13.4 f (R) theories in extra dimensions
Although f (R) theories have been introduced mainly in four dimensions, the same models
may appear in the context of braneworld [502, 501] in which our universe is described by a
brane embedded in extra dimensions (see [404] for a review). This scenario implies a careful
use of f (R) theories, because a boundary (brane) appears. Before looking at the real working
scenario in braneworld, it is necessary to focus on the mathematical description of f (R) models
through a sensible definition of boundary conditions for the metric elements on the surface of the
brane.
Some works appeared regarding the possibility of introducing f (R) theories in the context of braneworld
scenarios [499, 40, 96, 513, 97
]. In doing so one requires a surface term [222, 482, 69, 48, 49, 286],
which is known as the Hawking–Luttrell term [295] (analogous to the York–Gibbons–Hawking one for
General Relativity). The action we consider is given by
where
,
is the determinant of the induced metric on the
dimensional boundary,
and
is the trace of the extrinsic curvature tensor.
In this case particular attention should be paid to boundary conditions on the brane, that is, the Israel
junction conditions [323]. In order to have a well-defined geometry in five dimensions, we require that the
metric is continuous across the brane located at
. However its derivatives with respect to
can be
discontinuous at
. The Ricci tensor
in Eq. (2.4) is made of the metric up to the second
derivatives
with respect to
. This means that
have a delta-function dependence proportional
to the energy-momentum tensor at a distributional source (i.e., with a Dirac’s delta function centered on
the brane) [87, 86, 536]. In general this also leads to the discontinuity of the Ricci scalar
across the
brane. Since the discontinuity of
can lead to inconsistencies in f (R) gravity, one should add this
extra-constraint as a junction condition. In other words, one needs to impose that, although
the metric derivative is discontinuous, the Ricci scalar should still remain continuous on the
brane.
This is tantamount to imposing that the extra scalar degree of freedom introduced is continuous on the
brane. We use Gaussian normal coordinates with the metric
In terms of the extrinsic curvature tensor
for a brane, the l.h.s. of the equations
of motion tensor [which is analogous to the l.h.s. of Eq. (2.4) in 4 dimensions] is defined by
This has a delta function behavior for the
-
components, leading to [207
]
where
is the matter stress-energy tensor on the brane. Hence
is continuous, whereas its first
derivative is not, in general. This imposes an extra condition on the metric crossing the brane at
,
compared to General Relativity in which the condition for the continuity of
is not present. However,
it is not easy to find a solution for which the metric derivative is discontinuous but
is
not. Therefore some authors considered matter on the brane which is not universally coupled
with the induced metric. This approach leads to the relaxation of the condition that
is
continuous. Such a matter action can be found by analyzing the action in the Einstein frame
and introducing a scalar field
coupled to the scalaron
on the brane as follows [207]
The presence of the coupling
with the field
modifies the Israel junction conditions. Indeed, if
, then
must be continuous, but if
,
can have a delta function profile. This
method may help for finding a solution for the bulk that satisfies boundary conditions on the
brane.