Kiefer C. Quantum gravity

QUANTUM-GRAVITATIONAL ASPECTS 305
´Ýµ
Ý ¼
Ý
¦· ¦
Fig. 9.2. The situation in the Randall–Sundrum model with two branes. The
warp factor is a(y) =exp(−|y|/l).
This could provide a solution to the hierarchy problem in this model; see Randall
and Sundrum (1999). With the fine tuning (9.81) this solution exists for an
arbitrary brane separation d—the two flat branes stay in equilibrium. Their
flatness is the result of a compensation between the bulk cosmological constant
and the brane tensions. We note that the distance between the branes need not
be big, unlike the ADD scenario. This is because of the exponential decrease of
the gravitational ‘force’ between the branes.
More generally one considers the Randall–Sundrum model with small matter
sources for metric perturbations h AB (x, y) on the background of this solution,
ds 2 =dy 2 +e −2|y|/l η µν dx µ dx ν + h AB (x, y)dx A dx B , (9.83)
such that this five-dimensional metric induces on the branes two four-dimensional
metrics of the form
g ± µν(x) =a 2 ± η µν + h ± µν(x) . (9.84)
Here the scale factors a ± = a(y ± ) can be expressed in terms of the interbrane
distance,
a + =1, a − =e −2d/l ≡ a, (9.85)
and h ± µν (x) are the perturbations by which the brane metrics g± µν (x) differfrom
the (conformally) flat metric in the Randall–Sundrum solution (9.82).
Instead of using the Kaluza–Klein formalism with its infinite tower of modes,
one can employ an alternative formalism which captures more the spirit of the