478
M. C. Johnson et al.
r
–0.4
–0.2
0.2
0.6
0.8
1.0
1.2
0.4
Fig. 1 The potential V (r) for constant σ = 0.04 (red) and the modified potential ¯
V (r) (blue) with
σ (r) given by Eq. (11), and M = 0.045, Λ i /Λ = 0.994
σ (r) = 0.04 − 0.0035 tanh
25
r 0
(r − r 0 + 0.105)
(11)
where r 0 is the position of the maximum of the potential when σ = 0.04. We find
that the potential has a stable minimum as shown in Fig. 1. The potential V (r) for
constant σ is shown and the modified potential with the stable minimum is called
¯
V (r).
The dominant energy condition is obviously satisfied in the bulk. On the wall,
the spatial stress [4] is given by
ϑ(r) = σ (r) +
r
2
dσ (r)
dr
(12)
and we also find numerically that the dominant energy condition on the wall, σ
|ϑ|, is satisfied.
4 Conclusion
In conclusion, we report here that there exist dynamically stable, non-singular
solutions of the Einstein equations which correspond to negative mass bubbles in
an asymptotically de Sitter space-time. The import of such solutions to cosmology
must be examined.