where µ $^{2}$( z ) ≡ ∂ $^{2}$V ∂φ 2 ∣ ¯ φ$_{(}$z ) and where the actual value of ρ is that which extremizes the right hand side. This is a sufficient condition, but not a necessary one. In order to demonstrate how effective this weaker stability condition can be, we now turn to a simple example in which the condition can actually be evaluated. Consider a flat metric, where σ ( y ) ≡ 0, and a trivial background scalar configuration, i.e., ¯ φ ( y ) ≡ 0, but where the 5D scalar potential is allowed to have a tachyonic mass. In this case equation (16) becomes If ϕ$_{y}$ has Dirichlet boundary conditions, the solutions to this problem are where m 2 n − µ 2 = n $^{2}$π$^{2}$/L2 and L = b − a is the size of the extra dimension. The mass of the lightest mode is m 2 0 = µ 2 + π $^{2}$/L2 and so the condition for stability is m$_{0}$ ≤ 0 , which means that the bulk scalar mass µ 2 can be negative, but not arbitrarily so: Therefore in this case (where σ ′ = σ ′′ = 0), our sufficient condition (23) becomes which leads to The value of ρ that extremizes this bound is ρ = πL $^{2}$/e , and so our weaker bound is This result is a factor of 2 π/e weaker than the exact bound (26). Nevertheless this result is nontrivial as it clearly demonstrates that it is possible to have negative bulk masses and retain a stable system. 1 1 The stability conditions of the trivial vacuum in the presence of negative bulk mass terms in an extra- dimensional scalar field theory have been analyzed and generalized to general warped backgrounds in [46]. µ 2 ≥ − eπ 2 1 L 2 . (29) µ 2 ≥ 1 ρ ln( L e √ ρπ ) . (28) e − ρµ 2 ∫ b a dz ≤ e √ ρπ , (27) µ 2 ≥ − π $^{2}$/L2 . (26) ϕ$_{y}$ = sin( √ m 2 n − µ 2 y ) (25) ϕ ′′ y − ( µ 2 − m 2 n ) ϕ$_{y}$ = 0 . (24) 11