### 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 4+d-dimensional Poisson equation, whose
solution is the gravitational potential,
If the length scale of the extra dimensions is , then on scales , the potential is 4+d-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 [294] 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 [75, 87, 189, 194].