

3.3 The brane viewpoint: A 1 +
3-covariant analysis
Following [218
], a systematic
analysis can be developed from the viewpoint of a brane-bound
observer. The effects of bulk gravity are conveyed, from a brane
observer viewpoint, via the local (
) and nonlocal
(
) corrections to Einstein’s equations. (In the more
general case, bulk effects on the brane are also carried by
, which describes any 5D fields.) The
term cannot in general be determined from data on
the brane, and the 5D equations above (or their equivalent) need to
be solved in order to find
.
The general form of the brane energy-momentum
tensor for any matter fields (scalar fields, perfect fluids,
kinetic gases, dissipative fluids, etc.), including a combination
of different fields, can be covariantly given in terms of a chosen
4-velocity
as
Here
and
are the energy density and isotropic
pressure, respectively, and
projects into the comoving rest space orthogonal to
on the brane. The momentum density and anisotropic
stress obey
where angled brackets denote the spatially projected, symmetric,
and tracefree part:
In an inertial frame at any point on the brane, we have
where
.
The tensor
, which carries local bulk
effects onto the brane, may then be irreducibly decomposed as
This simplifies for a perfect fluid or minimally-coupled scalar
field to
The tracefree
carries nonlocal
bulk effects onto the brane, and contributes an effective “dark”
radiative energy-momentum on the brane, with energy density
, pressure
, momentum density
, and anisotropic stress
:
We can think of this as a KK or Weyl “fluid”. The brane “feels” the
bulk gravitational field through this effective fluid. More
specifically:
- The KK (or Weyl) anisotropic stress
incorporates the scalar or spin-0 (“Coulomb”), the
vector (transverse) or spin-1 (gravimagnetic), and the tensor
(transverse traceless) or spin-2 (gravitational wave) 4D modes of
the spin-2 5D graviton.
- The KK momentum density
incorporates spin-0 and spin-1 modes, and defines a
velocity
of the Weyl fluid relative to
via
.
- The KK energy density
, often called the “dark radiation”, incorporates the
spin-0 mode.
In special cases, symmetry will impose
simplifications on this tensor. For example, it must vanish for a
conformally flat bulk, including
,
The RS models have a Minkowski brane in an
bulk. This bulk is also compatible with an FRW
brane. However, the most general vacuum bulk with a Friedmann brane
is Schwarzschild-anti-de Sitter spacetime [249
, 32
]. Then it follows
from the FRW symmetries that
where
only if the mass of the black hole in
the bulk is zero. The presence of the bulk black hole generates via
Coulomb effects the dark radiation on the brane.
For a static spherically symmetric brane (e.g.,
the exterior of a static star or black hole) [73
],
This condition also holds for a Bianchi I brane [221
]. In these cases,
is not determined by the symmetries, but by the 5D
field equations. By contrast, the symmetries of a Gödel brane fix
[20].
The brane-world corrections can conveniently be
consolidated into an effective total energy density, pressure,
momentum density, and anisotropic stress:
These general expressions simplify in the case of a perfect fluid
(or minimally coupled scalar field, or isotropic one-particle
distribution function), i.e., for
, to
Note that nonlocal bulk effects can contribute to effective
imperfect fluid terms even when the matter on the brane has perfect
fluid form: There is in general an effective momentum density and
anisotropic stress induced on the brane by massive KK modes of the
5D graviton.
The effective total equation of state and sound
speed follow from Equations (98) and (99) as
where
and
. At very
high energies, i.e.,
, we can generally neglect
(e.g., in an inflating cosmology), and the effective
equation of state and sound speed are stiffened:
This can have important consequences in the early universe and
during gravitational collapse. For example, in a very high-energy
radiation era,
, the effective cosmological equation of
state is ultra-stiff:
. In late-stage gravitational
collapse of pressureless matter,
, the effective
equation of state is stiff,
, and the
effective pressure is nonzero and dynamically important.

