p2

membrane called a D-brane (D = Dirichlet boundary conditions). But the graviton, which
corresponds to a closed loop of string, can propagate in all the dimensions. It provides both
unification and quantization of gravity by assuming that there are n new spatial dimensions in
addition to the three infinite spatial dimensions we know about. And the extra space dimensions
are ≈ 10 −35 m in extent. A very recent alternative version of this model is called “3-brane” theory.
In this theory, each of the n extra space dimensions is of finite extent R ≈ 2×10 (32/n)–17 centimeters.
The space spanned by the new dimensions is called “the bulk.” In this theory, the particles of the
standard model live within our familiar realm of three spatial dimensions, which forms a threedimensional
(hypersurface) membrane or “3-brane” within the bulk. The propagation of
electroweak and strong nuclear forces is then confined to our 3-brane. However, at distances (r)
less than R, gravity (via gravitons) propagates in the full (3 + n)-dimensional space, whereby its
strength falls as r −(2+n) with increasing separation r. When r > R, the gravitational force reverts to
its normal Newtonian r −2 falloff because there is no longer any extra-dimensional space for it to
spread into. If n = 1, then the size of the extra-dimension would have to be R ≈ 2×10 15 cm (or
2×10 10 km = 133.3 AU; 1 AU = 1.5×10 8 km is the mean Earth-Sun distance) in order to account
for the weakness of gravity, but an extra space dimension this large would have already made
itself obvious in the observed dynamics of the solar system. For this reason, investigators have
discounted the possibility that n = 1. If n = 2, then the size of both extra space dimensions would
have to be R ≈ 0.2 cm (or 2 mm). In any case, inconspicuous neighboring 3-branes may be
separated from the 3-brane we live on by only a fraction of a millimeter, or even much smaller
distances, across the higher-dimensional bulk. Such neighboring 3-branes may be distant folds of
our own 3-brane, with the same physics, but able to influence us across shortcuts through the
bulk. Or they may be completely separate 3-branes possessing their own fundamental laws and
parameters of nature that are completely different from our own. Several tabletop Cavendishtype
experiments are now looking for sub-millimeter deviations from Newtonian gravitation as a
first step towards verifying 3-brane theory, and other experiments are now being planned or are
already underway (Pease, 2001). At present the preliminary experimental results have been
negative for the existence of extra space dimensions, and the experimental data suggests that two
extra space dimensions are now constrained to length scales