energy and Momentum

Energy and momentum are other relative concepts, depending on the state of motion of the observer. We did not need Einstein to tell us this. We already knew it from Newton. Consider a mass at rest. Newton says it has zero momentum and zero kinetic energy. Now move in a car at speed v, and it appears from your vantage point like the mass has non-zero momentum and kinetic energy. So, even for Newton, momentum and energy are not invariants. Motion is relative and, therefore, the associated energy and momentum are as well. The same is true for Einstein’s theory of special relativity, although the detailed formulas for how energy and momentum depend on the speed of the observer v are different. 2.8.2

Rest Mass – another Relativistic Invariant

Just like two observers disagree about measurements of space and time but agree on the combination (c × time)2 − (distance)2 , Einstein found that observers disagree about measurements of energy E and momentum p, but there is an invariant combination of energy and momentum on which all observers agree E 2 − (pc)2

(19)

For any object, this combination is equal to (mc2 )2 where m is the ”rest mass” of the object. Why do we call it the ”rest mass”? Just imagine an object that is at rest according to the observer, an object sitting on the table, for example. According to the observer, the particle has no momentum, so Equation (19) reduces (after you take the square root of both sides) to something you might recognize: E = mc2 .

(20)

Even at zero velocity, Einstein’s theory says the object has energy. In fact, it has lots of energy, an amount equal to the mass of the object multiplied by c2 . This is arguably the most famous consequence of Einstein’s special theory of relativity. (Something to check for yourself: What does the invariant tell us the energy of the object is if it is not at rest; for example, what if its momentum is p according to the observer? You don’t get such an easy-to-remember formula, do you?) 2.8.3

The Meaning of E = mc2 – Unifying Mass and Energy

”What I like about E = mc2 is not only its simplicity but [in] how many different environments in the universe the equation applies. It applies to what’s going on inside of stars, inside of our own sun. It applies to what’s going on in the center of the galaxy. It applies to what’s going on in the vicinity of black holes. It applies to all the events that took place at the big bang. Our


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fundamental knowledge of the formation and evolution of the universe would be practically zero were it not for the existence and understanding of that equation. And, as a recipe for converting matter into energy and back into matter, it’s something that doesn’t happen in your kitchen or in everyday life, because the energies required to make that happen fall far outside of anything that goes on in everyday life. Because, for example, visible light that you use to illuminate the page you read, you can calculate how much energy that light has. It’s not enough to make any particles with. You need more energetic light than visible light, than ultraviolet. You gotta get into X-rays. If you get high enough energy X-rays passing by your room, spontaneously, unannounced, unprompted, unscripted, they will make electrons. The whole suite of particles you learn about, all of those can be manufactured simply by entering a pool of energy where that energy is above the mass threshold for that particle. We are fortunately not bathed in that level of energy, because we would first get sterilized, then it would mess with our DNA, and then we would die. So we should be glad we don’t see E = mc2 happening in front of us. It would be a dangerous environment indeed. There are places in the universe where this equation is unfolding moment by moment. How else do you think the universe can be as big as it is now but start out with something smaller than a marble? E = mc 2 is cranking, converting matter into energy and back again. When you’re energy you don’t have to take up much space. You can get very small when you’re a pocket of energy. So I was once asked what do I think is the greatest equation ever. There are a lot in the running but I would have to put E = mc2 at the top. If you sit back, look at the universe and say, what equation holds all the cards, that would be E = mc2 . That’s all I gotta say.” Neil deGrasse Tyson Hear Einstein himself talk about E = mc 2 (in ”English”): http://www.aip.org/history/einstein/sound/voice1.mp3

The impact of E = mc2 onto our lives needs no explanation. E = mc2 has transformed our world. Boosted by the enormous value of the speed of light c, minuscule amounts of mass can be converted into incredible energies. E = mc2 powers nuclear fusion inside the Sun, so we certainly couldn’t exist without the mass-to-light conversion implied by Einstein’s formula. Radioactivity is E = mc2 in action. Particle accelerators use E = mc2 in reverse by smashing particles into each other with large enough energies to produce massive particles in the resulting debris. Pure energy in converted into mass, m = E/c2 . Einstein’s deep results about the true nature of space and time, have turned into something very concrete, which for good and bad have changed the course of the world forever.

2.9

The Absolute Speed Limit

Time stretches for fast moving objects as quantified by the relation t = γtrest .

(21)


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In particular, the gamma factor grows indefinitely as one approaches the speed of light (see Figure 2), so time appears to stand still for objects traveling at the speed of light (t → ∞). We just saw that energy also increases for fast moving objects. The relation of the total energy of a moving object to its rest energy bears a (non-accidental) similarity to equation (21) E = γErest = γmc2 . (22) The energy of an object in motion therefore also increases indefinitely as it approaches the speed of light. In other words, accelerating a particle to and beyond the speed of light requires an infinite amount of energy! Infinity is a lot plus more, so it is impossible to travel faster than the speed of light. One simply hasn’t got enough energy to get there. There is a loophole in this argument, though. What if the particle has zero mass. Then, this argument fails because γmc2 combines two factors, one of which is zero and one of which approaches infinity. This is ill-defined, so we cannot trust the energy formula we used. To find out what happens in this case, we go back to the invariant, E 2 − (pc)2 . We said that this must equal (mc2 )2 , but this is zero. But this still leaves a perfectly sensible possibility: the object may have zero mass, but it can still have non-zero energy E and non-zero momentum p provided E = pc (or −pc). Such ’particles’ do exist: one is called the photon, the quantum of light. The photon travels at the speed of light, of course. From Einstein’s analysis, we have just learned that it must, therefore, have zero mass and its energy must equal its momentum times the speed of light. This is indeed what is measured in the laboratory. Another example is the graviton, the quantum of gravity. The graviton is to a gravitational wave what the photon is to an electromagnetic wave. According to Einstein’s general theory of relativity (see below), the graviton has zero mass, so it also travels at the speed of light. So, the legal limits on the speed of light are rather subtle: An object with mass can travel at speeds up to the speed of light, but can never reach or exceed it. Massless objects can travel at the speed of light, but never slower or faster. And nothing can travel faster than the speed of light. 5

2.10

Simultaneity is Relative

Having established the essence of (special) relativity (time dilation and length contraction) we can now investigate some of the perplexing consequences of this new understanding of space and time. Let us begin by asking the following, seemingly innocent question: How do you know when is ”right now” some distance away? When are spatially separated events happening simultaneously? A simple way to define simultaneity is to synchronize two clocks locally and 5

Science fiction stories and, occasionally, physicists talk about ‘tachyons’, hypothetical particles that might travel faster than the speed of light. Such particles, if they existed, could wreak havoc because they could also travel backwards in time. For example, one could travel back in time and kill your parents before you were born. Most physicists believe this kind of time travel is impossible and that tachyons are impossible in nature. [At least one of your TAs however believes tachyons are not completely crazy and might be real. Try to find out who!] )


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then send one far away. Two events occurs simultaneously whenever the two clocks read the same time. Of course, you need good clocks! They must remain synchronized i.e. ”flow” at the same rate. Now, here is the problem: According to Einstein moving clocks don’t ”flow” at the same rate. Time is individual, not universal. Absolute synchronization is impossible! This is the essence of the strange result that simultaneity is relative. The basic idea is illustrated by the following thought experiment:

Figure 4: Simultaneity is relative. Left: Passenger on the train see events 1 and 2 happening simultaneously. Right: Stationary observers declare 1 to happen before 2. Imagine an explosion is suddenly set off in the middle of a moving train (you can picture the Dinky again if you want, but any train moving close to the speed of light will do). Since the explosion is set off at the center, light from the explosion reaches the front end and the back end of the train at the same time as judged by the passengers on the train. Stationary observers outside the train however will disagree with this observation! They will claim that the light from the explosion reaches the back first and then the front. To see this more clearly consider the following sketches (Figure 4) which show snapshots of the events from the point of view of observers outside the train. The back end of the train moves toward the light ray, whereas the front end moves away. Since the speed of light is constant, the back is reached first. What the moving observers on the train judged as simultaneous events is non- simultaneous for observers at rest outside the train. Simultaneity is not an absolute concept.