Conservation equations

3.4 Conservation equations

Conservation of $Tμν$ gives the standard general relativity energy and momentum conservation equations, in the general, nonlinear case:

⃗ μ μ μν ˙ρ + Θ(ρ + p) + ∇ qμ + 2A qμ + σ πμν = 0, (105 ) 4- ⃗ ⃗ν ν ν ν α q˙⟨μ⟩ + 3Θq μ + ∇ μp + (ρ + p)A μ + ∇ π μν + A πμν + σμνq − ɛμναω q = 0. (106 )

In these equations, an overdot denotes $ν u ∇ ν$ , $μ Θ = ∇ uμ$ is the volume expansion rate of the $μ u$ worldlines, $A μ = u˙μ = A⟨μ⟩$ is their 4-acceleration, $σμν = ∇⃗⟨μuν⟩$ is their shear rate, and $ω μ = − 12 curlu μ = ω⟨μ⟩$ is their vorticity rate.

On a Friedmann brane, we get

where $H = ˙a∕a$ is the Hubble rate. The covariant spatial curl is given by

where $ɛμαβ$ is the projection orthogonal to $μ u$ of the 4D brane alternating tensor, and $⃗ ∇ μ$ is the projected part of the brane covariant derivative, defined by

In a local inertial frame at a point on the brane, with $uμ = δμ0$ , we have: $0 = A = ω = σ = ɛ = curl V = curlW 0 0 0μ 0αβ 0 0μ$ , and

where $i,j,k = 1,2,3$ .

The absence of bulk source terms in the conservation equations is a consequence of having $Λ5$ 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 $uμ$ gives the nonlocal energy conservation equation, which is a propagation equation for $ρ ℰ$ . In the general, nonlinear case, this gives

4 ρ˙ℰ + --Θ ρℰ + ⃗∇μq ℰμ + 2A μqℰμ + σ μνπℰμν = 3 [ -1- 6π μν ˙π + 6(ρ + p )σ μνπ + 2Θ (2qμq + πμνπ ) + 2A μqνπ 4 λ μν μν μ μν μ]ν μ⃗ μ⃗ ν μν⃗ μν α μν − 4q ∇ μρ + q ∇ πμν + π ∇ μqν − 2σ παμ πν − 2σ qμqν . (111 )

Projecting into the comoving rest space gives the nonlocal momentum conservation equation, which is a propagation equation for $ℰ qμ$ :

4 1 4 ˙qℰ⟨μ⟩ + -Θq ℰμ + --⃗∇ μρℰ + -ρℰA μ + ⃗∇ νπℰμν + A νπℰμν + σμνqℰν − ɛμναωνqℰα = 3 [ 3 3 -1- ⃗ ⃗ν ν ν⃗ 4λ − 4(ρ + p)∇μρ + 6 (ρ + p)∇ πμν + q π˙⟨μν⟩ + πμ ∇ ν(2ρ + 5p) ( ) − 2-παβ ⃗∇ μπαβ + 3⃗∇ απβμ − 3πμα⃗∇ βπαβ + 28qν⃗∇ μqν 3 3 ν αβ 8- αβ αβ +4 ρA πμν − 3πμαA βπ + 3A μπ π αβ − π μασ qβ + σ π αβq + π ɛναβω q − ɛ ωαπβνq + 4(ρ + p)Θq μα β μν α β μαβ ] ν μ ν 14- ν αβ +6q μA qν + 3 A μq qν + 4qμσ π αβ . (112 )

The 1+3-covariant decomposition shows two key features:

Inhomogeneous and anisotropic effects from the 4D matter-radiation distribution on the brane are a source for the 5D Weyl tensor, which nonlocally “backreacts” on the brane via its projection $ℰ μν$ .
There are evolution equations for the dark radiative (nonlocal, Weyl) energy ( $ρ ℰ$ ) and momentum ( $ℰ qμ$ ) densities (carrying scalar and vector modes from bulk gravitons), but there is no evolution equation for the dark radiative anisotropic stress ( $π ℰμν$ ) (carrying tensor, as well as scalar and vector, modes), which arises in both evolution equations.

In particular cases, the Weyl anisotropic stress $πℰ μν$ may drop out of the nonlocal conservation equations, i.e., when we can neglect $μν ℰ σ πμν$ , $⃗ ν ℰ ∇ πμν$ , and $ν ℰ A πμν$ . 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 $π ℰ μν$ terms cannot be neglected. Even when we can neglect these terms, $ℰ πμν$ 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 $q μ$ and $π μν$ . For a single perfect fluid or scalar field, only the $⃗∇ ρ μ$ term on the right of Equation (112 ) survives, but in realistic cosmological and astrophysical models, further terms will survive. For example, terms linear in $πμν$ 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 $vμ$ generating a momentum density $qμ = ρvμ$ , 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 $ℰμν$ on the brane. What is missing is an evolution equation for $πℰμν$ , 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 $ℰ π μν$ encodes the nonlocality.

In special cases the missing equation does not matter. For example, if $π ℰ = 0 μν$ by symmetry, as in the case of an FRW brane, then the evolution of $ℰ μν$ is determined by Equations (111 ) and (112 ). If the brane is stationary (with Killing vector parallel to $μ u$ ), then evolution equations are not needed for $ℰμν$ , although in general $πℰμν$ 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

while the nonlocal conservation equations (111

) and (112

) reduce to

ρ˙ℰ + 4Θ ρℰ + ⃗∇ μqℰ+ 2Aμqℰ + σ μνπℰ = 0, (115 ) 3 μ μ μν ℰ 4- ℰ 1⃗ 4- ⃗ν ℰ ν ℰ ν ℰ να ℰ (ρ +-p)⃗ q˙⟨μ⟩ + 3 Θq μ + 3∇ μρℰ + 3 ρℰA μ + ∇ π μν + A πμν + σμ qν − ɛμ ω νqα = − λ ∇ μρ. (116 )

Equation (116

) shows that [291 ]

if $ℰμν = 0$ and the brane energy-momentum tensor has perfect fluid form, then the density $ρ$ must be homogeneous, $⃗∇ μρ = 0$ ;
the converse does not hold, i.e., homogeneous density does not in general imply vanishing $ℰμν$ .

A simple example of the latter point is the FRW case: Equation (116 ) is trivially satisfied, while Equation (115 ) becomes

This equation has the dark radiation solution

If $ℰμν = 0$ , then the field equations on the brane form a closed system. Thus for perfect fluid branes with homogeneous density and $ℰμν = 0$ , 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 $ℰ μν$ :

For example, stellar solutions on the brane necessarily have $ℰ μν ⁄= 0$ in the stellar interior if it is non-uniform. Perturbed FRW models on the brane also must have $ℰ μν ⁄= 0$ . Thus a nonzero $ℰμν$ , and in particular a nonzero $ℰ πμν$ , is inevitable in realistic astrophysical and cosmological models.