Hiding in the Mirror (26 page)

What is the net effect of all of this? Well, if
the size of the extra dimension is
R,
then
gravity falls off with one extra power of distance between the
Planck scale and
R,
compared to what it
would be if there were only three dimensions for gravity to
propagate in. Thus, by the time objects are separated by a distance
R,
the gravitational attraction between
them would be weaker by a much larger factor than they would
otherwise have been. If one is only measuring gravity on scales
larger than
R
, gravity would then be
measured to fall off with an inverse square law, just as Newton
argued, but by this time the apparent strength of gravity would be
much smaller than it would have been if the gravitational field had
not been able to fall, at least temporarily, faster than inversely
with the square of distance. Without knowledge of these extra
dimensions, this extra suppression factor would simply be
incorporated into the basic definition of the strength of gravity
itself. This strength is given by what we conventionally call
Newton’s constant, which appears in the formula for the inverse
square law gravitational force between two bodies.

By a similar argument, if there are two extra
hidden compactified dimensions, then gravity will fall off by an
even greater factor between the real Planck scale and the scale,
R,
of the extra dimensions, and so on for
yet more compactified dimensions.

Because it tells us how strong the
gravitational force is between measured bodies, we use Newton’s
constant to determine on what distance or energy scale we expect
that quantum mechanical gravitational effects should become
significant. Thus, it is Newton’s constant that determines the value
of what we conventionally define as the Planck scale. As a result,
because of the hidden effects of the extra dimensions on scales
between the Planck scale
R,
we would
“inaccurately” deduce the Planck scale to be much higher than it
actually is. This is because the strength of the gravitational
force would grow much faster as one decreases the distance between
massive objects, on scales smaller than
R,
than we would otherwise have expected. What Arkani-Hamed and
collaborators realized is that this would allow the possibility
that the real Planck scale might actually be equal to the distance
scale at the point where the weak and electromagnetic interactions
are unified, instead of seventeen orders of magnitude smaller, as we
would otherwise estimate based on our incorrect extrapolation of
the behavior of gravity on scales smaller than
R
. This would therefore naturally explain the
apparent large hierarchy between the electroweak scale and the
Planck scale. The hierarchy problem would therefore be a problem of
our own making; no such actual hierarchy would exist in nature.
There is an immediate concern, however. With just one extra
dimension, the extra falloff in the strength of gravity from the
real Planck scale to the size of the extra dimension,
R,
is sufficiently slow so that to yield the strength
of gravity that we actually do measure on large scales forces this
latter size
R
to be roughly equal to the
size of our solar system! This is clearly impossible, since this
would mean that gravity would be measured to fall off inversely
with the cube of distance throughout our solar system, when it was
precisely the measurements of gravity over our solar system that
led Newton to propose the inverse square law in the first place.
Clearly then, one such extra large dimension is ruled out by
observation. However, if there were two extra new dimensions, then
because gravity would fall off even faster with distance for
distances smaller than
R,
this would allow
the size of the extra dimension,
R,
to be
much smaller than it would be in the case of one extra dimension.
If one works out the numbers, the scale
R
would only have to be about one millimeter. In this case, the extra
dimensions would indeed be as large as small pebbles. Not only is
such a possibility not ruled out, but this is precisely the scale
at which new experiments had been designed to explore the inverse
square law behavior of gravity. If Arkani-Hamed and collaborators
were correct, and if there are only two extra large dimensions,
these experiments could measure something quite different,
revealing for the first time the hidden dimensions that have
otherwise remained within the realm of theorists’ imaginations for
all these years.

To recap: It is perfectly possible for extra
dimensions to exist and be relatively large, provided that the only
force that can propagate in these extra dimensions is gravity.
Moreover, one can resolve the hierarchy problem for gravity, making
the Planck scale, and with it presumably the string scale,
essentially equal to the electroweak scale if there are two extra
large dimensions into which gravity can propagate, both of which
are about a millimeter in size.

As I have indicated, the possibility that extra
dimensions, such as those that might be associated with string
theory, might be large enough to actually be measured sent a jolt
of excitement through much of the particle physics community that
was perhaps stronger than any that had been experienced since the
first string revolution of 1984. Suddenly a host of potential new
experimental probes—not only of quantum gravity, but also of string
theory and even extra dimensions—would become feasible. One of the
most exciting such exotic probes involves exploring strings and
extra dimensions at current or planned particle physics
accelerators. For if the Planck scale and with it the string scale
coincide with the electroweak scale, then machines designed to
explore weak interaction physics could uncover exotic new
phenomena. Higher-energy string excitations in extra dimensions
might be excited in high-energy particle collisions, which would be
manifested in precisely the same tower of new particle states
(albeit at now much higher energies) that had first been predicted
when strings had been proposed as a theory of the strong
interactions. Equally interesting would be the possibility that
some of the energy in these highenergy collisions might literally
disappear in gravitational waves that could move off into the extra
dimensions.

Finally, perhaps the most exciting prediction
of all would be that gravity itself would become strong enough at
the electroweak scale so that new quantum gravitational phenomena
might be directly observable there. For example, high-energy
collisions in new accelerators might produce primordial,
elementary, particlelike “black holes,” which might then
spontaneously decay in a burst of radiation, as predicted by
Hawking. Not only would such new signatures be striking, they would
allow us to confirm one of the key phenomena predicted to occur when
quantum mechanical effects are incorporated into gravity and
ultimately directly explore one of our most puzzling paradoxes, the
information loss paradox. Any of these experiments might be
exciting enough to get one’s juices flowing, even if they are long
shots, but for those who truly crave dimensions large enough to
hide aliens in, millimeter sizes, even if huge by comparison to
what had previously been assumed in string theory, just don’t cut
it.

Happily an even more exotic possibility was
independently proposed within a year of Arkani-Hamed and coworkers’
theory, by Lisa Randall, now at Harvard University, and a past
student of mine, Raman Sundrum, currently at Johns Hopkins.

Randall and Sundrum argued that there is
another way to resolve the hierarchy problem using extra dimensions
that is quite distinct, and certainly more subtle, than the
mechanism proposed by Arkani-Hamed and colleagues. They proposed
starting with a single compact extra dimension, but not one
completely independent of our own. In the true spirit of
Star
Trek
they introduced
what they called a “warp factor,” though theirs has nothing to do
with faster-than-light travel. Rather, it arises from the
suggestion that an extra dimension exists that is strongly curved
(or “warped”) as one moves away from the three-brane that makes up
the three-dimensional world we experience.

What Randall and Sundrum realized is that, in
this case, even if the size of the extra compactified dimension is
perhaps only of order of ten to fifty times larger than the Planck
scale, it is still possible to produce a natural large hierarchy,
of perhaps fifteen orders of magnitude, between this scale and the
scale of the elementary particle masses and interactions we
observe.

The effect is subtle and somewhat difficult to
directly picture physically without recourse to mathematics, as is
due to effects of curvature in the extra dimension. Remember that
general relativity tells us that the curvature of space is related
to the overall magnitude of the mass and energy of objects within
the space. Now the curvature associated with the warping of the
extra dimension near our three-brane, in the Randall-Sundrum
picture, could be rather large, characteristic of energies near the
Planck energy scale. But, if the extra dimension is perhaps fifty
times larger than the characteristic scale over which it curves,
then when one solves the full five-dimensional equations associated
with general relativity, a hierarchy appears. It turns out that
even if, in the five-dimensional theory, the fundamental mass and
energy parameters are all of the order of the Planck scale, in our
four-dimensional world all fundamental particle masses will instead
appear to be suppressed compared to the Planck scale by a factor of
1015.

Randall and Sundrum also showed that there was
another slightly more intuitive way of thinking of this problem, in
terms of the relative strength of the forces in nature. The
hierarchy problem can be recast as follows: Gravity is measured to
be more than a billion billion billion billion times weaker than
electromagnetism, and even weaker still when compared to the strong
force. It may not seem so weak, especially in the morning when you
try and pull yourself out of bed, but remember that you are feeling
the gravitational force of the entire earth acting on you. By
contrast, even a small excess of electric charge on an object such
as a balloon produces a large enough electric field to hold it up on
a wall against the gravitational pull of the entire earth. The
hierarchy problem involves the question of why there is this huge
discrepancy. In the Randall-Sundrum scheme, the warping of space
near our threebrane implies that gravity near our brane acts
effectively much more weakly than it does outside our brane. The
exponential warping, in fact, makes gravity appear exponentially
weaker on our brane than it is at the other side of five-dimensional
space. If we happened to live on a threebrane located there, which
I remind you is located merely a microscopic distance “away” from
our world in the extra spatial dimension, gravity would appear to
have the same strength as the other forces in nature. The observed
hierarchy in our world then becomes merely an environmental
accident. Gravity “leaks” into our dimension as surely as Buckaroo
Banzai’s extra-dimensional nemesis does. Like the Arkani and
coworkers scenario, Randall-Sundrum’s extradimensional solution of
the hierarchy problem would bring extra dimensions into the realm
of the testable. In this case, only the massless particle (the
graviton) that conveys the gravitational force would be weakly
coupled on our brane. As in all compactified theories, there would
also be a tower of higher-mass particles that could be produced if
one had sufficient energy. In the Randall-Sundrum model, however,
these higher-dimensional gravitational modes would have masses
characteristic of the electroweak scale and coupling strengths not
characteristic of gravity, but rather of electroweak physics. The
new extra-dimensional states would thus be produced in great
abundance, with observable decay modes, just like ordinary
particles, if one had an accelerator that could achieve the
necessary energies. And, interestingly, just such an accelerator is
being built at CERN (the European Center for Nuclear Research, in
Geneva, Switzerland) and is due to come online in 2007 or 2008.

In fact, not only would new higher-dimensional
gravitational excitations be produced at such an accelerator if
this idea is correct, but at slightly higher scales fundamental
strings could also be generated and explored. All the mysteries of
string theory or M-theory would be laid bare for experimentalists
to probe, even if theorists have remained, by that time, unable to
untangle their complexities.

Now, before you go out and buy CERN futures,
you might want to step back and note a few of the hidden, but
profound, problems with this model as it stands. First and foremost
is an issue that has plagued all Kaluza-Klein theories since their
origin: Why are the extra dimensions small, and our three-brane
possibly infinitely large? There has simply been no good answer to
this question in the past ninety years. While a great deal of work
has been devoted to trying to find physical mechanisms that would
allow such a possibility, no real progress has been made. It is
simply assumed that something happens so that the extra dimensions
remain hidden, whether they curl up on the size of the Planck scale
or are as large as a pebble on the road.

Actually, the situation is often even worse
than I have thus far described. In general, it turns out that the
dynamic equations of the theory tend to drive the size of the extra
dimension to be infinitely large, as presumably are the three
spatial dimensions in which we live, even if one initially starts
the extra dimensions off to be the Planck scale, say. This
embarrassment is solved in the way other similar confusing aspects
of string theory and M-theory are sometimes dealt with: Namely, it
is assumed that when we fully understand the ultimate theory,
everything will become clear.