content_image/360948.jpg content_image/360949.jpg content_image/360950.jpg content_image/360951.jpg content_image/360952.jpg content_image/360953.jpg content_image/360954.jpg
With these assumptions we arrive at the Einstein equations 

<img src='content_image/360948.jpg'>

<img src='content_image/360949.jpg'>

which fix the values of H and L in terms of n and the parameters of the original Lagrangian according to 

<img src='content_image/360950.jpg'>

<img src='content_image/360951.jpg'>

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. 

## B. The bubble of nothing as a de Sitter brane 

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: 

<img src='content_image/360952.jpg'>

Comparing with the right hand illustration in Fig. (2), consider a metric of the form 

<img src='content_image/360953.jpg'>

with boundary conditions 

for r → 0, and 

<img src='content_image/360954.jpg'>

