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1.3 Heuristics of KK modes

The dilution of gravity via extra dimensions not only weakens gravity on the brane, it also extends the range of graviton modes felt on the brane beyond the massless mode of 4-dimensional gravity. For simplicity, consider a flat brane with one flat extra dimension, compactified through the identification y ↔ y + 2πnL, where n = 0,1,2,.... The perturbative 5D graviton amplitude can be Fourier expanded as
a ∑ iny∕L a f (x ,y) = e fn (x ), (8 ) n
where fn are the amplitudes of the KK modes, i.e., the effective 4D modes of the 5D graviton. To see that these KK modes are massive from the brane viewpoint, we start from the 5D wave equation that the massless 5D field f satisfies (in a suitable gauge):
(5) 2 □f = 0 ⇒ □f + ∂ yf = 0. (9 )
It follows that the KK modes satisfy a 4D Klein–Gordon equation with an effective 4D mass mn,
2 n- □fn = m nfn, mn = L . (10 )
The massless mode f0 is the usual 4D graviton mode. But there is a tower of massive modes, −1 −1 L ,2L ,..., which imprint the effect of the 5D gravitational field on the 4D brane. Compactness of the extra dimension leads to discreteness of the spectrum. For an infinite extra dimension, L → ∞, the separation between the modes disappears and the tower forms a continuous spectrum. In this case, the coupling of the KK modes to matter must be very weak in order to avoid exciting the lightest massive modes with m ≳ 0.

From a geometric viewpoint, the KK modes can also be understood via the fact that the projection of the null graviton 5-momentum (5)pA onto the brane is timelike. If the unit normal to the brane is nA, then the induced metric on the brane is

g = (5)g − n n , (5)g nAnB = 1, g nB = 0, (11 ) AB AB A B AB AB
and the 5-momentum may be decomposed as
(5)pA = mnA + pA, pAnA = 0, m = (5)pA nA, (12 )
where pA = gAB (5)pB is the projection along the brane, depending on the orientation of the 5-momentum relative to the brane. The effective 4-momentum of the 5D graviton is thus p A. Expanding (5) (5)A (5)B gAB p p = 0, we find that
gABpApB = − m2. (13 )
It follows that the 5D graviton has an effective mass m on the brane. The usual 4D graviton corresponds to the zero mode, m = 0, when (5)pA is tangent to the brane.

The extra dimensions lead to new scalar and vector degrees of freedom on the brane. In 5D, the spin-2 graviton is represented by a metric perturbation (5) hAB that is transverse traceless:

(5)hA = 0 = ∂ (5)h B. (14 ) A B A
In a suitable gauge, (5)hAB contains a 3D transverse traceless perturbation hij, a 3D transverse vector perturbation Σi, and a scalar perturbation β, which each satisfy the 5D wave equation [88]:
(5)h − → h ,Σ ,β, (15 ) AB ij i hii = 0 = ∂jhij, (16 ) ∂ Σi = 0, (17 ) i ( ) β (□ + ∂2y)( Σi) = 0. (18 ) hij
The other components of (5)hAB are determined via constraints once these wave equations are solved. The 5 degrees of freedom (polarizations) in the 5D graviton are thus split into 2 (hij) + 2 (Σi) +1 (β) degrees of freedom in 4D. On the brane, the 5D graviton field is felt as

The massive modes of the 5D graviton are represented via massive modes in all 3 of these fields on the brane. The standard 4D graviton corresponds to the massless zero-mode of hij.

In the general case of d extra dimensions, the number of degrees of freedom in the graviton follows from the irreducible tensor representations of the isometry group as 12(d + 1 )(d + 4).

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