ͣ͡
and, at another time like this, at 6 cm:
The similarity with the example I presented when discussing
relativity is intentional. In the case of Plato’s cave dwellers, however,
we recognized that this length contraction occurred because the
cave dwellers were merely seeing two-dimensional shadows of an
underlying three-dimensional object. Viewed from above, it can
easily be seen that the shorter shadow projected on the wall results
because the ruler has been rotated at an angle to the wall:
And as another Greek philosopher, Pythagoras, taught us, when
seen this way, the length of the ruler is fixed, but the projections
onto the wall and a line perpendicular to the wall always combine
together to give the same length, as shown below:
This yields the famous Pythagorean theorem, L2 = x2 + y2, which
high school students have been subjected to for as long as high
schools have taught geometry. In three dimensions, this becomes L2
= x2 + y2 + z2.
Two years after Einstein wrote his first paper on relativity,
Minkowski recognized that perhaps the unexpected implications of
ͣ͢
the constancy of the speed of light, and the new relations between
space and time unveiled by Einstein, might also reflect a deeper
connection between the two. Knowing that a photograph, which we
usually picture as a two-dimensional representation of three-
dimensional space, is really an image spread out in both space and
time, Minkowski reasoned that observers who were moving relative
to each other might be observing different three-dimensional slices
of a four-dimensional universe, one in which space and time are
treated on an equal footing.
If we return to the ruler example in the case of relativity, where
the ruler of the moving observer is measured to be shorter by the
other observer than it would be in the frame in which it is at rest, we
should also remember that for this observer the ruler is also “spread
out” in time—events at either end that are simultaneous to the
observer at rest with respect to the ruler are not simultaneous for the
second observer.
Minkowski recognized that one could accommodate this fact, and
all the others, by considering that the different three-dimensional
perspectives probed by each observer were in some sense different
“rotated” projections of a four-dimensional “space-time,” where
there exists an invariant four-dimensional space-time “length” that
would be the same for all observers. The four-dimensional space,
which we now call Minkowski space, is a little different from its 3-D
counterpart, in that time as a fourth dimension is treated slightly
differently from the three dimensions of space, x, y, and z. The four-
dimensional “space-time length,” which we can label as S, is not
written, in analogy to the three-dimensional length, which we
denoted by L, above, as
S2 = x2 + y2 + z2 + t2
but rather as
ͣͣ
S2 = x2 + y2 + z2 - t2.
The minus sign that appears in front of t2 in the definition of
space-time length, S, gives Minkowski space its special
characteristics, and it is the reason our different perspectives of space
and time when we are moving relative to one another are not simple
rotations, as in the case of Plato’s cave, but something a little more
complicated.
Nevertheless, in one fell swoop, the very nature of our universe
had changed. As Minkowski poetically put it in 1908: “Henceforth
space by itself, and time by itself, are doomed to fade away into mere
shadows, and only a kind of union of the two will preserve an
independent reality.”
Thus, on the surface, Einstein’s Special Theory of Relativity
appears to make physical reality subjective and observer dependent,
but relativity is in this sense a misnomer. The Theory of Relativity is
instead a theory of absolutes. Space and time measurements may be
subjective, but “space-time” measurements are universal and
absolute. The speed of light is universal and absolute. And four-
dimensional Minkowski space is the field on which the game of
nature is played.
The depth of the radical change in perspective brought about by
Minkowski’s reframing of Einstein’s theory can perhaps best be
understood by considering Einstein’s own reactions to Minkowski’s
picture. Initially Einstein called it “superfluous learnedness,”
suggesting that it was simply fancy mathematics, devoid of physical
significance. Shortly thereafter he emphasized this by saying, “Since
the mathematicians have invaded relativity theory, I do not
understand it myself anymore.” Ultimately, however, as happened
several times in his lifetime, Einstein came around and recognized
that this insight was essential to understand the true nature of space
ͣͤ
and time, and he later built his General Theory of Relativity on the
foundation that Minkowski had laid.
It would have been difficult if not impossible to guess that
Faraday’s spinning wheels and magnets would eventually lead to
such a profound revision in our understanding of space and time.
With the spectacles of hindsight, however, we could have had at
least an inkling that the unification of electricity and magnetism
could have heralded a world where motion would reveal a new
underlying reality.
Returning to Faraday and Maxwell, one of the important
discoveries that started the ball rolling was that a magnet acts on a
moving electric charge with an odd force. Instead of pushing the
charge forward or backward, the magnet exerts a force always at
right angles to the motion of the electric charge. This force, now
called the Lorentz force—after Hendrik Lorentz, a physicist who
came close to discovering relativity himself—can be pictured as
follows:
The charge moving between the poles of the magnet gets pushed
upward.
But now consider how things would look from the frame of the
particle. In its frame, the magnet would be moving past it.
ͣͥ
But by convention we think of an electrically charged particle at
rest as being affected only by electric forces. Thus, since the particle
is at rest in this frame, the force pushing the particle upward in this
picture would be interpreted as an electric force.
One person’s magnetism is therefore another person’s electricity,
and what connects the two is motion. The unification of electricity
and magnetism reflects at its heart that uniform relative motion
gives observers different perspectives of reality.
Motion, a subject first explored by Galileo, ultimately provided,
three centuries later, a key to a new reality—one in which not only
electricity and magnetism were unified, but also space and time. No
one could have anticipated this saga at its beginning.
But that is the beauty of the greatest story ever told.
ͤ͜
C h a p t e r 6
&nb
sp; T H E S H A D O W S O F R E A L I T Y
As they were walking along and talking together, suddenly a
chariot of fire and horses of fire appeared and separated the two
of them.
—2 KINGS 2:11
One might have thought that, in 1908, following the
aftershock of the discovery of an unexpected hidden connection
between space and time, nature couldn’t have gotten much stranger.
But the cosmos doesn’t care about our sensibilities. And once again,
light provided the key to the door of the rabbit hole to a world that
makes Alice’s experiences seem tame.
While they may be strange, the connections unearthed by
Einstein and Minkowski can be intuitively understood—given the
constancy of the speed of light—as I have tried to demonstrate. Far
less intuitive was the next discovery, which was that on very small
scales, nature behaves in a way that human intuition cannot ever
fully embrace, because we cannot directly experience the behavior
itself. As Richard Feynman once argued, no one understands
quantum mechanics—if by understand one means developing a
concrete physical picture that appears fully intuitive.
Even many years after the rules of quantum mechanics were
discovered, the discipline would keep yielding surprises. For
example, in 1952 the astrophysicist Hanbury Brown built an
apparatus to measure the angular size of large radio sources in the
sky. It worked so well that he and a colleague, Richard Twiss, applied
the same idea to try to measure the optical light from individual stars
ͤ͝
and determine their angular size. Many physicists claimed that their
instrument, called an intensity interferometer, could not possibly
work. Quantum mechanics, they argued, would rule it out.
But it worked. It wasn’t the first time physicists had been wrong
about quantum mechanics, and it wouldn’t be the last. . . .
Coming to grips with the strange behavior of quantum mechanics
means often accepting the seemingly impossible. As Brown himself
amusingly put it when trying to explain the theory of his intensity
interferometer, he and Twiss were expounding the “paradoxical
nature of light, or if you like, explaining the incomprehensible—an
activity closely, and interestingly, analogous to preaching the
Athanasian Creed.” Indeed, like many of the stranger effects in
quantum mechanics, the Holy Trinity—Father, Son, and Holy Ghost
all embodied at the same time in a single being—is also seemingly
impossible. The similarity ends there, however.
Common sense also tells us that light cannot be both a wave and
a particle at the same time. However, in spite of what common sense
suggests, and whether we like it or not, experiments tell us it is so.
Unlike the Creed, developed in the fifth century, this fact is not a
matter of semantics or choice or belief. So we don’t need to recite
quantum mechanics creeds every week to make them seem less
bizarre or more believable.
One hears about the “interpretation of quantum mechanics” for
good reason, because the “classical” picture of reality—namely the
picture given by Newton’s laws of classical motion of the world as
we experience it on human scales—is inadequate to capture the full
picture. The surface world we experience hides key aspects of the
processes that underlie the phenomena we observe. So too Plato’s
philosophers could not discover the biological processes that govern
humans by observing just the shadows of humans on the wall. No
ͤ͞
level of analysis would be likely to allow them to intuit the full reality
underlying the dark forms.
The quantum world defies our notion of what is sensible—or
even possible. It implies that at small scales and for short times, the
simple classical behavior of macroscopic objects—baseballs thrown
from pitcher to catcher, for example—simply breaks down. Instead,
on small scales, objects are undergoing many different classical
behaviors—as well as classically forbidden behaviors—at the same
time.
Quantum mechanics, like almost all of physics since Plato, began
with scientists thinking about light. So it is appropriate to begin to
explore quantum craziness by starting with light, in this case by
returning to an important experiment first reported by the British
polymath Thomas Young around 1800—the famous “double-slit
experiment.”
Young lived in an era that is hard to appreciate today, when a
brilliant and hardworking individual could make breakthroughs in a
host of different fields. But Young was not just any brilliant
hardworking individual. He was a prodigy, reading at two, and by the
age of thirteen he had read the major Greek and Latin epic poems,
had built a microscope and a telescope, and was learning four
different languages. Later, trained as a medical doctor, Young was
the first to propose, in 1806, the modern concept of energy, which
now permeates every field of scientific endeavor. That alone would
have made him memorable, but in his spare time he also was one of
the first to help decipher the hieroglyphics on the Rosetta stone. He
developed the physics of elastic materials, associated with what is
now called Young’s modulus, and helped first elucidate the
physiology of color vision. And his brave demonstration of the wave
nature of light (which argued against Isaac Newton’s powerful claim
ͤ͟
that light was made of particles) was so compelling that it helped lay
the basis of Maxwell’s discovery of electromagnetic waves.
Young’s experiment is simple. Let’s return to Plato’s cave and
consider a screen placed in front of the back wall of the cave. Place
two slits in the screen as shown below (as seen from above):
If the light is made of particles, then those light rays that pierce
the slits would form two bright lines on the wall behind these two
slits:
However, it was well known that waves, unlike particles, diffract
around barriers and narrow slits and would produce a very different
pattern on the wall. If waves impinge on the barrier, and if each slit is
narrow, a circular pattern of waves is generated at each slit, and the
patterns from the two slits can “interfere” with each other,
sometimes constructively and sometimes destructively. The result is
a pattern of bright and dark regions on the back wall, as shown
below:
ͤ͠
Using just such an apparatus, with narrow slits, Young reported
this interference pattern, characteristic of waves, and so definitively
demonstrated the wave nature of light. In 1804, this was a milestone
in the history of physics.
One can try the same experiment that Young tried for light on
elementary particles such as electrons. If we send a beam of electrons
toward a phosphorescent screen, like the screen in old-fashioned
television sets, you will see a bright dot where the beam hits the
screen. Now imagine that we put two slits in front of the sc
reen, as
Young did for light, and aim a wide stream of electrons at the screen:
Here, based on the reasoning I gave when I discussed the
behavior of light, you would expect to see a bright line behind each
of the two slits, where the electrons could pass through to the screen.
However, as you have probably already guessed, this is not what you
would see, at least if the slits are narrow enough and close enough.
Instead, you see an interference pattern similar to that which Young
observed for light waves. Electrons, which are particles, seem to
behave in this case just like waves of light. In quantum mechanics,
particles have wavelike properties.
That the electron “waves” emanating from one slit can interfere
with electron “waves” emanating from the other slit is unexpected
ͤ͡
and strange, but not nearly as strange as what happens if we send a
stream of electrons toward the screen one at a time. Even in this case,
the pattern that builds up on the screen is identical to the
interference pattern. Somehow, each electron interferes with itself.
Electrons are not billiard balls.
We can understand this as follows: The probability of an
electron’s hitting the screen at each point is determined by treating
each electron as not taking a single trajectory, but rather following
many different trajectories at once, some of which go through one
slit and some of which go through the other. Those that go through
one slit then interfere with those that go through the other slit—
producing the observed interference pattern at the screen.
Put more bluntly, one cannot say the electron goes through either
one slit or the other, as a billiard ball would. Rather it goes through
neither and at the same time it goes through both.
Nonsense, you insist. So you propose a variant of the experiment
to prove it. Put an electron-measuring device at each slit that clicks
when an electron passes through that slit.
Sure enough, as each electron makes its way to the screen, only
one device clicks each time. So each electron apparently does go
through one and only one slit, not both.
However, if you now look at the pattern of electrons accumulating
at the screen behind the slits, the pattern will have changed from the
Lawrence Krauss - The Greatest Story Ever Told--So Far Page 8
