RS braneworlds do not rely on compactification to localize gravity at the brane, but on the curvature of the bulk (sometimes called “warped compactification”). What prevents gravity from ‘leaking’ into the extra dimension at low energies is a negative bulk cosmological constant,
where is the curvature radius of and is the corresponding energy scale. The curvature radius determines the magnitude of the Riemann tensor: The bulk cosmological constant acts to “squeeze” the gravitational field closer to the brane. We can see this clearly in Gaussian normal coordinates based on the brane at , for which the metric takes the form with being the Minkowski metric. The exponential warp factor reflects the confining role of the bulk cosmological constant. The symmetry about the brane at is incorporated via the term. In the bulk, this metric is a solution of the 5D Einstein equations, i.e., in Equation (2). The brane is a flat Minkowski spacetime, , with selfgravity in the form of brane tension. One can also use Poincare coordinates, which bring the metric into manifestly conformally flat form, where .The two RS models are distinguished as follows:
I will concentrate mainly on RS 1brane from now on, referring to RS 2brane occasionally. The RS 1brane models are in some sense the most simple and geometrically appealing form of a braneworld model, while at the same time providing a framework for the AdS/CFT correspondence [87, 125, 136, 210, 254, 259, 282, 289, 293]. The RS 2brane introduce the added complication of radion stabilization, as well as possible complications arising from negative tension. However, they remain important and will occasionally be discussed.
In RS 1brane, the negative is offset by the positive brane tension . The finetuning 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 [81, 106, 117, 265],
(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 (62), discussed below, and leads to , since the transverse traceless part of the perturbed energymomentum 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 extradimensional effects. These effects serve to slightly strengthen the gravitational field, as expected.Tabletop 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 1brane model:
These limits do not apply to the 2brane case.For the 1brane model, the boundary condition, Equation (38), admits a continuous spectrum of KK modes. In the 2brane 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 1brane model. The 2brane model by contrast, for suitable choice of and so that , does predict collider signatures that are distinct from those of the ADD models [132, 137].
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