The Casimir force on a piston in the spacetime with extra compactified dimensions

A one-dimensional Casimir piston for massless scalar fields obeying Dirichlet boundary conditions in high-dimensional spacetimes within the frame of Kaluza-Klein theory is analyzed. We derive and calculate the exact expression for the Casimir force on the piston. We also compute the Casimir force in the limit that one outer plate is moved to the extremely distant place to show that the reduced force is associated with the properties of additional spatial dimensions. The more dimensionality the spacetime has, the stronger the extra-dimension influence is. The Casimir force for the piston in the model excluding one plate under the background with extra compactified dimensions always keeps attractive. Further we find that when the limit is taken the Casimir force between one plate and the piston will change to be the same form as the corresponding force for the standard system consisting of two parallel plates in the four-dimensional spacetimes if the ratio of the plate-piston distance and extra dimensions size is large enough. PACS number(s): 03.70.+k; 11.10Kk ∗E-mail address: [email protected] 1 In 1948 a remarkable macroscopic quantum effect describing the attractive force between two conducting and neutral parallel plates was predicted by Casimir [1]. The Casimir effect appears due to the disturbance of the vacuum of the electromagnetic field induced by the presence of boundary. Twenty years later Boyer researched on the Casimir effect for a conducting spherical shell to find that this kind of Casimir force is repulsive [2]. This effect is more complicated than we thought. Afterwards more efforts have been paid for the problem and related topics. All results brought attention to the fact that whether the Casimir force is attractive or repulsive depends on the geometry of the configuration strongly [3]. However, there are several reasons to be suspicious of the analysis of the Casimir effect problems. Maybe their results are not perfect. For example, we always investigated a massless scalar field in a confined region such as parallel plates, rectangular box and so on to find the vacuum energy while we let the field satisfy the Dirichlet boundary conditions at the borders of the region [3-8]. Having regularized the vacuum energy, we obtain the Casimir energy. Certainly the Casimir force can be received by means of derivative of Casimir energy with respect to the distance between two edges. Here it should be pointed out that these former considerations on the topic have not involved the contribution to the vacuum energy from the area outside the confined region which depends on its dimensions while we discard the divergent terms related to the boundary also depending on the geometry and dimensions during the regularization process. In order to ignore the flaws mentioned above, a slightly different model called piston was put forward [9]. The system is a single rectangular box with dimensions L × b divided into two parts with dimensions a × b and (L − a) × b respectively by a piston which is an idealized plate that is free to move along a rectangular shaft. In ref. [9] the author calculated the Casimir force on a two-dimensional piston as a consequence of fluctuations of a scalar field obeying Dirichlet boundary conditions on all surfaces and found that the force on the piston is always attractive as L goes to the infinity, regardless of the ratio of the two sides. Immediately the issue attracted more attention. The Casimir force acting on a conducting piston with arbitrary cross section always keeps attractive although the existence of the walls weaken the force [10]. The three-dimensional Casimir piston for massless scalar fields obeying Dirichlet boundary conditions was also explored, and it was found that the total Casimir force is negative no matter how long the lengths of sides are [11]. In addition in the case of various boundary conditions the Casimir force on a piston may be repulsive [12, 13]. In a word the Casimir piston is a new important model revealing its own distinct effects and can be used to explore the related topics. This model is also simpler to be constructed as a device from the experimental point of view. The model of higher-dimensional spacetime is a powerful ingredient to be needed to unify the interactions in nature. More than 80 years ago Kaluza and Klein put forward the issue that our universe has more than four dimensions [14, 15]. The Kaluza-Klein theory introduced an extra compactified dimension to unify gravity and classical electrodynamics in our world. The theory has been generalized and developed greatly. Recently the quantum gravity such as string theory or brane-world scenario is developed to reconcile the quantum mechanics and gravity with the help