### 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 1+3 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 .