- Generalized Raychaudhuri equation (expansion propagation):
- Vorticity propagation:
- Shear propagation:
- Gravito-electric propagation (Maxwell–Weyl E-dot equation):
- Gravito-magnetic propagation (Maxwell–Weyl H-dot equation):
- Vorticity constraint:
- Shear constraint:
- Gravito-magnetic constraint:
- Gravito-electric divergence (Maxwell–Weyl div-E equation):
- Gravito-magnetic divergence (Maxwell–Weyl div-H equation):
- Gauss–Codazzi equations on the brane (with ): where is the Ricci tensor for 3-surfaces orthogonal to on the brane, and .

The standard 4D general relativity results are regained when and , which sets all right hand sides to zero in Equations (129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140). Together with Equations (119, 120, 121, 122), these equations govern the dynamics of the matter and gravitational fields on the brane, incorporating both the local, high-energy (quadratic energy-momentum) and nonlocal, KK (projected 5D Weyl) effects from the bulk. High-energy terms are proportional to , and are significant only when . The KK terms contain , , and , with the latter two quantities introducing imperfect fluid effects, even when the matter has perfect fluid form.

Bulk effects give rise to important new driving and source terms in the propagation and constraint equations. The vorticity propagation and constraint, and the gravito-magnetic constraint have no direct bulk effects, but all other equations do. High-energy and KK energy density terms are driving terms in the propagation of the expansion . The spatial gradients of these terms provide sources for the gravito-electric field . The KK anisotropic stress is a driving term in the propagation of shear and the gravito-electric/gravito-magnetic fields, and respectively, and the KK momentum density is a source for shear and the gravito-magnetic field. The 4D Maxwell–Weyl equations show in detail the contribution to the 4D gravito-electromagnetic field on the brane, i.e., , from the 5D Weyl field in the bulk.

An interesting example of how high-energy effects can modify general relativistic dynamics arises in the analysis of isotropization of Bianchi spacetimes. For a Binachi type I brane, Equation (140) becomes [307]

if we neglect the dark radiation, where and are the average scale factor and expansion rate, and is the shear constant. In general relativity, the shear term dominates as , but in the brane-world, the high-energy term will dominate if , so that the matter-dominated early universe is isotropic [307, 68, 67, 413, 373, 23, 93]. This is illustrated in Figure 4.Note that this conclusion is sensitive to the assumption that , which by Equation (121) implies the restriction

Relaxing this assumption can lead to non-isotropizing solutions [5, 94, 66].The system of propagation and constraint equations, i.e., Equations (119, 120, 121, 122) and (129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140), is exact and nonlinear, applicable to both cosmological and astrophysical modelling, including strong-gravity effects. In general the system of equations is not closed: There is no evolution equation for the KK anisotropic stress .

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