# Casimir effect in an axially symmetric spacetime with unparticles

## Abstract

We investigate the Casimir effect of the massless scalar field in a cavity formed by ideal parallel plates in the spacetime generated by a rotating axially symmetric distribution of vector or scalar (tensor) unparticles, around which the plates orbit. The presence of the unparticles is incorporated to the background by means of a correction to the Kerr solution of the Einstein equations, in which the characteristic length and the scale dimension associated to the unparticle theory are taken into account. We show that the Casimir energy density depends also on these parameters. The analysis of the “ungravity” limit for the Casimir energy density, in which the characteristic length is very large in comparison to the horizon radius, is made, too. At zero temperature, we show that such a limit implies the instability of the system, since the Casimir energy density becomes an imaginary quantity. The general result is compared to the current terrestrial experiments of the Casimir effect. Thermal corrections also are investigated and the ungravity limit again examined, with the aforementioned instability disappearing at high temperatures.

## 1 Introduction

The Standard Model (SM) of particles and fields seems to have reached the limit of its extraordinary predictive capacity. The 27-km-perimeter Large Hadron Collider (LHC) has so far successfully confirmed this model, with relatively few surprises revealed since its first operations started ten years ago. There are still a plethora of data to be processed and analyzed, which will take some time. Irrespective of this, one must search for more information through alternative experiments which probe other phenomena, in order to test theories which go beyond the SM, since there are several opened questions that this model does not answer.

In addition to leaving out a consistent quantum description of gravity, other unanswered questions by the SM are the dark matter origin and why matter survived annihilation with antimatter in different stages after the Big Bang [1, 2]. One of the proposals to explain the former involves the existence of new particles out of the SM, which includes WIMPs (Weakly Interacting Massive Particles), axion-like particles and sterile neutrinos [3]. On the other hand, extensions of the SM based on Supersymmetry may explain the asymmetry matter-antimatter by proposing the existence of new massive particles and interactions which break the time-reversal symmetry, endowing, in addition, common charged leptons with an electric dipole moment aligned with the particle’s spin [4].

Just as the breakdown of some symmetries of the SM implies the existence of new particles, the symmetry previously restricted to the massless sector of the model—the conformal invariance—may be extended to a new category of microscopic objects termed *unparticles*, proposed some time ago by Georgi [5, 6]. These entities also could account for both the dark matter nature and baryon asymmetry [7, 8]. They have undefined mass or continuous values for it [9] depending on the energy scale at which one detects them, i.e., the usual energy-momentum dispersion relation for a free particle is not longer valid, hence the name unparticle. An usual particle only owns scale invariance if it is massless. The unparticles, even endowed with mass, though indefinite, enjoys that property. Beside this, the scale dimension of the fields in the action can be fractionary, which leads to the representation of non-integral numbers of massless particles.

The unparticle proposal was inspired in the older theory of Bank-Zacks [10], where a conformal invariant high energy sector near a critical (fixed) point is possible, around which there would be fields of unknown nature weakly coupled to those ones of the SM. However, the very weak magnitude of the couplings would impose serious restrictions to the detection of the unparticles. Despite this fact, some events in the domain of high energies could indicate their presence [11, 12, 13] as well as in Astrophysics and Cosmology [14, 15, 16, 17, 18, 19, 20], and also in low energies phenomena [21, 22], including the Casimir Effect [23]. In this case, the fractionary character of the scale dimension of the fields reflects in the dimensionality of the plates: It is non-integral, presenting a fractal nature, therefore.

The Casimir effect was discovered in 1948 as the attractive force arising between two parallel and uncharged metallic plates placed in vacuum, which results from the modification of the zero point oscillations of the electromagnetic field induced by the material boundaries [24, 25, 26, 27]. Nowadays there are no doubt about the existence of this effect confirmed by many accurate experiments which have been performed during the last twenty years. Generically, the phenomenon also occurs when the vacuum of an arbitrary quantum field is disturbed by the presence of boundaries with different shapes and made of different materials, usually revealing itself through a force that arises on or between such boundaries [28]. The disturbance of the quantum vacuum can arise also on empty spaces with nontrivial topology [29, 30]. This phenomenon can still be associated to a quantum field with arbitrary spin describing baryonic or even exotic matter [31]. The progressive increment in the precision of the Casimir effect measurements show that they tend to be a relevant source of tests for both high energy physics and modified theories of gravity [32, 33, 34, 35].

In this paper, we will investigate the Casimir effect associated to a massless scalar field in a cavity formed by two ideal parallel plates, which are placed in the spacetime generated by a rotating axially symmetric gravitational source based on a distribution of vector or scalar (tensor) unparticles. This work extends the one that studied the phenomenon in the flat spacetime considering only scalar (tensor) unparticles [23]. Here, the computation of the Casimir energy density will follow the approach contained in [36] according to which it is made a coordinate transformation that will allow us to use a Cartesian coordinate system associated to the rectangular cavity. The presence of the unparticles is incorporated to the background by taking the axially symmetric solutions of the Einstein field equations obtained in [20]. The analysis of the ”ungravity” limit for the Casimir energy density, in which the characteristic length of the theory is very large in comparison to the horizon radius [16, 17], will be made, too. The obtained result will be compared to the current experiments about the Casimir effect. Thermal corrections also will be investigated and the aforementioned limit again examined.

This paper is organised as follows: In Sect. 2 we review the unparticle features in the gravitational scenario. In Sect. 3 we compute the Casimir energy density in the parallel plates configuration. In Sect. 3.1 we analyse the thermal corrections and, finally, in Sect. 4 we close the paper with the conclusions.

## 2 Unparticle static black holes

*s*,

*v*unparticles with gravity and \(M_{Pl}\) is the Planck mass. The plus signal is taken for the scalar (tensor) unparticle case (\(R_{s}\)) and the minus one for the vector case (\(R_{v}\)). These solutions have horizon curves defined by \(g_{rr}^{-1}=0\), and thus we get

### 2.1 Unparticles and quintessence in Kerr-like spacetimes

*et al.*) solution and the functions \(F_{1,2}\) are given by

## 3 Casimir effect

*x*,

*y*,

*z*) is centered on one of these plates such that the

*z*axis is tangential to the path of the circular orbit [36, 43]. In this case, the spherical coordinates centered on the source and the Cartesian axes of the orbiting system are related by \(dy = dr\), \(dz = r d\phi '\), and \(dx =-r d\theta \), where \(\phi ' = \phi - \Omega t\). Therefore, in the Cartesian frame, the metric given by Eq. (10) can be written as

On the other hand, based on Eq. (5), we can find the upper bound on the energy scale of the unparticles, \(\Lambda _U\), as a function of the scale dimension, \(d_U\). The graph in Fig. 2 depicts this, where we have taken the coupling constant as being \(\kappa _s=1\).

It is interesting to compare this result with that one given in [23], which considered the Casimir effect of scalar unparticles in the Minkowsky spacetime, with a relative error of the current measurements as being \(\delta _{\epsilon }=30\%\) and without dependence on the coupling constant. We obtain for the ungravity Casimir effect a very stronger bound.

### 3.1 Thermal corrections to the Casimir energy

*V*is given by [28]

*A*being the area between the plates and \(k_B\) is the Boltzmann constant. The term \(f_{bb}\) is calculated using the expression

*k*is the modulus of the wave vector \(\mathbf {k}\). The terms \({\mathcal {O}}(T^2)\) and \({\mathcal {O}}(T^3)\) are obtained by expanding the free energy. Defining the Helmholtz energy density by \({\tilde{f}}_0=\frac{{\tilde{F}_0}}{V_p}\) and calculating the internal energy density through the expression

## 4 Concluding remarks

We have studied the contribution to the Casimir effect by a massless scalar field in a cavity formed by ideal parallel plates orbiting a rotating distribution of vector or scalar (tensor) unparticles, according to the Georgi formalism [5]. The presence of these entities was considered in an axially symmetric Kerr-like solution of the Einstein equations obtained in [20]. We taken into account the different technics of obtaining rotating solutions via Ghosh [41] or Toshmatov et al. [42] prescriptions, which depend on the characteristic length, \(R_{s,v}\), and on the scale dimension, \(d_U\), both the parameters associated to the unparticle theory. It was made a transformation in the Kerr-like metric in order to associate a Cartesian coordinate system to the rectangular cavity. Thus, the computation of the vacuum energy, including its regularization, was done by following Sorge approach [36]. The obtained results show that the Casimir energy density depends on those unparticle parameters.

The analysis of the limit in which the characteristic length is very large in comparison to the horizon radius—the ungravity regime—was made, and we concluded that, at zero temperature, the system is unstable since the Casimir energy density becomes an imaginary quantity. The computed Casimir energy density was then compared to the result of the actual experiments realized with the Casimir effect, and from this we have graphically pointed out a set of allowed magnitudes for \(R_s\) and \(d_U\), as well as for \(\Lambda _U\) and \(d_U\), in the parameter space. In fact, the current measurements of the Casimir effect point to a scale dimension slightly different from unity, since otherwise we would live in a world in which predominates the ungravity regime, which does not seem to be the case. Furthermore, the proposed case of the plates on the Earth offers stronger limits on unparticles than the one registered in the Minkowsky spacetime according to [23]. Finally, thermal corrections to the Casimir energy density were investigated and the ungravity limit again examined, with the aforementioned instability disappearing at high temperatures.

## Notes

### Acknowledgements

The authors thank CNPq and FUNCAP for their partial support under the grant PRONEM PNE-0112-00085.01.00/16. H.S.V. is funded by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)-Finance Code 001.

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