Propagation and constraint equations on the brane

3.5 Propagation and constraint equations on the brane

The propagation equations for the local and nonlocal energy density and momentum density are supplemented by further 1+3-covariant propagation and constraint equations for the kinematic quantities $Θ$ , $Aμ$ , $ωμ$ , $σμν$ , and for the free gravitational field on the brane. The kinematic quantities govern the relative motion of neighbouring fundamental world-lines. The free gravitational field on the brane is given by the brane Weyl tensor $C μναβ$ . This splits into the gravito-electric and gravito-magnetic fields on the brane:

where $Eμν$ is not to be confused with $ℰμν$ . The Ricci identity for $μ u$

and the Bianchi identities

produce the fundamental evolution and constraint equations governing the above covariant quantities. The field equations are incorporated via the algebraic replacement of the Ricci tensor $R μν$ by the effective total energy-momentum tensor, according to Equation (63

). The brane equations are derived directly from the standard general relativity versions by simply replacing the energy-momentum tensor terms $ρ, ...$ by $ρtot,...$ . For a general fluid source, the equations are given in [218

]. In the case of a single perfect fluid or minimally-coupled scalar field, the equations reduce to the following nonlinear equations:

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 $ωμ = 0$ ):

where $R ⊥μν$ is the Ricci tensor for 3-surfaces orthogonal to $uμ$ on the brane, and $R ⊥ = hμνR ⊥μν$ .

Figure 4:

The evolution of the dimensionless shear parameter $Ωshear = σ2 ∕6H2$ on a Bianchi I brane, for a $V = 12m2 φ2$ model. The early and late-time expansion of the universe is isotropic, but the shear dominates during an intermediate anisotropic stage. (Figure taken from [221

].)

The standard 4D general relativity results are regained when $λ−1 → 0$ and $ℰμν = 0$ , which sets all right hand sides to zero in Equations (123 , 124 , 125 , 126 , 127 , 128 , 129 , 130 , 131 , 132 , 133 , 134 ). Together with Equations (113 , 114 , 115 , 116 ), 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 $ρ∕λ$ , and are significant only when $ρ > λ$ . The KK terms contain $ρℰ$ , $qℰμ$ , and $πμℰν$ , 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 $Θ$ . The spatial gradients of these terms provide sources for the gravito-electric field $E μν$ . The KK anisotropic stress is a driving term in the propagation of shear $σ μν$ and the gravito-electric/gravito-magnetic fields, $E$ and $H μν$ 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., $(E μν,Hμν)$ , 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 (134 ) becomes [221 ]

if we neglect the dark radiation, where $a$ and $H$ are the average scale factor and expansion rate, and $Σ$ is the shear constant. In general relativity, the shear term dominates as $a → 0$ , but in the brane-world, the high-energy $2 ρ$ term will dominate if $w > 0$ , so that the matter-dominated early universe is isotropic [18 , 47

, 48

, 64 , 221

, 280 , 308 ]. This is illustrated in Figure 4

Note that this conclusion is sensitive to the assumption that $ρℰ ≈ 0$ , which by Equation (115 ) implies the restriction

Relaxing this assumption can lead to non-isotropizing solutions [1 , 46 , 65 ].

The system of propagation and constraint equations, i.e., Equations (113 , 114 , 115 , 116 ) and (123 , 124 , 125 , 126 , 127 , 128 , 129 , 130 , 131 , 132 , 133 , 134 ), 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 $π ℰ μν$ .