Hiding in the Mirror (8 page)

Here, for the first time in human history, was
an empirical observation capable of disentangling the geometry of
the entire visible universe. And

you don’t have to be a rocket scientist to
discern the answer. As in the Goldilocks story, the lumps predicted
in the positively curved universe were too large compared to the
observations, while the lumps in a negatively curved universe were
too small. A precisely flat universe, however, would produce more or
less precisely what was observed. Just as Lobachevsky had inferred
150 years earlier, on a scale that we now recognize would have been
far too small to detect the minute curvature of space that might
have existed on these scales, observations of the cosmic microwave
background have now convincingly suggested that we live in a flat
universe.

One’s first response might be “How boring.” Of
all the interesting possible universes to live in that are allowed
by general relativity, why should we live in one that is precisely
flat on large scales?

Before I attempt to answer that, let me attempt
to clear up a possible misconception that you may have arrived at
from what you have just read. Remember that I described earlier how
Einstein’s theory of general relativity was first experimentally
confirmed in 1919 by witnessing the fact that light rays bent around
the sun. Yet I have now just argued that light rays that traverse
the universe travel in straight lines. These two facts are not
inconsistent. Matter can locally curve space in its vicinity, as
the sun, the earth, and even you do. However, the fundamental
question that has puzzled physicists since Einstein first proposed
his theory was whether the sum total of all the matter and energy
in the universe produces a net curvature of space on the largest
scales. If it did, one could imagine, for example, as Einstein first
did, that space could ultimately curve back upon itself so that one
could live in a finite universe, but one without end. Namely, if you
looked far enough in any direction, you would see the back of your
head! It is like a three-dimensional version of living on the
surface of an expanding balloon.

A finite but endless universe is fascinating,
but it does have one drawback. If matter and radiation are all that
make up such a universe, general relativity implies that it must
ultimately recollapse back into a hot, dense reverse of the big
bang. This provides a rather unpleasant end, and so it is fortunate
that other possible geometries for the universe exist that may
imply less violent finales. A negatively curved universe, like a
three-dimensional version of a horse’s saddle, can be infinite in
spatial extent, and such a universe containing matter and radiation
will expand indefinitely. With time the universe would cool down,
its stars would ultimately burn out, and it would become cold and
dark. This, too, is not a particularly pleasant future, but the
timeframe over which the darkness would fall is so
gradual—trillions of years—that such a universe, which ends with a
whimper rather than a bang, seems more hospitable, at least from a
human perspective. Falling right between these two extremes is a
flat universe. In such a universe containing matter and radiation,
our expansion will continue to slow with time, but it will never
quite stop. Like a negatively curved universe, it, too, can be
infinite in spatial extent. However, because the expansion rate
slows more quickly in this universe than in a negatively curved
space, the time it takes before such a universe becomes cold and
empty is far longer.

Longevity is not the reason that theorists
preferred a flat universe long before observations confirmed this to
be the case, however. The reason for their preference is partly
aesthetic and partly practical. Einstein’s equations from general
relativity establish a relationship between the curvature of the
universe, the rate of its expansion, and the total density of
matter and energy within it. Observations of the expansion of the
universe and measurements of the total matter density had long
established that these quantities were within an order of magnitude
or so of what was required to produce a flat universe.

Now, being within an order of magnitude is
certainly not compelling evidence, on its own, of equality. But a
remarkable mathematical relationship does exist that made believers
out of many theorists long before the appropriate experimental
evidence was amassed. It turns out that general relativity implies
that if the geometry of the universe is not flat, then, as the
universe expands, it quickly moves farther and farther away from
the mathematical equality implied by flatness. Since the universe is
over ten billion years old, it is difficult to imagine how the
relation between the expansion rate and the mass density could
still remain so close to that for a flat universe unless the
universe was, in fact, precisely flat. This puzzle was so significant
that cosmologists even gave it a name: the flatness problem. In 1981
a Stanford physicist (now at MIT) named Alan Guth proposed an
ingenious mechanism that would resolve this puzzle by producing,
independent of its initial conditions, a flat universe today. His
idea, called inflation, was that the universe underwent a rapid
early period of expansion, far faster than had previously been
envisaged. Like a balloon being blown up, as the universe inflated,
any original curvature of space would be progressively reduced,
ultimately producing a universe that was indistinguishable from a
precisely flat universe. What’s more, Guth demonstrated that
physical conditions that would lead to an early inflationary phase
could arise naturally in so-called grand unified models of particle
physics, which I shall later describe, in which the fundamental
forces in nature are unified into a single force at very early
times. Once Guth had shown that inflation could easily result in
these models, and how it could resolve a variety of fundamental
problems in cosmology beyond the flatness problem, it quickly became
the basis of what is now considered the standard model of
cosmology. Aside from Guth’s inflationary paradigm, there is,
however, another reason a flat universe is particularly attractive,
at least from a theorist’s perspective: The total gravitational
energy of a flat universe is precisely zero!

How can a universe full of matter and radiation
have zero total energy? While the energy associated with these
quantities, in the absence of considerations of gravity, is indeed
positive, it turns out that the gravitational energy of attraction
between objects is negative. This is another way of saying that it
takes energy to pull objects farther apart, so they have less
energy if they are close together. Hence, all objects of a finite
size have less gravitational energy than they would have if they
were dispersed over infinitely large distances. If we define such a
state in which matter is infinitely diluted as having zero energy,
then all other, smaller, configurations have negative energy. If
this negative energy precisely cancels the positive energy of
matter and radiation in the universe, then general relativity tells
us that the overall curvature of space vanishes.

Moreover, with zero net energy, the possibility
that the universe itself arose spontaneously out of nothing becomes
at least plausible, since one would imagine that “nothing” would
also have zero energy. As Guth put it: “There is such a thing as a
free lunch!” It was theoretical considerations such as these, which
are primarily mathematically aesthetic, that convinced most
theorists and ultimately even many observers, well in advance of
the cosmic microwave background observations, that the universe was
flat. In this case, as sometimes but not always happens in science,
nature cooperated.

However, it was premature to slap ourselves on
our collective backs and congratulate one another. For, what
actually makes the universe flat is something that no one, or at
least almost no one, anticipated. Perhaps the most puzzling
discovery in all of physics during the past century has been the
fact that the dominant form of energy in the universe is not
associated with matter or radiation at all. Rather, it appears that
empty space, devoid of any particles at all, carries energy—enough
energy, in fact, to overwhelm, by a factor of almost three, the
energy of everything else in the universe.

This energy of empty space, sometimes called
“vacuum energy” or “dark energy,” is the most mysterious form of
energy we know of. No one currently has a good explanation of why
empty space should have precisely this amount of energy, and, as we
shall see, trying to understand its nature is currently driving
much of our current scientific thinking about the nature of space
and time itself.

The discovery of a mysterious energy permeating
all of empty space also changed everything in the way we think
about cosmology. Even the original, vital connection between
geometry and destiny is now gone. If empty space can possess
energy, a positively curved universe need not ultimately collapse,
while a negatively curved or flat universe need not expand forever.
Still, as I have suggested, it could be that there might be some
deeper connection between the geometry of space and its energy
content, perhaps something that involves probing yet deeper into
the meaning of space and time. Certainly the puzzle of dark energy
is so revolutionary it motivates even extreme reconsiderations of
the nature of space and time. The resolution of this mystery may
not be as revolutionary as the question itself, but one never knows
until one explores the possibilities. But extraordinary claims
require extraordinary evidence, as Carl Sagan used to say. We shall
return to this mystery later in the book. First, however, we shall
explore how the collective creative imagination of the world
responded to our first revolution in the physics of space and time
inspired by Einstein and later Minkowski: Namely, the existence of
a fourdimensional space-time continuum associated with special
relativity.

Ever drifting down the
stream—

Lingering in the golden
gleam—

Life, what is it but a
dream?

—Lewis Carroll,
Through the
Looking Glass

W
hile life may imitate
art, it is nevertheless also true that art imitates life. One might
thus wonder whether the publication of Abbott’s
Flatland
within a decade following Maxwell’s
discovery about the nature of otherwise invisible electric and
magnetic fields and less than a decade before Michelson and Morley’s
experiments to probe the ether and Lorentz’s pioneering
speculations about the nature of space and time was purely a
coincidence. Was there something in the intellectual air at the
time that suggested something revolutionary was about to occur in
our understanding of nature?

In one sense the answer to this question is
clearly no. It was, after all, in 1900 that Lord Kelvin uttered his
famous remark that all laws of physics had already been discovered
and all that remained were more and more precise measurements. Yet
in spite of such hubris, scientific and mathematical puzzlement
about the nature of space and time had been spilling over to the
literary imagination for well over a century before Abbott wrote
his story. The notion that time might somehow be considered a
fourth dimension actually appeared in print as early as 1754, in an
article by Jean Le Rond d’Alembert on “Dimensions” in his
Encyclopédie,
although he attributed the idea to a
friend, possibly the French mathematician Joseph-Louise Lagrange. A
hundred years later German psychologist and spiritualist Gustav
Fechner wrote a satirical piece involving a “shadow man,” the
shadow projection of a three-dimensional image. Interestingly,
Fechner argued that such shadow figures would interpret the effects
of motion perpendicular to their plane of existence (which they, of
course, could not perceive as movement in space) as acting like
time. Fechner’s combined interest in extra dimensions and
spiritualism presaged, as we shall see, events that would unfold a
half a century later.

Ultimately the notion of time as a fourth
dimension was made famous within popular culture a full decade
before Einstein’s special relativity and thirteen years before
Minkowski clarified the dimensional relationship between space and
time by none other than H. G. Wells in his classic science fiction
epic,
The Time Machine,
published in 1895.
On the very first page of this novel, Wells’s hero, the Time
Traveller, has the following dialogue with an audience he has
invited for the occasion:

“You must follow me carefully. I shall have to
controvert one or two ideas that are almost universally accepted.
The geometry, for instance, they taught you at school is founded on
a misconception.”

“Is not that rather a large thing to expect us
to begin upon?” said Filby, an argumentative person with red
hair.

“I do not mean to ask you to accept anything
without reasonable ground for it. You will soon admit as much as I
need from you. You know of course that a mathematical line, a line
of thickness NIL, has no real existence. They taught you that?
Neither has a mathematical plane. These things are mere
abstractions.”

“That is all right,” said the Psychologist.

“Nor, having only length, breadth, and
thickness, can a cube have a real existence.”

“There I object,” said Filby. “Of course a
solid body may exist. All real things.”

“So most people think. But wait a moment. Can
an INSTANTANEOUS cube exist?”

“Don’t follow you,” said Filby.

“Can a cube that does not last for any time at
all, have a real existence?”

Filby became pensive. “Clearly,” the Time
Traveller proceeded, “any real body must have extension in FOUR
directions: it must have Length, Breadth, Thickness, and—Duration.
But through a natural infirmity of the flesh, which I will explain to
you in a moment, we incline to overlook this fact. There are really
four dimensions, three which we call the three planes of Space, and
a fourth, Time. There is, however, a tendency to draw an unreal
distinction between the former three dimensions and the latter,
because it happens that our consciousness moves intermittently in
one direction along the latter from the beginning to the end of our
lives.”

“That,” said a very young man, making spasmodic
efforts to relight his cigar over the lamp, “that . . . very clear
indeed.”

“Now, it is very remarkable that this is so
extensively overlooked,” continued the Time Traveller, with a
slight accession of cheerfulness. “Really this is what is meant by
the Fourth Dimension, though some people who talk about the Fourth
Dimension do not know they mean it. It is only another way of
looking at Time. THERE IS NO DIFFERENCE BETWEEN TIME AND ANY OF THE
THREE DIMENSIONS OF SPACE EXCEPT THAT OUR CONSCIOUSNESS MOVES ALONG
IT. But some foolish people have got hold of the wrong side of that
idea. You have all heard what they have to say about this Fourth
Dimension?”

“I have not,” said the Provincial Mayor.

“It is simply this. That Space, as our
mathematicians have it, is spoken of as having three dimensions,
which one may call Length, Breadth, and Thickness, and is always
definable by reference to three planes, each at right angles to the
others. But some philosophical people have been asking why THREE
dimensions particularly—why not another direction at right angles
to the other three?—and have even tried to construct a
FourDimension geometry. Professor Simon Newcomb was expounding this
to the New York Mathematical Society only a month or so ago. You
know how on a flat surface, which has only two dimensions, we can
represent a figure of a three-dimensional solid, and similarly they
think that by models of three dimensions they could represent one
of four—if they could master the perspective of the thing.
See?”

This passage is remarkable not merely because
of Wells’s anticipation of a connection between space and time in a
four-dimensional framework, but because he correctly recognized
that what fascinated writers and the public alike was not a
temporal
fourth dimension but a spatial
one. Wells also wrote several stories reminiscent of
Flatland,
in which he utilized four spatial
dimensions as plot devices. In no fewer than four tales Wells
exploited different manifestations of extra dimensions that would
be borrowed by a host of future science fiction writers. These
included a story involving a person being turned into his mirror
image through a four-dimensional rotation, the possibility of
connecting otherwise distant locations in three-dimensional space
via a four-dimensional portal, the mysterious appearance and
disappearance of a four-dimensional being (an angel, as it happens)
traveling through our three-dimensional plane of existence, and
finally an object achieving invisibility by sliding into the fourth
dimension.

About 150 years earlier, around the same time
as d’Alembert was writing, none other than Immanuel Kant was
pondering the possibilities of extra spatial dimensions. While he
may have felt that Euclidean geometry was an essential part of
existence, he was much more sanguine about variations beyond our
three-dimensional space, although he felt that while they could
exist, they must be separate from ourselves. He discussed this
possibility in his very first published work,
Thoughts on the True Estimation of
Living Forces,
concluding: “Spaces of this kind,
however, can not stand in connection with those of a quite
different constitution. Accordingly such spaces would not belong to
our world, but must form separate worlds.”

The German physicist and mathematician August
Möbius, father of the famous one-sided Möbius strip, followed up on
Kant’s earlier musings from the 1700s and came up with an
interesting suggestion. He argued in 1827 that a fourth dimension
would allow otherwise distinct threedimensional figures—such as a
right hand and a left hand—to coincide. Namely, just as a mirror
flips left and right, one could turn a right hand into a left hand
by twisting it into a fourth dimension and back again. Indeed,
Kant, himself, in his
Prolegomena to Any Future
Metaphysics
(1793) wondered explicitly about how a right hand
becomes a left hand when viewed in a mirror, and so two identical
objects can at the same time be completely different.

The premise inherent in
Flatland
was that we could simply be ignorant of an
ever-present fourth spatial dimension, which would appear as
foreign to our intuition as a third dimension would be to a
two-dimensional being. Abbott was, of course, not writing in a
vacuum, and there was a swirl of activity in England in the years
prior to 1884 surrounding attempts to understand physically and
mathematically what a fourth dimension might be like. As I have
mentioned, H. G. Wells himself wrote at least one tale in which
this very issue is central. His “The Plattner Story”(1896) focuses
on an individual who moves into the fourth dimension and returns
with left and right inverted. Almost eighty years later, a charming
rendition of this same apparent paradox was replayed in Lars
Gustafsson’s tale
The Death of
a Beekeeper.
The protagonist muses: “But, since I
moved outside the normal dimensions, right and left somehow got
exchanged. My right hand is now my left one, my left hand my right
one.” At the same time, this transition changes his previous,
pessimistic, view of our world: “Returned into the same world and
see it now as a happy one. The shreds of peeled paint on the door
belong to a mysterious work of art.” If only it were so. Numerous
authors before Abbott had exploited two-dimensional beings as an
allegory to help us imagine a fourth dimension. In England, the
mathematician J. J. Sylvester wrote a popular article using them in
1869. In it he quoted from the biography of the great mathematician
Gauss, in which the late mathematician was reported to have stated
that he had kept several geometric questions aside, waiting to pass
on so that he would have a better appreciation of four or more
dimensions!

Not only was Sylvester a bold advocate of
understanding four dimensions, he also firmly believed that higher
dimensions actually exist, and strongly asserted an “inner
assurance of the reality of transcendental space.”

Another mathematician who popularized
two-dimensional beings was Charles Dodgson, known to the world as
Lewis Carroll, the author of the Alice in Wonderland stories. In an
1865 story, entitled “Dynamics of a Particle,” he described a
romance between a pair of linear, one-eyed animals moving along on
a flat surface. When I first learned this fact I was particularly
intrigued, because
Through the Looking
Glass
(1872) was the first story I could remember that envisaged
a foreign world lying right beneath our eyes. Moreover it was a
world I had been fascinated with as a child—so much so that it
influenced the title of this book. What if the world hiding on the
other side of a mirror was real?

I have since learned, however, that Dodgson was
in fact parodying the British fascination of the time with the
literal idea of a fourth dimension. Dodgson’s mirror world of
talking chessmen and tiger lilies may not appear to a modern reader
to deal directly with such issues, but apparently the psyches of
nineteenth-century British readers were more attuned to his satire.
At least the white queen’s memories involved both the past and the
future, so time appeared to be heavily involved in the mix. Or
maybe it was the queen’s propensity for believing six impossible
things before breakfast that Dodgson employed to parody the fads of
the time. Actually, Dodgson later became interested in the occult,
and with that presumably his skeptical attitude toward extra
dimensions disappeared. In the late 1870s a more sinister
application of the fourth dimension appeared when a German
physicist and astronomer, J. C. F. Zöllner, who in his day job (or,
more appropriately, night job) actually invented a method of
accurately measuring the brightness of stars, became fascinated
with an American medium named Henry Slade. In séances carried out
for Zöllner and others Slade performed magic tricks—such as untying
a knotted cord without touching it, and transporting objects out of
a sealed container—that seemed to defy explanation unless somehow
he was reaching “into” an extra dimension. Like many of those who
followed (including Russell Targ and Harold Puthoff, who in the
1970s claimed to have found scientific evidence for remote
perception), Zöllner left scientific skepticism behind and fell for
Slade’s chicanery, becoming his ardent defender and writing
prolifically about Slade’s empirical demonstrations of the existence
of extra dimensions. Zöllner’s fascination with Slade and the
occult strikes to the heart, just as forcefully as Alice’s yearning
to disappear into the mirror, of why humans have always seemed to
want to believe in the possibility of extra dimensions. We seem to
need somewhere beyond the world of our experience, a place that’s
either better or just different. Part of this desire, I believe,
arises because, while science describes the workings of the natural
world, it does so without reference to “purpose,”

so that even those who adhere to scientific
principles may ultimately find its view of reality lacking. For
Zöllner and others, the fact that the possibility of the existence
of extra dimensions was at least allowed by science—even if no
direct evidence had been forthcoming—also meant it allowed a place
for a world of purpose, the spiritual world, to exist. This deep
yearning is undoubtedly associated with ubiquity of religion in the
human experience, of which I spoke at the beginning of this book.
The need for a hidden god to guide the universe of our experience
while existing outside of that universe, and the hoped-for
existence of a “better place” where we might go after we die, are
part and parcel of the same sense of longing for something
transcendent that is evident even in the fourteen-thousand-year-old
cave paintings in France.