

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 as
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 [88]:
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 2
(
) + 2 (
) +1 (
) 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 3 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
.

