Casimir densities for a boundary in Robertson-Walker spacetime
aa r X i v : . [ h e p - t h ] D ec Casimir densities for a boundary in Robertson-Walker spacetime
A. A. Saharian ∗ and M. R. Setare † Department of Physics, Yerevan State University,1 Alex Manoogian Street, 0025 Yerevan, Armenia Department of Science, Payame Noor University, Bijar, Iran
November 8, 2018
Abstract
For scalar and electromagnetic fields we evaluate the vacuum expectation value of theenergy-momentum tensor induced by a curved boundary in the Robertson–Walker spacetimewith negative spatial curvature. In order to generate the vacuum densities we use the con-formal relation between the Robertson–Walker and Rindler spacetimes and the correspond-ing results for a plate moving by uniform proper acceleration through the Fulling–Rindlervacuum. For the general case of the scale factor the vacuum energy-momentum tensor ispresented as the sum of the boundary free and boundary induced parts.
The influence of boundaries on the vacuum state of a quantum field leads to interesting physicalconsequences. Well known example is the Casimir effect [1, 2, 3, 4], when the modificationof the zero-point fluctuations spectrum by the presence of boundaries induces vacuum forcesacting on the boundaries. It may have important implications on all scales, from cosmologicalto subnuclear. The particular features of the resulting vacuum forces depend on the nature ofthe quantum field, the type of spacetime manifold, the boundary geometries and the specificboundary conditions imposed on the field.The Casimir effect can be viewed as a polarization of vacuum by boundary conditions. An-other type of vacuum polarization arises in the case of an external gravitational field. In thispaper, we study an exactly solvable problem with both types of sources for the polarization.Namely, we consider the vacuum expectation value of the energy-momentum tensor for bothscalar and electromagnetic fields induced by a curved boundary in background of Robertson-Walker (RW) spacetime with negative spatial curvature. In order to generate the vacuum densi-ties we use the well known relation between the vacuum expectation values in conformally relatedproblems (see, for instance, [5]) and the corresponding results for an infinite plane boundarymoving with uniform acceleration through the Fulling–Rindler vacuum.The latter problem for conformally coupled Dirichlet and Neumann massless scalar fields andfor the electromagnetic field in four dimensional Rindler spacetime was considered by Candelasand Deutsch [6]. These authors consider the region of the right Rindler wedge to the right of ∗ E-mail: [email protected] † E-mail: [email protected]
As a background geometry we shall consider the k = − ds = g ik dx i dx k = a ( η )( dη − γ dr − r d Ω D − ) , (1)where γ = 1 / √ r and d Ω D − is the line element on the ( D − x i = ( η, r, θ, θ , . . . , θ D − , φ ) → x ′ i = ( η, ξ, x ′ ) , (2)with x ′ = ( x ′ , ..., x ′ D ), defined by the relations (see Ref. [5] for the case D = 3) ξ = ξ Ω , x ′ = ξ r Ω sin θ cos θ , . . . , x ′ D − = ξ r Ω sin θ sin θ · · · sin θ D − cos θ D − ,x ′ D − = ξ r Ω sin θ sin θ · · · sin θ D − cos φ, x ′ D = ξ r Ω sin θ sin θ · · · sin θ D − sin φ, (3)2here ξ is a constant with the dimension of length and we use the notationΩ = γ/ (1 − rγ cos θ ) . (4)Under this transformation the RW line element takes the form ds = g ′ ik dx ′ i dx ′ k = a ( η ) ξ − (cid:0) ξ dη − dξ − d x ′ (cid:1) . (5)In this form the RW metric is manifestly conformally related to the metric in the Rindlerspacetime with the line element ds : ds = a ( η ) ξ − ds , ds = g R ik dx ′ i dx ′ k , g ′ ik = a ( η ) ξ − g R ik . (6)By using the standard transformation formula for the vacuum expectation values of theenergy–momentum tensor in conformally related problems (see, for instance, [5]), we can gener-ate the results for the RW spacetime from the corresponding results in the Rindler spacetime.First we shall consider the corresponding quantities in the coordinates ( η, ξ, x ′ ) with the lineelement (5). These quantities are found by using the transformation formula for conformallyrelated problems: h RW | T ki (cid:2) g ′ lm , ϕ (cid:3) | RW i = [ ξ/a ( η )] D +1 h R | T ki (cid:2) g R lm , ϕ R (cid:3) | R i + h T ki (cid:2) g ′ lm , ϕ (cid:3) i (an) , (7)where the second term on the right is determined by the trace anomaly. In odd spacetime dimen-sions the conformal anomaly is absent and the corresponding part vanishes: h T ki [ g ′ lm , ϕ ] i (an) = 0for even D . The vacuum expectation value of the energy-momentum tensor in coordinates(1) is obtained by the standard coordinate transformation formulae. For a second rank ten-sor A ik , which is diagonal in coordinates x ′ i = ( η, ξ, x ′ ), the transformation to coordinates x i = ( η, r, θ, θ , . . . , θ D ) has the form A = A ′ , A = A ′ + Ω sin θ ( A ′ − A ′ ) ,A = Ω sin θ cos θ − rγr ( A ′ − A ′ ) , (8) A = A ′ + Ω sin θ ( A ′ − A ′ ) , A ll = A ′ , l = 3 , . . . , D. In this paper, as a Rindler counterpart we shall take the vacuum energy–momentum tensorinduced by an infinite plate moving by uniform proper acceleration through the Fulling–Rindlervacuum. We shall assume that the plate is located in the right Rindler wedge and has thecoordinate ξ = b . In coordinates x i the boundary ξ = b is presented by the hypersurface p r − r cos θ = 1 /b , b = b/ξ . (9)The corresponding normal has the components n l = b ra ( η ) (0 , p r (1 − p r /b ) , − sin θ, , . . . , . (10)We consider the cases of scalar and electromagnetic fields separately.3 Vacuum expectation values for the energy-momentum tensor:Scalar field
In this section we consider a conformally coupled massless scalar field ϕ ( x ) on background ofspacetime with the line element (1). The corresponding field equation has the form (cid:18) ∇ l ∇ l + D − D R (cid:19) ϕ ( x ) = 0 , (11)where R is the Ricci scalar for the RW spacetime. We assume that the field satisfies the Robinboundary condition ( A + Bn l ∇ l ) ϕ ( x ) = 0 , (12)on the hypersurface (9).The expectation value of the energy-momentum tensor induced by the presence of an infiniteplane boundary moving with uniform acceleration through the Fulling–Rindler vacuum wasinvestigated in [6, 7]. For a scalar field ϕ R ( x ′ ) it is presented in the decomposed form: h R | T ki [ g R lm , ϕ R ] | R i = h ˜0 R | T ki [ g R lm , ϕ R ] | ˜0 R i + h T k (R) i i (b) . (13)In this formula, | R i are | ˜0 R i are the vacuum states for the Rindler spacetime in presence andabsence of the plate respectively and h T k (R) i i (b) is the part of the vacuum energy-momentumtensor induced by the plate. For the part without boundaries one has h ˜0 R | T ki [ g R lm , ϕ R ] | ˜0 R i = a D ξ − D − D − π D/ Γ( D/
2) diag ( − , /D, . . . , /D ) , (14)with the notation a D = Z ∞ ω D dωe πω + ( − D l m Y l =1 "(cid:18) D − − l ω (cid:19) + 1 , (15)where l m = D/ − D > l m = ( D − / D >
1, and the value for theproduct over l is equal to 1 for D = 1 , , ϕ R ( x ′ ) satisfying the Robin boundary condition (cid:16) A R + B R n ′ l R ∇ ′ l (cid:17) ϕ R ( x ′ ) = 0 , ξ = b, n ′ l R = δ l , (16)with constant coefficients A R and B R , the boundary induced part in the region ξ > a is givenby the formula [7] h T k (R) i i (b) = − − D δ ki b − D − π ( D +1) / D Γ (cid:0) D − (cid:1) Z ∞ dx x D Z ∞ dω ¯ I ω ( x )¯ K ω ( x ) F ( i ) [ K ω ( xξ/b )] . (17)Here the functions F ( i ) [ g ( z )] have the form F (0) [ g ( z )] = g ′ ( z ) + D − z g ( z ) g ′ ( z ) + (cid:20) − (2 D − ω z (cid:21) g ( z ) ,F (1) [ g ( z )] = − Dg ′ ( z ) − D − z g ( z ) g ′ ( z ) + D (cid:18) ω z (cid:19) g ( z ) , (18) F ( i ) [ g ( z )] = g ′ ( z ) + (cid:18) ω z − D + 1 D − (cid:19) g ( z ) , i = 2 , . . . , D.
4n Eq. (17), I ω ( z ) and K ω ( z ) are the modified Bessel functions and for a given function f ( z ) weuse the notation ¯ f ( z ) = A R f ( z ) + ( B R /b ) zf ′ ( z ) . (19)The expression for the boundary part of the vacuum energy-momentum tensor in the region ξ < a is obtained from formula (17) by the replacements I ω ⇄ K ω .The formulae given above allow us to present the RW vacuum expectation value in coordi-nates x ′ i in the form similar to (13): h RW | T ki (cid:2) g ′ lm , ϕ (cid:3) | RW i = h ˜0 RW | T ki (cid:2) g ′ lm , ϕ (cid:3) | ˜0 RW i + h T ki (cid:2) g ′ lm , ϕ (cid:3) i (b) , (20)where h ˜0 RW | T ki [ g ′ lm , ϕ ] | ˜0 RW i is the vacuum expectation value in the RW spacetime withoutboundaries and the part h T ki [ g ′ lm , ϕ ] i (b) is induced by the boundary (9). Conformally trans-forming the Rindler results one finds h ˜0 RW | T ki (cid:2) g ′ lm , ϕ (cid:3) | ˜0 RW i = [ ξ/a ( η )] D +1 h ˜0 R | T ki [ g R lm , ϕ R ] | ˜0 R i + h T ki (cid:2) g ′ lm , ϕ (cid:3) i (an) , (21) h T ki (cid:2) g ′ lm , ϕ (cid:3) i (b) = [ ξ/a ( η )] D +1 h T k (R) i i (b) . (22)Under the conformal transformation g ′ ik = [ a ( η ) /ξ ] g R ik , the field ϕ R is changed by the rule ϕ ( x ′ ) = [ ξ/a ( η )] ( D − / ϕ R ( x ′ ) . (23)Now by comparing boundary conditions (12), (16) and taking into account Eq. (23), one obtainsthe relation between the coefficients in the boundary conditions: a ( η ) A/B = bA R /B R + (1 − D ) / . (24)As it is seen from this relation, the Dirichlet boundary condition in the problem on the RW bulk( B = 0) corresponds to the Dirichlet boundary condition in the conformally related problemfor the Rindler spacetime. For the case of the Neumann boundary condition in the RW bulk( A = 0) the corresponding problem in the Rindler spacetime is of the Robin type with bA R /B R =( D − / x i in the form ofthe sum of purely RW and boundary parts: h RW | T ki [ g lm , ϕ ] | RW i = h ˜0 RW | T ki [ g lm , ϕ ] | ˜0 RW i + h T ki i (b) . (25)By using the relations (8) for the purely RW part one finds (for the vacuum polarization in RWspacetimes see [5, 15, 16] and references therein) h ˜0 RW | T ki [ g lm , ϕ ] | ˜0 RW i = 2 a D [ a ( η )] − D − (4 π ) D/ Γ( D/
2) diag ( − , /D, . . . , /D ) + h T ki [ g lm , ϕ ] i (an) . (26)In particular, for D = 3 we have [5] h T ki [ g lm , ϕ ] i (an) = (3) H ki − (1) H ki / π , (27)where the expressions for the tensors ( j ) H ki are given in [5]. Now it can be easily checked thatfor the static case, a ( η ) = const, one has h ˜0 RW | T ki [ g lm , ϕ ] | ˜0 RW i = 0. In the special case of thepower-law expansion, a ( t ) = αt c , with t being the synchronous time coordinate, we find h ˜0 RW | T ki [ g lm , ϕ ] | ˜0 RW i = c ( c − c + 3) (3 c − π t diag( 3 c c − , , , . (28)5he corresponding energy density is negative for | c − | < √ x ′ i the spatial part is notisotropic and the corresponding part in coordinates x i is more complicated (no summation over l ): h T ll i (b) = [ ξ/a ( η )] D +1 h T l (R) l i (b) , l = 0 , , . . . , D, h T ll i (b) = [ ξ/a ( η )] D +1 h h T l (R) l i (b) + ( − l Ω sin θ ( h T i (b) − h T i (b) ) i , l = 1 , , (29) h T i (b) = [ ξ/a ( η )] D +1 Ω sin θ cos θ − rγr ( h T i (b) − h T i (b) ) . As we see the resulting energy-momentum tensor is non-diagonal. Note that for the case of thepower-law expansion the ratio of the boundary induced and boundary free parts at a given spatialpoint behaves as t − c ) in the model with D = 3. Hence, at early stages of the cosmologicalexpansion the boundary induced part dominates for c >
1. In figure 1 we have plotted theboundary induced parts in the vacuum energy density ( l = 0) and -stress ( l = 3) as functionsof the ratio ξ/b for D = 3 scalar field with Dirichlet boundary condition. The correspondingenergy density is positive in the region ξ < b and negative for ξ > b . In the case of Robincondition the energy density can be either negative or positive in dependence of the coefficientin the boundary condition. l=0 33 00.0 0.5 1.0 1.5 2.0 2.5 3.0 - - Ξ (cid:144) b a H < T ll > L H b L Figure 1: Boundary induced parts in the vacuum energy density and -stress for D = 3 Dirichletscalar field. The electromagnetic field is conformally invariant in D = 3. The vacuum expectation value ofthe energy-momentum tensor induced by the presence of conducting plate moving with uniformacceleration through the Fulling–Rindler vacuum is investigated in Refs. [6, 7]. We will assumethat the plate is a perfect conductor with the standard boundary conditions of vanishing ofthe normal component of the magnetic field and the tangential components of the electric field,evaluated at the local inertial frame in which the conductor is instantaneously at rest. As in thecase of a scalar field, the expectation value of the energy-momentum tensor is presented in the6orm (13), where the boundary free part is given by the formula h ˜0 R | T ki [ g R lm , ϕ R ] | ˜0 R i = 11240 π ξ diag ( − , / , / , / . (30)In the region ξ > b for the boundary induced part one has the expression [6, 7] h T k (R) i i (b) = − δ ki π b Z ∞ dx x Z ∞ dω (cid:20) I ω ( x ) K ω ( x ) + I ′ ω ( x ) K ′ ω ( x ) (cid:21) F ( i )em [ K ω ( xξ/b )] , (31)with the notations F ( i )em [ g ( z )] = ( − i g ′ ( z ) + [1 − ( − i ω /z ] g ( z ) , i = 0 , ,F (2)em [ g ( z )] = F (3)em [ g ( z )] = − g ( z ) . (32)The corresponding formula in the region ξ < b is obtained from (31) by the replacements I ω ⇄ K ω . By taking into account that [ I ω ( x ) K ω ( x )] ′ <
0, we see that h T i (b) > ξ > b and h T i (b) < ξ < b . For the perpendicular stress one has h T i (b) > ξ < b / ξ > b .The vacuum expectation value of the energy-momentum tensor in the RW bulk is presentedin the form (25), where the boundary free part is given by the expression h ˜0 RW | T ki [ g lm , ϕ ] | ˜0 RW i = 11 a − ( η )240 π diag ( − , / , / , /
3) + 62 (3) H ki + 3 (1) H ki π . (33)As in the case for a scalar field, this expectation value vanishes for the static RW spacetime.For the power-law expansion, a ( t ) = αt c , from (33) we find h ˜0 RW | T ki [ g lm , ϕ ] | ˜0 RW i = c (31 c + 54 c − c − π t diag( 3 c c − , , , − c ( c − π a ( t ) t diag( 3 cc − , , , . (34)For c < D = 3 and with h T k (R) i i (b) givenby (31) for the region ξ > b . For points near the boundary the leading terms in the asymptoticexpansions for the components of the energy-momentum tensor have the form h T i (b) ≈ − h T i (b) b sin θ ≈ h T i (b) b sin θ − ≈ − h T i (b) ≈ (1 − ξ/b ) − π a ( η ) , h T i (b) ≈ rγ − cos θr a − ( η ) b sin θ π (1 − ξ/b ) . (35)In the asymptotic term for the off-diagonal component r and θ are related by (9). Near theboundary the total energy-momentum tensor is dominated by the boundary-induced part.In the limit ξ → l ) h T i (b) ≈ h T i (b) ≈ − h T ll i (b) ≈ − . ξ/b ) π a ( η ) ln(2 b/ξ ) , l = 2 , , h T i (b) ≈ − b sin θ . ξ/b ) sin ( θ/ π a ( η ) r ln(2 b/ξ ) . (36)7his limit corresponds to large values of the coordinate r with the relation ξ/ξ ≈ [2 r sin ( θ/ − .Now we turn to the limit ξ/b → ∞ . In terms of the coordinates r and θ , this limit correspondsto large values of r and small values of θ with ξ/ξ ≈ r/ (cid:0) r θ + 1 (cid:1) . To the leading order forthe boundary induced part we have h T ki i (b) = a − ( η )96 ln ( ξ/b ) diag (1 , − / , − / , − / . (37)In this limit the total energy-momentum tensor is dominated by the boundary free part (33). In the investigations of the Casimir effect the calculation of the local densities of the vacuumcharacteristics is of special interest. In particular, these include the vacuum expectation value ofthe energy–momentum tensor. In addition to describing the physical structure of the quantumfield at a given point, the energy–momentum tensor acts as the source of gravity in the Einsteinequations. It therefore plays an important role in modelling a self-consistent dynamics involvingthe gravitational field.In the present paper we have investigated the vacuum expectation value of the energy-momentum tensor for scalar and electromagnetic fields induced by the boundary, defined byEq. (9), on background of RW spacetime with negative spatial curvature. For a scalar field theRobin boundary condition is imposed and for the electromagnetic field we have assumed thatthe boundary is a perfect conductor. In order to obtain the vacuum expectation values we haveused the corresponding results for a plate moving with constant proper acceleration throughthe Fulling–Rindler vacuum and the conformal relation between the k = − h T k (R) i i (b) defined by formulae (17) and (31) for scalar and electromagnetic fields in the region ξ > b . Thecorresponding formulae for the region ξ < b are obtained by the replacements I ω ⇄ K ω . In thecase of the electromagnetic field the boundary induced energy density is positive (negative) in theregion ξ < b ( ξ > b ). For the power-law expansion with a ( t ) ∝ t c , c >
1, at a given spatial pointthe ratio of the boundary induced and boundary free parts in the vacuum energy-momentumtensor behaves as t − c ) in the model with D = 3 and at early stages of the cosmologicalexpansion the boundary induced part dominates. Acknowledgments
A.A.S. was supported by the Armenian Ministry of Education and Science Grant No. 119.
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