Physics 115: Future Physics, Fall 2005
Einstein’s Relativity
by Daniel Baumann and Paul Steinhardt
Einstein’s Relativity
p.2
Contents
1 Time and Space prior to Einstein
4
2 Special Relativity
How Einstein Transformed our Understanding of Time and Space
5
2.1
Einstein’s Dreams∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.2
Mr. Tompkins in Wonderland∗ . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.3
Two Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.3.1
Principle of Relativity . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.3.2
Constancy of the Speed of Light . . . . . . . . . . . . . . . . . . . . .
9
2.3.3
Just Two Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
Time Dilation – Moving Clocks go Slow . . . . . . . . . . . . . . . . . . . . .
11
2.4.1
A Light Clock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.4.2
Muon Decay in Cosmic Rays . . . . . . . . . . . . . . . . . . . . . . .
14
2.4
2.5
Space Contraction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
Muons Reanalyzed . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2.6
Take a deep breath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.7
”Not everything is relative!” . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.8
E = mc2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.8.1
Relative Energy and Momentum . . . . . . . . . . . . . . . . . . . . .
20
2.8.2
Rest Mass – another Relativistic Invariant . . . . . . . . . . . . . . .
20
2.8.3
The Meaning of E = mc2 – Unifying Mass and Energy . . . . . . . .
20
The Absolute Speed Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.10 Simultaneity is Relative . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.11 (Non-)Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2.11.1 The Twin Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
2.11.2 Barn and Pole Puzzle . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
2.12 Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.5.1
2.9
3 General Relativity
”Matter tells space how to curve. Space tells matter how to move.”
33
3.1
Gravity and Acceleration – The Principle of Equivalence . . . . . . . . . . .
34
3.2
Curved Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
Einstein’s Relativity
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3.3
Bending of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
3.4
”Gravity is not a Force” . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
3.5
Beautiful and True: The Spectacular Confirmation of GR . . . . . . . . . . .
42
3.6
Further Implications and Tests of GR . . . . . . . . . . . . . . . . . . . . . .
44
3.6.1
Gravitational Waves - Ripples in Spacetime . . . . . . . . . . . . . .
44
3.6.2
Down-to-Earth GR: GPS . . . . . . . . . . . . . . . . . . . . . . . . .
45
3.7
Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
3.8
The Big Bang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
4 Time and Space after Einstein
48
4.1
Warped Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
4.2
Speculations on the Breakdown of General Relativity . . . . . . . . . . . . .
48
4.2.1
Big Bang Singularity and Quantum Foam . . . . . . . . . . . . . . .
48
4.2.2
Dark Matter and Dark Energy . . . . . . . . . . . . . . . . . . . . . .
48
Einstein’s Relativity
1
p.4
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.
Einstein’s Relativity
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p.5
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”.
Einstein’s Relativity
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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.
Einstein’s Relativity
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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
Einstein’s Relativity
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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:
Einstein’s Relativity
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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.
Einstein’s Relativity
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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
p.11
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
Einstein’s Relativity
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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
d
d
2d
=
⇒ 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=
a)
c tA = d
b)
c tB
d
v
v tB
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
tB = q
≡ γtA ,
2
1 − vc2
(4)
(5)
Einstein’s Relativity
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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.
7
γ
2
1
0.2
0.4
0.6
0.8
1.0
v/c
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 .
(8)
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)
=√
2
2
1 − 0.992
1 − v /c
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.
Einstein’s Relativity
p.15
Einstein’s Relativity
p.16
Bob’s view
B
Dinky
v
Princeton
Tracks
Princeton
Junction
LB
Alice’s view
A
v
Tracks
LA
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
Einstein’s Relativity
p.17
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.
Einstein’s Relativity
p.18
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 invariants4 . 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’.
Einstein’s Relativity
p.19
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.
Einstein’s Relativity
2.8
2.8.1
p.20
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
Einstein’s Relativity
p.21
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 = mc2
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 = mc2 (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)
Einstein’s Relativity
p.22
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!] )
Einstein’s Relativity
p.23
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:
1
1
2
2
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.
2.11
(Non-)Paradoxes
If the theory of special relativity is applied imprecisely it leads to a number of apparent
paradoxes. All the paradoxes disappear if analyzed correctly. Learning about these famous
(non-)paradoxes however is fun and in fact helps our understanding of the subtleties of the
theory, so we will describe two representative examples.
Einstein’s Relativity
2.11.1
p.24
The Twin Paradox
A young astronaut, let’s call her Catha, takes a trip to a star 25 light-years away in a
spaceship that can travel at 99.98 percent the speed of light, giving her a Lorentz factor of
γ = 50. The astronaut has a twin sister, let’s call her Caro, who remains home on Earth.
Fifty years pass, and the Earthbound twin, grayhaired and showing clear signatures of age,
goes to the spaceport to welcome her adventurous sister. For Catha only one year has elapsed
and she is still young and lively! From the point of view of the twin on Earth, this is because
time itself slowed down on the ship. Clocks, and biological aging processes, are slowed to
one- fiftieth their normal rate on this fast spaceship. To the astronaut herself, however,
things seemed perfectly normal. From her point of view it was the Earth and the star that
were moving, so the distance between them shrank to half a light-year, for a one-year round
trip at close to the speed of light. But both agree that the astronaut is now 49 years younger
than her twin!
Early in the history of relativity, this story was offered as a refutation of the theory. Why
isn’t the twin on Earth younger? After all, from the point of view of the astronaut, it is the
Earthbound twin that is moving and therefore the Earthbound clock that ran slow! There
appears to be a contradiction.
The answer is that one can make a distinction between the astronaut and her sister. The
situation is not perfectly symmetric. The astronaut had to leave Earth, accelerate to a
stupendous speed, and turn around (another period of acceleration) at the star. Thus,
she is not in uniform motion at constant speed, and the symmetry, by which she feels the
Earthbound clock runs slow, does not apply. Analyzed in detail, the problem reveals that
from the point of view of the astronaut, most of the 50 years passed on Earth during the
short time she was turning around at the star. (See section 2.12 for a nice geometrical
explanation of the twin paradox).
Special relativity survives without contradiction.
2.11.2
Barn and Pole Puzzle
Another fun puzzle concerns a Harvard pole vaulter and a New Jersey farmer. The Harvard
student carries a pole that is longer than the farmer’s barn. Having studied some relativity in
his spare time the farmer proposes the following bet: He claims he can trap the pole vaulter
and his pole in the barn by closing the back and front door as the pole vaulter runs through
the barn (see Figures 5 to 7). The farmer gathers some witnesses to observe and judge the
event. He seems confident. Here is what the witnesses next to the barn observe: The pole
vaulter runs very fast. Due to length contraction, the pole is shortened and fits into the barn
(Figure 6). When the pole vaulter is entirely inside the barn, the farmer simultaneously shuts
the two doors, proving that the pole is shorter than the barn. The pole vaulter then bursts
through the second door. The witnesses agree: The farmer won! However, the Harvard
student starts complaining and claims he has been tricked. From his point of view the barn
was moving very fast; it appeared shorter and there was no way the pole could fit (Figure
7). According to the pole vaulter the farmer cheated. He closed the front door before the
back one! However, none of the witnesses agrees!
Einstein’s Relativity
p.25
a)
b)
Figure 5: When pole vaulter and barn are at rest relative to each other, the pole won’t fit
in the barn.
a)
v
b)
v
c)
v
Figure 6: Point of view of the farmer and the witnesses.
Einstein’s Relativity
p.26
a)
b)
c)
v
v
v
?
d)
v
Figure 7: Point of view of the upset pole vaulter.
2.12
Spacetime
”... 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 independence.”
Hermann Minkowski
The space-time diagram is a nice graphical way of understanding many features of special
relativity. The concept is very simple and powerful. We can describe everything that happens
in the universe as a series of events. Each event occurs at a particular point in space and at a
particular moment in time. So, we can indicate the event as E = (x, t) and plot it on a graph
with space plotted along one axis and time plotted along the other axis. The tradition is to
put space (x) along the horizontal axis and time along the vertical axis. Another tradition
is to plot c t instead of t along the vertical axis, where c is the speed of light. So, both the
horizontal axis and the vertical axis are measured in units of distance. (See Figure 8).
In addition to marking events, we can mark the history of a particle as a world-line.
Imagine taking a series of snapshots of the particle as it moves through space. Each snapshot shows the particle at a particular place and at a particular time: an event. Join the
events together with a curve (or line), and you have the world-line of the particle. If the
particle is at rest, then its world-line is just a vertical line because the particle remains at
the same point in space (or, equivalently, the same value of x) at all times. If the particle
is moving at constant speed, its world-line is a line with constant slope. The faster the
Einstein’s Relativity
p.27
ct
1
particle at rest
space-time event
massive particle
Time
(x c)
light ray
2
3
x
Space
Figure 8: Spacetime diagram showing events (1,2,3), the world-line of a person standing still
(vertical line), a person moving at speed v and a light ray.
particle, the more the line is tipped away from vertical. If the particle starts from the origin,
say, and is moving at the speed of light, its position at any later time t is x = c t. This
is a line that is at 45 degrees. If the particle is speeding up and slowing down, then the
world-line is no longer straight; it can curve, bend and wiggle as it proceeds forward in time.
A world-line can never have a slope greater than 45 degrees (relative to vertical) since this
would correspond traveling faster than light.
The result is called a ”space-time” diagram. Here we have only shown one spatial direction x. But we can imagine a more complicated graph showing all three spatial dimensions
(height, width, depth) plus time - four coordinates altogether to label each event. Physicists describe this as a ”four-dimensional space-time” and say that the universe has ”four
dimensions”. Sounds fancy, but it really is no more complicated than the graph we have just
discussed.
The diagram we have drawn so far represents the point-of-view of just one observer. That
is, the x and c t label the space and time coordinates for events and world-lines of particles
as measured by one observer only. So, we have not done any relativity yet. Next, we want
to compare these measurements to the space and time coordinates of a second observer who
is moving at speed v with respect to the first observer.
Before we do that, though, let’s consider a simple allegory. Let’s suppose you were asked
to make a quantitative map of the campus indicating the x and y coordinates of all the major
landmarks. You are told to use Fitzrandoph Gate as the origin. You choose Nassau Street
as the x axis and the line between the Gate and Nassau Hall as the y axis. You carefully
measure x and y for every building and landmark, record it in a table and draw it on a map,
just as indicated in Figure 9. You measure very carefully, so you are confident that your
measurements of x and y are accurate beyond question.
Einstein’s Relativity
p.28
y
y
y’
3
3
1
y2
1
2
2
x’
y’2
x’2
x
x
x2
Figure 9: Your map of campus landmarks using you choice of x and y directions (left) and
the x′ and y ′ directions chosen by the second student (rotated relative to yours).
So, you are shocked when a second student is given the same task, makes measurements
beginning from the same Gate, and yet obtains completely different results for x and y. You
say that you are confident that you have measured accurately; but the other student claims
the same. Who is right? Your Physics 115 AI is asked to review the records and judge who
is correct. After studying the two tables carefully, the AI concludes: both are right! The
discrepancy has arisen, the AI says, because the two used different coordinate systems for
measuring x and y. The second student chose the x axis to lie precisely east-west and the
y axis to like precisely north-south, and these directions are rotated compared to the axes
you chose, as shown in Figure 9. So, the second student’s coordinates for x and y are not
the same as yours because the choice of axes is different. In other words, x and y are not
the same in every frame of reference if you use different coordinates axes, you get different
measurements for each.
You are skeptical and challenge the AI: How can you be so sure? [Important life-lesson:
never question a Physics 115 AI.] Simple, is the reply. If observers measure carefully but
use different coordinate axes, there is a telltale signature. The values of x and y vary from
observer to observer, but the combination x2 + y 2 is the same for both. This quantity represents the square of the distance from the origin. The distance (or distance squared) is the
same no matter which coordinate axes you choose. Or, said more abstractly: The values of
x and the values of y are ”relative” (that is, they depend on the coordinate system), but the
distance is not relative6 . It has the same value for all observers. This allegory is very similar
to the story of space and time in Einstein’s theory of special relativity.
Let’s return to our space-time diagram and imagine two observers who are moving relative
to one another at speed v. We have already learned that the measurements of x and t are
”relative”: they differ for the two observers. We have further learned that there is a certain
combination, (ct)2 − x2 , which is not relative: it has the same value in both cases.
6
In the fancy language of physicists and mathematicians it is called an invariant.
Einstein’s Relativity
p.29
Sometimes you hear people say ”everything is relative”, and they point to Einstein’s
theory of special relativity as proof of their point of view. But you can now see that
this is a misinterpretation of Einstein’s theory. Einstein taught us that certain things
that we thought were absolute and the same for all observers like measurements of
space (x) and time (t) are in fact relative. The measurements depend on the observer,
just like the measurements of x and y in our allegory depended on which student did
the measurement. But other quantities, like (ct)2 − x2 , are not relative. So, Einstein
would not agree with the statement that ”everything is relative”. He would say that
there remain some quantities that are absolute, but they aren’t the things you might
have guessed.
This is very similar to our allegory, except that c t is substituted for y and there is an allimportant minus sign. This minus sign means that the measurements of the second observer
can not be obtained by rotating the coordinates about the origin. Instead, the coordinates
for the second observer correspond to squeezing the x and c t axes together towards the 45
degree line; with equal angles away from 45 degrees. See Figure 10. These new axes now
are separated by an angle of less than 45 degrees. We call such axes ”skew”. For ordinary
perpendicular axes, we measure x and c t by constructing a grid of lines parallel to the c t
and x axes. The grid is like graph paper and can be used to easily read off the coordinates
of any point drawn in the plane. We can do the analogous thing for x′ and c t′ : the only
strange thing is that now the grid consists of lines parallel to the skew axes and these lines
do not meet at ninety degree angles.
ct
c t1
c t2
Light Ray
ct’
ct
Light Ray
1
1
c t’1
2
2
c t’2
x1 x2
x
x’
x’1
x’2
x
Figure 10: For events 1 and 2 the diagram shows how to read off the x and c t values by
drawing lines parallel to one axis until it hits the other axis. For a second observer, the axes
c t′ and x′ are skew (each tilted away from vertical by the same amount). For either choice
of axes, the distance between any point on the world-line of a light ray at 45 degrees (point
2, for example) must be equidistant from the two axes.
Why are the axes skew? First, note that the (vertical) c t axis runs along the world-line
of the watch that measures t – the one held by the first observer who is at rest and
standing at x = 0. Similarly, the c t′ axis must run along the world-line of the watch
that measures t′ . This watch is held by the second observer who is moving at speed v
Einstein’s Relativity
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with respect to the first. We already mentioned that world-line of such an observer is
tilted away from vertical, by an amount that depends on v, and so the c t′ axis coincides
with this world-line. How about the x′ -axis? Light travels along x = c t (the 45 degree
line), which means that every point along the world-line of light must be equidistant
from the c t and x axes. So both axes must be equidistant from the 45 degree line.
Now, according to Einsteins relativity postulate, the speed of light is the same for all
observers. So, the same must be true for c t′ and x′ axes. That means that, if the c t′
axis is tilted towards the 45 degree line, so must the x′ axis. This leads to a situation
with the c t′ and x′ axes squeezed towards one another.
Once we have straightened out how to draw the space-time diagrams, they speak for
themselves and show us visually how relativity works.
First of all, consider the point labeled 1 in the two panels of Figure 10. The left-hand
side shows how to determine the coordinates as measured by the first observer; the second
shows the construction for the second observer. It is obvious that the two results differ. So,
now we see geometrically that x and t are relative: their values change depending on the
observer, simply because the coordinates differ. It is hard to see, but careful calculations
show that (ct)2 − x2 obtained from the original coordinates is equal to (ct′ )2 − x′2 obtained
from the skew coordinates, so the value of this quantity does not depend on the observer.
ct’
ct
t1 = t2
1
2
t’1
t’2
x’
x
Figure 11: Relativity of Simultaneity.
Secondly, compare points 1 and 2 in Figure 11. They lie along the same horizontal line,
so they have the same value of c t (i.e. t1 = t2 ). That is, according to the first observer,
they are simultaneous. But now consider the skew grid lines in the figure that are used to
measure c t′1 and c t′2 . The grid-lines are different, so c t′1 does not equal c t′2 (i.e. t′1 6= t′2 ).
That is, we see directly that two events that are simultaneous according to one observer are
not simultaneous for the other: this illustrates the relativity of simultaneity. In particular,
we can see from the Figure that t′2 is smaller than t′1 . This means the first observer says the
events are simultaneous, but the second says Event 2 occurred before Event 1.
Thirdly, we can use the space-time diagram to illustrate the solution to the twin paradox.
Einstein’s Relativity
p.31
Recall that the story appears to be a paradox because we cannot decide which twin is right.
The stay-at-home twin sees the traveling twin moving fast and aging slowly (due to time
dilation), but the traveling twin sees the stay-at-home twin moving fast and aging slowly.
Could they both be right? The answer is no. To explain why, we draw the world-lines of
the two observers. The world-line of the stay-at-home twin is straight because she is at rest
ct
Twin returns
constant speed
Turn around (acceleration)
constant speed
Twin leaves
x
Figure 12: Space-time diagram used to solve the Twin Paradox.
throughout. But the world-line of the second observer cannot be straight. In order to leave
and return to the same place, the second twin must travel on a world-line that curves or
bends. A curve or bend means that the second observer is not simply moving at constant
speed; the second twin must accelerate to be able to take off and decelerate to return to
Earth. So, immediately, our diagram shows pictographically that the two twins are distinguishable because one has a straight world-line and the other one does not.
Up to this point, we have only considered observers moving at constant speed. In that
case, relativity teaches us that all observers are equally valid. But, here, only the stay-athome twin has the straight world-line so only the stay-at-home twin’s clocks can be trusted.
Special relativity gives no rules for judging the clocks and rulers carried by accelerating or decelerating observers. So, the stay-at-home twin must be right and the traveling/accelerating
twin’s clock cannot be trusted.
Actually, the traveling twin’s clock is not completely useless. We just have to take account
of the effects of acceleration. However, the laws of special relativity, do not tell us how to
handle this case. In special relativity we only consider ’inertial observers’ those that are at
rest or travel at constant speed. The accelerating observers can be treated using an extension
of Einstein’s theory (developed by the Man himself of course), known as general relativity,
Einstein’s Relativity
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which we will discuss in a moment. This is a more complicated theory, but if you follow the
calculations through and include the acceleration of the traveling twin, you discover that,
indeed, the traveling twin gets the same result: namely,the stay-at-home twin has aged more.
We can summarize this according to a geometrical rule. Consider first our x − y plot and
imagine the same pair of curves, one straight and one bending, that intersect at two points.
One could ask: what is the shortest distance between two points? Euclid teaches us that
the answer is always the straight line. Mathematicians call a space obeying Euclid’s laws a
Euclidean space. Space-time is not Euclidean. It is what mathematicians call Lorentzian.
We have already seen one key difference: in Euclidean space, x2 + y 2 (with a plus sign) was
the same for all choices of coordinates and in Lorentzian space-time (ct)2 − x2 (with a minus
same) is the same for all choices of coordinates. As a result, one can show that Euclid’s
law (the shortest distance between two points is a straight line) turns into: the longest time
between two events is a straight line. So, the stay-at-home twin experiences the longest time
between take-off and return.
Einstein’s Relativity
3
p.33
General Relativity
”Matter tells space how to curve. Space tells matter how to move.”
“It is inconceivable, that inanimate brute matter, should, without the mediation of something else,
which is not material, operate upon and affect other matter without mutual contact. That Gravity
should be innate, inherent and essential to matter so that one body may act upon another at a
distance thro’ vacuum without the mediation of anything else, by and through which their action
and force may be conveyed, from one to another, is to me so great an absurdity that I believe no
Man who has in philosophical matters a competent faculty of thinking can ever fall into it. Gravity
must be caused by an agent acting constantly according to certain laws; but whether this agent be
material or immaterial, I have left to the considerations of my readers.”
Isaac Newton, Principia Mathematica
A central feature of special relativity is the absolute speed barrier set by light. It is important to realize that this limit applies not only to material objects but also to signals and
influences of any kind. There is simply no way to communicate information or a disturbance
from one place to another faster than light speed. This realization leads to a serious conflict
with Newton’s theory of gravity. In Newton’s theory one body exerts a gravitational pull
on another with a strength determined solely by the mass of the objects involved and the
magnitude of their separation. The strength has nothing to do with how long the objects
have been in each other’s presence. This means that if their mass or their separation should
change, the objects will, according to Newton, immediately feel a change in their mutual
gravitational attraction. This conclusion is in direct conflict with special relativity, since
the latter ensures that no information can be transmitted faster than the speed of light –
instantaneous transmission violates this precept maximally. Hence, Einstein sought a new
theory of gravity compatible with special relativity. This ultimately led him to the discovery
of general relativity, in which the character of space and time again went through a remarkable transformation.
The following table illustrates some of the conceptual conflicts between Newtonian gravity and special relativity:
Newton’s Gravity
- Force depends on mass
- Light does not gravitate
- Force depends on separation
- Inverse square law
- ”Action-at-a-distance”
- Force on mass B depends on
instantaneous position of mass A
Einstein’s Special Relativity
- Mass is just a form of energy
- Why not?
- Distance is relative
- Must be modified
- Causality and spacetime
- Instantaneous/simultaneous are
relative
Even before the discovery of special relativity, Newton’s theory of gravity was lacking in
one important respect. Newton himself realized that his theory of gravity was fundamentally
Einstein’s Relativity
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incomplete. Although F = GM m/r2 accurately described the motion of objects in the solar
system, his theory was lacking a mechanism for gravity. What transmits the gravitational
force? Newton’s theory offers no insight into what gravity is. Einstein once again revolutionized our understanding of space and time by showing in that they warp and distort to
communicate the force of gravity.
3.1
Gravity and Acceleration – The Principle of Equivalence
Figure 13: Gravity is universal: Objects fall the same way regardless of their mass and
composition. Photo: Julian Baumann
The historical importance of Galileo Galilei for the development of the modern scientific method cannot be overstated. Before Galileo most of science was in fact philosophy.
Galileo introduced the crucial idea of testing theories with experiments. One of Galileo’s
most famous experiments proved to be essential for the development of the general theory
of relativity. Legend has it that Galileo dropped different rocks off the leaning tower of Pisa
to test if and how the law of gravity depends on mass and composition of the object.7 What
Galileo found by doing careful experiments was the following:
7
There is significant doubt as to the historical accuracy of this story.
Einstein’s Relativity
p.35
Objects fall at the same rate irrespective of their mass and composition.
To understand the significance of this observation let us recall and clarify Newton’s laws
of motion and gravity:
The dynamical reaction of an object (e.g. Galileo’s rocks) to the forces exerted on it is
determined by its inertial mass, that is, the mass mI entering in Newtons first law
F = mI a .
(23)
If it is in the gravitational field of a spherical body (e.g. the Earth) with mass ME whose
center is situated a distance RE away, the resulting gravitational force on the object is
determined by its gravitational mass, that is the mass mG entering Newton’s gravitational
equation
ME m G
F =G
.
(24)
2
RE
Galileo and others observed a crucial feature of gravity: the gravitational and inertial masses
of any object are the same; that is
mG = mI ≡ m .
(25)
Combining the previous equations shows that at a distance RE from the center of the Earth,
the acceleration experienced by any small object due to the gravitational force exerted on it
is
2
a = GME /RE
(26)
independent of its mass m. Thus, different objects accelerate at the same rate in a gravitational field, irrespective of their mass or composition. This doesn’t hold for any other force
in nature (like the electrical force for example). Gravity is unique in that sense. Indeed, this
is the essential content of Galileo’s famous observation that bodies of all kinds fall at the
same rate when air resistance can be ignored. It also underlies the fact that we do no have to
know the composition or nature of a planet in order to calculate its orbit (the outer planets
such as Saturn and Jupiter, composed mainly of hydrogen-rich gases such as methane, move
on elliptic orbits, just as do the inner planets such as Mars and the Earth, made mainly of
rock and iron). Einstein took this as a serious hint about the true nature of the gravitational
force. He then came up with a stroke of genius:
In special relativity he was restricting himself to consider only uniform motion, that is,
observers that move at constant speed in straight lines. He formulated the principle of relativity only for such observers. In general relativity, he extended the principle of relativity
to all observers, whether moving uniformly or not. Thus in the general theory of relativity,
it is assumed that
The laws of physics are the same for all observers, no matter what their state
of motion.
Einstein’s Relativity
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Einstein then showed that this leads to a new understanding of the nature of gravity.
The following thought experiment is by now legendary:
Figure 14: Acceleration can cancel gravity. Left: Stationary elevator on Earth. Gravity
accelerates the ball downwards. Right: Freely falling elevator. Elevator and ball accelerate
at the same rate. Ball floats relative to the observers on the elevator.
Imagine carrying out experiments in an elevator. As long as the elevator is stationary
or in uniform motion, the results are identical to those in a stationary lab on the Earth’s
surface. For simplicity, consider the elevator when stationary; the Earth’s gravity acts on
the elevator and on the observer (in this case Einstein and Max Planck, see Figure 14) in it.
Tension exerted by the cable holding the elevator prevents it accelerating downwards at the
rate g = 9.81 m/s2 observed for every freely falling object. The reaction exerted by the floor
of the elevator on the Planck and Einstein prevent them from falling down the elevator shaft;
they experience this as their weight. If Planck releases a ball held in his hand, it accelerates
downward relative to him at the rate g and hits the ground. Because the equivalence of
gravitational and inertial mass, the same acceleration is experienced by all bodies no matter
how heavy they are or what they are made of (as in Galieo’s celebrated experiments at the
leaning tower of Pisa). However, if the cable attached to the elevator breaks, and we ignore
friction and air resistance, then relative to the Earth’s surface the elevator will accelerate
downwards at the rate g. Einstein and Planck also accelerate downwards relative to the
Earth at this rate, because the floor no longer prevents this from happening: it accelerates
away from them at just the free- fall rate, and so exerts no force to slow down their fall.
Since the reaction of the floor now vanishes, they will no longer feel their weight holding
them down on the floor. Thus, as far as they are concerned, the force of gravity now appears
to have no effect. If Planck releases a ball held in his hand, it will accelerate downwards
relative to the Earth at the rate g, precisely as he is doing, and so will float next to him at a
constant distance above the floor. Thus, because all freely falling bodies experience the same
acceleration in a gravitational field, any freely falling object will appear to be stationary in
the observer’s reference frame. The Earth’s gravitational field no longer causes objects to
Einstein’s Relativity
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accelerate towards the floor of the elevator. Its usual effects have been transformed away by
changing to an accelerating reference frame.
This example depends crucially on the equivalence of gravitational and inertial mass.
If this were note true, different bodies of the same inertial mass would experience different gravitational forces and so would accelerate at different rates, contrary to experiment;
transformation to an accelerating frame could remove the effective gravitational force from
some objects but not others (because the required rate of acceleration would be different
for different objects). As a result of this equivalence, there is a close relationship between
acceleration and gravity!
To understand this relationship more clearly, we follow Einstein in considering various
extensions of the previous thought experiment. First, suppose observer A is in an elevator which is at rest relative to the Earth (see Figure 15). The results of any experiments
done there will be those of everyday life on Earth; if an object is released, it will fall to
the ground. Secondly, consider observer B in a rocket moving with constant acceleration g
far from any massive body. For him, the results of experiments will be the same as for A.
An object released will fall to the floor (or, if you prefer, the floor will accelerate into it!)
with acceleration g. Suppose that observer C is in an elevator which is falling freely under
gravity because its cable has broken. The observer will fall at the same rate as any object released, and so will measure no relative acceleration; thus the results of all experiments
will be the same as for observer D in a stationary rocket far away from any gravitational field.
Einstein called this spectacular realization the ”happiest thought of [his] life” and summarized it in the Principle of Equivalence:
There is no way of distinguishing between the effects on an observer of a
uniform gravitational field and of constant acceleration.
3.2
Curved Space
It took Einstein five years to convert his ”happy thought” into a revolution of our understanding of gravity. The breakthrough was made by applying special relativity to the link
between gravity and accelerated motion and analyzing the striking consequences.8 To understand this step in Einstein’s reasoning we focus on a particular example of accelerated
motion.9 Recall that an object is accelerating if either the speed or the direction of its motion
changes. For simplicity we will focus on accelerated motion in which only the direction of
our object’s motion changes while its speed stays fixed. Specifically, we consider motion in a
circle such as what one experiences on the Tornado ride in an amusement park. You stand
8
This section is adapted from Brian Greene’s excellent book ”The Elegant Universe”
John Stachel, ”Einstein and the Rigidly Rotating Disk,” in General Relativity and Gravitation ed. A.
Held (New York: Plenum, 1980)
9
Einstein’s Relativity
p.38
g
B
A
g
g
D
C
g
Figure 15: Gravitational field vs. accelerating rocket. A: Stationary elevator on Earth. B:
Accelerating rocket. C: Elevator freely falling in a uniform gravitational field. D: Stationary
rocket in outer space, far from any gravitational influence.
Einstein’s Relativity
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with your back against the inside of a circular Plexiglas structure that spins at a high speed.
Like all accelerated motion, you can feel this motion – you feel the circular wall of Plexiglas
pressing on your back keeping you moving in a circle. If the ride is extremely smooth and
you close your eyes, the pressure of the ride on your back – like the support of a bed can
almost make you feel as if you are lying down. The ”almost” comes from the fact that you
still feel ordinary ”vertical” gravity, so your brain cannot be fully fooled. But if you were
to ride the Tornado in outer space, and if it were to spin at just the right rate, it would feel
just like lying in a stationary bed on Earth. Moreover, were you to ”get up” and walk along
the interior of the spinning Plexiglas, your feet would press against it just as they do against
an Earthbound floor. In fact, space stations are designed to spin in this manner to create
an artificial feeling of gravity in outer space.
Having used accelerated motion of the spinning Tornado to imitate gravity, we can now
follow Einstein and set out to see how space and time appear to someone on the ride. His
reasoning went as follow: We stationary observers can easily measure the circumference and
the radius of the spinning ride. For instance, to measure the circumference we can carefully
lay out a ruler – head to tail – alongside the ride’s spinning girth; for its radius we can
similarly use the head-to-tail method working our way from the central axle of the ride to
its outer rim. As we anticipate from high-school geometry, we find that their ratio is 2π –
just as for any circle drawn on a flat sheet of paper. But what do things look like from the
perspective of someone on the ride itself?
To get the clearest perspective let’s take a bird’s-eye-view of the ride, as in Figure 16.
Figure 16: Acceleration curves space.
This snapshot of the ride has been adorned with an arrow that indicates the momentary
direction of motion at each point. An observer on the ride is equipped with a ruler. As
he begins to measure the circumference, we immediately see from our bird’s-eye-perspective
that he is going to get a different answer than we did. As he lays the ruler out along the
circumference, we notice that the ruler’s length is shortened. This is noting but the length
contraction discussed in section 2.5, in which the length of an object appears shortened along
the direction of its motion. A shorter ruler means that he will have to lay it out – head to
tail – more times to traverse the whole circumference. Since he still considers the ruler to be
Einstein’s Relativity
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one foot long (since there is no relative motion between him and his ruler, he perceives it as
having its usual length of one foot), this means that our observer on the ride will measure a
longer circumference than we did.
What about the radius? If the observer on the ride uses the head-to-tail method to find the
length of a radial strut, we will see from our bird’s-eye-view that he get the same answer
as we did. The reason is that the ruler is not pointing along the instantaneous direction
of motion of the ride (as it is measuring the circumference). Instead, it is pointing at a
ninety-degree angle to the motion, and therefore it is not contracted along its length. He
will therefore find exactly the same radial length as we did. But now, if the observer on the
ride calculates the ratio of circumference to the radius he will get a number that is larger
than our answer of 2π, since the circumference is longer but the radius is the same. This is
weird. How in the world can something in the shape of a circle violate the ancient Greek
realization that for any circle this ratio is exactly 2π?
Here is Einstein’s explanation: The ancient Greek result holds true for circles drawn on a flat
surface. But just as the warped or curved mirrors in an amusement park fun-house distort
the normal spatial relationships of your reflection, if a circle is drawn on a warped or curved
surface, its usual spatial relationships will also be distorted: the ratio of its circumference to
its radius will generally not be 2π.
For instance, Figure 17 compares three circles whose radii are identical. Notice, however
Figure 17: Circles on curved spaces.
that their circumferences are not the same. The circumference of the circle drawn on the
curved surface of a sphere, is less than the circumference of the circle drawn on the flat
surface, even though they have the same radius. The curved nature of the sphere’s surface
causes the radial lines of the circle to converge toward each other slightly, resulting in a
small decrease in the circle’s circumference. The circle drawn on the saddle shaped surface is
greater than that drawn on the flat surface; the curved nature of the saddle’s surface causes
the radial lines of the circle to splay outward from each other slightly, resulting in a small
increase in the circle’s circumference. This lead Einstein to propose an idea – the curving
of space – as an explanation for the violation of ”ordinary” Euclidean geometry. The flat
geometry of the Greeks simply does not apply to someone on the spinning ride. Rather,
its curved space generalization (as schematically drawn in the saddle in Figure 17) takes its
place. Einstein realized that the familiar geometrical spatial relationships codified by the
Greeks, relationships that pertain to ”flat” space, do not hold from the perspective of an
accelerated observer. Of course, we have discussed only one particular kind of accelerated
motion, but Einstein showed that a similar result – the warping of space – holds in all instances of accelerated motion.
In fact, accelerated motion not only results in a warping of space, it also result in an analogous warping of time. Basically, this is the case because special relativity introduced a union
Einstein’s Relativity
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of space and time in the concept of spacetime and therefore declares, ”What’s true for space
is true for time.” The details are fascinating!
The final leap Einstein took now looks almost trivial. Since he had already shown gravity
and accelerated motion to be effectively indistinguishable, and since he now had shown
that accelerated motion is associated with the warping of space and time, he made the
following proposal for the unknown mechanism by which gravity operates. Gravity according
to Einstein, is the warping of space and time. Trumpets, please!
3.3
Bending of Light
There is another nice way to understand the curvature of spacetime and the effects gravity has
on light. Let’s go back to elevator example! Consider the situation of the spaceship floating
in space (labeled D in Figure 15). Imagine shining a laser across the spaceship (I don’t have
a figure of this, so make your own sketch to help you visualize the following argument). It
goes in a straight line as confirmed by the passengers of the spaceship. Now picture the
same situation if the spaceship is accelerated at g (labeled B in Figure 15). From an outside
view the laser is still going straight (since it is not in contact with the spaceship), but it
will appear to be curved downwards to the observers on the spaceship (since the spaceship
is accelerating towards the laser beam). However, as emphasized before this situation is
completely equivalent to the effect of a gravitational field of strength g (labeled A in Figure
15). Hence, gravity bends light! Lights follows a curved path mapping out the curvature of
space!
3.4
”Gravity is not a Force”
Matter causes space and time to curve! Space and time are not fixed and static, but flexible and dynamic. Space and time change in response to the distribution of matter in the
universe. A useful analogy is the following: Imagine space is represented by a flat rubber
sheet (like a trampoline). Now put a bowling ball on the sheet (see Figure 18). The sheet
will bend and curve under the weight of the bowling ball. Next put a small marble on the
curved surface of the rubber sheet and give it a kick. What path will the marble follow?
It will try to go on the ”straightest” possible path on the curved surface. This might lead
to a circular orbit of the marble around the bowling ball if the initial kick was in the right
direction. Let the bowling ball be the Sun, the marble be the Earth and the rubber sheet
be space and you have a picture of how gravity works in the solar system.
Notice however that there is no real force on the Earth or the marble. The reason that
the Earth still follows a circular path is because space is curved. That gives us the illusion
of a gravitational force, when really gravity is only the curvature of space (or spacetime to
be precise).
Einstein’s Relativity
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Figure 18: Rubber sheet analogy.
3.5
Beautiful and True: The Spectacular Confirmation of GR
”REVOLUTION IN SCIENCE
New Theory of the Universe
Newtonian Ideas Overthrown”
The London Times, November 7, 1919
Figure 19: The Man.
Two distinct aspects make Einstein’s theory of relativity such a fantastic achievement:
1. The theory is of an inner beauty and elegance that is unparalleled in the history of
science.
2. It is true! This simple statement is in fact the key to science. It doesn’t matter how
beautiful your theory is. If it disagrees with experiment it is wrong!
Einstein’s ideas are undoubtedly of a logical simplicity and elegance that is truly magical.
However, the consequences of his reasoning are very unfamiliar and very remote from ev-
Einstein’s Relativity
p.43
eryday experience. ”Time and space are different for different observers depending on their
state of motion.” ”They warp and stretch in the presence of mass and energy to transmit the
force of gravity.” ”Energy and matter are in fact equivalent.” How can this not be crazy?
I wish I could have witnessed how the theory was first tested and compared to experiment
in the 1920s. Imagine the amazement to realize that these crazy ideas in fact accurately portray reality. Nature is truly governed by the theory of relativity.
A clever experiment to test general relativity was in fact proposed by Einstein himself
(this guy didn’t need any help ...). We see stars at night, but of course they are also there
during the day. We usually don’t see them because their pinpoint light is overwhelmed by
the light emitted by the Sun. During a solar eclipse, however, the moon temporarily blocks
the light of the Sun and the distant stars become visible. Nevertheless, the presence of the
Sun still has an effect. Light from some of the distant stars must pass close to the Sun on the
way to Earth. Einstein’s general relativity predicts that the Sun will cause the surrounding
space and time to warp and as we saw such distortion will influence the path taken by the
starlight. After all, the photons of distant origin travel along the fabric of the universe; if
the fabric is warped, the motion of the photons will be affected. The bending of the path
of light is greatest for those light signals that just graze the Sun on their way to Earth. A
solar eclipse makes it possible to see such sun-grazing starlight without its being completely
obscured by sunlight itself.
The angle through which the light path is bent can be measured in a simple way. The
Figure 20: Deflection of starlight.
bending of the starlight’s path results in a shift in the apparent position of the star (see
Figure 20). The shift can be accurately measured by comparing this apparent position with
the star’s actual location known from observations of the star at night (in the absence of the
Sun’s warping influence), carried out when the Earth is at an appropriate position, some six
months earlier or later. In November 1915, Einstein used his new understanding of gravity to
calculate the angle through which starlight signals that just graze the Sun would be bent and
found the answer to be about 0.00049 of a degree. This tiny angle is equal to that subtended
Einstein’s Relativity
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by a quarter placed upright and viewed from nearly 2 miles away. The detection of such a
small angle was, however, within reach of the technology of the day. Sir Arthur Eddington10 ,
a well-known British astronomer, organized an expedition to the island of Principe off the
coast of West Africa to test Einstein’s prediction during the solar eclipse of May 29, 1919.
On November 6, 1919, after some five months of analysis of the photographs taken during
the eclipse at Principe it was announced at a joint meeting of the Royal Society and the
Royal Astronomical Society that Einstein’s prediction based on general relativity had been
confirmed. It took little time for word of his success - a complete overturning of previous
conceptions of space and time - to spread well beyond the confines of the physics community,
making Einstein a worldwide celebrity.
3.6
Further Implications and Tests of GR
3.6.1
Gravitational Waves - Ripples in Spacetime
The distribution of matter and energy in the universe determines the local curvature of
spacetime. Gravitational waves are disturbances in the curvature of spacetime caused by
the motions of matter. Propagating at the speed of light, gravitational waves do not travel
”through” spacetime as such – rather the fabric of spacetime itself is oscillating. Though
gravitational waves pass straight through matter, their strength weakens proportionally to
the distance traveled from the source. A gravitational wave arriving on Earth will alternately
stretch and shrink distances, though on an incredibly small scale – by a factor of 10−21 for
very strong sources. That’s roughly equivalent to measuring a change in the distance from
the Sun to Earth to the accuracy of the size of an atom! It is amazing that we can even
think about actually measuring such incredibly small signals.
Figure 21: Laser Interferometer Space Antenna - LISA, a proposed space-based gravitational
wave detector.
Detecting gravitational waves will be a triumph for modern science and reveal many
mysteries of the universe. Gravitational waves are produced most strongly when black holes
collide and during the violent expansion of the first split second in the history of the universe.
Gravitational waves therefore will allow us to get a glimpse at the very, very early universe,
as well as reveal the detailed physics of extreme gravitational effects close to black holes.
10
General relativity used to have the reputation of being a notoriously difficult subject. This is reflected in
the following anecdote involving Eddington: When told that only 3 men in the world understood Relativity,
Eddington asked ”I wonder who is the third?”
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3.6.2
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Down-to-Earth GR: GPS
GPS = a system of satellites using clocks to help you find exactly where you are, and locate
the nearest Starbucks. Clocks in orbit move more quickly than they would on Earth, due to
the curvature of spacetime. General relativity is necessary to make GPS work!
3.7
Black Holes
”Black holes are where God divided by zero.”
Steven Wright
The deflection of starlight, gravitational waves and GPS are tiny effects because the mass of
the Sun and the Earth is relatively small and spread over a relatively large region of space.
The effects of general relativity become more dramatic when things are very massive and
compressed into a small region of space, so that the warping of space and time is correspondingly severe.
One of the most famous predictions of general relativity is the possibility of black holes.
The mathematical theory of black holes was first developed by the German astronomer Karl
Schwarzschild while at the Russian front during World War I in 1916. Remarkably, just
months after Einstein has put the finishing touches on general relativity, Schwarzschild was
able to use the theory to gain a complete and exact understanding of the way space and
time warp in the vicinity of a perfectly spherical star. He showed that if the mass of a star
is concentrated in a small enough spherical region, so that its mass divided by its radius exceeds a particular critical value, the resulting spacetime warping is so radical that anything,
including light, that gets too closed to the star will be unable to escape its gravitational
grip. Since not even light can escape such ”compressed stars”, they were initially called
dark or frozen stars. It was the Princeton physicist John Wheeler who years later coined the
more catchy name black holes - black because they cannot emit light, holes because anything
getting too close falls into them, never to return.
The point of no return is called the black hole event horizon. Any objects that get too
close are doomed: they will be drawn inexorably toward the center of the black hole and
subject to an ever increasing and ultimately destructive gravitational strain. Imagine you
dropped feet first through the event horizon, as you approached the black hole’s center you
would find yourself getting increasingly uncomfortable. The gravitational force of the black
hole would increase so dramatically that its pull on your feet would be much stronger than
its pull on your head (since in a feet-first fall your feet are always a bit closer than your head
to the black hole’s center). Stephen Hawking likes to call this effect ”spaghettification”.
More likely however you would be stretched with a force that would quickly tear your body
to shreds.
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If you are more careful and make sure you always stay outside the black holes event
horizon, you could use the black hole for some fun. Imagine you are near a black hole of
about 1000 times the mass of the Sun. You lower yourself carefully to about an centimeter
above the black hole’s event horizon. Gravitational fields cause a warping of time, and this
means that your passage through time would slow down. In fact, since black holes have such
strong gravitational fields, your passage through time would slow way down. Your watch
would tick about ten thousand times more slowly than those of your friends back on Earth.
If you were to hover just above the black hole’s event horizon in this manner for a year, and
then travel back to Earth, upon arrival you would find that more than ten thousand years
have passed since your initial departure and your friends are all history. You would have
successfully used the black hole as a kind of time machine, allowing you to travel to Earth’s
distant future.
3.8
The Big Bang
”I’m astounded by people who want to ’know’ the universe when it’s hard enough to find your way
around Chinatown.”
Woody Allen
Figure 22: The Cosmic Microwave Background (CMB) – Echoes from the Big Bang.
When applied to the whole Universe Einstein’s theory gives new insight into its origin and
evolution. Einstein showed that space and time respond to the presence of mass and energy.
This distortion of spacetime affects the motion of other cosmic bodies moving in the vicinity
of the resulting warps. In turn, the precise way in which these bodies move, by virtue of
their mass and energy, has a further effect on the warping of spacetime, which further affects
the motion of the bodies, and on and on the interconnected cosmic dance goes. Through the
equations of general relativity, Einstein was able to describe the mutual evolution of space,
time, and matter quantitatively. To his great surprise, when the equations are applied beyond an isolated context within the Universe, such as a planet or a comet orbiting a star, to
the Universe as a whole, a remarkable conclusion is reached: the overall size of the spatial
Universe must be changing in time. That is, either the fabric of the Universe is stretching
or it is shrinking, but it is not simply staying put. The equations of general relativity show
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this explicitly.
All galaxies would be carried along on the substrate of stretching space, thereby speedily
moving away from all others. Remarkably, the American astronomer Edwin Hubble through
detailed measurements of distant galaxies experimentally established that the Universe is
expanding. If space is stretching, thereby increasing the distance between galaxies that are
carried along on the cosmic flow, we can imagine running the evolution backward in time to
learn about the origin of the Universe. In reverse, space shrinks, bringing all galaxies closer
and closer to each other. As the shrinking Universe compresses the galaxies together, the
temperature dramatically increases, stars disintegrate and a hot plasma of matter’s elementary constituents is formed. As space continues to shrink, the temperature rises unabated, as
does the density of the primordial plasma. As we imagined running the clock backward from
the age of the presently observed Universe, about 14 billion years, the Universe as we know
it is crushed to an ever smaller size. The matter making up everything - every car, house,
building, mountain on Earth; the Earth itself, the Moon; Saturn, Jupiter, and every other
planet; the Sun and every other star in the Milky Way; the Andromeda galaxy with its 100
billion stars and each and every other of the more than 100 billion galaxies - is squeezed by a
cosmic vise to astounding density. And as the clock is turned back to ever earlier times, the
whole of the cosmos is compressed to the size of an orange, a lemon, a pea, a grain of salt,
and to yet tinier size still. Extrapolating all the way back to ”the beginning”, the Universe
would appear to have begun as a point - in which all matter and energy is squeezed together
to unimaginable density and temperature. It is believed that a cosmic fireball, the big bang,
erupted from this volatile mixture spewing forth the seeds from which the Universe as we
know it evolved.
Einstein’s Relativity
4
Time and Space after Einstein
4.1
Warped Spacetime
4.2
Speculations on the Breakdown of General Relativity
4.2.1
Big Bang Singularity and Quantum Foam
4.2.2
Dark Matter and Dark Energy
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Einstein’s Relativity
References
[1] Alan Lightman, ”Einstein’s Dreams”
[2] George Gamow, ”Mr. Tompkins in Wonderland”
[3] David Mermin, ”Space and Time in Special Relativity”
[4] Brian Greene, ”The Elegant Universe”, ”Fabric of the Cosmos”
[5] Albert Einstein, ”Relativity - The Special and General Theory”
[6] George Ellis and Ruth Williams, ”Flat and Curved Space-Times”
[7] Richard Gott, ”Time Travel in Einstein’s Universe”
[8] Anthony Zee, ”Einstein’s Universe”
[9] Edwin Taylor and John Wheeler, ”Spacetime Physics”
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