The Casimir effect in the Fulling-Rindler vacuum

Abstract

The vacuum expectation values of the energy--momentum tensor are investigated
for massless scalar fields satisfying Dicichlet or Neumann boundary conditions,
and for the electromagnetic field with perfect conductor boundary conditions on
two infinite parallel plates moving by uniform proper acceleration through the
Fulling--Rindler vacuum. The scalar case is considered for general values of
the curvature coupling parameter and in an arbitrary number of spacetime
dimension. The mode--summation method is used with combination of a variant of
the generalized Abel--Plana formula. This allows to extract manifestly the
contributions to the expectation values due to a single boundary. The vacuum
forces acting on the boundaries are presented as a sum of the self--action and
interaction terms. The first one contains well known surface divergences and
needs a further regularization. The interaction forces between the plates are
always attractive for both scalar and electromagnetic cases. An application to
the 'Rindler wall' is discussed.

Full text

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arXiv:hep-th/0207073v2 3 Sep 2002
The Casimir effect in the Fulling–Rindler vacuum
R. M. Avagyan, A. A. Saharian∗, A. H. Yeranyan
Department of Physics, Yerevan State University,
1 Alex Manoogian St., 375049 Yerevan, Armenia
February 1, 2008
Abstract
The vacuum expectation values of the energy–momentum tensor are investigated for
massless scalar fields satisfying Dicichlet or Neumann boundary conditions, and for the elec-
tromagnetic field with perfect conductor boundary conditions on two infinite parallel plates
moving by uniform proper acceleration through the Fulling–Rindler vacuum. The scalar
case is considered for general values of the curvature coupling parameter and in an arbitrary
number of spacetime dimension. The mode–summation method is used with combination of
a variant of the generalized Abel–Plana formula. This allows to extract manifestly the con-
tributions to the expectation values due to a single boundary. The vacuum forces acting on
the boundaries are presented as a sum of the self–action and interaction terms. The first one
contains well known surface divergences and needs a further regularization. The interaction
forces between the plates are always attractive for both scalar and electromagnetic cases. An
application to the ’Rindler wall’ is discussed.
PACS number(s): 03.70.+k, 11.10.Kk
1Introduction
The imposition of boundary conditions on a quantum field leads to the modification of the spec-
trum for the zero–point fluctuations and results in the shift in the vacuum expectation values for
physical quantities such as the energy density and stresses. In particular, vacuum forces arise act-
ing on constraining boundaries. This is the familiar Casimir effect. The particular features of the
resulting vacuum forces depend on the nature of the quantum field, the type of spacetime manifold
and its dimensionality, the boundary geometries and the specific boundary conditions imposed on
the field. Since the original work by Casimir in 1948 [1] many theoretical and experimental works
have been done on this problem, including various types of boundary geometry and non-zero tem-
perature effects (see, e.g., [2, 3, 4, 5, 6, 7, 8] and references therein). Many different approaches
have been used: mode summation method with combination of the zeta function regularization
technique, Green function formalism, multiple scattering expansions, heat-kernel series, etc. An
∗E-mail address: saharyan@server.physdep.r.am
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interesting topic in the investigations of the Casimir effect is the dependence of the vacuum char-
acteristics on the type of the vacuum. It is well known that the uniqueness of vacuum state is
lost when we work within the framework of quantum field theory in a general curved spacetime
or in non–inertial frames. In particular, the use of general coordinate transformation in quantum
field theory in flat spacetime leads to an infinite number of unitary inequivalent representations
of the commutation relations. Different inequivalent representations will in general give rise to
different pictures with different physical implications, in particular to different vacuum states.
For instance, the vacuum state for an uniformly accelerated observer, the Fulling–Rindler vacuum
[9, 10, 11, 12], turns out to be inequivalent to that for an inertial observer, the familiar Minkowski
vacuum. Quantum field theory in accelerated systems contains many of special features produced
by a gravitational field avoiding some of the difficulties entailed by renormalization in a curved
spacetime. In particular, near the canonical horizon in the gravitational field, a static spacetime
may be regarded as a Rindler–like spacetime. Note that, as it has been shown in Ref. [13], there is
a class of solutions to the Einstein equations with a plane–symmetric matter distribution for which
the corresponding external geometry is described by the Rindler metric (’Rindler walls’). Another
motivation for the investigation of quantum effects in the Rindler space is related to the fact that
this space is conformally related to the de Sitter space and to the Robertson–Walker space with
negative spatial curvature. As a result the expectation values of the energy–momentum tensor
for a conformally invariant field and for corresponding conformally transformed boundaries on
the de Sitter and Robertson–Walker backgrounds can be derived from the corresponding Rindler
counterpart by the standard transformation (see, for instance, [14]).
In this paper we will consider the vacuum expectation values of the energy–momentum tensors
for a scalar and electromagnetic fields in the region between two parallel plates moving by constant
proper acceleration through the Fulling–Rindler vacuum. This problem for a single plate case
was considered by Candelas and Deutsch [15] and by one of us [16]. In Ref. [15] the cases
of conformally coupled Dirichlet and Neumann massless scalar and electromagnetic fields are
investigated in the region of the right Rindler wedge on the right from the barrier. In Ref. [16]
both regions, including the one between the barrier and Rindler horizon are considered for a
massive scalar field with general curvature coupling parameter and Robin boundary conditions in
arbitrary number of spacetime dimensions, and for the electromagnetic field. As in Ref. [16] (see
also [17, 18, 19, 20, 21]), our regularization scheme here is based on a variant of the generalized
Abel–Plana formula derived in Appendix A. This allows to extract form the vacuum expectation
values the single boundary parts and to present the ”interference” parts in terms of strongly
convergent integrals useful for numerical evaluations. We have organized the paper as follows. In
the next section we evaluate the vacuum expectation values of the energy–momentum tensor for the
Dirichlet scalar. The corresponding interaction forces between the plates are investigated in section
3. Section 4 is dedicated to the case of the Neumann boundary conditions. Then the vacuum
densities and interaction forces for the electromagnetic field are considered in section 5. Section
6 concludes the main results of the paper and an application to the ’Rindler wall’ is discussed.
In Appendix B we consider the case of the scalar field in two spacetime dimensions separately.
An alternate representation of the vacuum expectation values for the energy–momentum tensor
is obtained in Appendix C.
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2Vacuum energy-momentum tensor for a Dirichlet scalar
We consider a real massless scalar ϕ(x) field with general curvature coupling parameter ζ satisfying
the field equation
∇µ∇µϕ + ζRϕ = 0,
with R being the scalar curvature for a d + 1–dimensional background spacetime, ∇µ is the
covariant derivative operator associated with the metric gµν. For minimally and conformally
coupled scalars ζ = 0 and ζ = (d − 1)/4d, respectively. By using field equation (2.1) it can be
seen that the corresponding energy–momentum tensor (EMT) can be presented in the form
(2.1)
Tµν= ∇µϕ∇νϕ +
??
ζ −1
4
?
gµν∇ρ∇ρ− ζ∇µ∇ν− ζRµν
?
ϕ2,(2.2)
where Rµνis the Ricci tensor.
Let {ϕα(x),ϕ∗
equation (2.1), where α denotes a set of quantum numbers. Expanding field operator over these
eigenfunctions and using the commutation relations it can be easily seen that the vacuum expec-
tation values (VEV’s) of the EMT are presented in the form
α(x)} is a complete set of positive and negative frequency solutions to the field
?0 | Tµν| 0? =
?
α
Tµν{ϕα,ϕ∗
α},(2.3)
where for a scalar field the quadratic form Tµν{f,g} directly follows from the classical EMT given
by Eq. (2.2).
Our main interest in this paper will be the vacuum expectation values (VEV’s) of the EMT
in the Rindler spacetime induced by two parallel plates moving with uniform proper accelera-
tion when the quantum field is prepared in the Fulling-Rindler vacuum. For this problem the
background spacetime is flat and in Eqs. (2.1),(2.2) we have R = 0, Rµν = 0. As a result the
eigenmodes are independent on the curvature coupling parameter and the EMT VEV’s will de-
pend on this parameter through the expression (2.2) only. In the following it will be convenient
to introduce Rindler coordinates (τ,ξ,x) related to the Minkowski ones, (t,x1,x) by
t = ξ sinhτ,x1= ξ coshτ,(2.4)
where x = (x2,...,xd) denotes the set of coordinates parallel to the plates. In these coordinates
the Minkowski line element takes the form
ds2= ξ2dτ2− dξ2− dx2,(2.5)
and a wordline defined by ξ,x = const describes an observer with constant proper acceleration
ξ−1. Assuming that the plates are situated in the right Rindler wedge x1> |t| we shall let the
surfaces ξ = ξ1and ξ = ξ2, ξ2> ξ1represent the trajectories of these boundaries, which therefore
have proper accelerations ξ−1
1
and ξ−1
2
(see Fig. 1). First we will consider the case of a scalar field
satisfying Dirichlet boundary condition on the surface of the plates:
ϕ |ξ=ξ1= ϕ |ξ=ξ2= 0(2.6)
To evaluate the VEV’s of the EMT by Eq. (2.3) we need the form of the eigenfunctions ϕα(x). For
the geometry under consideration the metric and boundary conditions are static and translational
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Figure 1: The (x1,t) plane with the Rindler coordinates. The heavy lines ξ = ξ1 and ξ = ξ2
represent the trajectories of the plates.
invariant in the hyperplane parallel to the plates. It follows from here that the corresponding part
of the eigenfunctions has the standard plane wave structure:
ϕα= Cφ(ξ)exp[i(kx − ωτ)],α = (k,ω),
k = (k2,...,kd).(2.7)
The equation for φ(ξ) is obtained from field equation (2.1) on background of metric (2.5) and has
the form
ξ2φ′′(ξ) + ξφ′(ξ) +
?
where the prime denotes a differentiation with respect to the argument, and k = |k|. In the
region between the plates the corresponding linearly independent solutions to equation (2.8) are
the Bessel modified functions Iiω(kξ) and Kiω(kξ). The solution satisfying boundary condition
(2.6) on the plate ξ = ξ2is in form
ω2− k2ξ2?
φ(ξ) = 0,(2.8)
Diω(kξ,kξ2) = Iiω(kξ2)Kiω(kξ) − Kiω(kξ2)Iiω(kξ).
Note that this function is real, Diω(kξ,kξ2) = D−iω(kξ,kξ2). From the boundary condition on the
plate ξ = ξ1we find that the possible values for ω are roots to the equation
(2.9)
Diω(kξ1,kξ2) = 0.(2.10)
This equation has an infinite set of solutions. We will denote them by ω = ωDn, ωDn > 0,
n = 1,2,..., and will assume that they are arranged in the ascending order ωDn < ωDn+1.
The coefficient C in formula (2.7) is determined from the standard Klein-Gordon orthonormality
condition for the eigenfunctions which for metric (2.5) takes the form
(ϕα,ϕα′) = −i
?
dx
?ξ2
ξ1
dξ
ξϕα
↔
∂τϕ∗
α′ = δαα′.(2.11)
It can be easily seen that for any two solutions to equation (2.8), φ(m)
integration formula takes place
ω (ξ), m = 1,2 the following
?ξ2
ξ1
dξ
ξφ(1)
ω(ξ)φ(2)
v(ξ) =
ξ
ω2− υ2
?
φ(1)
ω(ξ)dφ(2)
υ(ξ)
dξ
− φ(2)
ν(ξ)dφ(1)
ω(ξ)
dξ
?ξ2
ξ1
. (2.12)
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Taking into account boundary condition (2.6) from Eq. (2.11) for the normalization coefficient
one finds
C2
(2π)d−1
Iiω(kξ2)∂Diω(kξ1,kξ2)
Now substituting the eigenfunctions
D=
1
Iiω(kξ1)
∂ω
|ω=ωDn. (2.13)
ϕD
α(x) = CDDiωDn(kξ,kξ2)exp[i(kx − ωDnτ)](2.14)
into Eq. (2.3) and integrating over the directions of k for the VEV’s of the EMT we obtain
diagonal form (no summation over i)
?0D|Tk
i|0D? = δk
iπAd
?∞
0
dkkd
∞
?
n=1
Iiω(kξ1)
Iiω(kξ2)∂Diω(kξ1,kξ2)
∂ω
f(i)[Diω(kξ,kξ2)] |ω=ωDn,(2.15)
where |0D? is the amplitude for the Dirichlet vacuum between the plates, and
Ad=
1
2d−2π(d+1)/2Γ(d−1
2).(2.16)
In formula (2.15) for a given function G(z) we use the notations
f(0)[G(z)] =
?1
2− 2ζ
??????
dG(z)
dz
?????
dz|G(z)|2+1
?
2
+ζ
z
d
dz|G(z)|2+
?1
2− 2ζ +ω2
1 −ω2
z2
z2
?1
2+ 2ζ
??
|G(z)|2, (2.17)
f(1)[G(z)] = −1
2
?????
dG(z)
dz
?????
2
−ζ
z
d
2
??
|G(z)|2,(2.18)
f(i)[G(z)] = −|G(z)|2
d − 1
−
?
2ζ −1
2

?????
dG(z)
dz
?????
2
+
?
1 −ω2
z2
?
|G(z)|2

;i = 2,...,d,(2.19)
where G(z) = Diω(z,kξ2), and the indices 0,1 correspond to the coordinates τ, ξ respectively. It
can be easily seen that for a conformally coupled scalar the EMT (2.15) is traceless.
For the further evolution of VEV’s (2.15) we will apply to the sum over n summation formula
(A.5) derived in Appendix A by making use of the generalized Abel-Plana formula [17]. This
yields
?0D|Tk
i|0D? = Adδk
i
?∞
0
dkkd
?∞
0
dω
?sinhπω
π
f(i)[˜Diω(kξ,kξ2)] −Iω(kξ1)
Iω(kξ2)
F(i)[Dω(kξ,kξ2)]
Dω(kξ1,kξ2)
?
,
(2.20)
where we have introduced the notation
˜Diω(kξ,kξ2) = Kiω(kξ) −Kiω(kξ2)
Iiω(kξ2)Iiω(kξ), (2.21)
and the functions F(i)[G(z)], i = 0,1,...,d are obtained from the functions f(i)[G(z)] (see Eqs.
(2.17)–(2.19)) replacing ω → iω:
F(i)[G(z)] = f(i)[G(z),ω → iω].
The vacuum energy density, ε, effective pressures in perpendicular, p, and parallel, p⊥, to the
plates directions are determined by relations (no summation over i)
(2.22)
ε = ?0D|T0
0|0D?,p = −?0D|T1
1|0D?,p⊥= −?0D|Ti
i|0D?,i = 2,...,d.(2.23)
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It can be easily checked from Eqs. (2.20), (4.6) and (2.17)–(2.19) that they satisfy the standard
continuity equation for the EMT, which for the geometry under consideration takes the form
d(ξp)
dξ
= −ε.(2.24)
For a conformally coupled scalar we have an additional zero–trace relation ε −p −(d− 1)p⊥= 0.
Let us consider the limit ξ2→ ∞ of general formula (2.20) for fixed ξ. It can be easily seen that
in this limit the VEV’s take the form
lim
ξ2→∞?0D|Tk
i|0D? = ?0R|Tk
i|0R? + ?Tk
i?(1b)
D(ξ1,ξ), ξ > ξ1, (2.25)
where
?0R|Tk
i|0R? =Adδk
i
π
?∞
0
dkkd
?∞
0
dω sinhπωf(i)[Kiω(kξ)] (2.26)
are the corresponding VEV’s for the Fulling–Rindler vacuum without boundaries, and the term
?Tk
i?(1b)
D(ξ1,ξ) = −Adδk
i
?∞
0
dkkd
?∞
0
dωIω(kξ1)
Kω(kξ1)F(i)[Kω(kξ)] (2.27)
is induced in the region ξ > ξ1 by the presence of a single plane boundary located at ξ = ξ1.
Expressions (2.27) are finite for all values ξ > ξ1and all divergences are contained in the purely
Fulling-Rindler part (2.26). These divergences can be regularized subtracting the corresponding
VEV’s for the Minkowskian vacuum. The subtracted VEV’s
?Tk
i?(R)
sub= ?0R|Tk
i|0R? − ?0M|Tk
i|0M?(2.28)
are investigated in a large number of papers (see, for instance, [15, 16, 22, 23, 24, 25, 26, 27,
28, 29, 30, 31, 32] and references therein). The most general case of a massive scalar field in an
arbitrary number of spacetime dimensions has been considered in Ref. [28] for conformally and
minimally coupled cases and in Ref. [16] for general values of the curvature coupling parameter
(for the corresponding Green function see [22]). The formulae relevant to this paper are given in
[16]. For a massless scalar VEV’s (2.28) can be presented in the form
?Tk
i?(R)
sub= −
δk
iξ−d−1
2d−1πd/2Γ(d/2)
?∞
0
ωdg(i)(ω)dω
e2πω+ (−1)d
(2.29)
(the expressions for the functions g(i)(ω) are given in Ref. [16]) correspond to the absence from
the vacuum of thermal distribution with standard temperature T = (2πξ)−1. As we see from Eq.
(2.29), in general, the corresponding spectrum has non-Planckian form: the density of states factor
is not proportional to ωd−1dω. The spectrum takes the Planckian form for conformally coupled
scalars in d = 1,2,3 with g(0)(ω) = −dg(i)(ω) = 1, i = 1,2,...d. It is interesting to note that for
even values of spatial dimension the distribution is Fermi-Dirac type (see also [33, 34]). For the
massive scalar the energy spectrum is not strictly thermal and the corresponding quantities do
not coincide with ones for the Minkowski thermal bath.
The boundary induced quantities (2.27) are investigated in Ref. [15] for a conformally coupled
d = 3 massless Dirichlet scalar and in Ref. [16] for a massive scalar with general curvature coupling
and Robin boundary condition in an arbitrary number of dimensions. The single boundary part
(2.27) diverges at the plate surface ξ = ξ1with leading terms proportional to (ξ − ξ1)−d−1for
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i = 0,2,...,d and to (ξ−ξ1)−dfor i = 1 (see below). These leading terms vanish for a conformally
coupled scalar, and for i = 0,2,...,d coincide with the corresponding quantities for a plane
boundary in the Minkowski vacuum [16].
Now we turn to the limit ξ1→ 0 in formula (2.20), when the left plate coincides with the right
Rindler horizon. In this limit in the second term on the right of formula (2.20) the subintegrand
behaves as ξ2ω
1
and tends to zero. As a result one obtains
lim
ξ1→0?0D|Tk
i|0D? =Adδk
i
π
?∞
0
dkkd
?∞
0
dω sinhπω f(i)[˜Diω(kξ,kξ2)].(2.30)
These quantities coincide with the corresponding ones induced in the region ξ < ξ2by a single
plate at ξ = ξ2. They are investigated in Ref. [16], where it has been shown that the VEV’s (2.30)
can be presented in the form similar to Eq. (2.25):
lim
ξ1→0?0D|Tk
i|0D? = ?0R|Tk
i|0R? + ?Tk
i?(1b)
D(ξ2,ξ),ξ < ξ2,(2.31)
where the expressions for the boundary part ?Tk
formulae (2.27) by replacing (see Ref. [16])
i?(1b)
D(ξ2,ξ) in the region ξ < ξ2are obtained from
Iω→ Kω,Kω→ Iω,ξ1→ ξ2,ξ2→ ξ1.(2.32)
By using Eqs. (2.20),(2.30),(2.31) the parts in the VEV’s induced by the existence of boundaries,
?Tk
i?(b)
D= ?0D|Tk
i|0D? − ?0R|Tk
i|0R?,(2.33)
can be written as
?Ti?(b)
D(ξ1,ξ2,ξ) = ?Tk
i?(1b)
D (ξ2,ξ) − Adδk
i
?∞
0
dk kd
?∞
0
dωIω(kξ1)
Iω(kξ2)
F(i)[Dω(kξ,kξ2)]
Dω(kξ1,kξ2)
. (2.34)
In Appendix C we show that the VEV’s (2.20) can be also presented in the form (C.4).
Substituting Eq. (C.11) into this formula, the boundary VEV’s can be also written in the form
?Tk
i?(b)
D(ξ1,ξ2,ξ) = ?Tk
i?(1b)
D(ξ1,ξ) − Adδk
i
?∞
0
dkkd
?∞
0
dωKω(kξ2)
Kω(kξ1)
F(i)[Dω(kξ,kξ1)]
Dω(kξ1,kξ2)
. (2.35)
This expression is obtained from Eq. (2.34) by replacements (2.32). The case d = 1 needs a
separate consideration and is investigated in Appendix B. It can be seen that the corresponding
formulae for the VEV’s are also obtained from the formulae given above in this section replacing
Ad
?∞
0
dkkd−2→1
π,
k → 0. (2.36)
Now let us present the VEV’s (2.20) in the form
?0|Tk
where
i|0?D= ?0R|Tk
i|0R?+?Tk
i?(1b)
D(ξ1,ξ)+?Tk
i?(1b)
D(ξ2,ξ)+∆?Tk
i?D(ξ1,ξ2,ξ),ξ1< ξ < ξ2, (2.37)
∆?Tk
i?D= −Adδk
i
?∞
0
dkkd
?∞
0
dωIω(kξ1)
?
F(i)[Dω(kξ,kξ2)]
Iω(kξ2)Dω(kξ1,kξ2)−F(i)[Kω(kξ)]
Kω(kξ1)
?
(2.38)
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is the ’interference’ term. The surface divergences are contained in the single boundary parts and
this term is finite for all values ξ1≤ ξ ≤ ξ2. An equivalent formula for ∆?Tk
Eq. (2.38) by replacements (2.32). In the limit ξ1→ ξ2expressions (2.38) are divergent and for
small values of ξ2/ξ1−1 the main contribution comes from the large values of ω. Introducing a new
integration variable x = k/ω and replacing Bessel modified functions by their uniform asymptotic
expansions for large values of the order (see Ref. [35]) at the leading order over 1/(ξ2− ξ1) one
receives (no summation over i)
i?Dis obtained from
?Ti
i?(1b)
D(ξj,ξ) ∼
d(ζc− ζ)Γ
2dπ(d+1)/2|ξ − ξj|d+1,
0?(1b)
?d+1
2
?
i = 0,2,...,d, (2.39)
?T1
1?(1b)
D(ξj,ξ) ∼ ?T0
D(ξj,ξ)ξj− ξ
dξj
,j = 1,2(2.40)
for the single boundary terms, and
∆?T0
0?D ∼ −1
d∆?T1
?∞
dζR(d + 1)Γ
(4π)(d+1)/2(ξ2− ξ1)d+1,
1?D+(ζ − ζc)(ξ2− ξ1)−d−1
22d−1πd/2Γ(d/2)
dttd
et− 1
?d+1
×
?
(2.41)
×
0
?
exp
?
tξ1− ξ
ξ2− ξ1
?
?
+ exp tξ − ξ2
ξ2− ξ1
??
∆?T1
1?D ∼
2
∆?Ti
i?D∼ ∆?T0
0?D,i = 2,3,...,(2.42)
for the ’interference’ terms. Here ζR(s) is the Riemann zeta–function. Expressions (2.39), (2.41),
(2.42) coincide with the corresponding formulae for two parallel plates geometry in d + 1 – di-
mensional Minkowski spacetime with separation ξ2−ξ1(see Ref. [36] for the conformally coupled
case and Ref. [18] for the general case of the curvature coupling parameter ζ). Note that in the
limit under consideration the ’interference’ term (2.42) for the vacuum perpendicular pressure
dominates the single boundary induced terms, given by Eq. (2.40).
3Interaction forces between the plates
Now we turn to the interaction forces between the plates. The vacuum force acting per unit
surface of the plate at ξ = ξiis determined by the1
The corresponding effective pressures can be presented as a sum of two terms:
1–component of the vacuum EMT at this point.
p(i)
D= p(i)
D1+ p(i)
D(int),i = 1,2.(3.1)
The first term on the right is the pressure for a single plate at ξ = ξiwhen the second plate is
absent. This term is divergent due to the well known surface divergences in the subtracted VEV’s.
The second term on the right of Eq. (3.1),
p(i)
D(int)= −?T1
1?(1b)
D(ξj,ξi) − ∆?T1
1?D(ξ1,ξ2,ξi),i,j = 1,2,j ?= i(3.2)
is the pressure induced by the presence of the second plate, and can be termed as an interaction
force. For the plate at ξ = ξ2the interaction term is due to the second summand on the right of
Eq. (2.20). Substituting into this term ξ = ξ2and using the Wronskian relation for the modified
Bessel functions one has
p(2)
D(int)(ξ1,ξ2) = −Ad
2ξ2
2
?∞
0
dkkd−2
?∞
0
dω
Iω(kξ1)
Iω(kξ2)Dω(kξ1,kξ2). (3.3)
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By a similar way from Eq. (2.35) for the interaction term on the plate at ξ = ξ1we obtain
p(1)
D(int)(ξ1,ξ2) = −Ad
2ξ2
1
?∞
0
dkkd−2
?∞
0
dω
Kω(kξ2)
Kω(kξ1)Dω(kξ1,kξ2).(3.4)
As the function Dω(kξ,kξ2) is positive for ξ1< ξ2, interaction forces per unit surface (3.3) and
(3.4) are always attractive. They are finite for all ξ1< ξ2, and do not depend on the curvature
coupling parameter ζ. In the limit ξ1→ ξ2these forces diverge due the contribution from the
large values ω and in this limit by introducing a new integration variable we can replace the Bessel
modified functions by their uniform asymptotic expansions for large values of the order. At the
leading order for the perpendicular vacuum pressures we obtain formula (2.42) which corresponds
to the standard Casimir attraction force for two parallel plates in Minkowski vacuum.
From expressions (3.3) and (3.4) it follows that
p(2)
D(int)(ξ1,ξ2) > p(1)
D(int)(ξ1,ξ2).(3.5)
This can be proved by using that the function z2Iω(z)Kω(z) is monotonic increasing. The latter
directly follows from the relations
?
1 +(ω + 1)2
z2
−1
z<I′
ω(z)
Iω(z)<
?
1 +ω2
z2
(3.6)
?
1 +(ω + 1)2
z2
+1
z> −K′
ω(z)
Kω(z)>
?
1 +ω2
z2. (3.7)
The proof for the right inequalities in Eqs. (3.6),(3.7) is presented in Ref. [15]. The left inequalities
are obtained from the recurrence relations for the Bessel modified functions. For instance, in the
case of the function Iω(z) one has:
I′
Iω(z)=Iω+1(z)
ω(z)
Iω(z)
+ω
z=
?I′
ω+1(z)
Iω+1(z)+ω + 1
−1
+ω
z
?−1
+ω
z>
>


?
1 +(ω + 1)2
z2
+ω + 1
z


z=
?
1 +(ω + 1)2
z2
−1
z,
(3.8)
where we have used the right inequality in Eq. (3.6). The left inequality in Eq. (3.7) can be
proved in a similar way.
To see the monotonicity properties of functions (3.3) and (3.4) note that
ξ1
∂p(1)
D(int)
∂ξ2
= −ξ2
∂p(2)
D(int)
∂ξ1
=
Ad
2ξ1ξ2
?∞
0
dkkD−2
?∞
0
dω
D2
ω(kξ1,kξ2).(3.9)
It follows from here that for a fixed value of ξ1 (ξ2) the quantity p(1)
increasing (decreasing) function on ξ2(ξ1). By taking into account that both this quantities are
negative we conclude that the modulus of the interaction force on the plate at ξ1(ξ2) is monotonic
decreasing (increasing) function on ξ2(ξ1) for a fixed value of ξ1(ξ2). From formula (3.3) it follows
that
∂p(i)
D(int)
∂ξi
D(int)(p(2)
D(int)) is monotonic
ξi
= −(d + 1)p(i)
D(int)− ξj
∂p(i)
D(int)
∂ξj
,i,j = 1,2,i ?= j.(3.10)
9

Page 10

For i = 2 the both terms on the right are positive and hence, the same is the case for the function
on the left. Therefore for a fixed ξ1the function p(2)
modulus of the corresponding interaction force is monotonic decreasing function on ξ2. In the
case i = 1 the terms on the right in this formula have different signs. For a fixed value of ξ2the
function p(1)
near the second plane, ξ1→ ξ2. It follows from here the modulus of the corresponding interaction
force acting on the plate at ξ1has minimum for some intermediate value.
In the limit ξ2≫ ξ1, introducing in Eq. (3.3) a new integration variable x = kξ2, and making
use the formula
Iω(y) =
2 Γ(ω)
and the standard relation between the functions Kωand I±ωone finds
D(int)is monotonic increasing on ξ2and the
D(int)is monotonic increasing on ξ1near the horizon, ξ1→ 0, and monotonic decreasing
?y
?ω
1
?
1 + O(y2)
?
,y = xξ1/ξ2,(3.11)
p(2)
D(int)≈ −
π2Ad
ln2(2ξ2/ξ1)
48ξd+1
2
?∞
0
dxxd−2
I2
0(x)
?
1 + O
?
lnx
ln(2ξ2/ξ1)
??
.(3.12)
The similar calculation for Eq. (3.4) yields
p(1)
D(int)≈ −
π2Ad
ξ2
24ξd−1
21ln3(2ξ2/ξ1)
?∞
0
dxxd−2K0(x)
I0(x)
?
1 + O
?
lnx
ln(2ξ2/ξ1)
??
. (3.13)
We have carried out numerical evaluations for the interaction forces by making use of formulae
(3.3) and (3.4). In Fig. 2 the corresponding results are presented for ξd+1
case d = 3 as functions on ξ1/ξ2.
2
p(i)
D(int), i = 1,2 in the
0.1 0.2 0.30.40.5 0.6
-1.75
-1.5
-1.25
-1
-0.75
-0.5
-0.25
0
a
b
Figure 2: The d = 3 vacuum effective pressures determining the interaction forces between Dirich-
let parallel plates, multiplied by ξ4
ratio ξ1/ξ2.
2, ξ4
2p(1)
D(int)(curve a) and ξ4
2p(2)
D(int)(curve b) as functions of the
4 VEV’s and the interaction forces for the Neumann scalar
In this section we will consider VEV’s for the EMT in the case of a scalar field satisfying the
Neumann boundary condition on the plates ξ = ξ1,ξ2:
∂ϕ
∂ξ|ξ=ξ1=∂ϕ
∂ξ|ξ=ξ2= 0. (4.1)
10

Page 11

The corresponding scheme is similar to that given above for the Dirichlet case. The eigenfunctions
to the field equation (2.1) have form (2.7) with
φ(ξ) = Niω(kξ,kξ2) = I′
iω(kξ2)Kiω(kξ) − K′
iω(kξ2)Iiω(kξ).(4.2)
As in the Dirichlet case this function is real. From the boundary condition on the plate ξ = ξ1we
obtain that the corresponding eigenfrequencies are solutions to the equation
N′
iω(kξ1,kξ2) = I′
iω(kξ2)K′
iω(kξ1) − K′
iω(kξ2)I′
iω(kξ1) = 0.(4.3)
We will denote them by ω = ωNn, n = 1,2,... , arranged in the ascending order ωNn< ωNn+1. The
normalization coefficient C can be found from orthonormality condition (2.11) using integration
formula (2.12):
1
(2π)d−1
I′
C2
N=
I′
iω(kξ1)
iω(kξ2)∂N′
iω(kξ1,kξ2)
∂ω
|ω=ωNn.(4.4)
Substituting the eigenfunctions into the mode sum formula (2.3) one obtains
?0N|Tk
i|0N? = πAdδk
i
?∞
0
dkkd
∞
?
n=1
I′
iω(kξ1)
I′
iω(kξ2)∂N′
iω(kξ1,kξ2)
∂ω
f(i)[Niω(kξ,kξ2)]|ω=ωNn,(4.5)
where |0N? is the amplitude for the Neumann vacuum state between the plates, and the functions
f(i)[G(z)] are defined in accordance with Eqs. (2.17)–(2.19). To sum the series over the eigen-
frequencies ωNnwe will apply the summation formula derived in Appendix A, Eq. (A.11). This
yields
?0N|Tk
i|0N? = Adδk
i
?∞
0
dkkd
?∞
0
dω
?sinhπω
π
f(i)[˜ Niω(kξ,kξ2)] −I′
ω(kξ1)
I′
ω(kξ2)
F(i)[Nω(kξ,kξ2)]
N′
iω(kξ1,kξ2)
?
(4.6)
,
with functions F(i)[G(z)] defined as in Eq. (2.22), and we use the notation
˜ Nω(kξ,kξ2) = Kiω(kξ) −K′
iω(kξ2)
I′
iω(kξ2)Iiω(kξ). (4.7)
To identify the terms in Eq. (4.6) let us consider limiting cases. In the limit ξ2→ ∞, from Eq.
(4.6) one obtains
lim
ξ2→∞?0N|Tk
i|0N? = ?0R|Tk
i|0R? + ?Tk
i?(1b)
N(ξ1,ξ), (4.8)
where the term
?Tk
i?(1b)
N(ξ1,ξ) = −Adδk
i
?∞
0
dkkd
?∞
0
dωI′
ω(kξ1)
K′ω(kξ1)F(i)[Kω(kξ)] (4.9)
is induced in the region ξ > ξ1by a single Neumann boundary located at ξ = ξ1. This quantities
for d = 3 case are investigated in Ref. [15]. In the limit ξ1 → 0 the left plate coincides with
the Rindler horizon and the second term in the figure braces in Eq. (4.6) vanishes. In this case
the VEV’s coincide with the corresponding expressions for a single plate at ξ = ξ2induced in the
region ξ < ξ2. They are investigated in Ref. [16], where it has been shown that the VEV’s (2.30)
can be presented in the form similar to Eq. (4.8):
lim
ξ1→0?0N|Tk
i|0N? = ?0R|Tk
i|0R? + ?Tk
i?(1b)
N(ξ2,ξ),ξ < ξ2,(4.10)
11

Page 12

where the expressions for the boundary part ?T
from formulae (4.9) by replacements (2.32).
By using Eqs. (4.6),(4.10) the parts in the VEV’s induced by the existence of boundaries,
k
Ni?(1b)
N(ξ2,ξ) in the region ξ < ξ2are obtained
?Tk
i?(b)
N= ?0N|Tk
i|0N? − ?0R|Tk
i|0R?,(4.11)
can be presented as
?Tk
i?(b)
N(ξ1,ξ2,ξ) = ?Tk
i?(1b)
N (ξ2,ξ) − Adδk
i
?∞
0
dkkd
?∞
0
dωI′
ω(kξ1)
I′ω(kξ2)
F(i)[Nω(kξ,kξ2)]
N′
ω(kξ1,kξ2)
.(4.12)
Similar to the Dirichlet case, the Neumann boundary VEV’s can be also written in the form
?Tk
i?(b)
N(ξ1,ξ2,ξ) = ?Tk
i?(1b)
N(ξ1,ξ) − Adδk
i
?∞
0
dk kd
?∞
0
dωK′
ω(kξ2)
K′ω(kξ1)
F(i)[Nω(kξ,kξ1)]
N′
ω(kξ1,kξ2)
, (4.13)
with ?Tk
As we see, this expression is obtained from (4.12) by replacements (2.32).
Now let us present the VEV’s (4.6) in the form
i?(1b)
N(ξ1,ξ) being the VEV’s induced by a single Neumann boundary located at ξ = ξ1.
?0N|Tk
where
i|0N? = ?0R|Tk
i|0R?+?Tk
i?(1b)
N(ξ1,ξ)+?Tk
i?(1b)
N(ξ2,ξ)+∆?Tk
i?N(ξ1,ξ2,ξ),ξ1< ξ < ξ2, (4.14)
∆?Tk
i?N= −Adδk
i
?∞
0
dkkd
?∞
0
dωI′
ω(kξ1)
?
F(i)[Nω(kξ,kξ2)]
I′ω(kξ2)N′
ω(kξ1,kξ2)−F(i)[Kω(kξ)]
K′ω(kξ1)
?
(4.15)
is the ’interference’ term. An equivalent formula for ∆?Tk
replacements (2.32).
’Interference’ term (4.15) is finite for all ξ1≤ ξ ≤ ξ2, ξ1< ξ2, and diverges in the limit ξ1→ ξ2.
In this limit the main contribution into the ω–integral comes from the large values ω. Introducing
a new integration variable x = kξ1/ω and using the uniform asymptotic expansions for the Bessel
modified functions in the leading order one obtains that the quantities ∆?Tk
VEV’s for two parallel plates in d + 1–dimensional Minkowski spacetime with separation ξ2− ξ1
[36, 18]. The corresponding expressions are given by formulae (2.41),(2.42) with the opposite sign
of the integral term on the right of formula (2.41).
Now we turn to the Neumann vacuum effective pressures determining the forces acting on
the plate due to the presence of the second plate (interaction forces). This force acting per unit
surface of the plate ξ = ξ2, p(2)
of formula (4.12) at ξ = ξ2. The nonzero contribution comes from the last term on the right
of Eq.(2.18) (with replacement (2.22)). Using the standard Wronskian relation for the Bessel
modified functions one obtains
i?N is obtained from Eq. (4.15) by
i?Ncoincide with the
N(int)is defined by the1
1–component of the second term on the right
p(2)
N(int)(ξ1,ξ2) =Ad
2ξ2
2
?∞
0
dkkd−2
?∞
0
dωI′
ω(kξ1)(1 + ω2/k2ξ2
I′ω(kξ2)N′
2)
ω(kξ1,kξ2).(4.16)
By a similar way for the interaction force per unit surface of the first plate from the second term
on the right of Eq. (4.13) at ξ = ξ1we receive
p(1)
N(int)(ξ1,ξ2) =Ad
2ξ2
1
?∞
0
dkkd−2
?∞
0
dωK′
ω(kξ2)(1 + ω2/k2ξ2
K′ω(kξ1)N′
1)
ω(kξ1,kξ2). (4.17)
12

Page 13

Note that pressures (4.16),(4.17) are independent on the curvature coupling parameter. It can be
seen that the function I′
ξ1< ξ2. In combination with Eqs. (4.16), (4.17) it follows from here that p(i)
hence, as in the Dirichlet case, the Neumann interaction forces are always attractive. By using
that the function z4I′
inequalities (3.6),(3.7) ) we see that
ω(z)/K′
ω(z) is monotonic decreasing, and as a result N′
ω(kξ1,kξ2) < 0 for
N(int)< 0, i = 1,2, and
ω(z)K′
ω(z)/(z2+ ω2) is monotonic decreasing (this can be proved by using
p(2)
N(int)(ξ1,ξ2) > p(1)
N(int)(ξ1,ξ2).(4.18)
In the limit ξ1→ ξ2replacing the Bessel modified functions by their uniform asymptotic expansions
we can see that to the leading order from Eqs. (4.16),(4.17) the standard Casimir interaction force
is obtained for two parallel plates with separation ξ2− ξ1in the d + 1–dimensional Minkowski
spacetime.
From formulae (4.16), (4.17) one has
ξ1
∂p(1)
N(int)
∂ξ2
= −ξ2
∂p(2)
N(int)
∂ξ1
=
Ad
2ξ1ξ2
?∞
0
dkkD−2
?∞
0
dω(1 + ω2/k2ξ2
1)(1 + ω2/k2ξ2
ω(kξ1,kξ2)
2)
N′2
.(4.19)
As seen from here for a fixed value of ξ2(ξ1) the modulus of the interaction force acting on the
plate at ξ = ξ2(ξ1) is a monotonic increasing (decreasing) function on ξ1(ξ2). For the other partial
derivatives, similar to the Dirichlet case, one has the relation (3.10) with replacement D → N.
In particular, we can see that ∂p(2)
the interaction forces per unit surface given by Eqs. (4.16), (4.17) are plotted in Fig. 3 as
functions of ξ1/ξ2for the case d = 3. As seen from Fig. 2 and Fig. 3 the Dirichlet and Neumann
N(int)/∂ξ2 > 0. The Neumann effective pressures determining
0.10.20.30.4 0.50.6
-1.75
-1.5
-1.25
-1
-0.75
-0.5
-0.25
0
a
b
Figure 3: The d = 3 vacuum effective pressures determining the interaction forces per unit surface
between Neumann parallel plates, multiplied by ξ4
functions of the ratio ξ1/ξ2.
2, ξ4
2p(1)
D(int)(curve a) and ξ4
2p(2)
D(int)(curve b) as
vacuum interaction forces are numerically close to each other. This is a consequence of that the
subintegrands in formulae (3.3) and (4.16) and in formulae (3.4) and (4.17) are numerically close.
This can be also seen analytically by using relations (3.6),(3.7).
13

Page 14

5Electromagnetic field
We now turn to the case of the electromagnetic field in the region ξ1< ξ < ξ2. We will assume
that the mirrors are perfect conductors with the standard boundary conditions of vanishing of
the normal component of the magnetic field and the tangential components of the electric field,
evaluated at the local inertial frame in which the conductors are instantaneously at rest. By
considerations similar to those given in Ref. [15] for d = 3, it can be seen that the corresponding
eigenfunctions for the vector potential Aµmay be resolved into one transverse magnetic (TM)
and d − 2 transverse electric (TE) (with respect to ξ-direction) modes Aµ
α = (k,ω):
Aµ
1α
= (−ξ∂/∂ξ,−iω/ξ,0,...0)ϕ0α,
Aµ
σα
= ǫµ
σ = 0,2,...,d − 2,
where the polarization vectors ǫµ
ǫ0
ǫσµǫµ
σα, σ = 0,1,...,d − 2,
σ = 1,
TE modes,
TM mode,(5.1)
(5.2)
σϕσα,
σobey the following relations
σ= ǫ1
σ= 0,
σ′ = −k2δσσ′,ǫµ
σkµ= 0. (5.3)
From the perfect conductor boundary conditions one has the following conditions for the scalar
fields ϕσα:
ϕσα|ξ=ξ1= ϕσα|ξ=ξ2= 0,
As a result the TE/TM modes correspond to the Dirichlet/Neumann scalars. In the correspond-
ing expressions for the eigenfunctions Aµ
orthonormality relation
?
On the base of this normalization condition for the separate scalar modes one has
ϕσα=2π1/2
k
where Z = D for σ = 0,2,...,d − 2 and Z = N for σ = 1, and the coefficients CDand CN are
defined in accordance with Eqs. (2.13),(4.4). Substituting the eigenfunctions (5.1), (5.2) into the
mode sum formula
d−2
?
with the standard bilinear form for the electromagnetic field EMT one finds
π(d−1)/2
?d−1
where β0and β1are the numbers of the independent polarization states for TE and TM modes
respectively. In Eq. (5.8) for a given function G(z) the following notations are introduced
σ = 0,2,...,d − 2,
∂ϕ1α
∂ξ
|ξ=ξ1=∂ϕ1α
∂ξ
|ξ=ξ2= 0.(5.4)
σαthe normalization coefficient is determined from the
dx
?ξ2
ξ1
dξ
ξAµ
σαA∗
σ′α′µ= −2π
ωδαα′δσσ′. (5.5)
CZZiωZn(kξ,kξ2)exp[i(kx − ωZnτ)],(5.6)
?0|Tk
i|0? =
σ=0
?
dk
?
ωZn
Tk
i{Aσαµ,A∗
σαµ},(5.7)
?0|Tk
i|0? = δk
i
Γ
2
?
?∞
0
dkkd?
σ=0,1
βσ
∞
?
n=1
C2
Zf(i)
em[ZiωZn(kξ,kξ2)],β0= d − 2,β1= 1,(5.8)
f(0)
em[G(z)] =
?????
dG(z)
dz
?????
2
+
?
1 +ω2
z2
?
|G(z)|2,
f(1)
em[G(z)] = −
?????
dG(z)
dz
?????
2
+
?
1 −ω2
z2
?


|G(z)|2,(5.9)
f(i)
em[G(z)] =
d − 5
d − 1|G(z)|2+d − 3
d − 1
?????
14
dG(z)
dz
?????
2
−ω2
z2|G(z)|2

,i = 2,3,...,d.

Page 15

By making use of the summation formulae derived in the Appendix A the VEV’s are presented
in the form
?0|Tk
i|0? = δk
i
Ad
2
?∞
0
dkkd
?∞
0
dω
?
σ=0,1
βσ
?sinhπω
π
f(i)
em[˜Ziω(kξ,kξ2)] −I(σ)
ω(kξ1)F(i)
I(σ)
em[Zω(kξ,kξ2)]
ω (kξ2)Z(σ)
ω (kξ1,kξ2)
?
,
(5.10)
where I(0)
are obtained from Eqs. (5.9) replacing ω → iω:
ω
= Iω, I(1)
ω
= I′
ω, and the same notations for the functions Kω, Zω. The functions F(i)
em
F(i)
em[G(z)] = f(i)
em[G(z),ω → iω]. (5.11)
It can be easily checked that the components (5.10) obey the covariant conservation equation and
the corresponding EMT is traceless for d = 3. The first term in the figure braces of Eq. (5.10)
corresponds to the VEV induced by a single plate at ξ = ξ2in the region ξ < ξ2. For the case
d = 3 they are investigated in Ref. [16]. The generalization for an arbitrary d is straightforward
and these quantities are presented in the form
?0|Tk
i|0?(1b)(ξ2,ξ) = ?0R|Tk
i|0R? −1
2δk
iAd
?∞
0
dkkd
?∞
0
dω
?
σ=0,1
βσK(σ)
I(σ)
ω(kξ2)
ω (kξ2)
F(i)
em[Iω(kξ)],(5.12)
where ?0R|Tk
aries. By the way similar to that given in Ref. [16] for the case of a scalar field, it can be seen
that
i|0R? are the VEV’s for the Fulling–Rindler electromagnetic vacuum without bound-
?0R|Tk
i|0R? = ?0M|Tk
i|0M? −δk
i(d − 1)ξ−d−1
2d−1πd/2Γ(d/2)
?∞
0
ωdf(i)
e2πω+ (−1)d
0em(ω)dω
lm
?
l=1


?d − 1 − 2l
2ω
?2
+ 1

, (5.13)
where lm= d/2−1 for even d > 2 and lm= (d−1)/2 for odd d > 1, and the value for the product
over l is equal to 1 for d = 1,2,3. In Eq. (5.13) we have introduced notations
f(0)
0em(ω) = −df(1)
f(i)
0em(ω) = 1 +(d − 1)2
4ω2
, (5.14)
0em(ω) = f(1)
0em(ω) +(d − 1)(d − 3)
4ω2
,i = 2,...,d.
For physically most important case d = 3, formula (5.13) leads to the standard result derived by
Candelas and Deutsch in Ref. [15].
An alternative form for the vacuum EMT in the region between two plates is
?0|Tk
i|0? = ?0R|Tk
−
i|0R? + ?0|Tk
?∞
i|0?(1b)(ξ1,ξ) −
?∞
σ=0,1
1
2δk
iAd
0
dkkd
0
dω
?
βσK(σ)
ω(kξ2)F(i)
K(σ)
em[Zω(kξ,kξ1)]
ω (kξ1)Z(σ)
ω (kξ1,kξ2)
,(5.15)
where ?0|Tk
ξ > ξ1. The latter is obtained from (5.12) by replacements (2.32). For the interaction force p(i)
i = 1,2 per unit area of the plate at ξ = ξifrom Eqs. (5.10) and (5.15) one obtains
i|0?(1b)(ξ1,ξ) is the vacuum EMT induced by a single boundary at ξ = ξ1in the region
em(int),
p(1)
em(int)
= −Ad
2ξ2
1
?∞
?∞
0
dkkd−2
?∞
?∞
0
dω
?
σ=0,1
(−1)σβσK(σ)
ω(kξ2)
K(σ)
ω (kξ1)
(1 + ω2/k2ξ2
Z(σ)
(1 + ω2/k2ξ2
Z(σ)
1)σ
ω (kξ1,kξ2)
,(5.16)
p(2)
em(int)
= −Ad
2ξ2
2
0
dkkd−2
0
dω
?
σ=0,1
(−1)σβσI(σ)
ω(kξ1)
I(σ)
ω (kξ2)
2)σ
ω (kξ1,kξ2)
.(5.17)
15