Stable, Thin Wall, Negative Mass Bubbles in de Sitter Space-Time
477
and
1 −
m − (r)
r
− m
− (r) + ˙
r 2 + r ¨
r
1 −
2m − (r)
r
+ ˙
r 2
1/2
−
1 +
M
r −
2Λr 2
3 + ˙
r 2 + r ¨
r
1 +
2M
r −
Λr 2
3 + ˙
r 2
1/2 = 8πϑr.
(5)
Writing Eq. (4) as
a + ˙
r
2
1/2 −
b + ˙
r
2
1/2 = c
1/2
(6)
we can solve for ˙
r as
˙
r 2
2
+ V (r) = 0.
(7)
with
V (r) = −
1
2
(a − b) 2
4c
−
(a + b)
2
+
c
4
,
(8)
which is the potential for the radius of the wall and explicitly it is
V (r)
= −
1
2
⎛
⎜
⎝
m − (r) + M −
Λr 3
6
2
16π 2 σ 2 r 4
−
1 +
(m − (r) − M)
r
+
Λr 2
6
+ 4π
2 σ
2 r
2
⎞
⎟
⎠ .
(9)
3 Stable Negative Mass de Sitter Bubbles
If we take m − (r) =
Λ i r 3
6 with σ → (
√
Λ/3/4π)σ , r →
√
3/Λ r and M →
(1/
√
3Λ)M, we find
V (r) = −
M 2
18σ 2 r 4 +
M
(1 − (Λ i /Λ)) + σ 2
6σ 2 r
−
r 2
2(Λ i /Λ)σ 2 + (1 − (Λ i /Λ)) 2 + σ 4 + 2σ 2
8σ 2
+
1
2
.
(10)
Then if we generalize σ → σ (r) we can obtain a stable minimum. We start with
σ = 0.04 and then we modify it as