- The simplest example arises from considering a
charged bulk black hole, leading to the Reissner-Nordström
bulk metric [17
]. This has the form
of Equation (178), with
where
is the “electric” charge parameter of the bulk black
hole. The metric is a solution of the 5D Einstein-Maxwell
equations, so that
in Equation (44) is the
energy-momentum tensor of a radial static 5D “electric” field. In
order for the field lines to terminate on the boundary brane, the
brane should carry a charge
. Since the RN
metric is 4-isotropic, it is still possible to embed
a FRW brane in it, which is moving in the coordinates of
Equation (178). The effect of the
black hole charge on the brane arises via the junction conditions
and leads to the modified Friedmann equation [17],
The field lines that terminate on the brane imprint on the brane an
effective negative energy density
, which
redshifts like stiff matter (
). The negativity of this
term introduces the possibility that at high energies it can bring
the expansion rate to zero and cause a turn-around or bounce (but
see [144] for problems with such
bounces).
Apart from negativity, the key difference
between this “dark stiff matter” and the dark radiation term
is that the latter arises from the bulk Weyl
curvature via the
tensor, while the former arises from
non-vacuum stresses in the bulk via the
tensor in
Equation (63). The dark stiff
matter does not arise from massive KK modes of the graviton.
- Another example is provided by the Vaidya-
metric, which can be written after transforming to a
new coordinate
in Equation (178), so that
are null surfaces, and
This model has a moving FRW brane in a 4-isotropic bulk (which is
not static), with either a radiating bulk black hole (
), or a radiating brane (
) [53, 198, 197, 199]. The metric satisfies
the 5D field equations (44) with a null-radiation
energy-momentum tensor,
where
. It follows that
In this case, the same effect, i.e., a varying mass parameter
, contributes to both
and
in the brane field equations. The modified Friedmann
equation has the standard 1-brane RS-type form, but with a dark
radiation term that no longer behaves strictly like radiation:
By Equations (68) and (228), we arrive at the
matter conservation equations,
This shows how the brane loses (
) or gains
(
) energy in exchange with the bulk black hole. For an
FRW brane, this equation reduces to
The evolution of
is governed by the 4D contracted Bianchi
identity, using Equation (229):
For an FRW brane, this yields
where
.
- A more complicated bulk metric arises when
there is a self-interacting scalar field
in the
bulk [225
, 16
, 236
, 97
, 194
, 98
, 35
]. In the simplest
case, when there is no coupling between the bulk field and brane
matter, this gives
where
satisfies the 5D Klein-Gordon equation,
The junction conditions on the field imply that
Then Equations (68) and (235) show that matter
conservation continues to hold on the brane in this simple case:
From Equation (235) one finds that
where
so that the modified Friedmann equation becomes
When there is coupling between brane matter and
the bulk scalar field, then the Friedmann and conservation
equations are more complicated [225
, 16
, 236
, 97
, 194
, 98
, 35
].