

2 Randall-Sundrum Brane-Worlds
RS brane-worlds 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 self-gravity 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:
- RS
2-brane:
- There are two branes in this
model [266], at
and
, with
-symmetry identifications
The branes have equal and opposite tensions
, where
The positive-tension brane has fundamental scale
and is “hidden”. Standard model fields are confined
on the negative tension (or “visible”) brane. Because of the
exponential warping factor, the effective scale on the visible
brane at
is
, where
So the RS 2-brane model gives a new approach to the hierarchy
problem. Because of the finite separation between the branes, the
KK spectrum is discrete. Furthermore, at low energies gravity on
the branes becomes Brans-Dicke-like, with the sign of the
Brans-Dicke parameter equal to the sign of the brane
tension [105
]. In order to
recover 4D general relativity at low energies, a mechanism is
required to stabilize the inter-brane distance, which corresponds
to a scalar field degree of freedom known as the radion [120, 305
, 248
, 202
].
- RS
1-brane:
- In this model [265
], there is only one,
positive tension, brane. It may be thought of as arising from
sending the negative tension brane off to infinity,
. Then the energy scales are related via
The infinite extra dimension makes a finite contribution to the 5D
volume because of the warp factor:
Thus the effective size of the extra dimension probed by the 5D
graviton is
.
I will concentrate mainly on RS 1-brane from
now on, referring to RS 2-brane occasionally. The RS 1-brane models
are in some sense the most simple and geometrically appealing form
of a brane-world model, while at the same time providing a
framework for the AdS/CFT correspondence [87
, 254
, 282
, 136
, 289
, 293
, 210
, 259
, 125
]. The RS 2-brane
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 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 [265, 105
, 117, 81],
(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 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 [132, 137].

