

3.1 Field equations on the
brane
Using Equations (44) and (48), it follows that
where
, and where
is the projection of the bulk Weyl tensor orthogonal to
. This tensor satisfies
by virtue of the Weyl tensor symmetries. Evaluating
Equation (54) on the brane
(strictly, as
, since
is not defined on the brane [291
]) will give the
field equations on the brane.
First, we need to determine
at the brane from the junction conditions. The total
energy-momentum tensor on the brane is
where
is the energy-momentum tensor of particles and
fields confined to the brane (so that
). The 5D
field equations, including explicitly the contribution of the
brane, are then
Here the delta function enforces in the classical theory the string
theory idea that Standard Model fields are confined to the brane.
This is not a gravitational confinement, since there is in general
a nonzero acceleration of particles normal to the brane [218
].
Integrating Equation (58) along the extra
dimension from
to
, and taking the
limit
, leads to the Israel-Darmois junction
conditions at the brane,
where
. The
symmetry means that when you approach the brane from
one side and go through it, you emerge into a bulk that looks the
same, but with the normal reversed,
.
Then Equation (46) implies that
so that we can use the junction condition Equation (60) to determine the
extrinsic curvature on the brane:
where
, where we have dropped the
, and where we evaluate quantities on the brane by
taking the limit
.
Finally we arrive at the induced field equations
on the brane, by substituting Equation (62) into
Equation (54):
The 4D gravitational constant is an effective coupling constant
inherited from the fundamental coupling constant, and the 4D
cosmological constant is nonzero when the RS balance between the
bulk cosmological constant and the brane tension is broken:
The first correction term relative to Einstein’s theory is
quadratic in the energy-momentum tensor, arising from the extrinsic
curvature terms in the projected Einstein tensor:
The second correction term is the projected Weyl term. The last
correction term on the right of Equation (63), which generalizes
the field equations in [291
], is
where
describes any stresses in the bulk
apart from the cosmological constant (see [225
] for the case of a
scalar field).
What about the conservation equations? Using
Equations (44), (49) and (62), one obtains
Thus in general there is exchange of energy-momentum between the
bulk and the brane. From now on, we will assume that
so that
One then recovers from Equation (68) the standard 4D
conservation equations,
This means that there is no exchange of energy-momentum between the
bulk and the brane; their interaction is purely gravitational. Then
the 4D contracted Bianchi identities (
), applied
to Equation (63), lead to
which shows qualitatively how
spacetime
variations in the matter-radiation on the brane can source KK
modes.
The induced field equations (71) show two key
modifications to the standard 4D Einstein field equations arising
from extra-dimensional effects:
is the high-energy
correction term, which is negligible for
, but dominant
for
:
is the projection of the bulk Weyl
tensor on the brane, and encodes corrections from 5D graviton
effects (the KK modes in the linearized case). From the
brane-observer viewpoint, the energy-momentum corrections in
are local, whereas the KK corrections in
are nonlocal, since they incorporate 5D gravity wave
modes. These nonlocal corrections cannot be determined purely from
data on the brane. In the perturbative analysis of RS 1-brane which
leads to the corrections in the gravitational potential,
Equation (41), the KK modes that
generate this correction are responsible for a nonzero
; this term is what carries the modification to the
weak-field field equations. The 9 independent components in the
tracefree
are reduced to 5 degrees of freedom by
Equation (73); these arise from the
5 polarizations of the 5D graviton. Note that the covariant
formalism applies also to the two-brane case. In that case, the
gravitational influence of the second brane is felt via its
contribution to
.

