tum mechanics to produce their extraordinarily accurate results.
Relativity, the best theory we have on gravitation, is a classical field theory
that makes no mention of quantum mechanics. The experimental levels
at which the theories have been confirmed differ remarkably. We can
probe subatomic structure at distances of 10^18 meter and have found
nothing that would contradict the existing electroweak and chromody-
namic theories. However, the best we can do to measure the effect of
gravity is to confirm it at distances of one-tenth of a millimeter, or 10^4
meters. Part of the difficulty is the extraordinary weakness of gravity
when compared to the other forces; the gravity of Earth cannot overcome
the static electric force when you run a comb through your hair on a cold
winter day, and it requires the gravity of a star to tear apart an atom.
Extra Dimensions Resurrected
The advent of string theory resurrected the Kaluza-Klein theory of addi-
tional space dimensions—but in a way that seems almost impossible to
grasp. After decades of work, string theorists have realized that there is
only one possible extra-dimensional space-time that will result in equa-
tions compatible with the known universe—but that extra-dimensional
space-time requires ten spatial dimensions and one time dimension. If
we have as yet been unable to see any evidence of the one extra spatial
dimension of the Kaluza-Klein theory, what possible chance do we have
of seeing the seven extra ones required by space-time theorists? And what
of the extent of these extra dimensions: Are they large, in the sense that
the normal three spatial dimensions are large, or are they small—and if
so, how small?
Ever since Newton developed the mathematics of calculus to help for-
mulate his theories of mechanics and gravitation, advances in physics
have gone hand in hand with advances in mathematics—but there are
times that each leads the other. When Maxwell developed his theory of
electromagnetism, he used off-the-shelf vector calculus that had been
around for nearly a century; and when Einstein came up with general
relativity, he discovered that the differential geometry worked out decades
previously by Italian mathematicians was just the right tool for the job.
However, string theory has been forced to develop much of its own math-
ematics, and consequently the mathematics—the language in which the
results of string theory is expressed—is incompletely understood.
Compounding this problem is one that has affected physics ever since it
began relying upon mathematics to phrase its results—the necessity of
approximation. When equations cannot be solved exactly—and as we have
Space and Time: Is That All There Is? 149