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We examine the possibility of generating net forces on concave isolated objects from backgrounds consisting of randomly created waves carrying momentum. This issue is examined first for waves at the surface of a liquid, and second for quantum vacuum electromagnetic waves, both in relation with a one-side-open rectangular structure whose interior embodies a large number of parallel reflecting plates. Using known results about the Casimir-like effect and the original Casimir effect for parallel plates, we explain why and how such rectangular hollow structures should feel net oriented forces. We briefly describe real systems that would allow testing these theoretical results.

It is commonly believed that isolated objects in any background consisting of a spectrum of randomly created waves carrying momentum will not be subject to net forces due to this background. This belief rests upon the idea that such forces would then derive from a potential associated with the background energy. The energy of any isolated object in such a potential being independent of its position, no net forces should result from any configuration of the objects. The Casimir effect (see e.g. [

In the original Casimir effect two ideal rigid conducting and uncharged parallel plates attract each other due to the quantum vacuum energy spectrum of the electromagnetic radiation at zero temperature. The effect results from an imbalance of the radiation forces on the inside and the outside surfaces of the plates, which both equal infinity. The plates discretize the spectrum of electromagnetic waves between and transverse to the plates. In the region between the plates the energy density is smaller than outside the plates, which causes the imbalance of the radiation forces. The net force is then proportional to the difference in the local energy density. Using a regularization procedure needed to evaluate the above indeterminate form, one can show that the force of attraction between the plates is inversely proportional to the fourth power of their separation distance [

The Casimir effect is not restricted to the electromagnetic radiation of quantum vacuum. Casimir-like effects also exist. Here likeness means that the force between bodies results from a wave background different from that of the electromagnetic radiation of quantum vacuum. Such Casimir-like effects have been treated theoretically and observed experimentally for liquid surface waves [

In this paper, we shall consider the action of two different wave backgrounds on concave structures made up of many parallel and equally spaced plates. In each region between two adjacent plates the energy density is lower than the one outside the whole structure of plates. We are interested in the possibility of transforming this difference of energy density into a kinetic energy applying to the whole structure. The first wave background examined is that of liquid surface waves. The concave structure considered will then consist of many partly submerged parallel plates floating perpendicularly to the liquid surface and whose form seen from top would look like a comb. The second wave background will be that of the original Casimir effect. In this case, the concave structure will have the form of a rectangular parallelepiped made up of many parallel conducting plates, with one of its faces open showing one edge of each of its interior parallel plates. For other approaches to the project of extracting energy through the Casimir effect, see e.g. [

Consider two flat and square plates of macroscopic area a^{2} facing each other at a distance

Let us examine the collective behavior of a set of waves first between two infinite-size square plates a relatively short distance apart. We start with a single wave between the plates. Because of the boundary conditions imposed by the plates, this wave can effectively be created only if its half-wavelength fits a whole number of times into the distance b between the plates. Due to interference, a plurality of waves of the same wavelength but with different phases will gradually cancel out each other while moving between the plates. Therefore, only in-phase waves having as wavelength an integer divisor of 2b can exist between the two parallel plates. We now consider the set of in-phase waves characterized by a same wavelength which is an integer divisor of 2b. Using the Huygens-Fresnel principle, it is straightforward to show that between the plates these waves will form a wavefront moving perpendicularly to the plates [

We now determine what happens in the case of finite-size square plates. For the original Casimir effect, it can be shown that taking into account the edges of the two identical plates results into a slight increase of the effective area of the plates [

Consider two rigid square plates of area a^{2} kept at a small distance b apart by an a × b rigid plate of the same material as for the square plates. Seen from top this structure would have the shape of a long square subset sign. The structure is partially but deeply submerged in a liquid whose surface is disturbed by waves of small amplitude, which are everywhere created with frequencies randomly selected within a limited band. The origin of the waves could be natural or artificial. Since both gravity and surface tension of the liquid contribute to the restoring force, these driven oscillations of the liquid surface are sometimes called deep gravity-capillary waves. The aim of the deep placement is twofold. First, it is to prevent the wave motion from passing under the structure and into the region between the plates. We thus assume that the plates are sufficiently deep so that they are subject to essentially all of the wave motion. Second, it is to increase the viscous damping in order to reduce the motion of the structure due to fluctuations of random waves. In practice, the distance of separation b is chosen so that the waves motion between the parallel plates is negligible compared with the one outside the square subset sign structure.

The asymmetry of the four lateral sides of the square subset sign structure causes an imbalance in the momentum it receives in the direction going through its open face and its opposite closed one. We have seen that almost all momentum of the waves created outside the structure with the possibility of propagating into the region between the square plates undergoes elastic isotropic diffuse reflections by the free edge of each plate cutting the liquid surface, a small part being absorbed by the structure. However, the waves created outside the square subset sign structure which meet its opposite closed face are reflected by it. It thus follows a net force acting on the structure in the sense going from its closed face toward its opposite open one. This force results from the asymmetry due to the concavity of the square subset sign structure.

The value of the above force can be estimated directly from [_{r} of the force due to the difference in the local energy density related to the liquid surface waves on the two sides of one of two parallel square and perfectly reflective plates of area a^{2} at a distance b apart, when they are free to move, is

where g is the acceleration due to gravity, _{rms} is the root-mean-square of the wave amplitude and a must be interpreted as the plate width [

We now consider a device D made up of N side-by-side square subset sign structures, N being such that

By keeping the distance b at the value which minimizes the wave motion between the parallel plates of D, and by increasing the number N of adjacent square subset sign structures, (1) shows that it would be possible, in principle, to generate an oriented net force having a value as large as one wishes.

In the above model, if the structure of parallel plates is first held fixed with respect to an observer and then released, it would in principle undergo a sustained acceleration. But this acceleration will fade out in practice. Indeed, as the structure of parallel plates moves with its open side ahead, the level of liquid between the plates and in front of its open side rises, becoming higher than elsewhere on the liquid surface. Combined with its friction with the liquid, this local accumulation of liquid will exert an extra force opposed to the structure movement, which in turn will gradually reduce the acceleration, until it completely disappears. The speed of the structure will then reach a constant value. This value is non-zero because otherwise the structure will undergo a new acceleration due to the appearance of a new difference in the local energy density between its inside and its outside regions. The movement of the structure generates waves on the liquid surface whose total energy compensates in part to the hollow of energy existing between the parallel plates. In principle, such a device should allow to extract a part of the surface wave energy.

To illustrate the above situation, let us use the physical characteristics of the experiment described in [^{3}), and that its root- mean-square surface vertical displacement is A_{rms} = 0.1 cm. According to (1), the net force starting value for a structure with b = 1.7 cm and N = 6 is approximately 9.9 dynes.

We are now interested in the interaction of the quantum vacuum electromagnetic waves with a rectangular one-side-open parallelepiped P made up of N + 1 identical conducting uncharged parallel square plates of area a^{2} kept at a same small distance b apart by three a × Nb rectangular plates of the same material as the square plates. One face of P is open toward its interior showing one edge of each square plate and the whole structure is electrically continuous. The local energy density of the quantum vacuum background of electromagnetic waves throughout the region between these square plates is lower than the one outside P. The electromagnetic waves stemming from the quantum vacuum which are created outside P and reach its open side, or the exterior closed face opposite to the open side, are respectively absorbed and reflected. A net force results from this difference of momentum on the two sides of this face of P.

Let us determine the value of the above net force. First, we know that the value of the force due to the difference in the local energy density related to the background of quantum vacuum electromagnetic waves on the two sides of one of two free to move parallel square and perfectly reflective plates of sides a, when they are at a distance b apart, is given by (see e.g. [

where h is Planck’s constant, c is the speed of light in vacuum and

The value of the force on P due to the absorption by P of the electromagnetic waves created outside P and incident on its open face equals half of

Equation (3) shows that by increasing N it is possible, in principle, to generate an oriented net force having a value as large as wanted.

If the parallelepiped P is first held fixed with respect to an observer and then released, it would in principle undergo a sustained acceleration. But, similar to the model with liquid surface waves this acceleration will fade out in practice. Indeed, the movement of P deforms inward the regions of lower energy density between its parallel plates, because some electromagnetic waves created outside P then enter between the plates. The faster P goes, the more marked becomes the deformation. Consequently, the quantity of negative energy between the plates gradually decreases until the acceleration of P reduces to zero. The speed of P then stabilizes at a constant value. This value is non-zero because otherwise the structure would undergo a new acceleration due to a new difference in the local energy density between its inside and its outside regions. The value of this constant speed will be proportional to the depth a of the inter-plate spaces between the open and opposite closed faces of P.

The above device seems to allow the extraction of a part of the quantum vacuum electromagnetic wave energy. Let us illustrate this by assuming that the number of spaces between facing square plates is such that Nb = a, where b is sufficiently small to generate a noticeable attracting force between two adjacent parallel plates. The corresponding value of (3) is

Hence, for a cube consisting of 5000 parallel square plates of sides a = 1 cm, of thickness τ = 10^{−}^{4} cm, and at distance b = 10^{−}^{4} cm apart, (4) yields a net force of 6.5 × 10^{−}^{3} dyne as starting value.

We have shown that in a background of randomly created waves a structure with a rectangular concave interior embodying parallel reflecting plates can be subject to net forces. This possibility has been put forward in two cases. The first one is related to a background of waves at the surface of a liquid which partially submerged the structure of reflecting plates. The second one involves a one-side-open parallelepiped of ideal parallel conducting plates in the background of quantum vacuum electromagnetic waves. The fact that a net force results from the latter concave configuration may be related to a similar theoretical outcome, which follows from calculations of the energy-momentum tensor expectation value due to the curvature of a perfectly conducting boundary [

Let us now discuss the meaning of the very existence of such a net force acting on an isolated object in a background of randomly created waves. This violates the ordinary principle of conservation of energy-momen- tum. As shown by the Casimir effect, the level of energy of the background must not necessarily be referred to a potential of energy. Physically realistic phenomena related to the energy background of quantum field theory actually fail to satisfy various local energy conditions [

Within General Relativity, the original Casimir effect means that no electromagnetic vacuum state associated with two separated neutral and conducting plates corresponds to a stable space-time. The net forces put forward in this paper in relation first with a Casimir-like effect, and second with the original Casimir effect show that the backgrounds of the energy density of these settings are also unstable in the sense that they allow energy of these backgrounds to be transferred to a body in the form of kinetic energy.

The author is grateful to Normand Beaudoin, Roby Gauthier, Pierre Gravel and Doris LaChance for interesting discussions and useful comments.