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Physical Principles

Unlike what would be expected from elementary electromagnetism, two parallel, neutral planes at a given distance do exert a force upon one another. This effect, referred to as the Casimir effect, is often introduced as a purely quantum electrodynamical effect in the published literature. Such is not the case, however, as this surface force can be understood as a completely classical phenomenon present in acoustics and observed in the past centuries between ships at sea under particular conditions. In the electromagnetic case, even space completely devoid of matter contains energy in the form of random fluctuations of the electric and magnetic fields. Such fluctuations, which are collectively referred to as the zero-point field, can be described classically by arbitrarily including random fields into Maxwell's equations, or quantum mechanically, as a natural consequence of Heisenberg's uncertainty principle. Any sophisticated technological use of the Casimir force requires the ability to control it so that complex tasks may be executed by means of it. Being able to modulate the Casimir force at will allows us to design devices that can carry out engine cycles during which a net work is done by converting energy from one form to another. Such a Casimir engine, invented by Dr. Pinto, is referred to as a Transvacer™ (acronym of TRANSducer of VACuum energy) device. A Transvacer™ device is thus defined as a Casimir force-based engine designed to carry out a complete cycle to convert a type of energy into another by appropriately modulating the zero-point field. Developing and commercializing Casimir force control, zero-point field engineering, and short term technological applications of the Transvacer™ device all represent focal areas of activity of InterStellar Technologies Corporation. Furthermore, the company investigates the possibility that, under some circumstances, the zero-point field itself may become an exploitable energy resource.

1. A Mysterious Force
2. The Acoustic Casimir Effect
3. Stochastic Electrodynamics (SED)
4. The Development of Quantum Mechanics
5. Quantum-Electro-Dynamics (QED)
6. The Casimir Effect
7. Energy and the Casimir Effect
8. The Transvacer™ Device

1. A Mysterious Force

In order to appreciate the physical principles at the core of the applications being explored at InterStellar Technologies Corporation, it is useful to first focus on a particular manifestation of so called dispersion forces, a phenomenon referred to as the Casimir effect. In its typical manifestation, the Casimir effect is a force between two infinite, parallel planes of given optical properties at a given distance from one another- usually on the order of a micrometer or less (one micrometer is equal to one millionth of a meter). For instance, we can consider two highly conducting metal plates, or any other combination of two conducting, insulating, or magnetic plates facing one another.

It is an experimental fact that such two surfaces, although completely neutral, will exert a force upon one another. This force is in addition to, and completely different from, the expected gravitational attraction between them. It is also quite unexpected. We are taught in basic physics classes that any two objects, such as plates, particles, or planets, can only interact because of one or more of the four fundamental interactions of nature: (1) gravitation, (2) the weak force, (3) electromagnetism, and (4) the strong force, listed in order of increasing intensity. Of these, gravitation and electromagnetism are referred to as long-range forces, whereas the weak and the strong force are referred to as short-range forces.

We are all of course familiar with gravitation, which is the force that dominates all aspects of our Earth-bound lives. In the description given to us by Sir Isaac Newton, the gravitational force between (for instance) two homogenous, spherical objects is proportional to the product of their masses and inversely proportional to the distance between their centers squared. Albert Einstein of course radically changed this view of gravitation, from that of a force between objects to the effect that their mass and energy has on the space-time where other objects move. In other words, whereas Newton would have said that his famous apple was being attracted by the Earth all the way down from the apple tree, Einstein would have envisioned an apple in free-fall under the action of absolutely no force, but in a space-time curved by the presence of Earth's mass.

The weak force is a much more exotic interaction than the gravitation we are familiar with. For instance, it is ultimately responsible for some very important radioactive processes, such as the beta-decay of the neutron, which causes this particle to be unstable. Interestingly, in recent years a unified theory of electromagnetism and of the weak force has been developed and many of its predictions have been experimentally verified. This theory of the electro-weak force, referred to as the Standard Model, is one of the most successful theories of the fundamental nature of the Universe ever developed.

The first step towards this process of force unification was taken by James Clerk Maxwell in the 19th century, when he introduced his famous set of Maxwell equations, which showed that electricity and magnetism are not separate phenomena, but are actually deeply intertwined as different manifestations of the same electromagnetic interaction.

In some way, electromagnetism is also a force with which we have familiarity. It is clearly around us, although it does not appear to dominate our interaction with our surroundings as decisively as gravitation does. In fact, electromagnetism is the strongest long-range force in the Universe. The reason that gravitation appears to play a more important role in our daily lives is that, most often, different regions of the Universe are neutral. However, whenever objects around us become electrically charged or an electric current flows, electromagnetism easily displays its immense superiority to gravitation - for instance when an entire car is lifted from the ground with an electromagnet or lightning briefly turns the night into day. The importance of this interaction is further understood if we recall that light itself is an entirely electromagnetic phenomenon

The strong force is responsible for the very existence of atomic nuclei, as it is the "glue" that binds protons and neutrons together. As one would expect, a force capable to bring together protons within distances that are extremely small fractions of the size of one atom must indeed be incredibly intense, so as to overwhelm the electric repulsion between the positively charged protons. However, the range of action of the strong force is not much larger than the size of an atomic nucleus itself. Therefore, although matter as we know it could not be possible without the existence of the strong force, we never directly experience the most intense interaction in the Universe.

If we now return to our neutral parallel plates, it is easy to see that the mysterious force between them shown by the experiments does not immediately appear to be related to any of the four fundamental forces of nature. This interaction between the plates does not fit the description of a short-range force, it is not of a gravitational nature, and it does not appear to be electromagnetic because the surfaces are uncharged. This would appear to leave us only with the option to consider a new, previously unknown force to explain why our neutral planes exert a measurable force upon one another, so giving rise to the Casimir effect.

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2. The Acoustic Casmir Effect

The Casimir effect is an example of a fascinating phenomenon historically first predicted within a very complex and counterintuitive theory (quantum-electro-dynamics) that was later found to be a much more common occurrence in nature. The fact that this effect was initially suggested as an unexpected consequence of a theory that is by itself complex and completely outside of our daily experiences still permeates most popular descriptions of it available in the literature. In fact, unlike many even contemporary claims, the Casimir effect does not exclusively manifest itself within quantum field theories.

To appreciate the more "ordinary" nature of this phenomenon, let us consider a very different experiment, which was also recently carried out. In this case, two plates similar to the ones we have considered were again placed parallel to one another inside an acoustically insulated tank containing air, but at a distance of several centimeters from one another (one centimeter is equal to one hundredth of a meter). Inside the tank, a number of speakers generated random noise of assigned intensity within a specified frequency range.

Once again, a measurable force was detected between the plates. Interestingly, this force was proportional to the intensity of the noise generated and depended in a complex way on the upper and lower boundaries of the frequency range chosen. Finally, everything else being the same, the dependence of this "acoustic Casimir force" on the distance between the plates was quite complex, oscillating between repulsive and attractive while overall decreasing inversely proportional to the distance. How can all this be explained?

In order to answer this question, it is useful to visualize the tank as a volume full of sound waves traveling in all directions as they are emitted by the speakers and are reflected by any wall they happen to collide against, including the two parallel plates. In order to simplify the modeling, let us assume that all such boundaries act as perfect reflectors of acoustic waves and that their positions do not change with time. It is also helpful to assume that both the overall size of the tank and the sides of the plates (assumed rectangular) are much larger than the distance between the plates themselves.

The key concept to understand the fundamental origin of the Casimir force is that of radiation pressure. Consider a tennis ball traveling perpendicularly to a wall. Upon colliding with the wall, and if energy losses can be neglected, the ball will commence to travel away from it at the same speed it had just before impact, but of course in the opposite direction. What happens to the wall in this process can be understood in terms of Newton's action-reaction law, or his Third Law of Mechanics. Since the tennis ball is being acted upon by a force due to the wall, the tennis ball will react by exerting an equal and opposite force upon the wall during the collision.

Similarly, when a wave travels perpendicularly to a wall and is reflected by it, it exerts a pressure upon that surface. This pressure, due not to a finite object, but to a wave, is referred to as radiation pressure (a pressure is the force per unit area upon which the force is exerted). Therefore, we can envision the interior walls of the tank as undergoing a continuous pressure due to the randomly reflecting waves generated by the speakers at all frequencies within the assigned range. But what about the parallel plates inside the tank?

Clearly the two sides of such plates facing away from one another are being exposed to radiation pressure of all frequencies, much as the interior walls of the tank. However, the situation in the volume of space between the parallel plates is quite different. This is so because we have assumed that the plates are quite close to one another and that they are absolutely static. Only some waves out of all those generated inside the tank will be able to satisfy the condition that no motion of the fluid must take place at both inner boundaries of the two plates. All waves that do not satisfy these boundary conditions will not exist in this small volume of space.

Therefore, it is very reasonable to expect that the total radiation pressure exerted by all frequencies within the assigned range inward will not equal the total outward pressure exerted by the subset of frequencies that can exist between the plates. Since these two pressures are not equal, there will exist a net force acting on both plates. Such net force must be identified with the Casimir force found experimentally.

It is of great importance to realize that, unlike what is stated in the overwhelming popular literature in this subject, there is no a priori reason that such net force should be directed inward or, equivalently, that the Casimir force between two parallel plates in this example should be expected to be attractive. Typically, such mistaken conclusion is justified by the fact that the number of waves available to exert a pressure in the volume between the plates is smaller than that available to exert a pressure on the outside. In fact, this also is mathematically incorrect, as one can show that the number of frequencies available is the same, but that the density of states is different in the two regions of space.

In fact, experiments on the Casimir effect show that the "acoustic Casimir force" can be attractive or repulsive at different distances, thus drastically contradicting the above simplistic reasoning. It is only in the limit of an infinite range of frequencies (from zero to infinity) that the Casimir force becomes attractive at all distances. Yet, even in this regime, it is possible to find situations in which the Casimir force is always repulsive and not attractive.

The acoustic Casimir effect is a splendid demonstration of the fact that such interaction can be explained in completely classical terms. By classical, we mean that no appeal to quantum field theory is needed to explain the experiments. From this standpoint, the acoustic Casimir effect is a simple dynamical consequence of the concept of radiation pressure in the presence of multiple boundaries and it can be explained by means of Newtonian physics alone.

Since we are now realizing that the Casimir effect can manifest itself under entirely classical circumstances, it is reasonable to wonder whether any observation of it has ever been reported on a macroscopic scale. The answer is not only affirmative but also fascinating. As reported in the professional physics literature in 1995, an entirely classical, that is, non-quantum, treatment of the Casimir force is all is needed to explain the mysterious attractive force between two parallel, rolling ships at sea with no wind and a building swell sailors were being warned about as far back as the early 19th century. By treating the two ships similarly to two atoms immersed in an ocean of random waves, one finds that, in analogy to the van der Waals force of chemical physics, such two vessels will also attract.

What have we learned? We have learned that whenever two parallel plates are immersed in a field of random waves, a net force will generally act upon them due to their effect on the traveling waves in the volume between the inner boundaries of the plates. The details of this interaction are quite difficult to predict without an explicit calculation and the Casimir force may be found to be attractive or repulsive and depend on the distance between the plates in a complicated manner. However, most importantly, if the field can be described classically, as the above acoustic field, the Casimir effect still exists and can also be entirely described by means of classical physics.

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3. Stochastic Electrodynamics (SED)

In order to understand what happens to the two electrically neutral plates we introduced in the very beginning, let us now simply replace the random noise field of the acoustic Casimir effect with a random electromagnetic field. That is, instead of visualizing the radiation pressure due to random sound waves impinging upon all boundaries, we shall now consider a spectrum of electromagnetic waves contained in a large box within which our two reflecting boundaries are placed.

It is well known that electromagnetic waves, of which visible light is but an example, also exert radiation pressure upon a reflecting surface. For instance, light coming from the Sun appreciably alters the trajectory of orbiting satellites. Therefore, with little or no change in principle to accommodate some basic differences between electromagnetic waves and sound waves, we are certainly justified in expecting that a Casimir effect will exist in this case as well.

Of course, the fundamental question now becomes: "In the electromagnetic case at hand, what is the equivalent of the speakers used to generate the random noise of the acoustic Casimir effect?" In order to start answering this question, we must remember that the two reflecting plates facing one another are now separated by a vacuum gap and not by a fluid medium such as air or sea water (the electromagnetic Casimir effect exists even if a substance fills the gap, but initially it is best to make the present assumption).

In principle at least, this seems easy enough. We can remove anything possible left in the gap between the plates until we have only empty space. To the extent that we are able to deal with all technical difficulties, it would appear that we can actually produce a total void between these two boundaries, with the exception, of course, of the electromagnetic radiation itself. But why would there be such a field?

As it turns out, this is not the right question. In fact, a critical analysis of Maxwell's equations reveals that his equations can be recast to accommodate an electromagnetic field of random fluctuations that is always present everywhere in the Universe, even in the complete absence of sources (hence the name zero-point field). What type of frequency spectrum such zero-point field should have away from all boundaries, that is, how its energy should be distributed among its frequencies, is actually a consequence of the principle of relativity, which is at the basis of Einstein's special theory of relativity.

According to the principle of relativity, no experiment can be carried out by an unaccelerated observer to determine his or her absolute velocity with respect to any other unaccelerated reference in the Universe. That is, all unaccelerated motion is relative. This statement is so powerful that one must discard an otherwise attractive law of physics simply if that law is proven to violate it. The mathematical condition imposed upon any physical law to determine whether it satisfies the principle of relativity is referred to as Lorentz invariance. By requiring Lorentz invariance of the classical zero-point field, one can determine its frequency spectrum within a multiplicative constant, which is found from direct experimentation.

So, our critical analysis of Maxwell's equations shows that the right question is not why a classical zero-point field should exist, but why it should not. Stimulated by these considerations, in the recent past a minority of researchers has aggressively pursued the study of a theory of electromagnetism which includes a random zero-point field, a theory which is referred to as stochastic electrodynamics (or SED).

As to the ultimate origin of the stochastic zero-point field, this is quite irrelevant to its relationship to the Casimir effect. One may very well accept it by fiat as a cosmological condition at the time of creation, or, as has been suggested, one may link its existence to the dynamics of all classical charges in the Universe. What matters is that, as far as any mathematical detail is concerned, a Casimir effect is predicted as a consequence of our assumption of the existence of such classical zero-point field.

Notice that this Casimir effect, that is, the "mysterious force" we discussed at the very beginning, is actually an electromagnetic effect. Therefore, classical electromagnetism, which is founded upon the experimental premise that there exists a force between charges (Coulomb's Law), predicts that two neutral objects will also exert a force upon one another.

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4. The Development of Quantum Mechanics

Although the Casimir effect, as well as many other phenomena, can be quantitatively correctly predicted by simply assuming that everything in the Universe is immersed in a bath of classical zero-point radiation, of course such is not at all the view of physical reality held by the majority of practitioners. In fact, as indicated above, only a minority of scientists works within, or perhaps even knows of the existence of, the framework of stochastic electro-dynamics. This is of course a very understandable position to hold, given the unquestioned success of quantum physics from the early days of its development at the beginning of the past century and until today.

At the beginning of the 20th century, a host of problems confronted scientists that appeared to completely defy classical electromagnetism and Newtonian mechanics. The calculation of the black body spectrum, such as the thermal radiation emitted by the surface of a hot furnace, represented the crucial test that caused Max Planck to realize that classical physics simply could not reproduce the experimental black body data and an entirely new hypothesis was needed: by assuming that the energy of the assembly of microscopic oscillators in a furnace wall responsible for the emission of light could only take on some discrete, and not continuous, values, Planck succeeded in producing a theory that could fit the experimental data. The oscillators could only transition between two contiguous, discrete energy values by absorbing a fundamental energy packet, referred to as a "quantum" of energy.

Albert Einstein greatly contributed to the development of these revolutionary ideas when he proved that not only was the energy of Planck's oscillators quantized, but the electromagnetic radiation itself could be envisioned as a collection of massless elementary particles, which he called the photons. In the emerging quantum theory, photons would be absorbed and emitted as whole units, causing oscillators to change their energy by corresponding energy quanta.

In the span of just a few years, these ideas were applied to atomic physics and Niels Bohr was able to provide the first quantum model of the atom and to explain, for instance, the spectral properties of the hydrogen atom. His atomic model, which is still today of great intellectual value, reflected Rutherford's revolutionary discovery that the hydrogen atom resembles a microscopic solar system, with a proton in the center, and one electron orbiting around it. However, unlike in typical celestial mechanics, not all orbits would be available to the orbiting electron, but only those that satisfied some specific quantization conditions. Of course, in order for the electron to transition from an orbit to another, it would need to either absorb or emit quanta of the electromagnetic field, or photons. Very importantly, the electron could never orbit closer to the proton than its innermost orbit, referred to as the ground state.

Although this intermediate step, referred to as the "old quantum theory," represented a gigantic leap in our understanding of the atomic world, it became immediately clear that it still left researchers unable to answer many questions. For instance: "What causes an electron orbiting in a higher energy orbit of a hydrogen atom to spontaneously emit one or more photons and to decay to lowest possible state?"

In order to understand the next evolutionary step of quantum theory, we must remember that the electromagnetic radiation had by the 1920s been experimentally proven to display a dual nature, both that of particles and of waves. For instance, such classical observations as interference and diffraction patterns could be explained by classical electromagnetism by means of Maxwell's equations. On the other hand, atomic interactions with light had to be described by means of the photon concept. Such was the case in Einstein's Nobel Prize winning treatment of the photoelectric effect.

In the photoelectric effect, a light beam is directed at a metal surface. Under some circumstances, it is possible to observe electrons escape from the metal by absorbing electromagnetic energy from the beam. Such added energy enables the electrons to overcome the forces that usually would contain them within the material. The puzzling experimental finding of the time was not that such process would occur, but that it would occur only if the wavelength of the impinging light wave was smaller than a very specific value, characteristic of the metal surface used. If light could be described as an electromagnetic wave, what kept electrons from building up enough energy to escape by absorbing it from low energy, long wavelength waves over time, but allowed them to do so when illuminated by high energy, short wavelength light?

Again Einstein provided an explanation for this mystery, by showing that the electrons in the material do not interact with an electromagnetic wave, but with photons, or quanta of the electromagnetic field. Therefore, either a photon has an energy equal to or larger than the minimum needed to cause the absorbing electron to escape, or it does not. Since it was known from Planck's and Einstein's studies that the energy of photons is proportional to their frequency and inversely proportional to their wavelength, evidently the particle interpretation of the electromagnetic field solved the mystery of the photoelectric effect.

The discovery of the dual, particle-wave nature of light, counterintuitive and challenging as it is, actually stimulated de Broglie to go a step further. His speculation could be described as follows: "If light can behave as both a wave and as particles, can matter display the same duality?" Of course we are completely familiar with the particle nature of matter as that is what we experience every day. But can matter display a wave-like behavior? The answer is affirmative, as was shown by revolutionary experiments in which electrons traveling through matter created diffraction patterns completely similar to those caused by light.

The confirmation of de Broglie's speculation led Schrödinger to a search for the appropriate wave equation to describe the newly discovered matter-waves, which resulted in what is today referred to as the Schrödinger equation. Such equation is appropriate to describe, for instance, electrons in atoms when their energies are not too high - or, in Bohr's language, when the electron speeds are negligible compared to the speed of light. The meaning of the intensity of the wave itself is related to the probability that the particle will be found at a particular location and it does not imply that the particle is "spread" over a volume of space.

At approximately the same time as Schrödinger was developing wave mechanics, Heisenberg approached the problem from a completely different angle and developed an equivalent scheme for the calculation of those quantities that could be observed in an experiment. Most important in our reasoning, Heisenberg expressed quantitatively a most important relationship between specific pairs of quantities in quantum mechanics that cannot, even in principle, be both measured at the same time with infinite precision. Such negative statement is known as the uncertainty principle.

The best-known example of the uncertainty principle is given by position and momentum (momentum in this case can taken to be the product of the mass of particle by its speed towards a particular direction) of a particle. The principle states that the product of the uncertainty on the measurement of these quantities at any time must be greater than a natural constant. Therefore, if our knowledge of position is extremely good (small uncertainty) than the uncertainty on momentum must be great for the principle to hold.

Interestingly, the uncertainty principle allows us to understand why an electron in a hydrogen atom cannot, on average, approach the proton at distances smaller than that of the ground state. If the electron could be confined within an infinitesimal region of space around the proton, then its momentum, and therefore its energy, would become infinite because of the uncertainty principle. Similarly, if we had perfect knowledge of the momentum of the electron by bringing it to rest, its position would be completely unknown. Therefore, the ground state is an energetic compromise, which allows the electron to rest into the lowest energy level possible that satisfies the uncertainty principle.

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5. Quantum-Electro-Dynamics (QED)

By means of the Schrödinger equation, it is possible to determine all possible states of the electron in the electric field of the proton in a hydrogen atom. Such states are described by mathematical objects referred to as the wave functions, which describe the probability of finding the electron at any position in space. Despite further progress from the old quantum theory, we are still unable to determine why the electron should transition from one state of higher energy to one of lower energy.

In order to do so, we must implement the rules of quantum physics not only in our description of the electron, but also in that of the electromagnetic field itself. For as long as we keep our description of the electromagnetic field classical, it is impossible to show that the higher energy states of the hydrogen atom are unstable and, in time, they will decay into the ground state with the emission of one or more photons.

The theory that describes not only matter, but all fields as well, by means of quantum principles is referred to as quantum electrodynamics (QED). In its most complete form, it naturally includes Einstein's special theory of relativity and it is therefore more advanced than even the non-relativistic Schrödinger equation. In much the same way as the position and momentum of a particle represent a pair of quantities that cannot be both measured at the same time with infinite precision, so also in QED the components of the electric field and of the magnetic field represent such a pair in the sense given by the uncertainty principle.

Intuitively, this means that, even in a state of vacuum (absence of all sources) in any volume of space, the uncertainty principle, applied now to the electromagnetic field itself, implies the existence of a "ground state" for such vacuum. In other words, we must visualize the vacuum not as an absolutely empty region of space, but as one where, in accordance with the uncertainty principle, the electromagnetic field randomly changes form place to place. According to QED, it is impossible to ever obtain a state "emptier" than such vacuum in free space. Perhaps the most provocative concept about this quantum vacuum is that, if we attempt to compute its total energy density, we obtain an infinite number.

This shocking finding is traditionally interpreted as meaning that, in order to extract information from QED, we have to somehow eliminate, subtract, or renormalize our results so as to avoid its infinities. Since the structure of the theory allows for this to be done, the diverging energy density of the quantum vacuum has not represented an insurmountable obstacle to use it in practice. However, this procedure of course does not mean that this infinite energy simply does not exist and, in fact, a long-standing debate has been taking place as to whether its appearance is simply due to mathematical gadgetry or to its actual physical existence.

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6. The Casimir Effect

Let us now once again consider our two parallel plates. In the QED framework, any volume of empty space both within and without the gap between the plates actually contains electromagnetic zero-point energy (ZPE) due to the electric and magnetic fields fluctuating because of the uncertainty principle. Such fluctuating fields correspond to photons that appear and disappear continuously. Unlike photons we experience in our daily lives, referred to as real photons, these photons cannot be directly detected nor can they exist for an infinitely long time because their existence violates the principle of the conservation of energy. They are referred to as virtual photons.

Because of the same arguments we discussed in the case of the acoustic Casimir effect, the presence of the two boundaries alters the energy density in the region between the two surfaces. Although this number is infinite, it is possible to devise techniques to subtract it from the energy density outside of the gap, which is also infinite. The result so obtained is a finite value.

This important finding shows that the presence of the two surfaces causes a change in the zero-point energy of the system that depends on the distance between the plates: the smaller the distance, the larger the change. This is the ultimate origin of the Casimir force.

It is natural to wonder whether it is possible to view the Casimir force in QED as a result of radiation pressure, given the fact that, in this case, no real photon field actually exists. The answer is affirmative, as it has been shown that the Casimir effect can be explained as the result of radiation pressure of the virtual photons upon the boundaries.

From the mathematical standpoint, there is absolutely no difference between the SED and QED treatments of the problem. Also, depending on the boundary conditions, the Casimir force may be repulsive - as is the case for instance in the interaction between a perfectly conducting and an infinitely magnetic plate.

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7. Energy and the Casimir Effect

In this section we shall carry out a type of experiment that is based exclusively upon logical reasoning and not actual measurement. Of course such thought (or gedanken) experiments, although they cannot produce new experimental data, are extremely powerful, as they allow one to extract a wealth of information from known physical laws.

Initially, let us restrict ourselves to a Casimir system consisting of two perfectly conducting plates, placed vertically one in front of the other at some relatively large, initial distance (Fig. 1). In the case of perfectly conducting plates, the Casimir force can be shown to be attractive at all distances. Our gedanken experiment consists of attaching a string to the left of the two plates, which will be free to move, while the plate on the right will be fixed in a permanent position. The string will be stretched horizontally to a pulley and then run down to a mass hanging in the gravitational field of the Earth.

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Let us now slowly let the two plates come together to a smaller, final distance (Fig. 2). As the Casimir plates come together, the Casimir force increases. Therefore, in order to maintain a state of quasi-equilibrium during this process, we must constantly add extra mass in addition to what was attached to the string in the beginning. The end result of this process will be that a total mass has been raised to a distance equal to the change in separation between the two plates. What has happened to the total energy of the system, including the plates, the mass, the Earth, and the vacuum?

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We know from basic mechanics that, whenever a force moves its application point, it is said to have done work. If the force is constant in magnitude and direction, the work done by this force is simply the product of the component of the force along the distance traveled multiplied by the force itself. In the case of the Casimir force, however, the force is not a constant and this calculation is not as trivial as carrying out a multiplication.

Regardless of mathematical details, this work done by the Casimir force upon the total mass raised manifests itself as an increased gravitational potential energy of the mass. Potential energy is so called because it can "potentially" be converted for instance into kinetic energy, or energy of motion, by dropping the object back to its initial position above the ground. Therefore, as the Casimir plates were drawing closer and closer, work was being done upon the mass, and its potential energy increased. Where did this increased potential energy come from?

Let us look at the total zero-point energy between the two Casimir plates. Since the Casimir force in this case is attractive, both the energy density and the total energy in such volume are negative (of course we are counting such value from the infinite offset of zero-point energy in free space). We also know that, as the plates become closer, the energy density, and therefore the total energy, becomes larger in magnitude while remaining negative in sign. Therefore, at the end of the lifting process, the total energy in the gap volume is larger in magnitude but still negative in sign than at the beginning. Mathematically, this means the total zero-point energy is smaller than previously. Of course, as one would expect, the increase in the gravitational potential energy of the mass exactly equals the decrease of the zero-point energy in the gap volume. Therefore, the total energy of the system is conserved.

Let us now imagine that we want to continue such process and raise more mass from the initial to the final height. In order to do so, we need to "reload" the Casimir system by again pulling the plates apart to their initial separation. How can we do that? The only way is to apply an outward force equal to or larger than the Casimir force and pull the plates away from one another. Again as expected, in order to do so we must provide exactly the same amount of energy as we obtained from the Casimir force system to lift the mass in the first place -- for instance by lowering the mass we just raised back to its initial position.

It is clear to see that, as predicted by the laws of mechanics, the total energy of the system is always rigorously conserved. However, it is also evident that this Casimir system does not represent a useful engine to lift masses from the ground up as such masses must then be lowered back down to reload the device. It rather resembles a car engine whose pistons can move down but must be pulled back up by hand -- definitely not a desirable situation.

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8. The Transvacer™ Device

In order to design an idealized, yet physically admissible, engine able to carry out a complete cycle, we need a process to manipulate the value of the Casimir force at any given separation of the plates. This is similar to what can be done, for instance, in the cylinders of a steam engine. If, for a fixed position of the piston, we change the temperature of the gas contained in the cylinder, the gas pressure will change. Likewise, again for a fixed position of the piston, we can open a valve in the cylinder and let air in or out and again change the gas pressure. This ability to modulate the force that is doing mechanical work in an engine at constant volume is what permits the cycle to be closed.

Can this be done in the case of the Casimir force? The answer is affirmative, and it represents one of the cornerstones of the applications we are pursuing at InterStellar Technologies Corporation.

The critical concept at the core of our idealized Casimir engine is the well-established fact that, in the realistic case of a material that is not a perfect conductor, the magnitude of the Casimir force at any distance between the plates depends on the detailed optical properties of the boundaries. That is, any process that can alter the reflectivity of the material, also affects the value of the Casimir force at any distance. This can be achieved in a wide variety of ways.

Let us for instance consider two plates made of a semiconducting material. The reflectivity of a semiconductor at any given wavelength is determined by several factors, including of course its lattice structure and its density of charge carriers, such as for instance electrons. By altering the density of charge carriers, one is effectively altering the reflectivity of the material in a range of wavelengths and, consequently, of the Casimir force. In a semiconductor, the density of charge carriers can be modified in any of several ways, such as by illuminating the surface with a beam of light of appropriate wavelength or by changing its absolute temperature.

The fact that the Casimir force can be modulated by acting on the charge carrier density of a semiconductor such as silicon is clearly predicted by available theories and very precise calculations of such have been carried out at InterStellar Technologies and have appeared in the refereed literature. Early experimental evidence that such effect does indeed occur was produced independently almost thirty years ago, although this finding was never used to implement any technological applications and quantitative agreement with theory was lacking because of the lack of precise computations.

With this important tool at our disposal, we are now ready to describe a realistic Casimir engine cycle to continuously raise mass from an initial to a final height.
At the beginning of the cycle, we shall assume that the Casimir force has been set to its maximum value by, for instance, transferring heat to the two surfaces from a large higher temperature heat reservoir thereby increasing their own temperature (Fig. 3). This will increase the charge carrier density and, in turn, the Casimir force. Let us again set the two plates at an initial distance and connect the system to an appropriate mass to find a position of quasi-equilibrium.

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The first leg of the engine cycle closely resembles what we have already discussed in the previous section. However, once the mass has been raised to its final height
(Fig. 4), we now cause the Casimir force to decrease while the plates are kept in a constant position, for instance by transferring heat from the plates to a lower temperature heat reservoir thereby decreasing their temperature and charge carrier concentration (Fig. 5). This induces a decrease of the Casimir force intensity, which requires us to remove some mass brought to final height in the first leg of the cycle. This process of decrease of the Casimir force at constant volume of the space between the plates represents our second leg of the cycle.

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At this point, again similarly to what was done in our initial example, we lower the remaining mass attached to the string back down in quasi-equilibrium. However, the mass being moved downward is now smaller than what was lifted, because of the overall decrease of the Casimir force between the plates. This is the third leg of our cycle (Fig. 6).

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Finally, the cycle is closed by again connecting the Casimir plates to the higher temperature heat reservoir and by causing the Casimir force to increase. This requires us to connect extra mass to the string to retain equilibrium at constant volume. At the end of this fourth leg of the cycle, the system appears exactly as initially, although a finite mass has been permanently raised to the final height (Fig. 7).

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In the case of the Casimir force-based engine cycle just described, zero-point energy is transformed, for instance, into mechanical energy. Of course variants are very numerous, in the way the Casimir force is modulated as well as in the type energy into which the zero-point energy is transformed. However, all such implementations have in common the fact that energy associated with the zero-point field is transformed into usable energy of some type. For this reason, the discoverer of the cycle, Dr. Pinto, introduced the term Transvacer™ device, from the acronym of TRANSducer of VACuum energy device, to describe it.

The background provided in this section therefore justifies the following general definition:

The Transvacer™ device is a Casimir force-based engine designed to carry out a complete cycle to convert a type of energy into another by appropriately modulating the zero-point field.

The short-term commercialization of an extremely broad variety of Transvacer™ device applications, as well as that of a host of advanced applications in zero-point field and Casimir force manipulation, represent the central interest of InterStellar Technologies Corporation. Furthermore, InterStellar Technologies Corporation investigates possible regimes in which the zero-point field energy itself may become available in addition to that exchanged with other sources during the engine cycle. In such regimes, although all laws of thermodynamics can be satisfied, the zero-point field (ZPF) becomes an energy resource from the standpoint of the user.

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