Hiding in the Mirror (11 page)

So, once again, life imitates art.

We are such stuff

As dreams are made on, and
our little life

Is rounded with a
sleep

—William Shakespeare,
The
Tempest

I
t is one thing for a
writer to dream up a new hypothetical universe in which to stage a
drama, and quite another to propose that such a universe might
really exist. This requires a different kind of chutzpa—the kind
that arises following a period of such great success building new
pictures of reality that one becomes emboldened in one’s
predictions. I first experienced this kind of hubris when I was a
graduate student at MIT in 1980. This was a heady era in particle
physics and an exciting time to be a student. In less than a decade
physicists had gone from clearly understanding only one of the four
known forces in nature (i.e., electromagnetism) in a way that was
consistent with quantum mechanics and relativity to understanding
in detail all the known forces except for gravity.

It was easy to feel that we were witnessing the
emergence of an astonishing new picture of the natural world. A
year earlier, Sheldon Glashow and Steven Weinberg (two faculty
members at nearby Harvard, where I took most of my graduate
courses) had won the Nobel Prize (along with Abdus Salam) for their
development in the 1960s—confirmed by experiments in the 1970s—of a
theory that unified two of the four forces in nature: the
electromagnetic and weak forces. The latter is the force that is
responsible for many nuclear reactions that turn protons into
neutrons and vice versa, and is an integral part of the process of
“nuclear fusion” that powers the sun. Shortly after that a graduate
student at Harvard, David Politzer, had discovered
contemporaneously with a Princeton graduate student, Frank Wilczek,
and his advisor David Gross, a key mathematical characteristic of a
theory that was soon recognized to describe the third
nongravitational force in nature, the so-called strong force
between quarks, the fundamental building blocks of protons and
neutrons. The theory in question, called quantum chromodynamics
(QCD), provided predictions about the interactions between quarks
that were previously unthinkable, and that were ultimately verified
to be in agreement with experiment, leading to a Nobel Prize thirty
years later for this trio.

Everywhere we turned, it seemed that the new
tools of elementary particle physics—based on combining special
relativity, quantum mechanics, and Maxwell’s electromagnetism—were
opening up doors. Emboldened by their success, physicists began to
seriously consider whether they might soon be able to unify not
just two forces in nature, but perhaps three or maybe even all
four, within a single theoretical mathematical framework, the holy
grail of “Grand Unification.”

I will return to grand unification and its
predictions later in this book, but took the liberty of jumping
ahead chronologically here to present a brief contemporary
perspective on how the excitement of discovery can be contagious
and can breed the kind of confidence that allows one to address
problems one would never have had the boldness to even consider
otherwise. A comparable situation occurred in the second decade of
the twentieth century, following the development of special and
then general relativity by Einstein.

Remarkably, just as the discoveries by Faraday,
Maxwell, Oersted, Ampère, and others about the relations between
electricity and magnetism led Einstein and Minkowski to propose the
existence of an underlying four-dimensional space-time continuum,
and just as the mathematical form of electromagnetism provided the
key that allowed the physicists mentioned above to solve the
mysteries surrounding the strong and weak interactions, so, too,
did electromagnetism play a central role in the first serious
scientific proposal that other dimensions, beyond the four we
experience, might actually exist. This proposal, like grand
unification some sixty years later, was motivated by a desire to
unify the forces of nature, and, as would be true of grand
unification, the specific mathematical form of electromagnetism
provided the direction. However, unlike the case of grand
unification, the direct trigger was the remarkable discovery by
Einstein that the force we feel as gravity could instead be
understood in terms of the curvature of space-time.

In retrospect, it is perhaps not surprising
that the advent of general relativity led physicists to consider
the possibility that extra dimensions might allow for a unification
of what were then the two known forces in nature: gravity and
electromagnetism. Einstein’s theory implied that local observers
could interpret the forces they felt as either due to gravity or
the effects of acceleration, depending upon their frame of
reference; similarly, Maxwell’s relations between electric forces
and magnetic forces also imply that observers can interpret the
forces they feel as either electric or magnetically induced,
depending upon their own state of motion. If gravity, then, could
be interpreted as being due to an underlying local curvature of
three-dimensional space, then could electromagnetism be somehow due
to some other sort of underlying local curvature? And since
curvature in an observable three-dimensional space resulted in
gravity, could curvature in some unperceived new dimension be
responsible for the extra force of electromagnetism?

The Finnish physicist Gunnar Nordström actually
developed the first physical theory that incorporated an extra
dimension in 1914, slightly before Einstein’s fully developed
general relativity appeared. His version of unification was in
spirit the opposite of the approach outlined above, as he tried to
derive gravity from electromagnetism, rather than vice versa.
Nordström had in fact developed his own theory of gravity, which
attempted to generalize special relativity, just as general
relativity would successfully do several years later. In
Nordström’s theory, the universe was five-dimensional, with one
extra spatial dimension, and Maxwell’s electromagnetism was a force
felt in every one of the dimensions. But if, for some reason, all
the electromagnetic fields were independent of the extra spatial
dimension (i.e., the fields were of a constant fixed magnitude in
that extra dimension, but could vary in strength over the three
spatial dimensions we are used to), then those of us sensitive to
only the three dimensions in which electromagnetic fields could vary
would measure not only electromagnetism, but an additional remnant
of the fourth spatial dimension. That additional remnant was
precisely Nordström’s gravitational field. Of course, once
Einstein’s general relativity was unveiled, interest in Nordström’s
ideas waned—especially interest by Einstein, who was known to have
had a less than cordial relationship with Nordström. In fact, in
all the subsequent proposals involving extradimensional unifications
in physics up through the early 1980s, there is not a single
reference to Nordström. Such was, I suppose, the danger of
competing with Einstein, at least where gravity was concerned.

The person generally credited with introducing
the idea of extra dimensions into mainstream physics was the German
mathematician Theodor Kaluza, in a beautiful paper entitled “On the
Unity Problem in Physics,” in which he argued that searching for a
unified worldview was “one of the great favorite ideas of the human
spirit.”

Kaluza also proposed a five-dimensional
universe, with four spatial dimensions plus time. He was motivated
in his efforts by an earlier proposal by Hermann Weyl to unify
electromagnetism and gravity in a purely geometric manner, as
Einstein had done for gravity alone. Thus, instead of considering
an electromagnetic field as fundamental, Kaluza imagined only a
gravitational field, described by a five-dimensional version of
general relativity (i.e., his theory described the curvature of
four spatial dimensions in terms of a gravitational field that
operated in four spatial dimensions plus time).

T
The fundamental
quantity that determines the nature of gravity in Einstein’s
general relativity is something called the
metric
. This is actually a set of quantities that
tell you at any point in space exactly how physical distances
between nearby points are related to any local coordinate system
(e.g.,
x, y,
and
z
coordinates that describe length, width, and height) that a local
observer may set up. If space is flat, then the relation between
physical distances and coordinates such as
x,
y,
and
z
is generally simple. In two
dimensions, for example, the square of the physical distance
between two points separated by coordinates
x
and
y
is, as Pythagoras
taught us, simply
x
²
+
y
²
.

But on a curved space such as the surface of a
sphere, the relation between physical distances and coordinates can
get strange. If one maps out points on this surface by latitudes
and longitudes, for example, as one does on Earth, then near the
poles, where the longitudes draw closer together, the physical
distances between them are very different than they are near the
equator. Thus, on a map in which latitudes and longitudes are
represented by perpendicular coordinate grids, Greenland looks
huge. It turns out that all of the geometric information about the
sphere is precisely encoded in the metric quantities that describe
the changing relation between distances as a function of latitude
and longitude, and that tells us how to find out the actual size of
Greenland from the difference in longitude between one side of it
and the other. In a five-dimensional space, more quantities are
needed to describe all the possible coordinates for any given
point. If one does the mathematics, it turns out that there are five
more quantities needed at every point to completely specify the
geometry of such a five-dimensional space. Kaluza the mathematician
argued as follows: Imagine a fivedimensional space that has one
dimension that is periodic, such as a circle, so that when you
travel in this direction, you return to your starting point. A
simple example of this in two dimensions is a cylinder. Further
imagine that the other four dimensions in the five-dimensional space
are just like the four dimensions of space and time that we
experience. The force we feel as gravity is related to the geometry
of these four dimensions, described completely by the metric
quantities I described earlier. Now, imagine that all the metric
quantities that describe the distances between nearby points along
the four-dimensional slices of five-dimensional space do not change
as you move around the circular fifth spatial dimension (as would
also be the case for a cylinder in two dimensions). This is the
same as saying that all metric quantities that describe the
five-dimensional space (there are a total of fifteen of them at any
point) are independent of this circular fifth dimension.

We who can only move around in three spatial
dimensions sense gravity in a way that depends upon ten of the
fifteen quantities that vary from point to point in our
four-dimensional slice of this five-dimensional “cylinder.” So what
do the other quantities determine? Kaluza was able to show that
four of the extra five quantities satisfy equations that are
precisely those discovered by Maxwell to describe the electric and
magnetic fields. In this way, the two known forces in nature
appeared to be unified in a beautiful and remarkable way, thereby
suggesting that what we measure as electromagnetic fields might be
merely a remnant of an underlying curvature in an invisible fifth
dimension. This is a truly amazing possibility that sounds almost
too ideal not to be true. So, why did Einstein vacillate for almost
two years before finally sponsoring its publication after receiving
Kaluza’s manuscript in 1919?

Well, in the first place the astute reader may
have noticed that I spoke of “four of the extra five quantities”
that describe the geometry of the fivedimensional universe. What
about the extra quantity? It turns out that Kaluza essentially
ignored it, for no good reason. If one does not do this, then it
turns out that the theory one arrives at in four dimensions is not
quite electromagnetism plus general relativity. There is an extra
term, which changes the nature of gravity. In modern language this
could be described as being due to the existence of an extra
massless particle in nature, which we have not observed. We shall
return to this issue later. The other question that Kaluza’s work
completely begs is one that resembles that question young children
are required to ask on the Jewish holiday of Passover: “Why is this
day different from all other days?” In this case, one would ask:
“Why is the fifth dimension different from all other dimensions?” To
this, Kaluza provided no concrete answer. Such was the luxury,
perhaps, of being a mathematician.

To be fair, it is worth noting that Kaluza
himself introduced a fifth dimension as a purely mathematical
convenience, and did not necessarily ascribe any physical
significance to it. Indeed, in his analysis he was apologetic in
tone, calling the decision to introduce such a possibility a
“strongly alienating decision.” He was driven to do so by a
mathematical similarity pointed out by Weyl between the way in
which electromagnetic fields could be written and the way in which a
certain mathematical quantity, called a “connection tensor” or just
“connection,” based on the metric in a curved space, could be
written. Because this quantity in our fourdimensional space-time is
used to describe the effects of gravity, he was forced, as he put
it, to consider an extra dimension that would allow additional
pieces of the connection to be interpreted as electromagnetic fields
in space-time. This would only be the case as long as the extra
dimension itself was rather impotent, with all physical quantities
(i.e., the metric) being independent of the circular fifth
dimension.