Finite Temperature Casimir Effect in Randall-Sundrum Models

The finite temperature Casimir effect for a scalar field in the bulk region of the two Randall-Sundrum models, RSI and RSII, is studied. We calculate the Casimir energy and the Casimir force for two parallel plates with separation $a$ on the visible brane in the RSI model. High-temperature and low-temperature cases are covered. Attractiveness versus repulsiveness of the temperature correction to the force is discussed in the typical special cases of Dirichlet-Dirichlet, Neumann-Neumann, and Dirichlet-Neumann boundary conditions at low temperature. The Abel-Plana summation formula is made use of, as this turns out to be most convenient. Some comments are made on the related contemporary literature.


Introduction
Inspired by the Randall-Sundrum models [1], there has recently been considerable interest in the Casimir effect in higher-dimensional space. We may recall the characteristic features of this model: in the first variant, called RSI, one assumes that we are living on a (3+1)-dimensional subspace called a 3-brane, separated from an additional hidden brane by a bulk region. Only gravity is assumed to propagate in the bulk. The extra dimension is a circle S 1 with radius r c , represented by a coordinate φ in the range −π ≤ φ ≤ π. The hidden and the visible branes are located at φ = 0 and φ = π respectively. Imposition of Z 2 symmetry means that the points (x µ , φ) and (x µ , −φ) are identified. In the second variant of the model, RSII, the hidden brane pertaining to the RSII case may be derived by letting r c → ∞ in the RSI expressions. The use of a scalar field makes of course the situation more unphysical that what would be the case by assuming an electromagnetic field in the bulk. But we avoid the complications arising from photon spin in higher-dimensional spacetimes (for some recent papers in that direction see, for instance, Refs. [22,23,24,25,26,27]). The conflict is not even resolved when we have only one extra spatial dimension and spacetime is flat. On one hand we have e.g. Poppenhaeger et al. [27] and Pascoal et al. [26]. They find the electromagnetic Casimir force by multiplying the scalar field expressions by a factor p (to account for the possible polarisations of the photon) and subtract the mode polarised in the direction of the brane. On the other hand we have Edery and Marachevsky [25] who start out with decomposition of the five-dimensional Maxwell action. This conflict in not the central issue of this paper, and we avoid it by consider scalar fields only.
Our main purpose will be to calculate the Casimir free energy and the Casimir force for the RSI model, when there are two parallel plates with separation a on the visible brane. This is the piston model. Our main focus will be on the following points: 1) The calculation is given for arbitrary temperature T , and the lowtemperature and high-temperature limits are thereafter considered. The attractiveness versus the repulsiveness of the temperature corrections for different boundary corrections at low temperature are of definite physical interest and are therefore pointed out. We regularize infinite expressions by using zeta functions and the Abel-Plana summation formula, as this formula turns out to be better suited to the problem than the more commonly used Euler-Maclaurin formula.
2) We assume Robin boundary conditions on the physical plates at x = 0 and x = a. Usually, one has been considering the more simple Dirichlet conditions when working on this kind of problems, although very recently the Robin considerations have begun to attract attention [28,21].
As an introductory step, we consider in the next section the partition function and the free energy of a bulk scalar field. We discuss the distinction between even and odd fields, and consider also the mode localization problem. After a brief survey of the Abel-Plana formula in Section 3 we consider in Section 4 the Dirichlet-Dirichlet (DD), Neumann-Neumann (NN), and Dirichlet-Neumann (DN) boundary conditions in flat space, at finite temperature without extra dimensions. Our main topic, the temperature RSI case, is covered in Section 5, where the Casimir free energy and force are calculated for the different boundary conditions. A brief treatment of the RSII case is given in Section 6.
It should be recognized that the warp factor is an important element in the present problem. One might analyze instead the analogous higherdimensional cases taking spacetime to be flat. Considerable interest has been devoted to this simpler variant of higher-dimensional Casimir theories in recent years. See, for instance, Refs. [19,29,25,30,31,32,26], and further references therein.

Free energy of a bulk scalar field
To find the partition function for a non-minimally coupled scalar field Φ with mass m in the RSI model, we follow a Kaluza-Klein reduction approach [33], starting from the Lagrangian density Here G=det G M N (with M, N = 0, 1, 2, 3, 5) is the determinant of the 5D metric, R is the 5D Ricci scalar, ζ is the conformal coupling, and c hid/vis are the boundary mass terms on the branes. Throughout the article we use = c = k B = 1. We have introduced above a new position coordinate z such that |z| = (e k|rcφ| − 1)/k, implying that z r = (e kπrc − 1)/k. It is convenient to introduce also the quantity A(z) = 1/(1 + k|z|). The partition function can now be calculated, making use of the whereη µν = −δ µν is the metric in the coordinatesx µ , and where (τ = it)p The eigenfunctions are normalized as The partition function now takes the form (an unimportant factor omitted) where the sum goes over all eigenvalues of M N and p. Our next step is now to identify M N and p.

N
We start from Eq. (8), assuming Robin boundary conditions on the physical walls, with constants β 0 and β a referring to x = 0 and x = a. The forms above are as in Ref. [28]. Dirichlet and Neumann boundary conditions correspond to β = 0 and β = ∞, respectively. We assume eigenfunctions of the form with eigenvalues For a bosonic field at temperature T the Matsubara frequencies are Equation (13) leads to the following constraint on α, whereas Eq. (14) yields Consider next the eigenfunctions ψ N (z). We insert the expression (6) into Eq. (9), take into account that the Ricci scalar for the RS metric is and change the position coordinate in the bulk back to y using d/dz = A(y)d/dy. Then, The solution is (we consider the region 0 < y < πr c only) where ν = 4 + (m/k) 2 − 20ζ and C N a normalization constant. This is the same result as in Ref. [5], except that we include curvature (ζ = 0) in our model. One should now distinguish between even fields satisfying ψ N (−y) = ψ N (y), and odd fields satisfying ψ N (−y) = −ψ N (y). The behavior may be summarized as follows: • Even scalar fields obey the Robin BC on the branes. If the field is minimally coupled (ζ = 0) and there is no mass boundary term (c brane = 0), the boundary condition reduces to the Neumann BC, ψ ′ N (y)| brane = 0.
• Odd scalar fields obey the Dirichlet BC on the branes.
These two cases may be combined: we introduce the two functions and let now z mean z = e kπrc M N /k (not to be mixed up with the coordinate z in Sect. 2), and d = e −kπrc . Then we can write the general BC as This is in accordance with Ref. [35] in the case of minimal coupling, if we choose c hid = −c vis = 2α/k. Special attention ought to be given to the massless case, M N = 0. For fields with m 2 − 20ζk 2 = 0 there is no solution of Eq. (21) with M N = 0 satisfying the Robin BC on both branes. For an even field with m 2 −20ζk 2 = 0 with no boundary mass term the situation is different, as ψ 0 =const is a solution of Eq. (21) and also satisfies the boundary condition which in that case is the Neumann BC. The M N = 0 case has important consequences for the Casimir force from a bulk scalar field. This is related to the localization problem for the Kaluza-Klein modes in general. In RSI, the massless mode is localized near the hidden brane at y = 0. In RSII the situation is reversed, as the massless mode is localized near the visible brane at y = 0 and the massive modes are delocalized. The reader may consult Refs. [12,36] for a discussion as to what weight to be given to the massless modes in RSI due to the fact that it is localized near the hidden brane only.

Approximate expressions for the masses
We assume d = e −kπrc ≪ 1 but keep z = e kπrc M N /k arbitrary, to find convenient approximative expressions for the Kaluza-Klein masses. As in this case j brane The situation can be divided into two classes: (i) For Dirichlet BC (β brane = 0) it follows that we need the zeros of J 2 (z). Making use of the large-z approximation J ν (z) ∼ (2/π) 1/2 cos(z − 1 2 νπ − 1 4 π) we find that the expression is useful for practical purposes.
(ii) For non-Dirichlet BC (β brane = 0) we obtain from Eq. (23) leading approximately to The formula is good for kβ brane > 1 and becomes better for higher N. As an example, choosing kβ brane = 10 3 , d = 10 −12 , ν = 2, we find the numerical error of the zeros to be about 4% when N = 3 and around 1% when N = 5. As we will see later the first (i.e. smallest) values of M N are the most significant for the Casimir force in RSI.

Two expressions for the free energy
From Eq. (12) we obtain for the free energy where k 2 ⊥ = k 2 y + k 2 z , V ⊥ is the transverse volume, ǫ l = 2πT l, and the summations over k x and M N go over all real zeros of the functions F x (k x ) (Eq. (19)) and F N (z) (Eq. (24)).
By making use of the zeta function following [34], we can re express F as where µ is an arbitrary parameter with dimension mass. We now derive the classical expression for F using that the Mellin trans- Applying the Poisson summation formula (details omitted) we can then derive Further manipulations lead us to the desired expression One may note here that a boson with energy E p contributes with (β = 1/T ) to the total partition function [35]. Summing over all energies we obtain the classical expression corresponding to Eq. (34). The expression of the free energy of a scalar bulk field is equal to that of bosons with energy where M N is the masses found in Section 2.1. For M N = 0 this is free energy of a scalar field in Minkowski (i.e. flat) spacetime without extra spatial dimensions. By letting T → 0 we find the zero-point energy Again, we observe that M N = 0-term in the sum correspond to the familiar expression for 3+1-dimensional Minkowski spacetime. Another expression for F can be derived which in our context is more useful, in view of our application of the Abel-Plana summation formula later on. We start from the expression (31), introduce a generalized polar coordinate transformation along the same lines as in Ref. [37], and integrate over all angles. We then obtain (the limit s → 0 is understood) where C is defined as The integral is solved using the variable change x = r 2 /C and leads essen- This is the finite-temperature form that we will use below. The corresponding zero-temperature form is found by a limiting procedure to be From now on we will set V ⊥ = 1. Hence E, F and P (force) refer to respectively energy, free energy and force per unit area of the physical plates.

The piston model
Before finding explicit expressions and specifying BCs we introduce the piston model. The model has attained a great deal of attention [38,39,29,40,41,21]. We introduce the piston (Figure 1) with the same notation as in Chapter 4.3 of [42]. Instead of only using the free energy F I of cavity I as the Casimir free energy, we use Initially the system is in an unstressed situation where the cavities have size X/η and X(1−1/η). Then we shift the middle plate so that the lengths of the two cavities are a and X − a; the system is now in a stressed situation. The Casimir free energy is the sum the free energies of two cavities in the stressed case (I and II) minus the free energies of the cavities in the unstressed case (III and IV). The constant η is ∼ 2, characterizing the unstressed situation.
In the end we let X → ∞ and effectively remove the rightmost plate from the setup. In the piston model all terms independent or linear in a vanish, hence from now on we will discard all such terms.
3 Casimir free energy and force: Initial remarks

The Abel-Plana formula
We want to find a more explicit expression for the Casimir free energy, one we can evaluate numerically. Thus all the summations over k x and M N from Eq. (39) need to be taken care of. Instead of using Eq. (39) we look at the complex function which reduces to the free energy in Eq. (39) when s = −1. The function F (s) is well-defined for large, positive Re(s) and we analytically continue it to the whole complex plane. Together with F (s) we will use a variant of the Abel-Plana formula [43,28] especially suited for plates with Robin BC. Here, z n denotes the n'th zero in the right half of the complex plane of the complex function F x (z = ak x ) in Eq. (19). From Eqs. (18) and (19) we can find the relation the left hand side of Eq. (43) matches the sum over k x in F (s). The notation j=0,a means there are contributions from both the left (j = 0) and the right (j = a) plate.

Application of the Abel-Plana formula
We can divide the free energy F at arbitrary temperature T into two separate parts, . For a massive scalar field there is no massless mode (M N = 0) at all. For a massless field, even and minimally coupled, there is an M N = 0 mode. Recall that F (M N = 0) yields the same expression as the free energy of the massless scalar in Minkowski spacetime. To find the Casimir energy and force for such a field one can simply add the massless mode term. It is natural therefore to analyze the M N = 0 mode separately. The formal expressions are divergent, and will be regularized by the use of zeta functions.
Let now M N be arbitrary. Insert the expression (45) into (43), and divide the sum into three separate parts as indicated by the underlines 1,2, and 3. We do not give the details here, as the formalism is analogous to that of Ref. [28], pertaining to the zero temperature case. The free energy can be written as the sum of three parts: one part F N P as the contribution when no plates are present, one part F j as the vacuum free energy along the transverse directions induced by the plates at x 0 = 0 and x a = a respectively, and a remaining part ∆F . Thus The two first terms do not refer to the gap width a, or are linearly dependent on a, and do not contribute to the free energy in the piston model. The last term ∆F , henceforth called simply F , is the term of physical importance. It is precisely the term corresponding to underline 3 in Eq. (43). We give this expression explicitly: We can now use this expression as basis for discussing special cases: DD, NN and DN boundary conditions. We first consider flat space with no additional spatial dimensions.

DD, NN and DN boundary conditions in flat space with no extra dimensions
To demonstrate the procedure used for finding the Casimir free energy and force we look at the well known case: A massless, scalar field in flat spacetime (Minkowski metric) and no extra spatial dimensions. An additional motivation for including this section is that F (M N = 0) = F M ink , as mentioned earlier.
Consider first the general formalism. With DD or NN boundary conditions we obtain, when making use of the substitution z = xa M 2 N + ǫ 2 l , We expand the denominator and use the relation In the limit s → −1 we use the property Γ(x) sin πx = π/Γ(1 − x) to get the free energy for arbitrary T The same expression follows if one makes use of zeta regularization. The Abel-Plana formula is powerful in the present context, as it is easily adjustable to different choices for the boundary conditions. Consider next flat space. With M N = 0 we obtain from Eq. (50) The first term corresponds to l = 0, and is derivable for instance by taking into account the properties of K ν (z) for small arguments. For high temperatures, aT ≫ 1, the expression (51) is suitable. The first term is the dominant one, as the K ν terms decrease for increasing temperatures.
For low temperatures, aT ≪ 1, some rewriting is however necessary. We go back to the complex function which corresponds to Eq. (39) when M N = 0, s = −1. Splitting off the l = 0 term and using again the Mellin transform (32) we can write F (s) as Here and S 2 (t) is the function possessing the property [42] S 2 (t) = − 1 2 The first of the three terms coming from the rhs of Eq. (56) cancels F l=0 , leaving The first term here is recognized as the zero-temperature energy, F (T = 0) = E. With s = −1 we find We can now make use of the Abel-Plana formula (43), choosing for the function f (z) the form This leads to, when omitting terms not contributing to the piston model, We once more use the Mellin transform, but this time choosing S 2 (4a 2 t) together with Eq. (56). Some calculation leads to the final expression where P DD,N N M ink (T = 0) = −π 2 /(480a 4 ). The Casimir energy and force are equal to the zero temperature expressions plus correction terms, the latter decaying exponentially as T → 0.
In Eq. (62) we may insert the asymptotic expansion for large arguments, K ν (z) = (π/2z) 1/2 e −z [1 + (4ν 2 − 1)/8z]. It is of interest to extract the dominant term in the correction, corresponding to n = l = 1. Approximately we then get The physically important point here is that the finite temperature term is positive, corresponding to a repulsive force correction (recall that we are considering aT ≪ 1). The situation is in some sense similar to that encountered in earlier studies when calculating the Casimir force between two parallel metallic slabs in physical space, assuming the Drude dispersion relation for the material: also in that case the finite temperature effect was found to weaken the attractive T = 0 force [44]. We now consider the third class of BC's mentioned above: assuming Dirichlet boundary conditions on one plate and Neumann on the other we find The steps are similar to those of the DD and NN calculations, only with a factor (−1) n due to the positive sign in the denominator and accordingly E DN M ink = −7/8E DD,N N M ink . The free energy density with DN boundary conditions becomes which is a convenient form for the case of high temperatures. For low temperatures we obtain by a similar reasoning as that given above, (−1) n n l Again extracting the dominant term by including only n = l = 1 we get approximately The correction term is the same as in Eq. (63), but with the opposite sign. The thermal correction is attractive.

DD, NN and DN boundary conditions in RSI
Consider first the high-temperature regime. Whereas in flat space this corresponds to aT ≫ 1, in RSI the natural choice for high temperatures is T ≫ ke −kπrc . Recall that the lowest values of M N are ∼ ke −kπrc ; this implies aT ≫ 1 since ake −kπrc ≫ 1 for all relevant distances in physical space. In this limit Eq. (50) is a suitable expression for the free energy and the Casimir force is After some rewriting we find this in accordance with Eq. (23) in [20]. We need only to include the E(M N = 0) term to get the Casimir force for a massless scalar instead of a massive. We find that the high temperature limit is valid for T ≫ 10 16 K. Only temperatures much less than these are expected to be of physical importance. It is most natural therefore to find the Casimir energy and force for T ≪ ke −kπrc . Note that the brane low-temperature condition does not fix the magnitude of the product aT relative to unity. With k ∼ 10 19 GeV, e −kπrc ∼ 10 −16 we only get the weak condition T ≪ 10 3 GeV. As an example, choose T = 300 K (2.6×10 −11 GeV), a = 1µm, from which it follows that aT = 0.15. In most cases of practical interest we will have aT ≪ 1, although one can easily consider cases where aT ≫ 1, still compatible with the condition T ≪ ke −kπrc .
Using the same procedure as in flat space we find the RSI equivalent to Eq. (58), (71) We can differentiate this expression to find the Casimir force. By assuming ∂k x /∂a = −k x /a we get Eq. (17) in [21], only missing the first term. The assumption holds for all k x proportional to 1/a, which is the case for DD, NN and DN BC. However, we are using the piston model, and a term like the first one of Eq. (17) in [21] should not occur in the Casimir force. The force term in question is independent of a, and thus corresponds to a free energy term that is linear in a. As a consequence of using the piston model, all terms independent or linear in a must be removed from the free energy. Although the metric in [21] does not include the warp factor e −2krcφ the expressions are the same, since the warp factor only affects the values of the M N s. In Eq. (71) E RSI is the zero temperature energy in RSI and is found from Eq. (40) using the Abel-Plana formula (43) with After some variable changes the energy reads (73) and for DD and NN boundary conditions it simplifies to The Casimir force at zero temperature is This is in accordance with [28] through that paper does not consider the Casimir effect rising from a bulk scalar in the RS model in particular. Inserting the approximation Eq. (28) for M N we see that the energy is essentially the same as in [10]. There are three minor differences. First of all the energy in [10] has some extra terms linear and independent of a since the piston model is not used. Secondly, factors p are included to make the expression hold for electromagnetic fields, where p is the polarizations of the photon. The last difference is a factor of 2 included to account for 'the volume of the orbifold'. Since we can not see how this factor occurs, it is not included. This is also equal to Eq. (26) in [20]; only it contains the E(M N = 0) term since a massless field is considered.
The summation over k x in Eq. (71) is still left and can be done using the Abel-Plana formula with (76) After inserting K 3 2 (z) = (π/2z) 1/2 e −z (1 + 1/z) the free energy reads We continue by inserting the β's for DD and NN boundary conditions, expansion of the denominator and the variable exchange x = z/a to get Integrals of this form is solved in Appendix A with the result giving the free energy and force This expression, belonging to the low-temperature regime T ≪ ke −kπrc , can be used both for aT ≪ 1 and for aT ≫ 1. The argument of the Bessel functions will always be large since ake −kπrc ≫ 1 for all relevant distances. The correction terms to the zero temperature energy and force expressions are small. The expression has to our knowledge not been given before. The leading term for the force in terms of T /M N is In contrast to flat space, both zero temperature Casimir force and the thermal correction is negative. Hence the Casimir effect in RSI is stronger in the low temperature limit (T ≪ ke −kπrc ) both for aT ≪ 1 and aT ≫ 1. The DN expressions deviate from the DD and NN expressions in RSI in the same way as in flat space. The factor of (−1) n must be included in sum over n, where the sum over n originates from the expansion of the denominator in Eq. (73) and (77).

Comparison to flat space
The reason for calculation the Casimir force in RSI is to find out where there are deviations from the Casimir force in flat spacetime without extra spatial dimensions. For an easier comparison we give the full expression for the Casimir force of a massless bulk scalar in RSI with DD/NN BCs. When can we see a deviation from the ordinary Casimir force? Looking at Eq. (83) we see that at zero temperature we need aM N ∼ 1 for noticeable difference. We know that M N ∼ ke −kπrc for low N, and k is usually set to ∼ M P l ≈ 10 19 GeV in RSI. In the original paper of Randall and Sundrum they propose to choose kr c ∼ 10 in order to solve the hierarchy problem. With these values we find that a is ∼ 10 −21 m. There is no point in looking at distances smaller than the size of an atom. Only distances of physical relevance (> 1nm) are of interest. In Figure 2 we keep k = 10 19 GeV, but choose e −kπrc = 10 −26 . The difference from RSI to ordinary Casimir force F M ink at the zero temperature is given in Fig. 2. By choise of parameters the magnitude of the correction in RSI is of the same order of magnitude as F M ink , given by the red line in Fig. 2 (a). With smaller value of kr c we will not se any difference at separations larger than 1nm. The corresponding size of the extra dimension is r c ≈ 10 −35 m. In Fig. 2 (b) we se the ratio of this difference to the Casimir force at zero temperature. We observe that the extra term we get in RSI, goes faster to zero than F M ink . Now we turn to relevant values of the temperature. The choices of r c , k and a are still the same, and after some testing it turns out that a max T ∼ 3 is suitable. To be sure that the sums have converged we let both n and l run to 30. The result is presented in Fig. 2. The green line is the P RSI − P M ink for T = 3 × 10 6 K and we see that this gives a stronger Casimir force than at zero temperature.

Comparison to flat space with one extra dimension
In the previous section we compared the Casimir force in RSI with the Casimir force for a massless scalar i ordinary flat 3+1-spacetime. In this section we will look at a higher dimensional spacetime which is flat, i.e. no warp factor in the metric. As mentioned in the introduction, this topic has gained a lot of interest lately. The extra dimension is a torus with circumfence 2πL. In this case M N = N/L, with N = 0, ±1, ±2, . . .. With these values for the M N s instead of the ones we have in RSI we see that Eq. (70) is equal to the high temperature expression in [19]. However we have not found an expression corresponding to Casimir force in Eq. (83). Hence, we will derive such an expression using the Chowla-Selberg formula in [45]. With the new values of the M N s, the function F (s) (for DD BCs) reads Rewriting this to homogenous Epstein zeta functions we find The notation ∞ k 1 ,k 2 ,...kp=−∞ means that for k 1 to k p we sum from −∞ to ∞. The prime ′ behind the sum means that the term k 1 = k 2 = . . . = k p = 0 is omitted. The Chowla-Selberg formula is Z E,p (s; a 1 , . . . , a p ) = Z E,m (s; a 1 , . . . , a m ) After use of this with m = 2, we put s = −1 and remove all terms linear and independent of a. Then we see that the free energy is The first term can be identified as F (M N = 0) and the second term is equal to Eq. (80) (when M N = N/L). From ∞ n,l=−∞ ′ we get the factor 4 when we let the sums over n and l run from 1 to ∞. The term l = 0, but n =, 0 gives the zero temperature expression and n = 0 with l = 0 is independent of a and should be removed in the piston model.
Since Eq. (80) leads to Eq. (83) we can conclude that we get the same answer with the Chowla-Selberg formula as with the Abel-Plana formula in flat spacetime with one extra spatial dimension. However, the Abel-Plana formula can be used regardless of the values of M N , while the Chowla-Selberg formula is only useful when we can rewrite our expressions to homogenous Epstein zeta functions. The second advantage of the Abel-Plana formula is that different boundary condititions can easily be obtained.

DD, NN and DN boundary conditions in RSII
In RSII the Kaluza-Klein modes are continuous and we must replace the sum over M N with an integral, In Eq. (50) we get an integral on the from The derivation of this formula is given in Appendix A. We find that the free energy and force in RSII are and These expressions are convenient for the high temperature limit aT ≫ 1. In RSII there are only two temperature regimes, aT ≫ 1 and aT ≪ 1, since M N is continuous. To find the low temperature limit we insert Eq. (89) into Eq. (80) and use the integral The free energy reads where E DD,NN RSII = − 3ζ R (5) 128πka 4 is the zero temperature energy in RSII. We can find this energy from e.g. Eq. (74) by making use of Eq. (93). We use the Mellin transform as in low temperature, flat spacetime (with S 2 (4a 2 t)) and find The Casimir force is with P DD,NN RSII (T = 0) = − 3ζ R (5) 32πka 5 . In the low temperature limit aT ≪ 1 we get from Eq. (96) the dominant term corresponding to n = l = 1, The temperature correction term is repulsive. As in RSI the only difference between DD/NN BC and DN is a factor (−1) n in the sum over n.

Concluding remarks
Our main objective has been to calculate the finite temperature Casimir effect for a scalar field residing in the bulk in the two Randall-Sundrum models, RSI and RSII. Two parallel plates are envisaged, with gap a, located on one of the RS branes. We have given most attention to the RSI model. Robin boundary conditions, cf. Eqs. (13) and (14), are assumed on the two plates. The geometrical picture is the piston model, as illustrated in Fig. 1. We have made use of the Abel-plana summation formula throughout, as this turns out to be the most convenient choice in the present context.
In the case of flat space the basic expressions for Casimir free energy and force (per unit surface area) are worked out in the form of series in Sect. 4, both for high and for low temperatures. A characteristic feature for DD and NN boundary conditions on the two plates is that the dominant part of the finite temperature correction term for low temperatures (aT ≪ 1) is repulsive.
That is, the force becomes decreased slightly when the temperature increases from zero. In this sense the behavior is analogous to that encountered in the case of conventional Casimir theory for metallic slabs in physical space when the dispersive relation for the material is taken to have the Drude form [44].
The RSI model is covered in Section 5 in an analogous way. The dominant term in the Casimir force shows the characteristic property to strengthen the zero temperature effect instead of weaken it as in RSII. From Eq.(81) we can evaluate the Casimir force both for aT ≪ 1 and aT ≫ 1, provided T ≪ ke −kπrc . This section also covers a comparison to flat space with and without a compactified extra dimension.
In Section 6 the RSII model is considered. We have ∞ 0 dM/k = π/k ∞ −∞ dM/(2π) and thus the Casimir force has a characteristic π/k times the Casimir force of a 4+1-dimensional Minkowski spacetime. This is pointed out by Morales-Técotl et al. [12], but as an argument against using zeta functions in the regularization and rather use Green's functions. It is not only the regularization method that is different, the physical picture is also differs from this article and the work by Frank et al. [10,11]. While Frank et al. calculate the free energy of a slice of the bulk, Morales-Técotl et al. try to restrict the system to the brane by evaluating the Green's function at the visible brane (y = 0 for RSII). In this way they claim to incorporate the localization properties of the modes of the scalar bulk field (ψ N (y)). Note that the Green's function method includes a integral over y and by setting y = 0, Morales-Técotl et al. thus remove the y-dependence of a part of the integrand before integrating. However, keeping the y-dependence before the integration makes one obtain results different from those of Frank et al.. The delicate point is how to include the localization properties of the modes. In Ref. [36] it is proposed to resolve the issue by changing the boundary conditions. The problem with the localization properties of the modes needs to be resolved before the Casimir effect form an electromagnetic field can be considered.