With these assumptions we arrive at the Einstein equations which fix the values of H and L in terms of n and the parameters of the original Lagrangian according to We thus conclude from Eq. (17) that flux vacua exist for S 1 compactifications provided the 5 d cosmological constant is negative (Λ < 0). Eq. (18) then indicates that these com- pactifications always yield a 4 d anti-de Sitter spacetime, enumerated by the integer n = 0. Furthermore, it can be easily shown by studying the 4 d effective action associated with these models [12], that the above solutions are perturbatively stable. We will now investigate the possibility that there exist non-pertubative instabilities of this geometry $^{2}$, similar to the pure Kaluza – Klein bubble of nothing presented in the previous section. A particularly useful description of our AdS$_{4}$ × S 1 compactification is given by the fol- lowing metric which covers a Rindler-like portion of the 4 d spacetime shown in Fig. (2) below: Comparing with the right hand illustration in Fig. (2), consider a metric of the form with boundary conditions for r → 0, and 2 Flux-changing instantons [12] have previously been shown to exist. Their relation to the bubble of nothing presented here will be discussed below. 3 H 2 = κ 2 $^{(}$n $^{2}$η2 2 L 2 + Λ$^{)}$ , (15) 6 H 2 = − κ 2 $^{(}$n $^{2}$η2 2 L 2 − Λ$^{)}$ , (16) L 2 = − 3 n $^{2}$η2 2Λ , (17) H 2 = 2 κ $^{2}$Λ 9 . (18) ds 2 = dr 2 + H − $^{2}$sinh$^{2}$( Hr )( − dt 2 + cosh $^{2}$t d Ω 2 $_{2}$) + L $^{2}$dy2 . (19) ds 2 = dr 2 + B $^{2}$( r )( − dt 2 + cosh $^{2}$t d Ω 2 $_{2}$) + r $^{2}$C ( r ) $^{2}$dy2 (20) B ( r ) → ℓ C ( r ) → 1 (21) B. The bubble of nothing as a de Sitter brane 5