On a Friedmann brane, we get
The absence of bulk source terms in the conservation equations is a consequence of having as the only 5D source in the bulk. For example, if there is a bulk scalar field, then there is energy-momentum exchange between the brane and bulk (in addition to the gravitational interaction) [16, 35, 97, 98, 194, 225, 236].
Equation (73) may be called the “nonlocal conservation equation”. Projecting along gives the nonlocal energy conservation equation, which is a propagation equation for . In the general, nonlinear case, this gives
In particular cases, the Weyl anisotropic stress may drop out of the nonlocal conservation equations, i.e., when we can neglect , , and . This is the case when we consider linearized perturbations about an FRW background (which remove the first and last of these terms) and further when we can neglect gradient terms on large scales (which removes the second term). This case is discussed in Section 6. But in general, and especially in astrophysical contexts, the terms cannot be neglected. Even when we can neglect these terms, arises in the field equations on the brane.
All of the matter source terms on the right of these two equations, except for the first term on the right of Equation (112), are imperfect fluid terms, and most of these terms are quadratic in the imperfect quantities and . For a single perfect fluid or scalar field, only the term on the right of Equation (112) survives, but in realistic cosmological and astrophysical models, further terms will survive. For example, terms linear in will carry the photon quadrupole in cosmology or the shear viscous stress in stellar models. If there are two fluids (even if both fluids are perfect), then there will be a relative velocity generating a momentum density , which will serve to source nonlocal effects.
In general, the 4 independent equations in Equations (111) and (112) constrain 4 of the 9 independent components of on the brane. What is missing is an evolution equation for , which has up to 5 independent components. These 5 degrees of freedom correspond to the 5 polarizations of the 5D graviton. Thus in general, the projection of the 5-dimensional field equations onto the brane does not lead to a closed system, as expected, since there are bulk degrees of freedom whose impact on the brane cannot be predicted by brane observers. The KK anisotropic stress encodes the nonlocality.
In special cases the missing equation does not matter. For example, if by symmetry, as in the case of an FRW brane, then the evolution of is determined by Equations (111) and (112). If the brane is stationary (with Killing vector parallel to ), then evolution equations are not needed for , although in general will still be undetermined. However, small perturbations of these special cases will immediately restore the problem of missing information.
If the matter on the brane has a perfect-fluid or scalar-field energy-momentum tensor, the local conservation equations (105) and (106) reduce to
A simple example of the latter point is the FRW case: Equation (116) is trivially satisfied, while Equation (115) becomes
If , then the field equations on the brane form a closed system. Thus for perfect fluid branes with homogeneous density and , the brane field equations form a consistent closed system. However, this is unstable to perturbations, and there is also no guarantee that the resulting brane metric can be embedded in a regular bulk.
It also follows as a corollary that inhomogeneous density requires nonzero :
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