Hiding in the Mirror (25 page)

Before wondering what this idea might imply
regarding the actual meaning of extra dimensions, you might wonder
how it could be possible that four dimensions could contain
all
the physical information of a
fivedimensional universe? After all, if one has extra dimensions,
there are extra physical degrees of freedom available. In our own
world, for example, it is hard to ignore the extra freedom offered
by being able to access the third dimension to jump over obstacles
on the ground, or the second dimension to go around obstacles in
front of you. If four dimensions are somehow to encompass five, then
somehow the extra five-dimensional physical degrees of freedom have
to be encoded—obviously in a different form—in the
lower-dimensional space. Perhaps it is simplest to think of the
four-dimensional universe as the surface of a fivedimensional
volume. Then the question becomes: How could one encode all the
information associated with some volume on a surface bounding that
volume?

Framed in these terms, there is a well-known
example of precisely this phenomenon in three dimensions:
holograms. A hologram, stored on a piece of film or plate, is a
two-dimensional record of a three-dimensional scene. But when you
look at or through the holographic sheet, depending upon its type
and the source of light, you see the entire original
threedimensional image. If you move your head, you can look
around
foreground objects to see objects in
the background. Unlike a photograph, which simply stores a
two-dimensional projection of the three-dimensional image a
hologram stores
all
the information in an
image. The reason a hologram allows this degree of image
reconstruction is reminiscent of the information loss problem when
material falls into a black hole. If the black hole evaporates,
then all the energy that fell into it may be radiated away by
Hawking radiation, but the question of whether the information can
be retrieved comes down to delicate issues having to do with
measurement, and what can be reconstructed from subsequent
detection of this radiation. When an ordinary camera records an
image, it simply records the intensity of light of each color
impinging on the photographic film, or the electronic digital
recording media, in the case of digital cameras. Because light is a
wave, however, not only does it have an intensity, but its
electromagnetic fields at any point oscillate in time as the wave
passes by. Different light rays, associated with different
electromagnetic waves, will cause electromagnetic field oscillations
which will in general be out of phase with one another when they
pass different points. This phase information is not recorded when
the light intensity alone is measured at any given point. However,
holograms manage to use sophisticated techniques to capture this
additional information. When this phase data is stored on a
twodimensional piece of film, it turns out that a full
three-dimensional image can be reconstructed.

The idea that is central to the Maldacena
conjecture—that somehow all the physical information in a volume
can be encoded on its surface—has thus become known as the
holographic principle
. I stress that while
it has been applied in a variety of contexts by various theorists,
the actual Maldacena conjecture itself involved two very specific
spaces: a four-dimensional flat space with supersymmetry and quantum
Yang-Mills fields, and a fivedimensional space with classical
supergravity, along with a very weird specific source of gravity
throughout the five dimensions (empty space full of negative
energy—unlike anything we have measured in our own universe). Such
a space is called an Anti-de Sitter space.

In any case, if Maldacena’s conjecture is
correct—namely, that there is absolutely no physical difference
other than appearance between these two spaces—then the physical
distinction between different dimensions itself gets blurred. A
host of questions naturally seems to arise. What is the utility of
an extra hidden dimension if ultimately nothing is hidden except
the existence of the extra dimension? And what is the practical
meaning of extra dimensions if you can experience all there is to
experience without actually moving into them?

Moreover, we may find ourselves somewhat like
the holodeck characters in
Star Trek,
who
have no sense that they may be mere projections. Are we merely a
pale reflection of the real world behind the mirror? Or, if the
surface contains everything that is inside, is it the extra
dimension itself that is illusory? If the world of our experience
is a hologram, where does the illusion end and reality begin?
Ultimately, if Maldacena’s conjecture is correct, then it implies
that these questions, as fascinating or troubling as they may seem,
are moot. Reality is in the eye of the beholder. Both worlds are
real, and identical, as different as they may seem.

If your head is now spinning, it should be. In
one chapter, you have been treated, or perhaps subjected, to a
menagerie of mathematical marvels associated with strings and
D-branes in ten dimensions, M-theory in eleven dimensions, and
holography in five dimensions. New dimensions have magically
appeared and disappeared with more aplomb than the Cheshire Cat and
with an uncanniness that might appear to make Alice’s voyage in
Wonderland pale in comparison. Most importantly, you may be
wondering what all of this wizardry has wrought? Are these
imaginings of theoretical physicists any more real or of any more
utility than those of Lewis Carroll?

These are good and valid questions. Remember
what ostensibly caused all of this mathematical effort in the first
place. String theory, or rather the Theory Formerly Known as String
Theory, must, if it is to be useful to physicists, address some
concrete physical problems and make concrete physical predictions.
In its original form, it had simply failed to do so, all the hype
surrounding it notwithstanding.

So, as mathematically remarkable as M-theory
might be, or as useful as the Maldacena conjecture might be for
trying to solve difficult mathematical problems associated with
Yang-Mills theories, unless all of these ideas eventually help
resolve fundamental physical questions, it
is
all just mathematics.

Thankfully, however, there has been some
progress. In my mind it is not clear that it fully justifies the
periodic hubris associated with string theory, but we shall see. It
is at least an encouraging beginning. You will recall that a
central problem in quantum gravity, which early work on string
theory did not appear to address, was the “black hole information
loss paradox.” Do black holes violate quantum mechanics? And if
not, where does all the information that falls into black holes
go?

A new approach to this problem did become
possible once D-branes began to be explored. Recall that D-branes
allow a new connection between the strongly interacting phase of
some theories and the weakly interacting phase, where reliable
calculations might be performed. It turns out that in certain
limits one finds objects in string theory that resemble black holes,
with highly curved geometries (in the extra dimensions). These are
called black p-branes. Interestingly, if one explores a different
limit of the same theory, where strings are weakly interacting, one
can describe much of the physics in a calculable way using standard
D-branes. One can hope, then, that the results of calculations one
can explicitly perform in the one limit of the theory where such
calculations are feasible might also be applicable in the other
limit of the theory, where one cannot do direct calculations, and
where the strongly gravitating black p-brane description applies.
Now, if one examines a very special sort of five-dimensional
p-brane, then in the weakly interacting limit of the theory, where
D-brane calculations become reliable, it turns out that one can
explicitly count the number of fundamental quantum states that
could be occupied by an object that would, in the strong coupling
limit of the theory, be associated with a black p-brane.

The result is striking. The number of quantum
states turns out to be precisely the number of states needed to
encode the information that was supposedly hidden behind the event
horizon of a black hole—the so-called Hawking-Bekenstein entropy.
This would suggest that the information is not, in fact, lost down
the black hole, but is instead somehow preserved and if we had a
way of accurately treating the quantum mechanics of realistic black
holes (which, I remind you, are not to be confused with the very
special five-dimensional black p-branes in this idealized
calculation), we would uncover it.

Note that this result is far from a proof that
black holes in string theory must behave like sensible quantum
mechanical objects, nor does it provide any hint of what might
actually happen to the information stored in a black hole’s
interior as it evaporates. Moreover, the black p-branes in question
are actually very finely tuned objects, which wouldn’t themselves
even evaporate by Hawking processes because of their special
configuration. However, this calculation is at least very
encouraging. In the regime where D-branes, which are perfectly
well-behaved quantum mechanical objects, are the appropriate
description of string theory/M-theory, there are precisely the
correct number of states to account for what one might hope a
well-behaved quantum mechanical accounting of black holes might
require. This was a real computational success in string theory,
and it has generated tremendous enthusiasm.

Nevertheless, a host of caveats remain. As one
increases the strength of the interaction needed to move from the
D-brane to black p-brane picture, the physics could change, and
information could be lost. Until one can calculate precisely where
the information flows in the evaporation process of realistic black
holes, extrapolating the apparent success of this aspect of the
theory remains a conjecture.

Also, as I have mentioned, a few months before
this writing Stephen Hawking made headlines throughout the world by
retracting his claim that black hole evaporation destroys
information. He has claimed that a new computation he has performed
in the context of classical general relativity demonstrates
explicitly how the information that falls into black holes gets
preserved as they evaporate. He has spoken about this at several
meetings. Many physicists are skeptical. However, when it comes to
black holes, Stephen has a good track record.

If Hawking’s new claim is correct, then it will
have a profound implication for the apparent success of string
theory in potentially addressing the black hole information loss
problem in classical general relativity, because the problem will
have literally evaporated. This will not mean that the string
accounting of p-brane states is incorrect, just that string theory
would not have been needed to solve this fundamental problem that
otherwise appeared to suggest the need to move beyond general
relativity. String theorists will have to turn their attention to
other problems the theory might more uniquely address. Which brings
us back, finally, to Einstein’s revenge: the cosmological constant
problem. This, after all, remains the key mystery in theoretical
physics, and the clearest place where a theory of quantum gravity
should shed some light. And it is the place where, I think it is
fair to say, string theory had its biggest unmitigated lack of
success. Nothing in all the work following the first string
revolution, or even immediately following the discovery of the
importance of D-branes and the emergence of M-theory, had shed any
light on the question of how the energy of empty space could be
precisely zero.

So, when in 1998 cosmological observations led
to the discovery that the energy of empty space isn’t precisely
zero, just almost zero, everyone—including string theorists—stood
up and took notice. Maybe, just maybe, this finding might provide a
vital clue that could either vindicate the string revolution or
help us move beyond it.

The result was a sudden new explosion of
interest in—you guessed it—extra dimensions—but not the
hypothetical, aetherial, and perhaps illusory extra dimensions that
had so fixated the ten-or eleven-dimensional imaginations of string
theorists. Rather, they were concrete and even potentially
accessible extra dimensions that might literally be hiding behind
the looking glass or on the other side of the wardrobe.

The small man said to the
other:

“Where does a wise man hide
a pebble?”

And the tall man answered
in a low voice:

“On the beach.”

—G. K. Chesterton

I
t is easy, in the
midst of discussing such things as D-branes and supersymmetric
state counting, to forget precisely what we are really talking
about here. In order to understand what might otherwise be
considered a somewhat esoteric corner of physical theory—the
intersection of gravity and quantum mechanics—string theory or its
successor, M-theory, suggests that we need to believe that the
world of our experience is but a minor reflection of a
higher-dimensional reality. The tragedies of human existence may be
very poignant, and the evolution of our visible universe may be
remarkable, but actually, they are all fundamentally a cosmic
afterthought. Somehow the key to our existence lies in the poorly
understood, but remarkably rich, possibilities available to a
universe with perhaps seven extra dimensions, although one or more
of these may not behave like any dimensions we have experienced.
Moreover, the conventional wisdom, steeped in a tradition
established by Kaluza and Klein almost a century ago, suggests
these seven dimensions are “compactified,” bundled up for as-of-yet
unknown reasons into regions so small that a pebble lying on a
beach would be, by comparison, as large as our own galaxy is
compared to the pebble.

At the same time deeper questions arise, some
of which I have already considered. If the convolutions appropriate
to extra-dimensional physics really do ultimately lead to a picture
of the four-dimensional universe that accurately resembles the
reality we experience, but if at the same time these dimensions
remain forever hidden, ephemeral theoretical entities, inaccessible
to our experiments, if not our imagination, then in what sense are
these extra dimensions more than merely mathematical
constructs?

What, in this case, does it mean to be
real
?

There are times when I have wondered whether
Michael Faraday, as he developed his fantastic mental images of
hypothetical electric and magnetic fields permeating space in 1840
felt that they were so simple and beautiful that they had to exist.
Or did he consider them to be merely a convenient crutch, so that
someone like himself, unschooled in mathematics, could comprehend
in an intuitive way some sliver of the physical world?

As I have mentioned, there is, of course, a
noble tradition in physics of mathematical crutches turning out to
have a physical reality. Faraday’s electromagnetic fields are just
one example. Quarks, when they were first introduced, were also seen
primarily as a mathematical classification scheme, rather than as
real entities. So, too, were atoms, for that matter. Indeed, Ludwig
Boltzmann committed suicide in part because he felt he could not
convince his contemporaries that atoms had to be real. On the other
hand, many mathematical models that have been proposed have thus
far borne no relation to the real world—even mathematics that at
one time or another seemed to show great promise. So the questions
posed earlier remain relevant, and short of a theoretical
breakthrough that unambiguously allows a prediction of unique laws
of nature that match the ones we observe, the only way we may know
if any of these higher-dimensional imaginings are correct is if
somehow we can ultimately experimentally probe the extra
dimensions, either directly or indirectly. Traditionally in string
theory this has seemed like a colossal long shot. If the string
scale is comparable with the Planck scale—about 10–33 cm, where
quantum mechanical effects in gravity are presumed to become
important—it is far removed from anything remotely accessible in
the laboratory. Imagine you were looking at our galaxy through a
distant telescope from another galaxy far, far away. Say your
telescope could just barely resolve individual stars in the Milky
Way, as the Hubble Space Telescope can in the nearby Andromeda
galaxy, two million light-years away. The problem of measuring
extra dimensions on the Planck scale is for us, then, similar to
the problem of your trying to detect and probe individual atoms in
that distant galaxy using your telescope!

The past decade has, however, produced some
remarkable transformations in the way we think about fundamental
physics, driven largely, I am happy to say, by the surprises nature
has wrought.

Nature provided a cosmic wake-up call, in the
form of dark energy, that not even those fully immersed in
eleven-dimensional mathematics could ignore. In particular, the
discovery that dark energy dominates the expansion of the universe
is so shocking that it seems very likely that it is related to
something fundamentally profound about the structure of space and
time. And since string theory has taken as its mantra the
revelation of profound new truths in these areas, the unexpected
appearance of dark energy cried out for attention. Or, at the very
least, it was irritating to the point of distraction.

The distraction was key, however. It stood as a
stark reminder that, at the earliest moments of the big bang, what
is now the visible universe was of a size comparable to the
microscopically small scale of the purported extra dimensions. Thus
perhaps the universe itself could provide the experiment that might
ultimately reveal these extra dimensions for all to witness.

I remember David Gross’s telling me in 2002 why
string theorists had suddenly become so interested in cosmology.
The big bang, taken back to t = 0, inevitably leads to a
singularity (a point of infinite density) at the beginning of time.
There is clearly something physically implausible about such a
state of infinitely high density. One of the main virtues of string
theory, however, is its apparent ability to dispense with such
infinite singularities, at least those that seemed to plague general
relativity. Thus, string theory might be able to dispense with the
big bang singularity, and perhaps in the process explain the
mystery of dark energy. I confess that in a skeptical moment I
responded to David by expressing the concern that string theory
might instead do for observational cosmology what it has thus far
done for experimental elementary particle physics: namely,
nothing.

Sarcasm aside, however, in 1998 several
theoretical breakthroughs transformed the way much of modern
research is being performed, and have made the question of the
possible reality of extra dimensions something of immediate and
practical interest. They did not arise from cosmology, however,
although they opened up, literally, a whole new universe of
cosmological possibilities. Rather, they were inspired by a new
consideration, reflected in the glow of D-branes, of the very same
problem that first motivated many particle physicists to adopt
supersymmetry as a useful guiding principle in nature: the
hierarchy problem.

Recall that the hierarchy problem in particle
physics relates to the question of why the GUT energy scale, where
the three nongravitational forces in nature may be unified, could be
fifteen orders of magnitude larger than the scale at which the weak
and electromagnetic interactions are unified. Worse still, the
Planck energy scale, where quantum gravity should become important,
is seventeen orders of magnitude larger than this latter scale. Not
only are these large discrepancies of scale inexplicable, but it
turns out that formally, within the context of the standard model
of particle physics without supersymmetry, this hierarchy is
unstable. Namely, as I have described, the effects of high-energy
virtual particles will tend to lead to intolerably large
corrections in the low-energy theory.

In 1998 physicists Nima Arkani-Hamed, Savas
Dimopoulos, and Gia Dvali proposed a dramatic new way of avoiding
the hierarchy between the Planck scale and the weak scale. They
suggested that perhaps the Planck scale is not really where we
think it is. The group was motivated by considering the possible
existence of extra dimensions, and also indirectly by the
development of D-branes in string theory. As I shall describe,
their argument relied on the possibility that perhaps the extra
dimensions, or at least one of them, might in fact not be
microscopically small, but rather could be “almost”
visible—perhaps, in fact, the size of a small pebble lying on a
gravel road.

An immediate question that comes to mind when
this possibility is raised is: If the extra dimensions are that
big, why don’t we see them? A possible answer lies in the magic of
D-branes. Remember that in string theory, open strings can end on
D-branes, so that the charges on the ends of these strings, and the
Yang-Mills fields and forces associated with these charges, might
reside only on the D-branes. Remember, however, that gravitons, the
particles associated with gravitational fields, are associated with
closed string loops (i.e., objects without ends). These loops can
also move about in the space between the branes, and thus
gravitational fields are not restricted to exist only on the
D-branes, but can also exist in the “bulk,” as the space between
the branes is called.

Thus, imagine that the three spatial dimensions
of our experience lie on a three-brane “surface” in a
higher-dimensional space. If gravity is the only force that can
exist outside of our three-brane, then only gravity could probe
these extra dimensions.

How would gravity do so? Well, Newton’s theory
of gravity tells us that the gravitational force between two
objects falls off inversely as the square of the distance between
them. This is, after all, precisely the same behavior that
characterizes the electric force between charged objects. We now
return at long last to Michael Faraday, whose brilliant idea of
field lines helped to provide an intuitive understanding of why
electric forces actually fell off as the inverse square of
distance. Remember that if field lines move out in all directions
from a charged particle, the number of field lines per unit area
crossing any surface will fall off inversely with this area or,
equivalently, inversely as the square of the distance from the
source.

What Arkani-Hamed and his collaborators
proposed was that a similar argument would suggest that if
gravitational forces propagated in extra dimensions, as well as in
our three-dimensional space, then the strength of the gravitational
force measured between massive particles in our space would fall
off faster than the inverse of the square of the distance between
them. Imagine, for example, a single extra dimension. If field lines
could spread out in our three dimensions, but also in this extra
dimension, then the number of field lines per unit area would fall
off as the area of a threedimensional spherical surface (bounding a
four-dimensional volume), and not as the two-dimensional spherical
surface bounding a three-dimensional volume that we normally
picture when we draw field lines spreading out into space. Since the
area of a three-dimensional spherical surface increases with the
cube of its radius, and not the square of its radius, as in a
twodimensional spherical surface, this means that the strength of
gravity would fall off inversely with the
cube
of distance, not the
square
of distance. There is, of course, a slight
problem here. Newton achieved fame and fortune by demonstrating
that a universal gravitational force that fell off with the square
of distance could explain everything from falling apples to the
orbits of planets! So, what gives?

Well, it is true that gravity has been measured
with great precision to have an inverse square law on scales
ranging from human scales to galactic scales. But, as Arkani-Hamed
and collaborators pointed out, it hadn’t been so measured at scales
smaller than about a millimeter. Imagine, then, that the extra
spatial dimension has a size of a millimeter. Then for objects
separated in our space by less than about a millimeter, the force
of gravity will fall off with the cube of distance. But once
objects get separated by a larger amount than this, the
gravitational field lines from one particle cannot spread out any
more in the extra dimension as they can do in our three dimensions
of space. As the field lines can continue to spread out only in the
three remaining large dimensions on scales larger than a
millimeter, the gravitational field again begins to now fall off
inversely with the square of distance. An example starting with
three dimensions, one of which is small, is shown below: