In this case one includes the effects of the curved metric along the extra dimension on the scalar field solutions while still ignoring the dynamics of the gravitational sector. We therefore consider the action where the form of the metric is now taken to be and where σ ( y ) is the warp-factor and γ$_{µν}$ the 4D metric on slices of constant y . The purpose of considering scalar field configurations on a fixed background is to explore whether our previous results continue to hold in the presence of a warped background in a regime where we still have semi-analytical control over the solutions. We postpone a discussion of the full dynamical problem, including the backreaction on the metric due to the presence of the scalar field, until the next section. In the original Randall-Sundrum (RS) model [9], the metric takes the form (7) with σ ( y ) = k | y | and γ$_{µν}$ = η$_{µν}$ , where k has dimensions of mass and is related to the 5D cosmological constant of AdS$_{5}$ . In this background any static nontrivial field configurations ¯ φ ( y ) are solutions of Scalar perturbations around this kink background, ϕ ( x, y ) = φ ( x, y ) − ¯ φ ( y ), can be decom- posed as such that the normal modes ϕ ( n ) x ( x ) and ϕ ( n ) y ( y ) are solutions of where µ 2 ≡ ∂ $^{2}$V ∂φ 2 ∣ ¯ φ and $^{(4)}$□ ≡ η $^{µν}$∂$_{µ}$ ∂$_{ν}$ . Taking the derivative of the kink equation (8) gives ϕ ′′ M − 4 kϕ ′ M − µ $^{2}$( y ) ϕ$_{M}$ = 0 , (12) $^{(4)}$□ ϕ$_{x}$ + m 2 $_{n}$ϕ$_{x}$ = 0 (10) ϕ ′′ y − 4 kϕ ′ y − ( µ $^{2}$( y ) − m 2 $_{n}$e 2 $^{ky}$) ϕ$_{y}$ = 0 , (11) ϕ ( x, y ) = ∑ n ϕ ( n ) x ( x ) ϕ ( n ) y ( y ) (9) ¯ φ ′′ − 4 k ¯ φ ′ − ∂V ∂φ ∣ ¯ φ = 0 . (8) ds 2 = e − 2 σ ( y $^{)}$γ$_{µν}$ ( x ) dx $^{µ}$dxν − dy 2 , (7) S = ∫ d 5 $_{x}$√ − g [ 1 2 g $^{MN}$( ∂$_{M}$ φ ) ∂$_{N}$ φ − V ( φ ) ] , (6) A. Kink Scalar in an AdS$_{5}$ Background 8