Kiefer C. Quantum gravity

QUANTUM-GRAVITATIONAL ASPECTS 303
the Planck scale m P ≈ 10 19 GeV. This works as follows. 5 One starts with the
Einstein–Hilbert action in D =4+d dimensions,
∫
1
S EH = d 4 xd d y √ −g (D) D R, (9.77)
16πG D
where the index D refers to the corresponding quantities in D dimensions. We
denote by m ∗ the D-dimensional Planck mass, that is,
G D = 1
m D−2
∗
= 1
m d+2
∗
where d = D − 4 is the number of extra dimensions. Assuming that the D-
dimensional metric is (approximately) independent of the extra dimensions labelled
by y, one gets from (9.77) an effectively four-dimensional action,
S EH =
V ∫
d
d 4 x √ −g (4) 4 R, (9.78)
16πG D
where V d ∼ R d denotes the volume of the extra dimensions. Comparison with
the four-dimensional Einstein–Hilbert action (1.1) gives the connection between
m ∗ and the four-dimensional Planck mass,
,
m P ∼ m ∗ (m ∗ R) d/2 . (9.79)
The four-dimensional Planck mass is thus big (compared to the weak scale) because
the size of the extra dimensions is big. Thereby the hierarchy problem
is transferred to a different problem: why is R so big? This reformulation has
two advantages. First, a unified theory such as string theory might give an explanation
for the size of R. Second, it opens the possibility of observing the
extra dimensions, either through scattering experiments at colliders or through
sub-mm tests of Newton’s law; see Rubakov (2001) and the references therein.
Higher-dimensional theories generically predict a violation of the Newtonian 1/rpotential
at some scale. No sign of the extra dimensions, however, has been seen
up to now.
Both in the traditional Kaluza–Klein and the ADD scenario, the full metric
factorizes 6 into the four-dimensional part describing our macroscopic dimensions
and the (compact) part referring to the extra dimensions. Such an assumption is,
however, not obligatory. If factorization does not hold, one talks about a ‘warped
metric’. The extra dimensions can be compact or infinite in size. A warped metric
occurs, for example, if the gravitational field produced by the brane is taken into
5 It was also suggested that this huge discrepancy in scales may have a cosmological origin;
cf. Hogan (2000).
6 Factorization here means that the D-dimensional metric can be put into block-diagonal
form, in which one block is the four-dimensional metric, and where the various blocks do not
depend on the coordinates referring to the other blocks.