3.1 Field equations on the brane

Using Equations (50View Equation) and (54View Equation), it follows that
[ ( ) ] 1- 2-2 (5) A B (5) A B 1(5) G μν = − 2Λ5g μν + 3κ5 TABg μ gν + TABn n − 4 T gμν 1 [ ] +KK μν − K μαK αν + -- K αβK αβ − K2 gμν − ℰμν, (60 ) 2
where (5)T = (5)TAA, and where
(5) C D A B ℰμν = CACBDn n gμ gν , (61 )
is the projection of the bulk Weyl tensor orthogonal to nA. This tensor satisfies
ℰABnB = 0 = ℰ[AB ] = ℰAA, (62 )
by virtue of the Weyl tensor symmetries. Evaluating Equation (60View Equation) on the brane (strictly, as y → ±0, since ℰAB is not defined on the brane [388Jump To The Next Citation Point]) will give the field equations on the brane.

First, we need to determine K μν at the brane from the junction conditions. The total energy-momentum tensor on the brane is

T brane= T − λg , (63 ) μν μν μν
where Tμν is the energy-momentum tensor of particles and fields confined to the brane (so that TABnB = 0). The 5D field equations, including explicitly the contribution of the brane, are then
(5) (5) 2[(5) brane ] GAB = − Λ5 gAB + κ 5 TAB + TAB δ(y) . (64 )
Here the delta function enforces in the classical theory the string theory idea that Standard Model fields are confined to the brane. This is not a gravitational confinement, since there is in general a nonzero acceleration of particles normal to the brane [303Jump To The Next Citation Point].

Integrating Equation (64View Equation) along the extra dimension from y = − 𝜖 to y = + 𝜖, and taking the limit 𝜖 → 0, leads to the Israel–Darmois junction conditions at the brane,

g+μν − g−μν = 0, (65 ) [ ] K+ − K − = − κ2 Tbrane − 1-Tbranegμν , (66 ) μν μν 5 μν 3
where brane μν brane T = g T μν. The Z2 symmetry means that when you approach the brane from one side and go through it, you emerge into a bulk that looks the same, but with the normal reversed, A A n → − n. Then Equation (52View Equation) implies that
K −μν = − K+μν, (67 )
so that we can use the junction condition Equation (66View Equation) to determine the extrinsic curvature on the brane:
[ ] 1- 2 1- K μν = − 2 κ5 T μν + 3 (λ − T )gμν , (68 )
where T = T μμ, where we have dropped the (+ ), and where we evaluate quantities on the brane by taking the limit y → +0.

Finally we arrive at the induced field equations on the brane, by substituting Equation (68View Equation) into Equation (60View Equation):

2 κ2- κ2- G μν = − Λgμν + κ Tμν + 6 λ 𝒮μν − ℰμν + 4 λ ℱμν. (69 )
The 4D gravitational constant is an effective coupling constant inherited from the fundamental coupling constant, and the 4D cosmological constant is nonzero when the RS balance between the bulk cosmological constant and the brane tension is broken:
2 2 1- 4 κ ≡ κ 4 = 6λ κ5, (70 ) 1[ ] Λ = --Λ5 + κ2λ . (71 ) 2
The first correction term relative to Einstein’s theory is quadratic in the energy-momentum tensor, arising from the extrinsic curvature terms in the projected Einstein tensor:
-1- 1- α -1- [ αβ 2] 𝒮μν = 12 TT μν − 4TμαT ν + 24 gμν 3TαβT − T . (72 )
The second correction term is the projected Weyl term. The last correction term on the right of Equation (69View Equation), which generalizes the field equations in [388Jump To The Next Citation Point], is
[ ] (5) A B (5) A B 1(5) ℱμν = TABg μ gν + TABn n − 4 T gμν, (73 )
where (5) TAB describes any stresses in the bulk apart from the cosmological constant (see [311Jump To The Next Citation Point] for the case of a scalar field).

What about the conservation equations? Using Equations (50View Equation), (55View Equation) and (68View Equation), one obtains

ν (5) A B ∇ T μν = − 2 TABn g μ. (74 )
Thus in general there is exchange of energy-momentum between the bulk and the brane. From now on, we will assume that
(5) TAB = 0 ⇒ ℱμν = 0, (75 )
so that
(5) (5) GAB = − Λ5 gAB in the bulk, (76 ) 2 κ2- Gμν = − Λg μν + κ Tμν + 6 λ 𝒮μν − ℰμν on the brane. (77 )
One then recovers from Equation (74View Equation) the standard 4D conservation equations,
∇νT μν = 0. (78 )
This means that there is no exchange of energy-momentum between the bulk and the brane; their interaction is purely gravitational. Then the 4D contracted Bianchi identities (∇ νG μν = 0), applied to Equation (69View Equation), lead to
μ 6κ2 μ ∇ ℰμν = -λ--∇ 𝒮μν, (79 )
which shows qualitatively how 1+3 spacetime variations in the matter-radiation on the brane can source KK modes.

The induced field equations (77View Equation) show two key modifications to the standard 4D Einstein field equations arising from extra-dimensional effects:


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