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, (111 ) 4- ⃗ ⃗ν ν ν ν α q˙⟨μ⟩ + 3Θq μ + ∇ μp + (ρ + p)A μ + ∇ π μν + A πμν + σμνq − 𝜀μναω q = 0. (112 )
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

Aμ = ω μ = σμν = 0, Θ = 3H, (113 )
where H = ˙a∕a is the Hubble rate. The covariant spatial curl is given by
curlVμ = 𝜀μαβ ⃗∇αV β, curlW μν = 𝜀αβ(μ⃗∇ αW βν), (114 )
where 𝜀μαβ 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
⃗ α... α... ν α δ γ... ∇ μF ...β = (∇ μF ...β)⊥u = hμ h γ ...h β∇ νF ...δ. (115 )
In a local inertial frame at a point on the brane, with uμ = δμ0, we have: 0 = A = ω = σ = 𝜀 = curl V = curlW 0 0 0μ 0αβ 0 0μ, and
∇⃗μF α......β = δμiδαj ...δβk∇iF j......k (local inertial frame ), (116 )
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) [311Jump To The Next Citation Point, 21Jump To The Next Citation Point, 322Jump To The Next Citation Point, 143Jump To The Next Citation Point, 274Jump To The Next Citation Point, 144Jump To The Next Citation Point, 48Jump To The Next Citation Point].

Equation (79View Equation) 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- μ ℰ μ ℰ μν ℰ ρ˙ℰ + 3Θ ρℰ + ⃗∇ qμ + 2A qμ + σ πμν = 1 [ ---3π μν ˙πμν + 3(ρ + p)σμνπμν + Θ (2qμqμ + πμνπ μν) + 6A μqνπμν 2λ ] − 2qμ⃗∇ μρ + 3qμ⃗∇ νπμν + 3πμν∇⃗μqν + 3σμνπ αμπνα − 3σμνqμq ν . (117 )
Projecting into the comoving rest space gives the nonlocal momentum conservation equation, which is a propagation equation for qℰ μ:
q˙ℰ + 4-Θq ℰ+ 1⃗∇ ρ + 4-ρ A + ⃗∇ νπ ℰ + A νπℰ + σ νqℰ − 𝜀 ναω qℰ= ⟨μ⟩ 3 μ 3 μ ℰ 3 ℰ μ μν μν μ ν μ ν α 1 [ --- − 4(ρ + p)⃗∇ μρ + 6(ρ + p)⃗∇νπ μν + 6q ν ˙π⟨μν⟩ + 2πμν∇⃗νρ 4λ ( ) − 6π αβ ⃗∇ π − ⃗∇ π − 3π ⃗∇ παβ + 12qν∇⃗ q − 6qν⃗∇ q − 2qμ ⃗∇νq μ αβ α βμ μα β μ ν ν μ ν − 6π μασαβqβ + 6σ μα παβqβ + 6πμν𝜀ναβω αqβ − 6𝜀μαβωα πβνqν ] +4(ρ + p)Θq μ + 2qμA νqν + 6A μqνqν + 4qμσ αβπαβ . (118 )
The 1+3-covariant decomposition shows two key features:

In particular cases, the Weyl anisotropic stress ℰ πμν may drop out of the nonlocal conservation equations, i.e., when we can neglect σμνπℰμν, ⃗∇ νπℰμν, and A νπℰμν. 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 ℰ πμν terms cannot be neglected. Even when we can neglect these terms, πμℰν 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 (118View Equation), are imperfect fluid terms, and most of these terms are quadratic in the imperfect quantities qμ and πμν. For a single perfect fluid or scalar field, only the ⃗∇μ ρ term on the right of Equation (118View Equation) survives, but in realistic cosmological and astrophysical models, further terms will survive. For example, terms linear in π μν 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 (117View Equation) and (118View Equation) constrain 4 of the 9 independent components of ℰμν on the brane. What is missing is an evolution equation for ℰ πμν, 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 ℰμν = 0 by symmetry, as in the case of an FRW brane, then the evolution of ℰ μν is determined by Equations (117View Equation) and (118View Equation). If the brane is stationary (with Killing vector parallel to μ u), then evolution equations are not needed for ℰμν, although in general ℰ πμν 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 (111View Equation) and (112View Equation) reduce to

˙ρ + Θ (ρ + p) = 0, (119 ) ⃗∇ μp + (ρ + p)Aμ = 0, (120 )
while the nonlocal conservation equations (117View Equation) and (118View Equation) reduce to
ρ˙ℰ + 4Θ ρℰ + ⃗∇ μqℰ+ 2Aμqℰ + σ μνπℰ = 0, (121 ) 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 − 𝜀μ ω νqα = − λ ∇ μρ. (122 )
Equation (122View Equation) shows that [388]

A simple example of the latter point is the FRW case: Equation (122View Equation) is trivially satisfied, while Equation (121View Equation) becomes

ρ˙ℰ + 4H ρℰ = 0. (123 )
This equation has the dark radiation solution
( ) ρ = ρ a0- 4. (124 ) ℰ ℰ0 a

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:

⃗ ∇ μρ ⁄= 0 ⇒ ℰμν ⁄= 0. (125 )
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.
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