

1.1 Heuristics of
higher-dimensional gravity
One of the fundamental aspects of string theory is the need for
extra spatial dimensions. This revives the original
higher-dimensional ideas of Kaluza and Klein in the 1920s, but in a
new context of quantum gravity. An important consequence of extra
dimensions is that the 4-dimensional Planck scale
is no longer the fundamental scale, which is
, where
is the number of extra dimensions. This
can be seen from the modification of the gravitational potential.
For an Einstein-Hilbert gravitational action we have
where
, and
is the gravitational coupling constant,
The static weak field limit of the field equations leads to the
-dimensional Poisson equation, whose solution is the
gravitational potential,
If the length scale of the extra dimensions is
, then on scales
, the
potential is
-dimensional,
. By
contrast, on scales large relative to
, where the extra
dimensions do not contribute to variations in the potential,
behaves like a 4-dimensional potential, i.e.,
in the
extra dimensions, and
. This means that the usual Planck scale becomes an
effective coupling constant, describing gravity on scales much
larger than the extra dimensions, and related to the fundamental
scale via the volume of the extra dimensions:
If the extra-dimensional volume is Planck scale, i.e.,
, then
. But if the
extra-dimensional volume is significantly above Planck scale, then
the true fundamental scale
can be much less than the
effective scale
. In this case, we
understand the weakness of gravity as due to the fact that it
“spreads” into extra dimensions and only a part of it is felt in 4
dimensions.
A lower limit on
is given by null
results in table-top experiments to test for deviations from
Newton’s law in 4 dimensions,
. These
experiments currently [212] probe sub-millimetre
scales, so that
Stronger bounds for brane-worlds with compact flat extra dimensions
can be derived from null results in particle accelerators and in
high-energy astrophysics [51
, 58
, 132
, 137
].

