seen, this is a frequently occurring situation—one possibility is to solve
the exact equations approximately, but another is to replace the exact equa-
tions with equations that approximate them and solve the approximating
equations. Physicists have been doing this for centuries—for small angles,
the sine of the angle is approximately equal to its radian measure (just as
360 degrees constitute a full circle, 2 radians do as well), and for most
purposes using the angle in an equation rather than its sine results in a
much more tractable equation. The equations of string theory sometimes
utilize such approximations in order to be solved, and when dealing with
something as unknown as infinitesimally small strings and equally in-
finitesimally small dimensions, it is hard to be certain that the solutions
one obtains ref lect the way the universe really is.
So how can we tell if string theory, with its ten spatial dimensions, is on
the right track? There are two possible approaches, but both are long
shots. Confirmation of the existence of strings would constitute infer-
ential proof of the existence of extra dimensions, as mathematical anal-
yses have mandated that the string scenario holds true only in the
eleven-dimensional (ten spatial dimensions, one time dimension) uni-
verse described above. However, string theory does not unequivocally
mandate the size of the strings. Although some versions of string theory
place the size of the strings in the neighborhood of 10^33 meter, which
would render them undetectable by any conceivable equipment technol-
ogy as we know it could muster, there are versions in which strings are
(relatively) huge, and possibly detectable, inferentially if not directly, by
the next generation of particle accelerators.
The other approach relies upon the fact that the inverse power law that
gravity satisfies depends upon the number of spatial dimensions. We see
gravity as an inverse square law because, in our three-dimensional universe,
gravitons spread out over the boundary of a sphere, whose surface area var-
ies as the square of the radius. In a two-dimensional universe, gravitons
would spread out over the boundary of an expanding circle, whose circum-
ference varies directly (is a constant multiple of) the radius. In higher di-
mensions, the gravitational force would drop precipitously. The boundary of
a p-dimensional sphere varies in size as the (p1)st power of the radius, and
so we would see an inverse (p1)-power law for gravitation.
That is, if we could measure the gravitational force at distances for
which the extra spatial dimensions are significant. The bad news is that
the extra spatial dimensions are required by current theory to be no
larger than about 10^18 meter—and gravity so far can be measured accu-
rately only on scales of about 10^4 meter. Thirteen orders of magnitude is
150 How Math Explains the World