Casimir stress on parallel plates in de Sitter space with signature change

Abstract

The Casimir stress on two parallel plates in a de Sitter background corresponding to different metric signatures and cosmological constants is calculated for massless scalar fields satisfying Robin boundary conditions on the plates. Our calculation shows that for the parallel plates with false vacuum between and true vacuum outside, the total Casimir pressure leads to an attraction of the plates at very early universe. Comment: 9 pages, no figures, accepted for publication in IJMPA

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arXiv:gr-qc/0611151v1 29 Nov 2006
Casimir stress on parallel plates in de Sitter
space with signature change
M.R. Setare∗
Department of Science, Payame Noor University. Bijar. Iran
Abstract
The Casimir stress on two parallel plates in a de Sitter background corresponding
to different metric signatures and cosmological constants is calculated for massless
scalar fields satisfying Robin boundary conditions on the plates. Our calculation
shows that for the parallel plates with false vacuum between and true vacuum
outside, the total Casimir pressure leads to an attraction of the plates at very early
universe.
∗E-mail: rezakord@ipm.ir
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1 Introduction
The Casimir effect is one of the most interesting manifestations of non-trivial properties
of the vacuum state in quantum field theory [1,2]. Since its first prediction by Casimir
in 1948[2] this effect has been investigated for different fields having different boundary
geometries [3, 8, 9, 10, 11, 12, 13, 14]. The Casimir effect can be viewed as the polarization
of vacuum by boundary conditions or geometry. Therefore, vacuum polarization induced
by a gravitational field is also considered as a Casimir effect.
Casimir effect may have interesting implications for the early universe. In [15] the Casimir
effect of a massless scalar field with Dirichlet boundary condition in a spherical shell hav-
ing different vacua inside and outside, which represents a bubble in the early universe
with false/true vacuum inside/outside, has been investigated. The Casimir stress on two
concentric spherical shells with constant comoving radius having different vacua inside
and outside in de Sitter space, which corresponds to a spherically symmetric domain wall
with thickness, has been calculated in [16].
In the context of hot big bang cosmology, the unified theories of the fundamental interac-
tions predict that the universe passes through a sequence of phase transitions. Different
types of topological objects may have been formed during these phase transitions, these
include domain walls, cosmic strings and monopoles [17, 18, 19]. These topological defects
appear as a consequence of breakdown of local or global gauge symmetries of a system
composed by self-coupling iso-scalar Higgs fields Φa.
On the other hand, signature changing space-times have recently been of particular im-
portance as the specific geometries with interesting physical effects. The original idea of
signature change was due to Hartle, Hawking and Sakharov [20, 21]. This interesting idea
would make it possible to have both Euclidean and Lorentzian metrics in path integral
approach to quantum gravity. Latter it was shown that the signature change may happen
even in classical general relativity, as well [22]-[32]. The issue of propagation of quantum
fields on signature-changing space-times has also been of some interest [33]-[38]. For ex-
ample, Dray et al have shown that the phenomenon of particle production may happen
for scalar particles propagation in space-time with heterotic signature. They have also
obtained a rule for propagation of massless scalar fields on a two- dimensional space-time
with signature change. Dynamical determination of the metric signature in space-time of
nontrivial topology is another interesting issue which has been studied in Ref.[39].
To the author knowledge, no attempt has been done to study the Casimir effect within
the geometries with signature change. A relevant work to the present paper is Ref. [40].
In this work the Casimir effect for the free massless scalar field propagating on a two-
dimensional cylinder with a signature-changing strip has been studied. Motivated by this
new element in studying the Casimir effect we have paid attention to study such a non-
trivial effect in a model of two parallel plates in de Sitter space with different cosmological
constants and metric signatures between and outside.
2 Parallel Plates with Robin boundary conditions in
de Sitter Space
Consider a massless scalar field coupled conformally to a de Sitter background space. The
scalar field satisfies the following Robin boundary conditions on two parallel plates within
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an arbitrary space-time [41]:
(1 + βm(−1)m−1∂x)φ(xν)|x=am= 0,
Here we have assumed that the two plates are normal to the Cartesian x-axis at x = a1,2.
The Robin boundary condition may be interpreted as the boundary condition on a thick
plate [42]. Rewriting(1) in the following form:
m = 1,2, (1)
∂xφ(xν) = (−1)m1
βmφ(xν), (2)
where |βm|, having the dimension of a length, may be called the skin-depth parameter.
This is similar to the case of penetration of an electromagnetic field into a real metal,
where the tangential component of the electric field is proportional to the skin-depth
parameter.
The corresponding field equation has the form
(∇µ∇µ+ ξR)φ = 0,∇µ∇µ=
1
√−g∂µ(√−ggµν∂ν), (3)
where ∇µis the covariant derivative operator, and R is the Ricci scalar for de Sitter space.
In the conformally coupled case, the corresponding stress-energy tensor is defined as [43]
Tµν= (1 − 2ξ)∂µφ∂νφ + (2ξ − 1/2)gµν∂λφ∂λφ − 2ξφ∇µ∇νφ +1
12gµνφ∇λ∇λφ, (4)
where ξ =1
scalar field the perpendicular pressure, P, is uniform in the region between the plates and
is given by [41]
P = 3εc,
6. It is known that in the Minkowski space-time for the conformally coupled
(5)
where εcis the Casimir energy density. This Casimir energy has been calculated to be
ε = εc=
−A
8Γ(5/2)π3/2a4, (6)
where A depends on β1,2and a may be inferred from Eq.(4.15) of [41]. It has also been
shown that for β1= −β2
ε = εc=−ζR(4)Γ(2)
(4π)2a4
=
−π2
1440a4, (7)
which is the same as for the Dirichlet and Neumann boundary conditions.
Consider now two parallel plates in a de Sitter space-time. To make the maximum use of
the flat-space calculations we use the de Sitter metric in the conformally flat form:
ds2=α2
η2[dη2−
3
?
i=1
(dxi)2],(8)
where η is the conformal time:
−∞ < η < 0.(9)
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The relation between the parameter α and the cosmological constant Λ is given by
α2=3
Λ. (10)
The quantization of a scalar field on background of the metric (8) is straightforward.
Let {φk(x),φ⋆
solutions to the field equation (3), obeying boundary condition (1). By expanding the
field operator over these eigenfunctions, using the standard commutation rules and the
definition of the vacuum state for the vacuum expectation values of the energy-momentum
tensor one obtains [43]
?0|Tν
k
k(x)} be a complete set of orthonormalized positive and negative frequency
µ|0? =
?
Tν
µ{φk,φ∗
k}, (11)
where |0 > is the amplitude for the corresponding vacuum state, and the bilinear form
Tν
problem under consideration we have a conformally trivial situation, namely a conformally
invariant field on background of the conformally flat spacetime. Instead of evaluating
Eq.(11) directly on background of the curved metric, the vacuum expectation values
can be obtained from the corresponding flat spacetime results for a scalar field φ by
using the conformal properties of the problem under consideration. Under the conformal
transformation ˜ gµν= Ω2gµνthe φ field will be changed by the rule
µ{φ,φ∗} on the right is determined by the classical energy-momentum tensor (4). In the
˜φ(x) = Ω−1φ(x). (12)
Using the standard relation between the energy-momentum tensor for conformally coupled
situations [43]
< Tµ
ν[˜ gαβ] >= (g
˜ g)
1
2< Tµ
ν[g(M)
αβ] > −
1
2880π2[1
6
˜H(1)µ
ν
−˜H(3)µ
ν
], (13)
where gµνand ˜ gµνare conformal to each other, with their respective determinants g and
˜ g. We are going to assume that gµν is the Minkowski metric. Now, < Tµ
regulatized energy- momentum tensor for a conformally coupled scalar field in the case of
a parallel plate configuration in flat space-time is given by
ν[g(M)
αβ] >, the
< Tµ
ν[g(M)
αβ] >= diag(ε,−P,−P⊥,−P⊥) = diag(ε,3ε,−ε,−ε).
The second term in (13) is the vacuum polarization due to the gravitational field, without
any boundary conditions. The functions H(1,3)µ
ν
components (see [43]). For a massless scalar field in de Sitter space, the term is given by
[43, 46]
1
2880[16
From(5,7,10,13,14) one can obtain the vacuum pressure due to the boundary acting on
the plates:
(14)
are some combinations of curvature tensor
−
˜H(1)µ
ν
−˜H(3)µ
ν
] =
1
960π2α4δµ
ν. (15)
P(1,2)
B
= PB(x1,2) = (g
˜ g)
1
2(−3π2
1440a4) =η4
α4(−3π2
1440a4) =−η4Λ2
3
π2
1440a4,(16)
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which is attractive. It has been shown that this pressure is zero for x < a1and x > a2
[44, 45]. The gravitational part of the pressure according to (15) is equal to
PG= − < T1
1>=
−1
960π2α4. (17)
This is the same from both sides of the plates, and hence leads to zero effective force.
Therefore, the effective force acting on the plates is only given by the boundary part.
3 Parallel plates with different cosmological constants
and signatures between and outside
Now, assume that there are different vacua between and outside of the plates, correspond-
ing to αbetwand αoutin the metric(8). As we have seen in the previous section, the vacuum
pressure due to the boundary is only non-vanishing between the plates. Therefore, we
have for the pressure due to the boundary
P(1,2)
B
=
η4
α4
betw
−3π2
1440a4=−η4Λ2
betw
3
π2
1440a4. (18)
Now, we obtain the pure effect of vacuum polarization due to the gravitational field
without any boundary conditions in Euclidean (outside) region with the following metric
ds2= −α2
η2[dη2+
3
?
i=1
(dxi)2]. (19)
To this end, we calculate the renormalized stress tensor for the massless scalar field in de
Sitter space with Euclidean signature. We use Eq.(13), then we obtain < Tµ
and
(1)Hµ
ν[g(M)
αβ] >= 0,
ν= 0,
(3)Hµ
ν=
3
α4δµ
ν.
Therefore
< Tµ
ν[˜ gαβ] >=
1
960π2α4δµ
ν, (20)
which is exactly the same result for the Lorentzian case [43]. The corresponding effective
pressures for the Euclidean (outside) and Lorentzian (between) regions with αoutand αbetw,
due to pure effect of gravitational vacuum polarization without any boundary condition,
are given respectively by
PE
out= − < T1
1>out=
−1
960π2α4
out
=−Λ2
out
9
1
960π2. (21)
PL
betw= − < T1
1>betw=
−1
960π2α4
betw
=−Λ2
betw
9
1
960π2, (22)
The corresponding gravitational pressure on the plates is then given by
PG= PL
betw− PE
out=
−1
9 × 960π2(Λ2
betw− Λ2
out). (23)
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