Eur. Phys. J. C (2015) 75 :133 Page 3 of 11 133
Fig. 1 The quartic potential V (S) =
3V
f
β S
4
f
S
4
−
2(β+3)V
f
β S
3
f
S
3
+
3(β+1)V
f
β S
2
f
S
2
; expressed in terms of the false vacuum field S
f
=
√
4πφ
f
,
the vacuum energy of the false vacuum V
f
and a free parameter β.
The patterns of the potential are shown for β = 0.1, V
f
= 10
−4
and S
f
=
√
4πφ
f
= (1) 0.1, (2) 0.2, (3) 0.3, (4) 0.4 (credit: Hwang
et al. [18])
R
μν
−
1
2
Rg
μν
= 8π T
μν
, (2)
where the energy-momentum tensors on the right-hand side
are written as
T
μν
= φ
;μ
φ
;ν
−
1
2
φ
;ρ
φ
;σ
g
ρσ
g
μν
− V (φ)g
μν
. (3)
Also, the scalar field equation for the system reads
∇
2
φ =
dV
dφ
. (4)
The Einstein equations (2) and the scalar field equation (4)
constitute a scalar field model that effectively describes
the mechanism of two colliding bubbles in curved space-
time [18]. Given a potential V (φ), the scalar field solution φ
and the geometry solution g
μν
should be obtained by solving
Eqs. (2) and (4) simultaneously.
1
To this end, we prescribe an
ansatz for the geometry g
μν
with the hyperbolic symmetry,
using the double-null coordinates:
ds
2
=−α
2
h
(u,v)dudv + r
2
h
(u,v)dH
2
, (5)
where d H
2
= dχ
2
+ sinh
2
χdθ
2
with 0 ≤ χ<∞,
0 ≤ θ<2π [18], and α
h
(u,v)and r
h
(u,v)are to be deter-
mined by solving Eqs. (2) and (4) simultaneously in the coor-
dinates (u,v,χ,θ). In the flat spacetime limit, the double-
null coordinates are defined as u ≡ τ − x and v ≡ τ + x
with τ
2
≡ t
2
− y
2
− z
2
, t = τ cosh χ, y = τ sinh χ sin θ,
z = τ sinh χ cos θ: in our analysis, the x-axis of Cartesian
coordinates is chosen to coincide with a line adjoining the
centers of the two bubbles, and the y-axis and the z-axis lie
in a plane perpendicular to the x-axis.
2.2 Solving the scalar field equation
To build the simplest model of the CDL instanton for two
identical colliding bubbles, we consider the potential in
Sect. 2.1 to be
1
This inevitably results in the effects of radiation reaction being
included in the solutions, φ and g
μν
[18].
V (φ) =
p
4
4!
φ
4
+
p
3
3!
φ
3
+
p
2
2!
φ
2
, (6)
where p
2
, p
3
, and p
4
are constants which can be appropri-
ately chosen to tune the shape of the potential. Bubble col-
lisions are represented by the scalar field moving along this
potential: the field initially in the false vacuum state (at higher
local minimum of potential) tunnels quantum mechanically
to the true vacuum state (at lower local minimum of poten-
tial), repeating the transitions back and forth between the two
states, eventually settling down in the true vacuum state.
Following Ref. [18], we may rescale the scalar field, S ≡
√
4πφ for computational convenience, and can specify p
2
,
p
3
, and p
4
in terms of the false vacuum field S
f
=
√
4πφ
f
,the
vacuum energy of the false vacuum V
f
and a free parameter
β. The potential (6) can then be rewritten as
V (S) =
3V
f
β S
4
f
S
4
−
2(β + 3)V
f
β S
3
f
S
3
+
3(β + 1)V
f
β S
2
f
S
2
. (7)
With this potential the scalar field equation (4), which is now
rescaled, reads
∇
2
S =
12V
f
β S
4
f
S
3
−
6(β + 3)V
f
β S
3
f
S
2
+
6(β + 1)V
f
β S
2
f
S. (8)
This is a non-linear wave equation whose analytical solution
is not generally known: we normally approach this type of
problem with numerical methods.
Now, we solve the scalar field equation (8) simultaneously
with the Einstein equations (2), using the ansatz given by (5),
in the coordinates (u,v,χ,θ). However, it turns out that our
scalar field solution is independent of the coordinates χ and
θ and is expressed in the coordinates (u,v) only; namely,
S(u,v) =
√
4πφ(u,v) [15,18,19]. With the choice of the
constants, β = 0.1, V
f
= 10
−4
and S
f
= (1) 0.1, (2) 0.2, (3)
0.3, (4) 0.4inEq.(7), the potential takes the forms as given
by Fig. 1 [18]. With this potential, our numerical solution
S(u,v)is obtained as presented by Fig. 2 [18]. In each case
of S
f
, (1)–(4), the bubble wall has a different value of tension
due to a different value of S
f
as shown by Fig. 1.Inthetop
left of Fig. 2 the bubble has the lowest tension while in the
bottom right it has the highest tension among the four cases
123