Everything we see in our daily lives comes to rest. Everything, that
is, except the Moon and the planets, which is perhaps one reason
that these were felt to be special in antiquity, guided by angels or
gods.
However, every sense that we have that we are at rest is an
illusion. In the example I gave earlier of throwing a ball up and
catching it while in a moving plane, you will eventually be able to tell
that your plane is moving when you feel the bouncing of turbulence.
But even when the plane is on the tarmac, it is not at rest. The
airport is moving with the Earth at about 30 km/sec around the Sun,
and the Sun is moving about 200 km/sec around the galaxy, and so
on.
Galileo codified this with his famous assertion that the laws of
physics are the same for all observers moving in a uniform state of
motion, i.e., at a constant velocity in a straight line. (Observers at rest
are simply a special case, when velocity is zero.) By this he meant
that there is no experiment you can perform on such an object that
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can tell you it is not at rest. When you look up in the air at an
airplane, it is easy to see that it is moving relative to you. But, there is
no experiment you can perform on the ground or on the plane that
will distinguish whether the ground on which you are standing is
moving past the plane, or vice versa.
While it seems remarkable that it took so long for anyone to
recognize this fundamental fact about the world, it does defy most of
our experience. Most, but not all. Galileo used examples of balls
rolling down inclined planes to demonstrate that what previous
philosophers thought was fundamental about the world—the
retarding force of friction that makes things eventually settle at rest
—was not fundamental at all but rather masked an underlying
reality. When balls roll down one plane and up another, Galileo
noted, on smooth surfaces the balls would rise back to the same
height at which they started. But by considering balls rolling up
planes of ever-decreasing incline, he showed that the balls would
have to roll farther to reach their same original height. He then
reasoned that if the second incline disappeared entirely, the balls
would continue rolling at the same speed forever.
This realization was profoundly important and fundamentally
changed much about the way we think about the world. It is often
simply called the Law of Inertia, and it set up Newton’s law of
motion, relating the magnitude of an external force to the observed
acceleration of an object. Once Galileo recognized that it took no
force to keep something moving at a constant velocity, Newton
could make the natural leap to propose that it took a force to change
its velocity.
The heavens and the Earth were no longer fundamentally
different. The hidden reality underlying the motion of everyday
objects also made clear that the unending motion of astronomical
objects was not supernatural, setting the stage for Newton’s
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Universal Law of Gravity, further demoting the need for angels or
other entities to play a role in the cosmos.
Galileo’s discovery was thus fundamental to establishing physics
as we know it today. But so was Maxwell’s later brilliant unification
of electric and magnetic forces, which established the mathematical
framework on which all of current theoretical physics is built.
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As Albert Einstein began his journey in this rich intellectual
landscape, he quickly spied a deep and irreconcilable chasm running
through it: both Galileo and Maxwell could not be right at the same
time.
More than twenty years ago, when my daughter was an infant, I
first began to think about how to explain the paradox that young
Einstein struggled with, and a good example literally hit me on the
head while driving her in my car.
Galileo had demonstrated that as long as I am driving safely and
at a constant speed and not accelerating suddenly, the laws of
physics in our car should be indistinguishable from the laws of
physics that would be measured in the laboratories in the physics
building to which I was driving to work. If my daughter was playing
with a toy in the backseat, she could throw the toy up in the air and
expect to catch it without any surprises. The intuition her body had
built up to play at home would have served her well in the car.
However, riding in the car did not lull her to sleep like many
young children, but rather made her anxious and uncomfortable.
During our trip, she got sick and projectile-vomited, and the vomit
followed a trajectory well described by Newton, with an initial speed
of, say, fifteen miles per hour, and a nice parabolic trajectory in the
air, ending on the back of my head.
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Say my car was coasting to a red light at this time at a relatively
slow speed, say, ten miles per hour. Someone on the ground
watching all of this would see the vomit traveling at 25 miles per
hour, the speed of the car relative to them (10 mph) plus the speed
of the vomit (15 mph), and its trajectory would be well described by
Newton again, with this higher speed (25 mph) as it traveled toward
my (now moving) head.
So far so good. Here’s the problem, however. Now that my
daughter is older, she loves to drive. Say she is driving behind a
friend’s car and dials him on her cell phone (hands-free, for safety) to
tell him to turn right to get to the place they are both going. As she
talks into the phone, electrons in the phone jiggle back and forth
producing an electromagnetic wave (in the microwave band). That
wave travels to the cell phone of her friend at the speed of light
(actually it travels up to a satellite and then gets beamed down to her
friend, but let’s ignore that complication for the moment) and is
received in time for him to make the correct turn.
Now, what would a person on the ground measure? Common
sense would suggest that the microwave signal would travel from my
daughter’s car to her friend’s car at a speed equal to the speed of
light, as might be measured by a detector in my daughter’s car (label
it with the symbol c), plus the speed of the car.
But common sense is deceptive precisely because it is based on
common experience. In everyday life we do not measure the time it
takes light, or microwaves, to travel from one side of the room to
another or from one phone to a nearby phone. If common sense
applied here, that would mean someone on the ground (with a
sophisticated measuring apparatus) would measure the electrons in
my daughter’s phone jiggling back and forth and observe the
emanation of a microwave signal, which would be traveling at a
speed c plus, say, ten miles per hour.
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However, the great triumph of Maxwell was to show that he
could calculate the speed of electromagnetic waves emanated by an
oscillating charge purely by measuring the stre
ngth of electricity and
magnetism. Therefore if the person on the ground observed the
waves having speed c plus 10 mph, then for that person the strength
of electricity and magnetism would have to be different from the
values that my daughter would observe, for whom the waves were
moving at a speed c.
But Galileo tells us this is impossible. If the measured strengths of
electricity and magnetism differed between the two observers, then
it would be possible to know who was moving and who was not,
because the laws of physics—in this case electromagnetism—would
take on different values for each observer.
So, either Galileo or Maxwell had to be right, but not both of
them. Perhaps because Galileo had been working when physics was
more primitive, most physicists came down closer to the side of
Maxwell. They decided that the universe must have some absolute
rest frame and that Maxwell’s calculations applied in that frame only.
All observers moving with respect to that frame would measure
electromagnetic waves to have a different speed relative to
themselves than Maxwell had calculated.
A long scientific tradition gave physical support to this idea. After
all, if light was an electromagnetic disturbance, what was it a
disturbance of? For thousands of years, philosophers had speculated
about an “ether,” some invisible background material filling all of
space, and it became natural to suspect that electromagnetic waves
were traveling in this medium, just as sound waves travel in water or
air. Electromagnetic waves would travel with some fixed,
characteristic speed in this medium (the speed calculated by
Maxwell), and observers moving with respect to this background
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would observe the waves as faster or slower, depending on their
relative motion.
While intuitively sensible, this notion was a cop-out, because if
you think back to Maxwell’s analysis, it would mean that these
different observers in relative motion would measure the strength of
electricity and magnetism to be different. Perhaps it was deemed to
be acceptable because all speeds obtainable at the time were so small
compared to the speed of light that any such differences would have
been minute at best and would certainly have escaped detection.
The actor Alan Alda once turned the tables on conventional
wisdom at a public event I attended by saying that art requires hard
work, and science requires creativity. While both require both, what
I like about his version is that it stresses the creative, artistic side of
science. I would add to this statement that both endeavors require
intellectual bravery. Creativity alone amounts to nothing if it is not
implemented. Novel ideas generally stagnate and die without the
courage to implement them.
I bring this up here because perhaps the true mark of Einstein’s
genius was not his mathematical prowess (although, contrary to
conventional wisdom, he was mathematically talented), but his
creativity and his intellectual confidence, which fueled his
persistence.
The challenge that faced Einstein was how to accommodate two
contradictory ideas. Throwing one out is the easy way. Figuring out a
way to remove the contradiction required creativity.
Einstein’s solution was not complex, but that does not mean it
was easy. I am reminded of an apocryphal story about Christopher
Columbus, who got a free drink in a bar before departing to find the
New World by claiming he could balance an egg upright on top of
the bar. After the barman accepted the bet, Columbus broke the tip
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off the egg and placed it easily upright on the counter. He never
mentioned not cracking it, after all.
Einstein’s resolution of the Galileo-Maxwell paradox was not that
different. Because, if both Maxwell and Galileo were right, then
something else had to be broken to fix the picture.
But what could it be? For both Maxwell and Galileo to be right
required something that was clearly crazy: in the example I gave,
both observers would have to measure the velocity of the microwave
emitted by my daughter’s cell phone to be the same relative to them,
instead of measuring values differing by the speed of the car.
However, Einstein asked himself an interesting question, What
does it mean to measure the velocity of light, after all? Velocity is
determined by measuring the distance something travels in a certain
time. So Einstein reasoned as follows: it is possible for two observers
to measure the same speed for the microwave relative to each of
them, as long as the distance each measures the ray to travel relative
to themselves during a fixed time interval (e.g., say, one second, as
measured by each of them in their own frame of reference) is the
same.
But this too is a little crazy. Consider the simpler example of the
projectile vomit. Remember that in my frame it travels from her
mouth in the backseat to hit my head, say, three feet away, in about
one-quarter second. But for someone on the ground the car is
traveling at 10 miles per hour during this period, which is about 14.5
feet per second. Thus for the person on the ground, in one-quarter
second the vomit travels about 3.6 feet plus 3 feet, or a total 6.6 feet.
Hence for the two observers, the distances traveled by the vomit
in the same time is noticeably different. How could it be that for the
microwave the distances both observers measure could be the same?
The first hint that perhaps such craziness is possible is that
electromagnetic waves travel so fast that in the time it takes the
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microwaves to get from one car to another, each car has moved
hardly at all. Thus any possible difference in measured distance
traveled during this time for the two observers would be essentially
imperceptible.
But Einstein turned this argument around. He realized that both
observers had not actually measured the distances traveled by the
microwaves over human-scale distances, because the relevant times
appropriate for light to travel over human-scale distances were so
short that no one could have measured them at the time. And
similarly, on human timescales light would travel such large
distances that no one could measure those distances directly either.
Thus, who was to say that such crazy behavior couldn’t really
happen?
The question then became, What is required for it to actually
occur? Einstein reasoned that for this seemingly impossible result to
be possible, the two different observers must measure distances
and/or times differently from each other in just such a way that light,
at least, would traverse the same measured distance in the same
measured time for both observers. Thus, for example, it would be as
if the observer on the ground in the vomit case were to measure the
vomit traversing 6.6 feet, but would somehow also infer the time
interval over which this happened to be larger than I would measure<
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it inside my car, so that the inferred speed of the vomit would be the
same relative to him as I measure it to be relative to me.
Einstein then made the bold assertion that something like this
does happen, that both Maxwell and Galileo were correct, and that
all observers, regardless of their relative state of motion, would
measure any light ray to travel at the same speed, c, relative to them.
Of course, Einstein was a scientist, not a prophet, so he didn’t just
claim something outlandish on the basis of authority. He explored
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the consequences of his claim and made predictions that could be
tested to verify it.
In doing so he moved the playing field of our story from the
domain of light to the domain of intimate human experience. He not
only forever changed the meaning of space and time, but also the
very events that govern our lives.
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C h a p t e r 5
A S T I T C H I N T I M E
He stretcheth out the north over the empty place, and hangeth
the earth upon nothing.
—JOB 26:7
The great epic stories of ancient Greece and Rome revolve
around heroes such as Odysseus and Aeneas, who challenged the
gods and often outwitted them. Things have not changed that much
for more modern epic heroes.
Einstein overcame thousands of years of misplaced human
perception by showing that even the God of Spinoza could not
impose his absolute will on space and time, and that each of us
evades those imaginary shackles every time we look around us and
view new wonders amid the stars above. Einstein emulated artistic
geniuses such as Vincent van Gogh and reasoned with the
parsimony of Ernest Hemingway.
Van Gogh died fifteen years before Einstein developed his ideas
on space and time, but his paintings make it clear that our
perceptions of the world are subjective. Picasso may have had the
chutzpah to claim that he painted what he saw, even as he produced
representations of disjointed people with body parts pointing in
different directions, but van Gogh’s masterpieces demonstrate that
the world can look very different to different people.
So too, Einstein explicitly argued, for the first time as far as I know
in the history of physics, that “here” and “now” are observer-
dependent concepts and not universal ones.