HOW A UFO FLYS USE OF THE CASIMIR FORCE FOR CONTROLLED PROPULSION OF MACROBODILES Alexander Veniaminovich Antipin 1. In 1948, G. Casimir theoretically predicted the effect later named after him [1]. The effect consists of two flats, parallel, conducting plates placed opposite each other in a vacuum being subjected to forces normal to them that are not of gravitational origin, tending to draw them together (Fig. 1). Fig. 1. Classical Casimir effect. The modern explanation for these forces is that they are caused by the difference in pressure between virtual photons on the plates from the outside and inside. According to the laws of quantum mechanics, only photons with wavelengths that fit within the gap between the plates can exist between them. Thus, the gap "eats out" most of the virtual photons present in free space, which have arbitrary wavelengths. As a result, the pressure on the plates from the outside significantly exceeds the pressure from the inside, which causes the Casimir force. 2. The Casimir force for two flat conducting surfaces, per unit area, is equal to: (1) Where "-" denotes the attraction between the plates, h is Planck's constant, c is the speed of light, and d is the distance between the plates [2]. Numerically, Fc [dyn] = 1.3*10^(-18) * S/d^4 , where S and d are measured in [cm]. For example, for plates with an area of 1 cm^2 and d = 10 nm, the force will be approximately 10^6 dyn, i.e., the pressure on the plates will be on the order of atmospheric pressure! The magnitude of the Casimir force has been confirmed experimentally since 1958 [3] and coincides with the theoretical value for a wide range of geometries: flat plates, a plate and a sphere, two cylinders, nanostructures, etc. (see, for example, numbers 7-15 in the list of references to [4] and numbers 13-21 in the list of references to [5]). Today, experimental accuracy has been reduced to a percentage of theoretical values, which irrefutably confirms the existence of the Casimir force as a physical phenomenon, as well as the accuracy of its calculation. 3. To study the properties of the Casimir force, in particular, the "sphere + plane" geometry (Fig. 2) is actively used [4], [5], [6]. Fig. 2. Geometry of "sphere + plane". The theoretical value of the Casimir force for a sphere and a plane (for the case d << R) is given by the expression [5]: This formula can be obtained from (1) using the most general and natural approximations known as PFA (Proximity Force Approximation) or PAA (Pairwise Additive Approximation), a calculation method [5], [6]. Using the standard method of integration over a sphere, an infinitesimal element of its surface , we replace the infinitesimal quadrilateral dS’, considered flat due to its size, with the normal directed along the radius at an angle θ to the Z-axis. The entire sphere is considered as a body formed by an infinite number of such infinitesimal quadrilaterals. For natural reasons, only the lower hemisphere of the sphere "C" is considered, i.e., the range of angles: θ = [0…π/2] and φ = [0…2π). Each element dS’ is projected onto a plane parallel to the XY plane into element dS’’. Next, using expression (1), the force is calculated, considering element dS’’ and an element dSxy of equal area in the XY plane beneath it as plates. The distance dtek is equal to the distance between the XY plane and the current element dS’ and, thus, varies from d 0 to d 0 + R. The direction of the Casimir force is normal to the element dS’, i.e. at an outward angle θ (Fig. 3). Fig. 3. Calculating the Casimir force in «sphere + plane» geometry. After integration (and introducing the constraint d<<R), we obtain a simple analytical expression for the Casimir force between a sphere and a plane (2). It should be noted that the condition d<<R is used merely to obtain a compact analytical expression, i.e., to simplify calculations and further analysis, and not as a physical constraint affecting the properties or the very existence of the Casimir force in this geometry. We also note the important fact that of the infinite number of elements dS' of this hemisphere, only ONE, namely the element located at its lowest point, is parallel to the XY plane. Thus, only for one element is the plane-parallel plate geometry of the original Casimir effect observed. 4. Experiments with different geometries demonstrate the presence of forces comparable in order of magnitude to the Casimir forces for plane-parallel plates of comparable dimensions. This directly indicates that the Casimir force exists and has similar "intensities" between both parallel and non-parallel surfaces, i.e., essentially, always and everywhere. For both the "sphere + plane" geometry and other geometries, the measured Casimir forces were found to be equal to those calculated with an accuracy of 1%, for example [7]. Thus, based on experiments, we can summarize that the Casimir force between a sphere and a plane exists and can be correctly calculated using expression (2). Formula (2) itself is a direct consequence of applying (1) to surfaces with curvature and located at arbitrary angles to each other. This follows from the fact that in the "sphere + plane" geometry, the angle of the normal to the element dS' varies in the range: θ = [0…π/2], relative to the Z-axis, which is the normal to the XY-plane. Thus, the agreement between the results of a series of experiments and the calculations performed using (2) demonstrates the fundamental applicability of expression (1) for calculating Casimir forces in arbitrary geometries. 5. Now let us consider the direction of Casimir forces in the geometry of flat, but not parallel, plates. As noted above, expression (1) works for arbitrary geometry and curvature; therefore, it also works in the simplest case: for planes located at arbitrary angles to each other. Let us arrange the plates as follows: we bring them into contact along one of the same sides, and move the opposite sides apart (Figure 4). We have obtained a "corner" structure. This structure resembles the letter "V" in plan and has an arbitrary length "in depth" of the drawing. Fig. 4. The "corner" design. The Casimir force results from the impact of virtual photons on the surface dS. During a perfectly elastic collision (which is what photon reflection is), only the normal component of the momentum Pphoton changes, while the tangential component remains unchanged. Thus, the vector of the momentum Pc transferred to the surface dS is directed normal to the surface. Note also that the direction of the photon's motion—top-down or bottom- up—does not affect the direction of the momentum Pc (Fig. 5). Fig. 5. The impulse is always transmitted in one direction, regardless of the direction of movement of the photons: from bottom to top or from top to bottom. Taking into account all the facts considered and the conclusions drawn from them, we conclude that each plane forming the given "V" structure, the "corner," is: 1. A Casimir force acts—exactly the same as it acts on any element of the sphere dS' that is not parallel to the XY plane. 2. For the reasons stated above, the force acts on each plane normal to it and is directed inward toward the "corner." By decomposing the Casimir forces Fc (acting on each of the plates) into components Fx and Fz, we see that: - the x-components of the forces applied to the plates of the corner are equal and directed toward each other. Thus, they are a pure Casimir force and tend to bring the plates closer together. - the z-components of the forces are SUMMED, resulting in an uncompensated force along the z-axis (Fig. 6). Fig. 6. Decomposition of the Casimir force into components (for the left surface). Therefore, we came to the conclusion that a constant force acts on the “corner” along the z-axis, created by the pressure on this macro structure of virtual particles (in this case, photons), and this force is directed from the top of the “corner” to its solution. From now on we will call this force “traction force” (Fт). 6. New effects must be assessed for their compliance with conservation laws, so it must be immediately and clearly noted that the existence of a thrust force does not violate these laws. The point is that we are considering an absolutely OPEN system, for which the "corner" is only one of its parts and, by itself, does not create any forces. The appearance of Fт is due to the interaction of the corner with virtual photons, i.e., with the vacuum of photons in the Universe, which (virtual photons) always exist in space and cannot be completely shielded in principle. To remove any difficulties in understanding the essence of the obtained result, it is sufficient to point out the almost complete analogy in principle between the described structure and a conventional sail. Both of these structures are merely obstacles, specially designed and placed in space where there is movement of material elements external to them. These external elements possess energy and momentum, which are determined by global processes, laws, and interactions that are completely independent of such a particular phenomenon as the placement of the corner or sail at a given point in spacetime. Consequently, the resulting force applied to the obstacle (sail or corner) is a consequence of the pressure of external elements on the obstacle and does not violate any conservation laws. Thus, the corner is a structure that transforms the motion of virtual photons into vector- and thrust- controlled motion of a macroscopic body, i.e., a controlled propulsion device. 7. Calculating the thrust of the "corner" using (1), in the PFA approximation (point 3), we obtain: where b is the "length" of the corner (the letter V "into" the page), Lmin is the distance between the sides of the corner at the Zmin level, and Lmax is the distance between the sides of the corner at the Zmax level. The specific measurement of these quantities is shown in Fig. 7. Fig. 7. Towards the derivation of the formula for the traction force of the “corner”. This formula works in the range of angles: 0<α<(π/4). For α= 0, it transforms into expression (1) for plane-parallel plates, and for angles α>=(π/4), the PFA approximation does not work for this geometry. Due to the dependence of Fт on ) it is obvious that the value of the parameter Lmax, in fact, does not play a role, since Lmax >> Lmin. Thus, for practical calculations and estimates, we have the following expression (assuming α~0): Fт [dyn] ~ 217 * b / (Lmin)^3, where b is measured in [cm] and Lmin in [nm]. The value of Lmin is limited from below by the "cutoff" level, which is determined technologically: - the precision of the wafer manufacturing (their roughness, degree of flatness), and - the MINIMUM wavelength of photons that can be effectively reflected by the material from which the angle is made. Particular attention should be paid to the fact that, due to the dependence of Ft on ( ), the thrust force is EXTREMELY sensitive to even the slightest change in Lmin. Changing other technological parameters in (3), such as: - increasing the surface reflectivity and/or expanding the reflector's efficiency range into the high-frequency region and - increasing the total length of the "corner" (parameter "b" – the length of the letter V "into the page"), will increase Ft linearly. 8. To understand where we are (technologically) at the moment, it can be noted that advanced, but not unique, modern microelectronic technologies, with appropriate refinement, will likely be able to create thruster panels measuring one meter by one meter and of negligible thickness, with a thrust of units to tens of dynes, making them suitable for use as low-thrust thrusters for spacecraft. The panel (in plan) will most likely look like an assembly of corners: “VVV…VVV”, and the engine itself will look like a set of such panels, secured on controlled independent suspensions (Fig. 8). Fig. 8. Construction of a panel made of “corners”. Note that for full control of the vector and thrust of the created device, two identical panels will be sufficient (Fig. 9). Fig. 9. The control principle of a structure based on panels with “corners”: a – no movement, b – movement in any direction. To estimate the traction force of the corner, we use the following values: - material: aluminum (Al), density ρ = 2.69 [g/cm^3], - angle half-angle of the corner, α - minimum, units of angular degrees, - maximum angle, Lmax >> Lmin, - length of the side of the corner (the length of one of the segments forming the letter V), L>~ 100 [μm], - the corner fills the entire possible area of ​​a panel measuring 1 [m] x 1 [m] (Fig. 8) such that the distance between identical elements of parallel corners is 200 [μm]. Therefore, its total length is b = 500,000 [cm] (5 km), - minimum wavelength of photons effectively reflected by the surface of the corner (Al), λmin = 200 [nm], and, thus, Lmin = 200 [nm], - surface reflectivity (Al) at a wavelength of λmin = 200 [nm]: R = 0.8, As a result, we obtain Fthrust ~ 10 [dyn]. Reducing Lmin to 50 [nm] (with R ~ 0.2) will provide a thrust force of Fthrust ~ 170 [dyn]. If Lmin can be reduced to 10 [nm], while maintaining a normal reflectivity of R ~ 0.1, this will yield Fthrust ~ 11,000 [dyn]. Estimating the acceleration of an unloaded panel, we obtain the following values ​​(for a panel mass of ~700g, dimensions of 1m * 1m * 0.5mm, void ratio = 0.5, and material - Al): - Lmin = 200 [nm]: acceleration a = 0.016 [cm/s^2], - Lmin = 50 [nm]: acceleration a = 0.24 [cm/s^2], - Lmin = 10 [nm]: acceleration a = 16 [cm/s^2] = 0.016 [g]. 9. Qualitative experimental confirmation of the effect under consideration can be obtained quite simply and quickly by measuring the thrust of a "corner" fixed in different orientations on a torsion balance. 10. The Casimir effect is a macroscopic result of the existence of virtual photons. All other virtual particles, both massive and massless, possess the same existence status. In this regard, the experimental study of analogs of the Casimir effect for other fields and particles is of considerable interest. Of particular interest is the assessment of the feasibility of generating a thrust force and its technologically achievable magnitude. This interest is explained by the obvious assumption that the thrust force generated by massive particles could be significantly greater than that generated by massless virtual photons. Fig. 10. Using panels based on “corners” to rotate the generator shaft. 11. The practical application of thrusters based on panels with “corners” is obvious: from the generation of energy (when placed on the shaft of a conventional electric generator) (Fig. 10), to the creation of traction thrusters for moving structures and apparatus (airplanes, rockets, flying platforms, spaceships, ocean-going surface and underwater vessels, vehicles, including miniature personal vehicles). Bibliographic list 1. H.B.G. Casimir, On the attraction between two perfectly conducting plates // Proc. K. Ned. Akad. Wet. 51, p.793–796 (1948). 2. Y. Aulin, Casimir-Lifshitz forces // Univer. of Groningen, pp. 1-21, (May 2009) 3. M.Y. Spaarnay, Measurements of attractive forces between flat plates // Physica, V.24, p.751 (1958) 4. H.B. Chan et al., Measurement of the Casimir Force between a Gold Sphere and a Silicon Surface // Phys.Rev.Lett., 101 (2008) 030401 5. F. Intravaia et al., Strong Casimir force reduction through metallic surface nanostructuring // Nature Comm, art. 2515, 4 (Sep. 2013) pp. 1-20 6. A.W. Rodriguez, F. Capasso, S.G. Johnson, The Casimir effect in microstructured geometries // Nature Photonics, V.5, (Apr. 2011), p.211-221 7. G. Bressi et al., Measurement of the Casimir Force between Parallel Metallic Surfaces // Phys.Rev.Lett. 88 (2002) 041804 Mine: A.V. Antipov wrote this paper in 2014. Since then, nothing has been heard about experiments in this area. Currently, I know how to make a similar panel. The equipment costs $3,000. If anyone could help me with any amount, I would be grateful. It would serve as additional incentive for me to continue my research. If you want to help, please write to me: proxyda@hotmail.com All the best to you!

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