The brane viewpoint: A 1+3-covariant analysis

3.3 The brane viewpoint: A 1+3-covariant analysis

Following [218

], a systematic analysis can be developed from the viewpoint of a brane-bound observer. The effects of bulk gravity are conveyed, from a brane observer viewpoint, via the local ( $𝒮 μν$ ) and nonlocal ( $ℰμν$ ) corrections to Einstein’s equations. (In the more general case, bulk effects on the brane are also carried by $ℱ μν$ , which describes any 5D fields.) The $ℰμν$ term cannot in general be determined from data on the brane, and the 5D equations above (or their equivalent) need to be solved in order to find $ℰμν$ .

The general form of the brane energy-momentum tensor for any matter fields (scalar fields, perfect fluids, kinetic gases, dissipative fluids, etc.), including a combination of different fields, can be covariantly given in terms of a chosen 4-velocity $uμ$ as

Here $ρ$ and $p$ are the energy density and isotropic pressure, respectively, and

projects into the comoving rest space orthogonal to $u μ$ on the brane. The momentum density and anisotropic stress obey

where angled brackets denote the spatially projected, symmetric, and tracefree part:

In an inertial frame at any point on the brane, we have

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

The tensor $𝒮μν$ , which carries local bulk effects onto the brane, may then be irreducibly decomposed as

1 [ ] 1 [ ] 𝒮 μν = --- 2ρ2 − 3παβπαβ uμu ν +--- 2ρ2 + 4ρp + π αβπαβ − 4qαqα hμν 24 24 − 1-(ρ + 2p)πμν + πα⟨μπν⟩α + q⟨μqν⟩ + 1ρq (μuν) − -1-qαπα(μuν). (88 ) 12 3 12

This simplifies for a perfect fluid or minimally-coupled scalar field to

The tracefree $ℰ μν$ carries nonlocal bulk effects onto the brane, and contributes an effective “dark” radiative energy-momentum on the brane, with energy density $ρℰ$ , pressure $ρℰ∕3$ , momentum density $ℰ qμ$ , and anisotropic stress $ℰ πμν$ :

We can think of this as a KK or Weyl “fluid”. The brane “feels” the bulk gravitational field through this effective fluid. More specifically:

The KK (or Weyl) anisotropic stress $π ℰ μν$ incorporates the scalar or spin-0 (“Coulomb”), the vector (transverse) or spin-1 (gravimagnetic), and the tensor (transverse traceless) or spin-2 (gravitational wave) 4D modes of the spin-2 5D graviton.
The KK momentum density $qℰμ$ incorporates spin-0 and spin-1 modes, and defines a velocity $vℰμ$ of the Weyl fluid relative to $uμ$ via $qℰμ = ρℰvℰμ$ .
The KK energy density $ρ ℰ$ , often called the “dark radiation”, incorporates the spin-0 mode.

In special cases, symmetry will impose simplifications on this tensor. For example, it must vanish for a conformally flat bulk, including $AdS5$ ,

The RS models have a Minkowski brane in an $AdS5$ bulk. This bulk is also compatible with an FRW brane. However, the most general vacuum bulk with a Friedmann brane is Schwarzschild-anti-de Sitter spacetime [32

, 249

]. Then it follows from the FRW symmetries that

where $ρℰ = 0$ only if the mass of the black hole in the bulk is zero. The presence of the bulk black hole generates via Coulomb effects the dark radiation on the brane.

For a static spherically symmetric brane (e.g., the exterior of a static star or black hole) [72 ],

This condition also holds for a Bianchi I brane [221

]. In these cases, $ℰ π μν$ is not determined by the symmetries, but by the 5D field equations. By contrast, the symmetries of a Gödel brane fix $π ℰμν$ [20 ].

The brane-world corrections can conveniently be consolidated into an effective total energy density, pressure, momentum density, and anisotropic stress:

ρtot = ρ + -1-(2ρ2 − 3π μνπμν) + ρℰ, (94 ) 4λ -1-( 2 μν μ) ρℰ- ptot = p + 4λ 2ρ + 4ρp + π μνπ − 4qμq + 3 , (95 ) 1 qtoμt = qμ + ---(2ρqμ − 3πμνq ν) + qℰμ, (96 ) 2λ πtot = π + -1-[− (ρ + 3p)π + 3π π α + 3q q ] + πℰ . (97 ) μν μν 2λ μν α⟨μ ν⟩ ⟨μ ν⟩ μν

These general expressions simplify in the case of a perfect fluid (or minimally coupled scalar field, or isotropic one-particle distribution function), i.e., for $qμ = 0 = πμν$ , to

( ) ρ-- ρℰ- ρtot = ρ 1 + 2λ + ρ , (98 ) ρ ρℰ ptot = p +---(2p + ρ) + --, (99 ) tot ℰ 2 λ 3 qμ = qμ, (100 ) πtot= π ℰ . (101 ) μν μν

Note that nonlocal bulk effects can contribute to effective imperfect fluid terms even when the matter on the brane has perfect fluid form: There is in general an effective momentum density and anisotropic stress induced on the brane by massive KK modes of the 5D graviton.

The effective total equation of state and sound speed follow from Equations (98 ) and (99 ) as

ptot w-+-(1-+-2w-)ρ∕2λ-+-ρℰ∕3-ρ wtot ≡ ρ = 1 + ρ∕2λ + ρ ∕ ρ , (102 ) tot [ ℰ ] [ ]−1 2 ˙ptot 2 ρ +-p- -------4ρℰ-------- -------4ρℰ-------- ctot ≡ ˙ρtot = cs + ρ + λ + 9(ρ + p)(1 + ρ∕λ) 1 + 3(ρ + p)(1 + ρ∕λ) , (103 )

where $w = p∕ ρ$ and $2 cs = ˙p∕ρ˙$ . At very high energies, i.e., $ρ ≫ λ$ , we can generally neglect $ρℰ$ (e.g., in an inflating cosmology), and the effective equation of state and sound speed are stiffened:

This can have important consequences in the early universe and during gravitational collapse. For example, in a very high-energy radiation era, $w = 1∕3$ , the effective cosmological equation of state is ultra-stiff: $wtot ≈ 5∕3$ . In late-stage gravitational collapse of pressureless matter, $w = 0$ , the effective equation of state is stiff, $wtot ≈ 1$ , and the effective pressure is nonzero and dynamically important.