Casimir effect in PostNewtonian gravity with Lorentzviolation
Abstract
We study the Casimir effect in the framework of standard model extension. Employing the weak field approximation, the vacuum energy density \(\varepsilon \) and the pressure for a massless scalar field confined between two nearby parallel plates in a static spacetime background are calculated. In addition, through the analysis of \(\varepsilon \), we speculate a plausible constraint on the Lorentzviolating term \({\bar{s}}^{\,00}\) which is lower than the bounds currently available for this quantity. Finally, we remark that our outcome has an intrinsic validity that goes beyond the treated case of a pointlike source of gravity.
1 Introduction
It is well known that General Relativity (GR) exhibits serious incompatibilities when it comes to link its domain of validity with the realm of Quantum Field Theory. The hope is to overcome these obstacles, so that it would be possible to describe the behavior of any interaction including gravity even when quantum effects are not negligible. In this direction, many reasonable and solid proposals have been made (such as Loop Quantum Gravity and String Theory), but the sensation is that there is still a great amount of conceptual problems and obstacles to overcome. However, there is a shared and accepted awareness that allows us to look for a unified theory at Planck scale (namely \(m_P\simeq 10^{19}\) GeV); this fact may not be surprising from a theoretical point of view anymore, but practically it means that it is impossible to detect even the smallest signal of quantum gravity. In other words, experiments do not provide any criterion to discern whether a physical argument can be rejected or not at those energy levels.
Nevertheless, even in current laboratory tests, there is still the possibility to search for little traces that can be directly related to an underlying unified theory, and one of the most important concepts in this perspective is represented by Lorentz symmetry breaking. Such a violation is highly recurrent in many candidates of quantum gravity, and for this reason it is considered an essential notion to take into account for a natural extension of our knowledge in such an unknown domain. Actually, Lorentz violation has widely been used as one of the main bedrocks on which to develop physics beyond the Standard Model (SM). SME (Standard Model Extension [1, 2, 3, 4]) is therefore born within this environment, and it is considered one of the most important effective field theories that includes SM as a limiting case. The intuition at the basis of SME comes from the study of covariant string field theory [5]. The idea is to build all possible scalars of the SME Lagrangian by contracting SM and gravitational fields with suitable coefficients that induce Lorentz (and CPT) violation. Of course, we expect these coefficients to be heavily suppressed, and thus to be considered extremely small if analyzed at current scales. However, many focused experiments have been performed to put constraints on their values and to gather useful information on them [6]. For an accurate overview on SME, see Ref. [7].
In this paper, we consider the Casimir effect in curved spacetime, where the metric is deduced by the gravitational sector of the SME Lagrangian. Generally speaking, the Casimir effect [8, 9] arises when a quantum field is bounded in a finite space. Such a confinement reduces the modes of the quantum field producing, as a consequence, a measurable manifestation. The Casimir effect has been studied in flat spacetime in great detail [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], showing the robustness of the assumptions at the basis of its theoretical explanation. In this framework, there are already works that study the Casimir effect with the contribution of the SME coefficients for the fermion and photon sector [22, 23]. In some recent papers [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], instead, the analysis of the role of a gravitational field in the vacuum energy density of a quantum field inside a cavity has been performed. This opened the doors to many interesting developments and lines of research. Indeed, an interesting investigation on the consistency between the Casimir energy and the equivalence principle is conduced in Refs. [35, 36, 37, 38, 39]. Moreover, possible modifications in the vacuum energy could become relevant in the dynamics of the universe [26, 40]. Microscopically, modifications of Casimir’s energy could be crucial in the context of quark confinement based on string interquark potentials [41, 42, 43, 44]. Finally, the implications of gravity on Casimir effect deal with the open issue regarding the limits of validity of GR at small distances [45].

To see how the pressure between the Casimir plates changes in the above context;

To obtain a “heuristic” and plausible constraint on SME Lorentzviolating terms derived within a PostNewtonian expansion of the metric tensor describing the spacetime in proximity of a pointlike source of gravity [46]. Such a result is obtained by considering the vacuum energy density \(\varepsilon \), and it is then compared to a more physical bound that stems from the study of the pressure.
The paper is organized as follows: in Sect. 2 the metric tensor is presented, as derived in the PostNewtonian approximation of the purely gravitational sector of SME in Ref. [46]. In Sect. 3 the dynamics of a canonic massless scalar field is studied within the Casimir plates, and in Sect. 4 the bound and the expression for the pressure are obtained. Discussions and conclusions are given in Sect. 5.
2 Metric tensor for a pointlike source with Lorentzviolating terms
3 Dynamics of a massless scalar field
Let us now consider a conventional massless scalar field \(\psi \left( {\mathbf {x}},t\right) \) in curved background, i. e. we consider the SME parameters are only into gravity sector. In general, the \({\bar{s}}^{\mu \nu }\) parameters can be moved from the gravity sector into the scalar sector using a coordinate choice [3]. The choice does not change the physics, so although the calculation looks different it must give the same result.
As it can be seen in Fig. 1, the configuration is simple: the plates are set in such a way that the one nearer to the source of gravity is distant R from it, and hence we can choose Cartesian coordinates so that \(r=R+z\), where the variable z is free to vary in the interval \(\left[ 0,\,D\right] \), if we denote with D the separation between the plates. Clearly, the relation \(D\ll R\) holds.
At this point, it is clear that the interest is focused on the variation of the field along the radial direction (namely, along the zaxis). Because there is no explicit dependence on other coordinates, one can think of a solution of the form \(\psi \left( {\mathbf {x}},t\right) =Ne^{i\left( \omega t{\mathbf {k}}_{\perp }\cdot {\mathbf {x}}_{\perp }\right) }\varphi \left( z\right) \), where \({\mathbf {k}}_{\perp }=\left( k_{x},k_{y}\right) \), \({\mathbf {x}}_{\perp }=\left( x,y\right) \) and N is the normalization factor.
4 Derivation of the pressure and of the bound
Equation (16) gives us the expression of Casimir vacuum energy density at the second order \({\mathscr {O}}(R^{2})\) in the framework of SME. We note that the part related to GR (Eq. (18)) does not have contributions at the first order in \({\mathscr {O}}(R^{1})\), but only at higher orders, such as \({\mathscr {O}}(R^{2})\). The Lorentzviolating sector Eq. (19), instead, exhibits a first order factor in \({\mathscr {O}}(R^{1})\) connected to \({\bar{s}}^{\,00}\).
To obtain a plausible bound on \({\bar{s}}^{\,00}\), we make the assumption \(\varepsilon _{LV}\lesssim  \varepsilon _{GR}\). This agrees with several considerations and results expressed in Refs. [6, 22, 46] and ensures the fact that Lorentzviolating manifestations are small, as widely employed in Lorentzviolation phenomenology [46]. However, the reasonableness of the constraint we derive cannot be directly tested, since \(\varepsilon \) is still now an unmeasurable quantity. This is why we need to compare the heuristic constraint with a physical one, which can only be calculated using the pressure.
It must be pointed out that recent developments in nanotechnology can further strengthen the above bound by one or two orders of magnitude. In fact, in the near future, the value of \(D_p\) could reach scales even smaller than nanometers (as already contemplated, for example, in Ref. [56]), thus transforming Eq. (20) into a more stringent constraint, \({\bar{s}}^{\,00}\lesssim 10^{15}\).
Note that the result Eq. (20) is achievable also if we require the inequality \(\leftP_{LV}\right\lesssim \leftP_{GR}\right\) to hold. Indeed, we obtain exactly the same order of magnitude for the upper bound of \({\bar{s}}^{\,00}\), even when the heuristic approach carried out for the mean vacuum energy density applies to the pressure. However, the above consideration on the elusive manifestation of Lorentz violation acquires a substantial meaning for the case of P, since such a quantity is measurable. In this perspective, the expression of Eqs. (23) and (24) for the contribution to the pressure of GR and SME, respectively, further strengthen our plausible constraint on the Lorentzviolating factor.
The total absolute experimental error of the measured Casimir pressure [57] is 0.2% (\( \delta P/P_0\simeq 0.002\)). Typical values of the ratio \(\frac{R}{R_S}\) in the Solar System are included between \(10^7\div 10^{10}\). In particular, for the Earth we have \(7.2\times 10^{8}\), which means that the term on the r.h.s. of Eq. (25) is of order \(10^{6}\). The comparison of such a result with Eq. (20) clearly shows that we cannot use^{4} the Casimir experiment to measure the pressure in order to significantly constrain the parameter \({\bar{s}}^{\,00}\). To to this, we need to enhance the experimental sensitivity on Earth by at least six order of magnitude, in such a way that \(\frac{\delta P}{P}\lesssim 10^{9}\).
Finally, a word must be spent on the choice of pointlike source of gravity. Although it has been considered only to simplify the treatment of the problem, the relevance of the outcome does not depend on it. In fact, even if we considered a rotating spherical object instead of a point, the value of the constraint would basically be the same. Lorentzviolating factors will very likely appear as combinations of the \({\bar{s}}^{\mu \nu }\) already introduced in this work, but the line of reasoning would exactly be the same. Consequently, we expect Eqs. (20) and (25) to be modified only in its l.h.s., for example with contributions such as \({\bar{s}}^{\,00}+\sum _{\mu ,j}{\bar{s}}^{\mu j}\) at the lowest order.
5 Conclusions
In the context of SME, working in the weak field approximation, we have studied the dynamics of a massless scalar field confined between two nearby parallel plates in a static spacetime background generated by a pointlike source. In order to obtain a reasonable constraint on Lorentzviolating terms in the context of the Casimir effect, we have derived the corrections to the flat spacetime Casimir vacuum energy density Eq. (16), in the framework of SME. We have found that, both in the energy density and in the pressure, GR gives us only contributions at the second order \({\mathscr {O}}(R^{2})\), while Lorentzviolating corrections occur at first order \({\mathscr {O}}(R^{1})\). After that, we have evaluated the pressure Eq. (21) to observe how it changes from the usual expression in flat spacetime, Eq. (22), in the presence of gravity (see Eq. (23)) and with SME coefficients (see Eq. (24)).
By requiring \(\varepsilon _{LV}\lesssim  \varepsilon _{GR}\) but also \(\leftP_{LV}\right\lesssim \leftP_{GR}\right\), we have then been able to find a significant bound on the SME coefficient \({\bar{s}}^{\,00}\). Such an assumption is related to the fact that manifestations of Lorentz violation in nature are expected to be extremely evanescent. If the above inequality did not hold true, it would have been possible to detect traces of Lorentzviolating terms in the tests proposed in Ref. [46] and in other experiments involving the intertwining between SME and gravity, but this is not the case.
We remark that, as already pointed out, the true measurable physical quantity in the context of the Casimir effect is the pressure P. One can possibly extract a constraint for \({\bar{s}}^{\,00}\) also with P as done in Eq. (25), but its order of magnitude would be extremely high if compared to Eq. (20) and especially to the data of Ref. [6].
We also point out that, following the same analysis of Refs. [35, 36, 37, 38, 39], no violation of equivalence principle arises in our framework, i.e. the parameter space here analyzed leads to the conclusion that the coefficients \({\bar{s}}^{\mu \nu }\) do not allow to discriminate between inertial mass and gravitational mass.
Footnotes
 1.
See Ref. [53] for a detailed explanation of this concept.
 2.
See “Appendix A” for a complete treatment.
 3.
The presence of Lorentzviolating terms allows for a nonvanishing scalar curvature. However, details of its form will not be necessary in the next steps, since it contributes to the mean vacuum energy density only at higher orders.
 4.
Unless we believe the heuristic bound to be true and thus physically consistent.
 5.
In this step, we have neglected an assumption of Ref. [46] which is irrelevant for our purposes.
 6.
Without affecting the generality of the studied problem, as already remarked in the Conclusions.
Notes
Acknowledgements
The authors would like to thank V. A. Kostelecký and E. Calloni for useful discussions and comments, and the anonymous referee for observations that improved the manuscript.
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