R. M. Avagyan, A. A. Saharian and A. H. Yeranyan- The Casimir effect in the Fulling–Rindler vacuum

8/3/2019 R. M. Avagyan, A. A. Saharian and A. H. Yeranyan- The Casimir effect in the FullingRindler vacuum

1/23

a r X i v : h e p - t h / 0 2 0 7 0 7 3 v 2 3 S e p 2 0 0 2

The Casimir effect in the FullingRindler vacuum

R. M. Avagyan, A. A. Saharian , A. H. YeranyanDepartment of Physics, Yerevan State University,1 Alex Manoogian St., 375049 Yerevan, Armenia

February 1, 2008

Abstract

The vacuum expectation values of the energymomentum tensor are investigated formassless scalar elds satisfying Dicichlet or Neumann boundary conditions, and for the elec-tromagnetic eld with perfect conductor boundary conditions on two innite parallel platesmoving by uniform proper acceleration through the FullingRindler vacuum. The scalarcase is considered for general values of the curvature coupling parameter and in an arbitrarynumber of spacetime dimension. The modesummation method is used with combination of a variant of the generalized AbelPlana formula. This allows to extract manifestly the con-tributions to the expectation values due to a single boundary. The vacuum forces acting onthe boundaries are presented as a sum of the selfaction and interaction terms. The rst onecontains well known surface divergences and needs a further regularization. The interactionforces between the plates are always attractive for both scalar and electromagnetic cases. Anapplication to the Rindler wall is discussed.

PACS number(s): 03.70.+k, 11.10.Kk

1 Introduction

The imposition of boundary conditions on a quantum eld leads to the modication of the spec-trum for the zeropoint uctuations 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 theresulting vacuum forces depend on the nature of the quantum eld, the type of spacetime manifoldand its dimensionality, the boundary geometries and the specic boundary conditions imposed onthe eld. Since the original work by Casimir in 1948 [1] many theoretical and experimental workshave 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 approacheshave been used: mode summation method with combination of the zeta function regularizationtechnique, Green function formalism, multiple scattering expansions, heat-kernel series, etc. An

E-mail address: [email protected]

1
http://arxiv.org/abs/hep-th/0207073v2http://arxiv.org/abs/hep-th/0207073v2http://arxiv.org/abs/hep-th/0207073v2http://arxiv.org/abs/hep-th/0207073v2http://arxiv.org/abs/hep-th/0207073v2http://arxiv.org/abs/hep-th/0207073v2http://arxiv.org/abs/hep-th/0207073v2http://arxiv.org/abs/hep-th/0207073v2http://arxiv.org/abs/hep-th/0207073v2http://arxiv.org/abs/hep-th/0207073v2http://arxiv.org/abs/hep-th/0207073v2http://arxiv.org/abs/hep-th/0207073v2http://arxiv.org/abs/hep-th/0207073v2http://arxiv.org/abs/hep-th/0207073v2http://arxiv.org/abs/hep-th/0207073v2http://arxiv.org/abs/hep-th/0207073v2http://arxiv.org/abs/hep-th/0207073v2http://arxiv.org/abs/hep-th/0207073v2http://arxiv.org/abs/hep-th/0207073v2http://arxiv.org/abs/hep-th/0207073v2http://arxiv.org/abs/hep-th/0207073v2http://arxiv.org/abs/hep-th/0207073v2http://arxiv.org/abs/hep-th/0207073v2http://arxiv.org/abs/hep-th/0207073v2http://arxiv.org/abs/hep-th/0207073v2http://arxiv.org/abs/hep-th/0207073v2http://arxiv.org/abs/hep-th/0207073v2http://arxiv.org/abs/hep-th/0207073v2http://arxiv.org/abs/hep-th/0207073v2http://arxiv.org/abs/hep-th/0207073v2http://arxiv.org/abs/hep-th/0207073v2


2/23

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 islost when we work within the framework of quantum eld theory in a general curved spacetimeor in noninertial frames. In particular, the use of general coordinate transformation in quantumeld theory in at spacetime leads to an innite number of unitary inequivalent representationsof the commutation relations. Different inequivalent representations will in general give rise todifferent pictures with different physical implications, in particular to different vacuum states.For instance, the vacuum state for an uniformly accelerated observer, the FullingRindler vacuum[9, 10, 11, 12], turns out to be inequivalent to that for an inertial observer, the familiar Minkowskivacuum. Quantum eld theory in accelerated systems contains many of special features producedby a gravitational eld avoiding some of the difficulties entailed by renormalization in a curvedspacetime. In particular, near the canonical horizon in the gravitational eld, a static spacetimemay be regarded as a Rindlerlike spacetime. Note that, as it has been shown in Ref. [13], there isa class of solutions to the Einstein equations with a planesymmetric matter distribution for whichthe 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 thatthis space is conformally related to the de Sitter space and to the RobertsonWalker space withnegative spatial curvature. As a result the expectation values of the energymomentum tensorfor a conformally invariant eld and for corresponding conformally transformed boundaries onthe de Sitter and RobertsonWalker backgrounds can be derived from the corresponding Rindlercounterpart by the standard transformation (see, for instance, [14]).

In this paper we will consider the vacuum expectation values of the energymomentum tensorsfor a scalar and electromagnetic elds in the region between two parallel plates moving by constantproper acceleration through the FullingRindler vacuum. This problem for a single plate casewas considered by Candelas and Deutsch [15] and by one of us [16]. In Ref. [15] the casesof conformally coupled Dirichlet and Neumann massless scalar and electromagnetic elds areinvestigated 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 amassive scalar eld with general curvature coupling parameter and Robin boundary conditions inarbitrary number of spacetime dimensions, and for the electromagnetic eld. As in Ref. [16] (seealso [17, 18, 19, 20, 21]), our regularization scheme here is based on a variant of the generalizedAbelPlana formula derived in Appendix A. This allows to extract form the vacuum expectationvalues the single boundary parts and to present the interference parts in terms of stronglyconvergent integrals useful for numerical evaluations. We have organized the paper as follows. Inthe next section we evaluate the vacuum expectation values of the energymomentum tensor for theDirichlet 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 vacuumdensities and interaction forces for the electromagnetic eld are considered in section 5. Section6 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 eld in two spacetime dimensions separately.An alternate representation of the vacuum expectation values for the energymomentum tensoris obtained in Appendix C.

2


3/23

2 Vacuum energy-momentum tensor for a Dirichlet scalar

We consider a real massless scalar (x) eld with general curvature coupling parameter satisfyingthe eld equation

+ R = 0 , (2.1)

with R being the scalar curvature for a d + 1dimensional background spacetime, is thecovariant derivative operator associated with the metric g . For minimally and conformallycoupled scalars = 0 and = ( d 1)/ 4d, respectively. By using eld equation (2.1) it can beseen that the corresponding energymomentum tensor (EMT) can be presented in the form

T = + 14

g R 2, (2.2)where R is the Ricci tensor.

Let { (x), (x)}is a complete set of positive and negative frequency solutions to the eldequation (2.1), where denotes a set of quantum numbers. Expanding eld operator over theseeigenfunctions and using the commutation relations it can be easily seen that the vacuum expec-tation values (VEVs) of the EMT are presented in the form

0 | T | 0 = T { , }, (2.3)

where for a scalar eld the quadratic form T {f, g }directly follows from the classical EMT givenby Eq. (2.2).Our main interest in this paper will be the vacuum expectation values (VEVs) of the EMT

in the Rindler spacetime induced by two parallel plates moving with uniform proper accelera-tion when the quantum eld is prepared in the Fulling-Rindler vacuum. For this problem thebackground spacetime is at and in Eqs. (2.1),(2.2) we have R = 0, R = 0. As a result theeigenmodes are independent on the curvature coupling parameter and the EMT VEVs will de-pend on this parameter through the expression (2.2) only. In the following it will be convenientto introduce Rindler coordinates ( ,, x ) related to the Minkowski ones, ( t, x 1, x ) by

t = sinh , x1 = cosh , (2.4)

where x = ( x2, . . . , x d) denotes the set of coordinates parallel to the plates. In these coordinatesthe Minkowski line element takes the form

ds2 = 2d 2 d2 dx 2, (2.5)and a wordline dened by , x = const describes an observer with constant proper acceleration 1. Assuming that the plates are situated in the right Rindler wedge x1 > |t| we shall let thesurfaces = 1 and = 2, 2 > 1 represent the trajectories of these boundaries, which thereforehave proper accelerations 11 and 12 (see Fig. 1). First we will consider the case of a scalar eldsatisfying Dirichlet boundary condition on the surface of the plates:

|= 1 = |= 2 = 0 (2.6)To evaluate the VEVs of the EMT by Eq. (2.3) we need the form of the eigenfunctions (x). Forthe geometry under consideration the metric and boundary conditions are static and translational

3


4/23

Figure 1: The ( x1, t ) plane with the Rindler coordinates. The heavy lines = 1 and = 2represent the trajectories of the plates.

invariant in the hyperplane parallel to the plates. It follows from here that the corresponding partof the eigenfunctions has the standard plane wave structure:

= C()exp[i (kx )] , = ( k , ), k = ( k2, . . . , kd). (2.7)The equation for () is obtained from eld equation (2.1) on background of metric (2.5) and hasthe form

2 () + () + 2 k22 () = 0 , (2.8)where the prime denotes a differentiation with respect to the argument, and k = |k |. In theregion between the plates the corresponding linearly independent solutions to equation (2.8) arethe Bessel modied functions I i (k) and K i (k). The solution satisfying boundary condition(2.6) on the plate = 2 is in form

D i (k,k2) = I i (k2)K i (k) K i (k2)I i (k). (2.9)Note that this function is real, D i (k,k2) = D i (k,k2). From the boundary condition on theplate = 1 we nd that the possible values for are roots to the equation

D i (k1, k2) = 0 . (2.10)

This equation has an innite set of solutions. We will denote them by = Dn , Dn > 0,n = 1 , 2, . . ., and will assume that they are arranged in the ascending order

Dn<

Dn +1.

The coefficient C in formula (2.7) is determined from the standard Klein-Gordon orthonormalitycondition for the eigenfunctions which for metric (2.5) takes the form

( , ) = i dx 2

1

d

= . (2.11)

It can be easily seen that for any two solutions to equation (2.8), (m ) (), m = 1 , 2 the followingintegration formula takes place

2

1

d

(1) ()(2)v () =

2

2

(1) ()d(2) ()

d (2) ()

d(1) ()d

2

1. (2.12)

4


5/23

Taking into account boundary condition (2.6) from Eq. (2.11) for the normalization coefficientone nds

C 2D =1

(2)d 1I i (k1)

I i (k2) D i (k 1 ,k 2 ) |= Dn. (2.13)

Now substituting the eigenfunctionsD (x) = C D D iDn (k,k2)exp[i (kx Dn )] (2.14)

into Eq. (2.3) and integrating over the directions of k for the VEVs of the EMT we obtaindiagonal form (no summation over i)

0D |T ki |0D = ki Ad

0dkkd

n =1

I i (k1)I i (k2) D i (k 1 ,k 2 )

f (i)[D i (k,k2)] |= Dn , (2.15)where |0D is the amplitude for the Dirichlet vacuum between the plates, and

Ad =1

2d 2 (d+1) / 2( d 12 ) . (2.16)

In formula (2.15) for a given function G(z) we use the notations

f (0) [G(z)] =12 2

dG(z)dz

2

+ z

ddz |G(z)|

2 +12 2 +

2

z212

+ 2 |G(z)|2, (2.17)

f (1) [G(z)] = 12

dG(z)dz

2

z

ddz |G(z)|

2 +12

1 2

z2 |G(z)|2, (2.18)

f (i)[G(z)] = |G(z)|2d

1 2

12

dG(z)dz

2

+ 1 2

z2 |G(z)|2 ; i = 2 , . . . , d , (2.19)

where G(z) = D i (z,k2), and the indices 0,1 correspond to the coordinates , respectively. Itcan be easily seen that for a conformally coupled scalar the EMT (2.15) is traceless.

For the further evolution of VEVs (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]. Thisyields

0D |T ki |0D = Adki

0dkkd

0d

sinh

f (i) [D i (k,k2)] I (k1)I (k2)

F (i) [D(k,k2)]D(k1, k2)

,

(2.20)where we have introduced the notation

D i (k,k2) = K i (k) K i (k2)I i (k2)

I i (k), (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 i:

F (i) [G(z)] = f (i)[G(z), i]. (2.22)The vacuum energy density, , effective pressures in perpendicular, p, and parallel, p , to theplates directions are determined by relations (no summation over i)

= 0D

|T 0

0 |0D , p =

0D

|T 1

1 |0D , p =

0D

|T i

i |0D , i = 2 , . . . , d . (2.23)

5


6/23

It can be easily checked from Eqs. (2.20), (4.6) and (2.17)(2.19) that they satisfy the standardcontinuity equation for the EMT, which for the geometry under consideration takes the form

d(p)d

= . (2.24)For a conformally coupled scalar we have an additional zerotrace relation p(d 1) p = 0.Let us consider the limit 2 of general formula (2.20) for xed . It can be easily seen thatin this limit the VEVs take the form

lim2

0D |T ki |0D = 0R |T ki |0R + T ki(1b)D (1, ), > 1, (2.25)

where0R |T ki |0R =

Adki

0dkkd

0d sinh f (i)[K i (k)] (2.26)

are the corresponding VEVs for the FullingRindler vacuum without boundaries, and the term

T ki(1b)D (1, ) = Adki

0dkkd

0d

I (k1)K (k1)

F (i)[K (k)] (2.27)

is induced in the region > 1 by the presence of a single plane boundary located at = 1.Expressions (2.27) are nite for all values > 1 and all divergences are contained in the purelyFulling-Rindler part (2.26). These divergences can be regularized subtracting the correspondingVEVs for the Minkowskian vacuum. The subtracted VEVs

T ki(R )sub = 0R |T ki |0R 0M |T ki |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 eld in anarbitrary number of spacetime dimensions has been considered in Ref. [28] for conformally andminimally 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 VEVs (2.28) can be presented in the form

T ki(R )sub =

ki d 1

2d 1d/ 2(d/ 2)

0

dg(i)()de2 + ( 1)d

(2.29)

(the expressions for the functions g(i)() are given in Ref. [16]) correspond to the absence from

the vacuum of thermal distribution with standard temperature T = (2 ) 1. As we see from Eq.(2.29), in general, the corresponding spectrum has non-Planckian form: the density of states factoris not proportional to d 1d. The spectrum takes the Planckian form for conformally coupledscalars in d = 1 , 2, 3 with g(0) () = dg(i)() = 1, i = 1 , 2, . . . d. It is interesting to note that foreven values of spatial dimension the distribution is Fermi-Dirac type (see also [33, 34]). For themassive scalar the energy spectrum is not strictly thermal and the corresponding quantities donot coincide with ones for the Minkowski thermal bath.

The boundary induced quantities (2.27) are investigated in Ref. [15] for a conformally coupledd = 3 massless Dirichlet scalar and in Ref. [16] for a massive scalar with general curvature couplingand Robin boundary condition in an arbitrary number of dimensions. The single boundary part(2.27) diverges at the plate surface = 1 with leading terms proportional to (

1) d 1 for

6


7/23

i = 0 , 2, . . . , d and to ( 1) d for i = 1 (see below). These leading terms vanish for a conformallycoupled scalar, and for i = 0 , 2, . . . , d coincide with the corresponding quantities for a planeboundary in the Minkowski vacuum [16].

Now we turn to the limit 1 0 in formula (2.20), when the left plate coincides with the rightRindler horizon. In this limit in the second term on the right of formula (2.20) the subintegrandbehaves as 21 and tends to zero. As a result one obtains

lim1 0

0D |T ki |0D =Adki

0dkkd

0d sinh f (i)[D i (k,k2)]. (2.30)

These quantities coincide with the corresponding ones induced in the region < 2 by a singleplate at = 2. They are investigated in Ref. [16], where it has been shown that the VEVs (2.30)can be presented in the form similar to Eq. (2.25):

lim1 0

0D |T ki |0D = 0R |T ki |0R + T ki(1b)D (2, ), < 2, (2.31)

where the expressions for the boundary part T ki(1b)D (2, ) in the region < 2 are obtained from

formulae (2.27) by replacing (see Ref. [16])

I K , K I , 1 2, 2 1. (2.32)By using Eqs. (2.20),(2.30),(2.31) the parts in the VEVs induced by the existence of boundaries,

T ki(b)D = 0D |T ki |0D 0R |T ki |0R , (2.33)

can be written as

T i (b)D (1, 2, ) = T ki (1b)D (2, ) Adki

0dk kd

0dI (k1)I (k2)

F (i)

[D(k,k2)]D(k1, k2). (2.34)

In Appendix C we show that the VEVs (2.20) can be also presented in the form (C.4).Substituting Eq. (C.11) into this formula, the boundary VEVs can be also written in the form

T ki(b)D (1, 2, ) = T

ki

(1b)D (1, ) Adki

0dk kd

0d

K (k2)K (k1)

F (i) [D(k,k1)]D(k1, k2)

. (2.35)

This expression is obtained from Eq. (2.34) by replacements (2.32). The case d = 1 needs aseparate consideration and is investigated in Appendix B. It can be seen that the correspondingformulae for the VEVs are also obtained from the formulae given above in this section replacing

Ad

0dk kd 2

1

, k 0. (2.36)Now let us present the VEVs (2.20) in the form

0|T ki |0 D = 0R |T ki |0R + T ki(1b)D (1, ) + T

ki

(1b)D (2, ) + T

ki D (1, 2, ), 1 < < 2, (2.37)

where

T ki D = Adki

0dkkd

0dI (k1)

F (i)[D(k,k2)]I (k2)D(k1, k2)

F (i)[K (k)]K (k1)

(2.38)

7


8/23

is the interference term. The surface divergences are contained in the single boundary parts andthis term is nite for all values 1 2. An equivalent formula for T ki D is obtained fromEq. (2.38) by replacements (2.32). In the limit 1 2 expressions (2.38) are divergent and forsmall values of 2/ 11 the main contribution comes from the large values of . Introducing a newintegration variable x = k/ and replacing Bessel modied functions by their uniform asymptoticexpansions for large values of the order (see Ref. [35]) at the leading order over 1/ (2 1) onereceives (no summation over i)

T ii(1b)D ( j , )

d( c ) d+122d (d+1) / 2| j |d+1

, i = 0 , 2, . . . , d , (2.39)

T 11(1b)D ( j , ) T 00

(1b)D ( j , )

j d j

, j = 1 , 2 (2.40)

for the single boundary terms, and

T 00 D

1

d T 11 D +

( c)(2 1) d 122d 1d/ 2(d/ 2)

(2.41)

0

dtt d

et 1exp t

1 2 1

+ exp t 22 1

T 11 Dd R (d + 1) d+12

(4)(d+1) / 2(2 1)d+1, T ii D T

00 D , i = 2 , 3, . . . , (2.42)

for the interference terms. Here R (s) is the Riemann zetafunction. 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 2 1 (see Ref. [36] for the conformally coupledcase and Ref. [18] for the general case of the curvature coupling parameter ). Note that in thelimit under consideration the interference term (2.42) for the vacuum perpendicular pressuredominates the single boundary induced terms, given by Eq. (2.40).

3 Interaction forces between the plates

Now we turn to the interaction forces between the plates. The vacuum force acting per unitsurface of the plate at = i is determined by the 11component of the vacuum EMT at this point.The corresponding effective pressures can be presented as a sum of two terms:

p(i)D = p(i)D 1 + p

(i)D (int) , i = 1 , 2. (3.1)

The rst term on the right is the pressure for a single plate at = i when the second plate isabsent. This term is divergent due to the well known surface divergences in the subtracted VEVs.The second term on the right of Eq. (3.1),

p(i)D (int) = T 11(1b)D ( j , i) T 11 D (1, 2, i), 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 interactionforce. For the plate at = 2 the interaction term is due to the second summand on the right of Eq. (2.20). Substituting into this term = 2 and using the Wronskian relation for the modiedBessel functions one has

p(2)D (int) (1, 2) = Ad22

2

0dkkd 2

0d

I (k1)I

(k

2)D

(k

1, k

2). (3.3)

8


9/23

By a similar way from Eq. (2.35) for the interaction term on the plate at = 1 we obtain

p(1)D (int) (1, 2) = Ad221

0dkkd 2

0d

K (k2)K (k1)D(k1, k2)

. (3.4)

As the function D(k,k2) is positive for 1 < 2, interaction forces per unit surface (3.3) and(3.4) are always attractive. They are nite for all 1 < 2, and do not depend on the curvaturecoupling parameter . In the limit 1 2 these forces diverge due the contribution from thelarge values and in this limit by introducing a new integration variable we can replace the Besselmodied functions by their uniform asymptotic expansions for large values of the order. At theleading order for the perpendicular vacuum pressures we obtain formula (2.42) which correspondsto 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) (1, 2) > p(1)D (int) (1, 2). (3.5)

This can be proved by using that the function z2I (z)K (z) is monotonic increasing. The latterdirectly follows from the relations

1 + ( + 1) 2z2 1z < I (z)

I (z)< 1 + 2z2 (3.6)

1 + ( + 1) 2z2 + 1z > K (z)

K (z)> 1 + 2z2 . (3.7)

The proof for the right inequalities in Eqs. (3.6),(3.7) is presented in Ref. [15]. The left inequalitiesare obtained from the recurrence relations for the Bessel modied functions. For instance, in the

case of the function I (z) one has:I (z)I (z)

=I +1 (z)I (z)

+z

=I +1 (z)I +1 (z)

+ + 1

z

1

+z

>

> 1 + ( + 1) 2z2 + + 1z 1

+z

= 1 + ( + 1) 2z2 1z , (3.8)where we have used the right inequality in Eq. (3.6). The left inequality in Eq. (3.7) can beproved in a similar way.

To see the monotonicity properties of functions (3.3) and (3.4) note that

1p(1)D (int)

2= 2

p(2)D (int)1

=Ad

212

0dkkD 2

0

dD 2(k1, k2)

. (3.9)

It follows from here that for a xed value of 1 (2) the quantity p(1)D (int) ( p

(2)D (int) ) is monotonic

increasing (decreasing) function on 2 (1). By taking into account that both this quantities arenegative we conclude that the modulus of the interaction force on the plate at 1 (2) is monotonicdecreasing (increasing) function on 2 (1) for a xed value of 1 (2). From formula (3.3) it followsthat

ip(i)D (int)

i= (d + 1) p

(i)D (int) j

p(i)D (int) j

, i, j = 1 , 2, i = j. (3.10)

9


10/23

For i = 2 the both terms on the right are positive and hence, the same is the case for the functionon the left. Therefore for a xed 1 the function p

(2)D (int) is monotonic increasing on 2 and the

modulus of the corresponding interaction force is monotonic decreasing function on 2. In thecase i = 1 the terms on the right in this formula have different signs. For a xed value of 2 thefunction p(1)

D (int)is monotonic increasing on 1 near the horizon, 1

0, and monotonic decreasing

near the second plane, 1 2. It follows from here the modulus of the corresponding interactionforce acting on the plate at 1 has minimum for some intermediate value.In the limit 2 1, introducing in Eq. (3.3) a new integration variable x = k2, and making

use the formulaI (y) =

y2

1()

1 + O(y2) , y = x1/ 2, (3.11)and the standard relation between the functions K and I one nds

p(2)D (int) 2Ad

48d+12 ln2(22/ 1)

0

dxxd 2

I 20 (x)1 + O

ln xln(22/ 1)

. (3.12)

The similar calculation for Eq. (3.4) yields

p(1)D (int) 2Ad

24d 12 21 ln3(22/ 1)

0

dxxd 2K 0(x)I 0(x)

1 + Oln x

ln(22/ 1). (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 d+12 p

(i)D (int) , i = 1 , 2 in the

case d = 3 as functions on 1/ 2.

0.1 0.2 0.3 0.4 0.5 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 42 , 42 p(1)D (int) (curve a) and 42 p(2)D (int) (curve b) as functions of theratio 1/ 2.

4 VEVs and the interaction forces for the Neumann scalar

In this section we will consider VEVs for the EMT in the case of a scalar eld satisfying theNeumann boundary condition on the plates = 1, 2:

|= 1 =

|= 2 = 0 . (4.1)

10


11/23


12/23

where the expressions for the boundary part T kNi(1b)N (2, ) in the region < 2 are obtained

from formulae (4.9) by replacements (2.32).By using Eqs. (4.6),(4.10) the parts in the VEVs induced by the existence of boundaries,

T ki(b)N = 0N

|T ki

|0N

0R

|T ki

|0R , (4.11)

can be presented as

T ki(b)N (1, 2, ) = T

ki

(1b)N (2, ) Adki

0dk kd

0d

I (k1)I (k2)

F (i)[N (k,k2)]N (k1, k2)

. (4.12)

Similar to the Dirichlet case, the Neumann boundary VEVs can be also written in the form

T ki(b)N (1, 2, ) = T

ki

(1b)N (1, ) Adki

0dk kd

0d

K (k2)K (k1)

F (i)[N (k,k1)]N (k1, k2)

, (4.13)

with T ki(1b)N (1, ) being the VEVs induced by a single Neumann boundary located at = 1.

As we see, this expression is obtained from (4.12) by replacements (2.32).Now let us present the VEVs (4.6) in the form

0N |T ki |0N = 0R |T ki |0R + T ki(1b)N (1, )+ T ki

(1b)N (2, )+ T ki N (1, 2, ), 1 < < 2, (4.14)

where

T ki N = Adki

0dkkd

0dI (k1)

F (i)[N (k,k2)]I (k2)N (k1, k2)

F (i)[K (k)]K (k1)

(4.15)

is the interference term. An equivalent formula for T ki N is obtained from Eq. (4.15) by

replacements (2.32).Interference term (4.15) is nite for all 1 2, 1 < 2, and diverges in the limit 1 2.In this limit the main contribution into the integral comes from the large values . Introducinga new integration variable x = k1/ and using the uniform asymptotic expansions for the Besselmodied functions in the leading order one obtains that the quantities T ki N coincide with theVEVs for two parallel plates in d + 1dimensional Minkowski spacetime with separation 2 1[36, 18]. The corresponding expressions are given by formulae (2.41),(2.42) with the opposite signof the integral term on the right of formula (2.41).

Now we turn to the Neumann vacuum effective pressures determining the forces acting onthe plate due to the presence of the second plate (interaction forces). This force acting per unitsurface of the plate = 2, p

(2)N (int) is dened by the

11component of the second term on the right

of formula (4.12) at = 2. The nonzero contribution comes from the last term on the rightof Eq.(2.18) (with replacement (2.22)). Using the standard Wronskian relation for the Besselmodied functions one obtains

p(2)N (int) (1, 2) =Ad222

0dkkd 2

0d

I (k1)(1 + 2/k 222)I (k2)N (k1, k2)

. (4.16)

By a similar way for the interaction force per unit surface of the rst plate from the second termon the right of Eq. (4.13) at = 1 we receive

p(1)N (int) (1, 2) =Ad22

1

0dkkd 2

0d

K (k2)(1 + 2/k 221 )K

(k

1)N

(k

1, k

2)

. (4.17)

12


13/23


14/23

5 Electromagnetic eld

We now turn to the case of the electromagnetic eld in the region 1 < < 2. We will assumethat the mirrors are perfect conductors with the standard boundary conditions of vanishing of the normal component of the magnetic eld and the tangential components of the electric eld,evaluated at the local inertial frame in which the conductors are instantaneously at rest. Byconsiderations similar to those given in Ref. [15] for d = 3, it can be seen that the correspondingeigenfunctions for the vector potential A may be resolved into one transverse magnetic (TM)and d 2 transverse electric (TE) (with respect to -direction) modes A , = 0 , 1, . . . , d 2, = ( k , ):

A1 = ( /, i/, 0, . . . 0) 0 , = 1 , TM mode, (5.1)A =

, = 0 , 2, . . . , d 2, TE modes , (5.2)

where the polarization vectors obey the following relations0 = 1 = 0 ,

=

k2 , k = 0 . (5.3)

From the perfect conductor boundary conditions one has the following conditions for the scalarelds :

|= 1 = |= 2 = 0 , = 0 , 2, . . . , d 2, 1

|= 1 = 1

|= 2 = 0 . (5.4)As a result the TE/TM modes correspond to the Dirichlet/Neumann scalars. In the correspond-ing expressions for the eigenfunctions A the normalization coefficient is determined from theorthonormality relation

dx 2

1

d

A A = 2

. (5.5)

On the base of this normalization condition for the separate scalar modes one has

=21/ 2

kC Z Z iZn (k,k2)exp[i (kx Zn )] , (5.6)

where Z = D for = 0 , 2, . . . , d 2 and Z = N for = 1, and the coefficients C D and C N aredened in accordance with Eqs. (2.13),(4.4). Substituting the eigenfunctions (5.1), (5.2) into themode sum formula

0|T ki |0 =d 2

=0 dk Zn T ki {A , A }, (5.7)with the standard bilinear form for the electromagnetic eld EMT one nds

0

|T ki

|0 = ki

(d 1)/ 2

d 1

2

0

dk kd

=0 ,1

n =1

C 2Z f (i)em [Z iZn (k,k2)], 0 = d

2, 1 = 1 , (5.8)

where 0 and 1 are the numbers of the independent polarization states for TE and TM modesrespectively. In Eq. (5.8) for a given function G(z) the following notations are introduced

f (0)em [G(z)] =dG(z)

dz

2

+ 1 +2

z2 |G(z)|2,

f (1)em [G(z)] = dG(z)

dz

2

+ 1 2

z2 |G(z)|2, (5.9)

f (i)em [G(z)] =d 5d

1|G(z)|

2 +d 3d

1

dG(z)dz

2

2

z2 |G(z)|2 , i = 2 , 3, . . . , d .

14


15/23

By making use of the summation formulae derived in the Appendix A the VEVs are presentedin the form

0|T ki |0 = kiAd2

0dkkd

0d

=0 ,1

sinh

f (i)em [Z i (k,k2)] I () (k1)F (i)em [Z (k,k2)]

I () (k2)Z () (k1, k2)

,

(5.10)where I (0) = I , I (1) = I , and the same notations for the functions K , Z . The functions F (i)emare obtained from Eqs. (5.9) replacing i:

F (i)em [G(z)] = f (i)em [G(z), i]. (5.11)

It can be easily checked that the components (5.10) obey the covariant conservation equation andthe corresponding EMT is traceless for d = 3. The rst term in the gure braces of Eq. (5.10)corresponds to the VEV induced by a single plate at = 2 in the region < 2. For the cased = 3 they are investigated in Ref. [16]. The generalization for an arbitrary d is straightforwardand these quantities are presented in the form

0|T ki |0 (1b) (2, ) = 0R |T ki |0R 12ki Ad 0 dkkd 0 d =0 ,1 K () (k2)I () (k2) F (i)em [I (k)], (5.12)

where 0R |T ki |0R are the VEVs for the FullingRindler electromagnetic vacuum without bound-aries. By the way similar to that given in Ref. [16] for the case of a scalar eld, it can be seenthat

0R |T ki |0R = 0M |T ki |0M ki (d 1) d 12d 1d/ 2(d/ 2)

0

df (i)0em ()de2 + ( 1)d

lm

l=1

d 1 2l2

2

+ 1 , (5.13)

where lm = d/ 21 for even d > 2 and lm = ( d1)/ 2 for odd d > 1, and the value for the productover l is equal to 1 for d = 1 , 2, 3. In Eq. (5.13) we have introduced notations

f (0)0em () = df (1)0em () = 1 +

(d 1)242

, (5.14)

f (i)0em () = f (1)0em () +

(d 1)(d 3)42

, i = 2 , . . . , d .

For physically most important case d = 3, formula (5.13) leads to the standard result derived byCandelas and Deutsch in Ref. [15].

An alternative form for the vacuum EMT in the region between two plates is

0|T ki |0 = 0R |T ki |0R + 0|T ki |0 (1b)(1, ) 12ki Ad

0dkkd

0d

=0 ,1 K

() (k2)F

(i)em [Z (k,k1)]

K () (k1)Z () (k1, k2)

, (5.15)

where 0|T ki |0 (1b)(1, ) is the vacuum EMT induced by a single boundary at = 1 in the region > 1. The latter is obtained from (5.12) by replacements (2.32). For the interaction force p

(i)em(int) ,

i = 1 , 2 per unit area of the plate at = i from Eqs. (5.10) and (5.15) one obtains

p(1)em(int) = Ad221

0dkkd 2

0d

=0 ,1(1)

K () (k2)K () (k1)

(1 + 2/k 221 )

Z () (k1, k2), (5.16)

p(2)em(int) = Ad222

0dkkd 2

0d

=0 ,1(1)

I () (k1)I () (k2)

(1 + 2/k 222 )

Z () (k1, k2). (5.17)

15


16/23

Recalling that ( 1) Z () > 0 we see the electromagnetic interaction forces are attractive. Notethat p(i)em(int) = ( d 2) p

(i)D (int) + p

(i)N (int) . In the limit 1 2 and to the leading order over 1 / (2 1) from these expressions the electromagnetic Casimir interaction force between plates in the

Minkowski spacetime is obtained.

6 Conclusion

It is well known that the uniqueness of vacuum state is lost when we work within the frameworkof quantum eld theory in a general curved spacetime or in noninertial frames. In this paperwe have considered vacuum expectation values of the energy-momentum tensor for scalar andelectromagnetic elds between two innite parallel plates moving by uniform proper acceleration,assuming that the elds are prepared in the Fulling-Rindler vacuum state. As the boundariesare static in the Rindler coordinates no Rindler quanta are created and the only effect of theimposition of boundary conditions on quantum elds is the vacuum polarization. For the scalar

case the both Dirichlet and Neumann boundary conditions are investigated. The VEVs arepresented in the form of mode sums involving series over zeros = Dn or = Nn of thefunctions D i (k1, k2) and N i (k1, k2) respectively. To sum these series we derive in AppendixA summation formulae for these types of series using the generalized Abel-Plana formula. Theapplication of these formulae allows to extract from the VEVs the parts due to the single plate.The latters are investigated previously in Refs. [15, 16]. The boundary induced parts are presentedin two alternative forms, Eqs. (2.34), (2.35), for the Dirichlet case, and Eqs. (4.11),(4.12) for theNeumann case. Various limiting cases are studied. In particular, in the limit when the left platecoincides with the Rindler horizon the corresponding VEVs are the same as for a single plategeometry. The vacuum forces acting on boundaries contain two terms. The rst ones are theforces acting on a single boundary then the second boundary is absent. Due to the wellknown

surface divergences in the VEVs of the energy-momentum tensor these forces are innite and needan additional regularization. The another terms in the vacuum forces are nite and are inducedby the presence of the second boundary and correspond to the interaction forces between theplates. These forces per unit surface do not depend on the curvature coupling parameter andare determined by formulae (3.9),(3.10) for the Dirichlet scalar and by formulae (4.16),(4.17) forthe Neumann scalar, and are always attractive for both plates. In particular, they are the same forconformally and minimally coupled scalars. For given 1, 2 the modulus of the interaction forceis larger for the plate at = 1 (see inequalities (3.5) and (4.18)). For small distances between theplates at the leading order the standard Casimir result on background of the Minkowski vacuum isrederived. The case of the electromagnetic eld is considered with the perfect conductor boundaryconditions in the local inertial frame in which the boundaries are instantaneously at rest. Thecorresponding eigenmodes are superposition of TE and TM modes with Dirichlet and Neumannboundary conditions respectively. The VEVs of the electromagnetic EMT in the region betweenthe plates are given by formulae (5.10) and (5.15). The corresponding vacuum interaction forcesper unit surface, Eqs. (5.16),(5.17), are obtained by summing the Dirichlet and Neumann d = 3scalar forces, and are attractive for all values of the proper accelerations for the plates.

The results obtained in this paper can be applied to the geometry of two parallel plates nearthe Rindler wall. With the x coordinate perpendicular to the wall and with the ( x2, x3) planelocated at the centre of the wall, x = 0, the static planesymmetric line element can be written as

ds2 = e (x) dt2 dx2 e (x) dx 2, x = ( x2, x3), (6.1)

16


17/23

where (x) and (x) are even functions. For this metric the Einstein equations with the diagonalmatter energy-momentum tensor T (m )ki = diag( (m ) , p(m ) , p(m ) , p(m ) ) admit two classes of solutions. For the rst one (0) > 0, and the corresponding external solution (the solution in theregion x > x s , where T

(m )ik = 0, with x = xs being the boundary of the wall) is described by the

standard Taub metric [37]. For the second class of internal solutions

(0) < 0, and the externalsolution is presented by the metric

ds2ext = e s [1 + 2 s (x xs )]2dt2 dx2 e s dx 2, (6.2)

where s = (xs ), s = (xs ), and

s = 2 e s / 2 s x s

0(m ) + 3 p(m ) e/ 2+ dx (6.3)

is the mass per unit surface of the wall. For a given equation of state p(m ) = p(m ) ((m )) theparameters s , s , xs are functions of the central pressure p(m )

|x=0 , and are determined by the

internal solution of the Einstein equations (see Ref. [13] for the case of the equation of statecorresponding to the incompressible liquid). Now redening

(x) = x xs +1

2 s, = 2 s e s / 2t, e s / 2xi xi , i = 2 , 3, (6.4)

from Eq. (6.2) we obtain the Rindler metric in the form (2.5). Hence, the VEVs for the EMT inthe region between two plates located at x = x1 and x = x2, xi > x s near the Rindler wall areobtained from the results given above substituting i = (xi), i = 1 , 2 and = (x). Note that fors > 0, x xs one has (x) (xs ) > 0 and the Rindler metric is regular everywhere.

Acknowledgements

We are grateful to Professor E. Chubaryan and Professor A. Mkrtchyan for general encouredgementand suggestions, and to L. Grigoryan and R. Davtyan for useful discussions. This work wassupported by the Armenian National Science and Education Fund (ANSEF) Grant No. PS14-00and by the Armenian Ministry of Education and Science Grant No. 0887.

A Summation formulae over zeros of D iz and N izIn this section we will derive a summation formula over zeros z =

Dkof the function

D iz (x, y) = I iz (y)K iz (x) I iz (x)K iz (y), y > x. (A.1)As we saw in section 2 the VEVs of the EMT for the Dirichlet scalar between two plates in theFulling-Rindler vacuum are expressed in the form of series over these zeros. To derive a summationformula we use the generalized Abel-Plana formula [17]. Let us choose in this formula

f (z) =2i

sinh z F (z), (A.2)

g(z) =I iz (y)I iz (x) + I iz (x)I iz (y)

D iz (x, y)F (z),

17


18/23

with a meromorphic function F (z) having poles z = zk in the right half-plane Re z 0. The sumand difference of functions (A.2) are presented in the formg(z) f (z) =

2I iz (x)I iz (y)D iz (x, y)

F (z). (A.3)

By taking into account that the zeros Dk are simple poles of the function g(z) for the functionR[f (z), g(z)] in the generalized Abel-Plana formula one obtains

R[f (z), g(z)] = 2i

k=1

I iz (y)I iz (x)

z D iz (x, y)F (z)|z= Dk + (A.4)

+k

Resz= zkF (z)

D iz (x, y)I isgn(Im zk )z (y)I isgn(Im zk )z(x) ,

where the zeros Dk are arranged in ascending order. As a result we obtain the following summa-tion formula

k=1

I iz

(y)I iz

(x)

z D iz (x, y) F (z)|z= Dk =12

0 sinh z F (z)dz (A.5)

k Resz= zkF (z)

D iz (x, y)I isgn(Im zk )z(y)I isgn(Im zk )z(x)

1

2

0dz

F (zei/ 2) + F (ze i/ 2)I z(y)K z(x) I z(x)K z (y)

I z(x)I z(y).

Here the condition for the function F (z) is easily obtained from the corresponding condition inthe Generalized AbelPlana formula by using the asymptotic formulae for the Bessel modiedfunction and has the form

|F (z)| < (|z|)e zyx

2|Im z|

, Re z > 0, |z| , (A.6)where |z| (|z|) 0 when |z| .A similar formula can be obtained for the series over zeros z = Nk , k = 1 , 2, . . . of the function

N iz (x, y) = I iz (y)K

iz (x) K iz (y)I iz (x), y > x. (A.7)

For this let us substitute in the Generalized Abel-Plana formula [17]

f (z) =2i

sinh z F (z) (A.8)

g(z) =I iz (y)I iz (x) + I iz (x)I iz (y)

N iz (x, y)

F (z).

Using these expressions it can bee easily seen that

g(z) f (z) =2I iz (x)I iz (y)

N iz (x, y)F (z). (A.9)

For the function R[f (z), g(z)] now one obtains

R[f (z), g(z)] = 2i

k=1

I iz (y)I iz (x)

z N iz (x, y)

F (z)|z= Nk + (A.10)

+

k

Resz= zkF (z)

N iz (x, y)I isgn(Im zk )z(y)I isgn(Im zk )z (x) .

18


19/23

As a result we obtain the following summation formula

k=1

I iz (y)I iz (x)

z N iz (x, y)

F (z)|z= Nk =12

0sinh z F (z)dz (A.11)

k Resz= zkF (z)

N iz (x, y) I isgn(Im zk )z(y)I isgn(Im zk )z(x)

1

2

0dz

F (zei/ 2) + F (ze i/ 2)I z (y)K z(x) I z(x)K z(y)

I z (x)I z(y),

where the corresponding condition for the function F (z) has the form (A.6).

B d = 1 case: Direct evoluationFor d = 1 case the linearly independent solutions to equation (2.8) are e i ln . The normalized

eigenfunctions satisfying Dirichlet boundary conditions (2.6) are in form

Dn =

e i

n sin(n ), =n

ln(2/ 1), n = 1 , 2, . . . , (B.1)

where we use the notation =

ln(2/ )ln(2/ 1)

. (B.2)

Substituting eigenfunctions (B.1) into modesum formula (2.3) and applying to the sum over nthe AbelPlana summation formula one nds

0D |T ki |0D 0M |T ki |0M = T ki (R )(sub) + T ki (1b)D (2, ) + 2

/ sin2

12 2 ln2(2/ ) diag(1, 0)

12 2

2

12ln2(2/ 1)+

ln(2/ )

( cot 1) diag(1, 1).(B.3)

Here the subtracted purely FullingRindler part without boundaries, T ki(R )(sub) , and the part in-

duced by a single boundary at = 2 are given by formulae [16]

T ki(R )(sub) =

12 2

112

diag(1, 1), (B.4)

T ki (1b)D (2, ) = 2 2 ln(/ 2)diag(1 + 1 / ln(/ 2), 1). (B.5)

Note that the expression (B.5) for a single boundary part is valid for both regions < 2 and > 2. Now for the vacuum interaction forces between the plates one obtains

p(i)D (int) =

242i ln2(2/ 1)

, i = 1 , 2. (B.6)

In the limit 1 2 to the leading order we recover the standard Casimir result on backgroundof the 2D Minkowski spacetime.

19


20/23

For the case of the Neumann boundary conditions (4.1) the normalized eigenfunctions havethe form

N n =

e i

n cos(n ), n = 0 , 1, 2, . . . , (B.7)

where and are given by the same relations (B.1) and (B.2) as in the Dirichlet case. Thesubstitution of these eigenfunctions into the modesum formula shows that the VEVs of theEMT for the Neumann boundary conditions can be obtained from the corresponding formula forthe Dirichlet case, Eq. (B.3), replacing in the boundary part .

C Alternative representation for the VEVs

As a solution to equation (2.8) satisfying rst boundary condition (2.6) one could take the function

D i (k,k1) = I i (k1)K i (k) K i (k1)I i (k). (C.1)Now from the boundary condition on the plate = 2 (2.6) we nd the possible values for beingroots to the equation (2.10). For the normalization coefficient we receive

C 2D =1

(2)d 1I i (k2)

I i (k1) D i (k 1 ,k 2 ) |= Dn. (C.2)

The VEVs of the energy - momentum tensor are obtained in a diagonal form

0D |T ki |0D = Adki

0dkkd

n =1

I i (k2)I i (k1) D i (k 1 ,k 2 )

f (i)[D i (k,k1)]|= Dn . (C.3)

For the further evoluation of VEVs (C.3) we can apply to the sum over n summation formula(A.5). This gives

0D |T ki |0D = Adki

0dkkd

0d

sinh

f (i)[D i (k,k1)]

I (k2)F (i)[D(k,k1)]

I (k1)D(k1, k2). (C.4)

This form of the VEVs is equivalent to Eq. (2.20). To see this let us consider the quantities

q(i) j = 1

0 d sinh f (i)[D i (k,k j )]

0 dI (k1)I (k2)D(k1, k2) F (i)

[D(k,k j )]I (k j )I (k j ) , (C.5)

where j = 1 , 2. Two representations (2.20) and (C.4) will be equivalent if

q(i)1 = q(i)2 . (C.6)

To prove this let us consider the difference

q(i)2 q(i)1 =

1

0d sinh s(i)

0d

I (k1)I (k2)D(k1, k2)

S (i) , (C.7)

20


21/23

where we have introduced the notations

s(i) = f (i)[D i (k,k2)] f (i)[D i (k,k1)], (C.8)S (i) =

j =1 ,2

(

1) j

F (i)[D(k,k j )]

I (k j )I (k j ).

By using the standard relation between the Bessel modied functions it can be seen that the rstintegral in formula (C.7) can be presented as

i2

0

I i (k1)I i (k2) I i (k1)I i (k2)I i (k2)K i (k1) I i (k1)K i (k2)

s (i)d, (C.9)

where the function s (i) / (I i (k2)K i (k1) I i (k1)K i (k2)) has no poles. For the term with therst (second) summand in the numerator rotating the integration contour by angle / 2 (/ 2)in complex plane and noting that the integrals over arcs with large radius vanish (subintegrandbehaves as (/ 2)

2|Im |

) we see thati2

0

I i (k1)I i (k2) I i (k1)I i (k2)I i (k2)K i (k1) I i (k1)K i (k2)

s (i)d =

0d

I (k1)I (k2)D(k1, k2)

S (i) . (C.10)

Hence, the difference (C.7) is equal to zero, which directly proves Eq. (C.6)By taking into account Eq. (C.5) from Eq. (C.6) in the limit 2 one obtains the followinguseful relation

1

0d sinh f (i)[D i (k,k1)] =

1

0d sinh f (i) [K i (k)] + (C.11)

+

0 dF (i)[D(k,k1)]I (k1)K (k1)

I (k1)K (k1) F

(i)

[K (k]

Substituting Eq. (C.11) into Eq. (C.4) one nds formula (2.35).

References

[1] H. B. G. Casimir, Proc. Kon. Nederl. Akad. Wet. 51 , 793 (1948).

[2] V. M. Mostepanenko and N. N. Trunov, The Casimir effect and its applications (OxfordUniversity Press, Oxford, 1997).

[3] G. Plunien, B. Muller and W.Greiner, Phys. Rep. 134 , 87 (1986).

[4] K. A. Milton, The Casimir Effect: Physical Manifestation of ZeroPoint Energy (WorldScientic, Singapore, 2002).

[5] S. K. Lamoreaux, Am. J. Phys. 67 , 850 (1999).

[6] M. Bordag (Ed.), The Casimir Effect. 50 years later (World Scientic, Singapore, 1999).

[7] M. Bordag, U. Mohidden, and V. M. Mostepanenko, Phys. Rep. 353 , 1 (2001).

[8] K. Kirsten, Spectral functions in Mathematics and Physics . CRC Press, Boca Raton, 2001.

21


22/23


23/23

[33] S. Tagaki, Prog. Theor. Phys. 74 , 142 (1985).

[34] H. Ooguri, Phys. Rev. D33 , 3573 (1986).

[35] M. Abramowitz and I. A. Stegun, Handbook of Mathematical functions (National Bureau of

Standards, Washington D.C.,1964).[36] J. Ambjrn and S. Wolfram, Ann. Phys. (N.Y.) 147 , 1 (1983).

[37] A. H. Taub, Ann. Math. 53 , 472 (1951).

R. M. Avagyan, A. A. Saharian and A. H. Yeranyan- The Casimir effect in the Fulling–Rindler vacuum

Documents