### 2.1 KK modes in RS 1-brane

In RS 1-brane, the negative is offset by the positive brane tension . The fine-tuning in
Equation (25) ensures that there is a zero effective cosmological constant on the brane, so that the
brane has the induced geometry of Minkowski spacetime. To see how gravity is localized at
low energies, we consider the 5D graviton perturbations of the metric [358, 155, 171, 122],
(see Figure 3). This is the RS gauge, which is different from the gauge used in Equation (15), but which
also has no remaining gauge freedom. The 5 polarizations of the 5D graviton are contained in the 5
independent components of in the RS gauge.
We split the amplitude of into 3D Fourier modes, and the linearized 5D Einstein equations
lead to the wave equation ()

Separability means that we can write
and the wave equation reduces to
The zero mode solution is
and the solutions are
The boundary condition for the perturbations arises from the junction conditions, Equation (68), discussed
below, and leads to , since the transverse traceless part of the perturbed energy-momentum
tensor on the brane vanishes. This implies
The zero mode is normalizable, since
Its contribution to the gravitational potential gives the 4D result, . The
contribution of the massive KK modes sums to a correction of the 4D potential. For , one obtains
which simply reflects the fact that the potential becomes truly 5D on small scales. For ,
which gives the small correction to 4D gravity at low energies from extra-dimensional effects. These effects
serve to slightly strengthen the gravitational field, as expected.
Table-top tests of Newton’s laws currently find no deviations down to , so that
in Equation (41). Then by Equations (25) and (27), this leads to lower limits on the brane
tension and the fundamental scale of the RS 1-brane model:

These limits do not apply to the 2-brane case.
For the 1-brane model, the boundary condition, Equation (38), admits a continuous spectrum
of KK modes. In the 2-brane model, must hold in addition to Equation (38). This leads to
conditions on , so that the KK spectrum is discrete:

The limit Equation (42) indicates that there are no observable collider, i.e., , signatures for
the RS 1-brane model. The 2-brane model by contrast, for suitable choice of and so
that , does predict collider signatures that are distinct from those of the ADD
models [189, 194].