Wave function of the radion in a brane world

Abstract

We calculate the linearized metric perturbation corresponding to a massless four-dimensional scalar field, the radion, in a five-dimensional two-brane model of Randall and Sundrum. In this way we obtain relative strengths of the radion couplings to matter residing on each of the branes. The results are in agreement with the analysis of Garriga and Tanaka of gravitational and Brans-Dicke forces between matter on the branes. We also introduce a model with infinite fifth dimension and "almost" confined graviton, and calculate the radion properties in that model. PACS numbers: 04.50.+h, 11.25.Mj hep-th/9912160 Recently, considerable interest has been raised by a five-dimensional model with an S 1 /Z 2 orbifold extra dimension with two 3-branes residing at its boundaries. 1This model and its non-compact analogues 2-5 (see Ref. 6for an account of earlier works) provide a novel setting for discussing various conceptual and phenomenological issues related to compactification of extra dimensions in models motivated by M-theory. In the two-brane Randall-Sundrum model, 1 the branes have tensions +σ and −σ, and the bulk cosmological constant is chosen in such a way that the classical solution describes fivedimensional space-time whose four-dimensional slices are flat,Here a(z) = e −k|z| , the fifth coordinate z runs from z + = 0 to z − = r c and k = (4π/3)G 5 σ where G 5 is Newton's constant in five dimensions. The orbifold symmetry, a local reflection symmetry at each brane, is assumed to hold for all fields in this space-time.The excitations above the background metric (1) contain a massless four-dimensional graviton (whose wave function is peaked at the positive tension brane) and the corresponding Kaluza-Klein tower.3 This is not the whole story, however. In general when one has a wall in spacetime, one might expect a translational zero mode giving rise to free motion of the wall. In the case of antide Sitter spacetime, ∂ z is not a translational Killing vector but a conformal Killing vector, nonetheless we can identify solutions to the perturbation equations which correspond to proper motion of the wall (although these will be singular on the AdS horizon). In the conventional application of the Israel equations, one identifies the extrinsic curvature, K µν , on each side of the wall, and then applies a Z 2 symmetry across the wall leading toK µν = (K + µν + K − µν )/2 = 0; geometrically this means that the wall is locally 'flat' i.e. totally geodesic. To describe proper dynamical motion of the wall, we require a nonzeroK µν , which is possible if the Z 2 -symmetry is not imposed. The appropriate solution for such a motion then turns out to bewhereK µ µ = 0, which is recognised as the 'Nambu' equation for a brane. Since this solution blows up at large z, it does not correspond to a small perturbation of the spacetime, and is indicative that in the presence of such free motion, the asymptotic structure of the spacetime is altered, similar to the difference between the metrics of a straight cosmic string and a crinkly...

Nonlocal braneworld action: An alternative to the Kaluza-Klein description

“…Now, one can use them in Eq. (20) for the linearized Ricci tensor in order to obtain the equation for H µν that can be solved by iterations in powers of R µν and ξ [34]. In the linear approximation this solution reads…”

“…Therefore, let us introduce two coordinate systems which are Gaussian normal relative to their respective branes and mark them by (±) [20]. Metric perturbations in these coordinate systems, denoted correspondingly by h…”

“…Their flatness is the result of compensation between the bulk cosmological constant and brane tensions. Now consider the Randall-Sundrum model with small matter sources for metric perturbations h AB (x, y) on the background of this solution [13,14,16,20],…”

“…In an arbitrary curvilinear coordinate system the expression (23) should be appropriately covariantized by the technique of the covariant perturbation theory of [34,35]. always go to the coordinate system with h 55 = h µ5 = 0, h ν µ , ν = 0, h µ µ = 0 [16,20] 7 , in which the linearized equations in the bulk simplify to a single equation for transverse-traceless h µν ,…”

“…Sundrum model, where the boundary is associated with the brane Σ located at y brane , this identification has led to the understanding that the recently observed localization of the graviton zero mode [13] and the recovery of the four-dimensional Einstein gravity on the brane [13,14,20] can be interpreted in terms of the AdS/CFT-correspondence [15,16,17]. This conclusion was reached in the language of the effective action of the brane field -the four-dimensional metric induced on the brane from the bulk geometry.…”