1.3 Heuristics of KK modes
The dilution of gravity via extra dimensions not only weakens gravity on the brane, it also extends the
range of graviton modes felt on the brane beyond the massless mode of 4-dimensional gravity. For
simplicity, consider a flat brane with one flat extra dimension, compactified through the identification
, where . The perturbative 5D graviton amplitude can be Fourier expanded
where are the amplitudes of the KK modes, i.e., the effective 4D modes of the 5D graviton. To see that
these KK modes are massive from the brane viewpoint, we start from the 5D wave equation that the
massless 5D field satisfies (in a suitable gauge):
It follows that the KK modes satisfy a 4D Klein–Gordon equation with an effective 4D mass ,
The massless mode is the usual 4D graviton mode. But there is a tower of massive modes,
, which imprint the effect of the 5D gravitational field on the 4D brane. Compactness of the
extra dimension leads to discreteness of the spectrum. For an infinite extra dimension, , the
separation between the modes disappears and the tower forms a continuous spectrum. In this case, the
coupling of the KK modes to matter must be very weak in order to avoid exciting the lightest massive
modes with .
From a geometric viewpoint, the KK modes can also be understood via the fact that the projection of
the null graviton 5-momentum onto the brane is timelike. If the unit normal to the brane is ,
then the induced metric on the brane is
and the 5-momentum may be decomposed as
where is the projection along the brane, depending on the orientation of the 5-momentum
relative to the brane. The effective 4-momentum of the 5D graviton is thus . Expanding
, we find that
It follows that the 5D graviton has an effective mass on the brane. The usual 4D graviton corresponds
to the zero mode, , when is tangent to the brane.
The extra dimensions lead to new scalar and vector degrees of freedom on the brane. In 5D, the
spin-2 graviton is represented by a metric perturbation that is transverse traceless:
In a suitable gauge, contains a 3D transverse traceless perturbation , a 3D transverse vector
perturbation , and a scalar perturbation , which each satisfy the 5D wave equation :
The other components of are determined via constraints once these wave equations are solved. The
5 degrees of freedom (polarizations) in the 5D graviton are thus split into degrees of
freedom in 4D. On the brane, the 5D graviton field is felt as
- a 4D spin-2 graviton (2 polarizations),
- a 4D spin-1 gravi-vector (gravi-photon) (2 polarizations), and
- a 4D spin-0 gravi-scalar .
The massive modes of the 5D graviton are represented via massive modes in all three of these fields on the
brane. The standard 4D graviton corresponds to the massless zero-mode of .
In the general case of extra dimensions, the number of degrees of freedom in the graviton follows
from the irreducible tensor representations of the isometry group as .