Preliminary analysis of a Nano Relativistic Motor

: (1) Background ; In a recent paper discussing Newton’s third law in the framework of 1 special relativity for charged bodies, it was suggested that one can construct a practical relativistic 2 motor provided high enough charge and current densities are available. As on the macroscopic 3 scale charge density is limited by the phenomena of dielectric breakdown, it was suggested to take 4 advantage of the high charge densities which are available on the microscopic scale. (2) Methods ; 5 We use standard physical theories such as Maxwell electrodynamics and Quantum mechanics, 6 supplemented by tools from vector analysis and numerics. (3) Results ; We show that a hydrogen 7 atom either in the ground state or excited state will not produce a relativistic engine effect, but 8 by breaking the symmetry or putting the electron in a wave packet state may produce relativistic 9 motor effect. (4) Conclusions ; A highly localized wave packet will produce a strong relativistic 10 motor effect. The preliminary analysis of the current paper suggests new promising directions of 11 research both theoretical and experimental. 12


Introduction
Relativity describes space-time structure. In Einstein's famous 1905 article: "On the 16 Electrodynamics of Moving Bodies" [9] the theory was introduced for the first time. This 17 result was a consequence of observations and electromagnetism laws, which were written 18 in the middle of the nineteenth century by Maxwell in his famous differential equations 19 [10][11][12] which owe their modern form to Oliver Heaviside [13]. Those equations imply 20 that an electromagnetic wave travels at the speed of light c, which led the scientific 21 community to believe that light is electromagnetic. Albert Einstein [9] used this to 22 formulate his theory of relativity, which underlines that c is the maximal velocity in 23 nature. According to relativity, any object, message, signal (even if not electromagnetic), 24 or field can not travel in as speed greater than the speed of light in vacuum. Thus 25 retardation is established, if a phenomena occurs at a distance R from an observer, it will 26 not be noticed for at least a time of R c . This means that action and reaction cannot be 27 generated simultaneously because of propagation speed. 28 It was thus suggested to take advantage of the high charge densities that are 94 available in the microscopic realm, for example in ionic crystals. We will further pursue 95 this idea in this paper in which we calculate the high charge densities and current 96 densities in the atomic level. We will also deduce a preliminary form for the optimized 97 wave function in terms of relativistic engine performance. We shall not derive the basic 98 equations of the relativistic engine here the interested reader is referred to [20], only the 99 principle results will be quoted. Let us consider two charged sub systems having the charge densities ρ 1 , ρ 2 and current densities J 1 , J 2 then according to the analysis presented in [20] the following force is acting on the physical system composed of those two sub systems: in the above µ 0 = 4π 10 −7 is the vacuum permeability in MKS units, ∂ t is a partial 102 temporal derivative. d 3 x 1 and d 3 x 2 are integrals of the volume of the first and 103 second subsystems respectively. We define R = x 1 − x 2 , R = | R|,R = R R .

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In the next subsections, we analyze examples of the implications of this formula. 105 We remark that the it is based on a second order approximation, in some fast changing 106 systems the second order correction will not be sufficient and higher order correction 107 will be requited. 108 2.1. Some basic preliminary observations 109 According to Newton's second law, a system with a non zero total force in its center of mass, must have a change in its total linear momentum P(t): F [2] T = d P dt (2) It can be shown that gravity alone has no "force" in free space and in case where gravity has a considerable effect the right hand side should be replaced by a geodesic equation of motion. However, in most cases electromagnetism is orders of magnitude stronger. Hence the right hand side should be considered a valid approximation to the geodesic form. Assuming that P(−∞) = 0 and null current and charge densities at the same time, it follows that: Thus we obtain finite linear momentum for stationary charge and current densities: it follows that the charged relativistic engine can produce forward linear momentum without interacting with any external system except it own electromagnetic field. The above expression can be simplified using the instantaneous (non retarded) potentials: in which we are reminded that µ 0 0 = 1 c 2 . Another important result is that in a charged relativistic motor we do not need both subsystems to be charged, that is we can take ρ 2 = 0: provided that the system has a non vanishing current density J 2 . The same result will be 110 obtained if sub systems 1 and 2 are charged but system 1 lacks current density: In [4] we showed that the forward linear momentum gained by the mechanical 112 system will be balanced by a backward linear momentum gained by the electromagnetic 113 system. We now study the challenges involved in constructing a powerful charged 114 relativistic engine. The charge that can be put in a volume or two dimensional surface is restricted due to the phenomena of electrical breakdown. In this case the surrounding medium is separated into electron and ions and becomes a conducting plasma. Thus a discharge results and the charge density is reduced and even nullified. The typical dielectric strength E max of air is 3 MV/m [21], for high vacuum one can achieve 20-40 MV/m [22] and for a diamond 2000 MV/m [23]. For an infinite surface the surface density σ is: in which r is the relative susceptibility. For air σ max 53 µC/m 2 . To calculate the charge Q which one can maintain in a given volume we notice that for a spherical symmetric charge ball we obtain at a distance r the radial field E: the stronger field is on the ball itself, that is at r = r s we must have: heating. This shortcoming can be circumvented using a superconductor, however, this 131 will require cooling to low temperatures, making the system cumbersome. But, even 132 a superconductor has a critical current density and superconduction properties losses 133 above that. Jung, S. G. et al. [25] have reported critical current densities as high as 134 5 kA/cm 2 . Coil windings will enable to reuse the current, and thus the winding number 135 in a given area is critical for performance. Proximity of current and charge will affect the amount of generated linear momenta according to equation (7) . However, installing a 137 conductor close to the charge may result in discharge, hence a balance is required. 138

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It is obvious from equation (7) that a larger relativistic motor is more powerful. An engine of mass M will acquire a kinetic energy of: P is calculated according to equation (

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Notice also that if not configured correctly a relativistic motor may radiate. This 158 was shown in previous works for an uncharged relativistic motor [7,8]. The total force 159 generation given by equation equation (1) includes also radiating field configurations, 160 hence in this respect it will not affect the reported results. In a previous paper [20] we have shown that intrinsic parameter limitations, es- average this leads to a null charge density. We also notice that a smaller lattice constant 172 will cause a still higher charge density, but it seems that the lattice constant cannot be 173 smaller than the relevant atom hence it is bounded from below. To see how we can circumvent this unfortunate reality we will investigate equation ( 175 7), calculating the spatial Fourier transform of the scalar potential and the current density 176 such that: Using the theorem of Parseval [26], we may now write equation (7) in the form: In this form it is obvious, that a microscopic distribution of periodic charge density will be beneficial if it is a accompanied by a current density distribution of the same period. How can we obtain such a microscopic charge density distribution? To this end we remind the reader that microscopic currents are associated with the electronic motion and electronic spin. The magnetization M is related to the magnetization current J M by the formulae [2,3]: We may replace in equation (6), equation (7) and equation (13) J by J M to obtain a similar effect. Furthermore, the magnetization M is related to a the microscopic dipole moments m i through: in which we sum over all dipoles and divide by the sample volume V. In known involve an ionic lattice in which one species of atom (say the positively charged) will 185 involve free spins that can be manipulated by an external magnetic field thus creating 186 a relativistic engine effect which can be easily manipulated in three axis. In the next 187 section we discuss the properties of a relativistic engine in the microscopic scale. this will not concern us here. Matter at the microscopic scale is described by quantum 194 mechanics. In this preliminary work it will suffice to discuss a one electron system in 195 which the nuclei effecting the motion of the electron are modeled as point charges. Schrödinger's electron is an electron in which the spin property is ignored. The earliest appearance of the non-relativistic probability density and current density satisfying a continuity equation is due to Schrödinger himself [27], obtained from his time-dependent wave-equation: in the above i = √ −1 and ψ is the complex wave function.ψ = ∂ψ ∂t is the partial time derivative of the wave function.h = h 2π is Planck's constant divided by 2π and m is the particles mass, V is the potential of a force acting on the particle. If now the modulus a and phase φ are introduced through: The following continuity equation is satisfied provided that ψ satisfies equation (16): in which the probability density is defined as: and is of course normalized, that is: The probability current density is: As the charge density of an electron with charge −e is: it follows that the current density is: Satisfying the charge density continuity equation: 4.2. Pauli's electron 198 Schrödinger's quantum mechanics is limited to the description of spin less particles. If spin is to be taken into account we must use the Pauli equation (for a non-relativistic particle) : ψ here is a two dimensional complex column vector (also denoted as spinor),Ĥ is a two dimensional hermitian operator matrix, µ is the magnetic moment of the particle, c is the velocity of light in vacuum. The electromagnetic interaction is described by the vector A and scalar V potentials and the magnetic field B = ∇ × A. σ is a vector of two dimensional Pauli matrices which can be represented as follows: A spinor ψ satisfying equation (25) must also satisfy a continuity equation of the form: In the above: The symbol ψ † represents a row spinor (the transpose) whose components are equal to the complex conjugate of the column spinor ψ. Holland [28] has suggested the following representation of the spinor: In terms of this representation the probability density is given as: The charge density is given as: The spin density can be calculated using the representation given in equation (29) as: This gives an easy physical interpretation to the variables θ , φ as angles which describe the projection of the spin density on the axes. θ is the elevation angle of the spin density vector and φ is the azimuthal angle of the same. The probability current density can now be calculated by inserting ψ given in equation (29) into equation (28): if the vector potential contribution is not significant, we may write: Hence, the electric current density is: If the electron is in a definitive spin state (say spin up) and the vector potential is null, 199 equation (25) is the same as equation (16). Thus for preliminary order of magnitude  The Hydrogen atom is one of the simplest quantum mechanical systems, and one of the few quantum mechanical systems that can be solved exactly. The Hydrogen atom is composed of a proton and an electron, the proton is modeled as a positive point charge of charge +e located conveniently at the origin of the axis. V in equation (16) is thus equal to −eΦ which is felt by an electron of charge −e, in which Φ is given in equation (  5) for a positive point charge +e located at the origin of axis, that is: The electron wave function can be of the form: In which ψ n is a spatial eigenfunction of the HamiltonianĤ with eigenenergy E n . The electron can be in a definite energy state or in a superposition of states. The functional form of ψ n is well known and is given in terms of the spherical coordinates r, θ, ϕ as follows: In the above L 2l+1 n−l−1 is the generalized Laguerre polynomial of degree n − l − 1, and Y m l (θ, ϕ) is a spherical harmonic function of degree l and order m defined as: In which P m l are associated Legendre polynomials. The definition also contains the reduced Bohr radius: in which m e and m p are the masses of the electron and proton respectively. Finally we use a normalized radial coordinate r defined as: The eigen energies defined in equation (37)  Hence the function ψ nlm are degenerate in the sense that different functions have the same energy, the degeneracy can be lifted if there is a perturbation that changes the potential V to a form that is not spherically symmetric. For a given energy the different eigen functions are listed by their azimuthal quantum number l = 0, 1, 2, . . . , n − 1 and their magnetic quantum number m = −l, . . . , l. Notice that the amplitude of the wave function given in equation (37) is a function of r and θ only, but not of ϕ and the time t.
Notice, also that the phase is a function of time and ϕ but not of r and θ: We are now in a position to calculate the current density given in equation (23) J =h m e ρ ∇φ = −m eh m e |ψ nlm | 2φ r sin θ (the reader should not confuse the magnetic number m with the mass m used in previous sections). In the above we have used the spherical representation of the nabla operator which is given in terms of the unit vectorsr,θ,φ as: We notice that the current density is linear in the magnetic number m, in particular if m = 0 there is no current density and thus no relativistic motor effect. We conclude that for an isolated hydrogen in the ground state n = 1, l = 0, m = 0 there is no relativistic motor effect. But also in excited states in which the current density does not necessarily vanish there will be no relativistic motor effect if the potential acting on the electron is spherically or cylindrically symmetric as is evident from equation (7). To see this notice that:φ = − sin ϕx + cos ϕŷ (47) x andŷ are constant unit vectors in the x and y directions respectively. In spherical coordinates the volume element is: since Φ|ψ nlm | 2 is cylindrically symmetric it does depend on ϕ and the result follows 203 immediately. We also notice that apparent singular terms in the current density of the 204 form 1 r sin θ (see equation (45)) are cancelled out by identical terms in the volume element.

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How can we use an hydrogen atom as a component in a relativistic motor despite 206 the fact that it is useless either in the ground state or in an excited state? In the following 207 section we will suggest two approaches, in one the electron is not in an energy eigen 208 state but in a superposition of states and in the other the potential is not cylindrically 209 symmetric. Both approaches will yield a finite relativistic engine effect.

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But before we proceed we make some orders of magnitude estimates to justify our endeavour. The volume charge density of an electron in a hydrogen atom has an order of magnitude:ρ = e  Let us assume an idealized wave packet of the form: in the above k andρ c are constants. As the wave function must be normalized it follows from equation (20) thatρ c must take the following value: hence this wave function has a linear phase and a uniform amplitude which is confined inside a sphere of radius R max . It is certainly not an eigen state of the Hydrogen atom Hamiltonian but may be approximated by a superposition of the eigenstates: The desired functions a nlm (t) can be calculated using the orthonormality properties of the eigen functions: the preparation of such a state will require a suitable electromagnetic pulse in accordance 214 with equation (25), the shape of the pulse for creating and maintaining the wave packet 215 is beyond the scope of the current paper. 216 We can now calculate the current density using equation (23) as follows: v has the units of velocity. Let us now take system 1 in equation (7) to be the proton and system 2 to be electron. The proton is conveniently modelled as a point charge located at the origin of axis. That is: It thus follows: Plugging equation (48) and equation (57) into the above expression and integrating will result in: Taking into account normalization according to equation (54) we obtain the following expression for P(t): It follows that the momentum gained by a relativistic motor is linearly proportional to the electron's "velocity" v and inversely proportional to the wave packet spatial extension R max . As the maximal speed generated by the hydrogen relativistic motor is obtained when the engine is not loaded and need to carry only its own mass it follows that: the electron mass m e is ignored as it is much smaller than the proton mass m p . Thus to obtain a predefined velocity v max we need a wave packer of the radius: According to equation (52) the typical "velocity" of the electron is 0.03 c in the hydrogen atom. Here we will assume that v c for the purpose of obtaining the maximal R max required, taking into account that in reality we will need a smaller R max to achieve the desired velocity. We also remark that for a relativistic electron the Schrödinger formalism is not adequate and one should use a Dirac equation instead. Moreover, the relativistic engine considered so far assumes slow moving components where the relativistic effect is due to the retardation of the electromagnetic signal, a different mathematical treatment is needed if the components of the engine move with relativistic speeds. Taking this into account we make the following preliminary observations: for a typical car a velocity of v max = 50 m/s = 180 km/h we obtain: in which r p = 8.4 10 −16 m is the proton charge radius. Thus for such velocities the wave packet is of a typical nuclear size rather than an atomic size. Finally if we imagine that the relativistic engine will reach the maximal speed available in a Lorentzian space-time for a particle which is initially subluminal that is v max c it follows that: that is the wave packet must be of sub nuclear dimensions. 217 We conclude this section by calculating the standard deviation of the wave packet defined in equation (53).
In which the expectation value of position is: It follows thus that: Similarly: Thus the electron is expected to be found in coincidence with the proton with a standard deviation which is about half the size of the wave packet. Calculating the expectation value for the momentum operatorsp x = −ih ∂ ∂x we find that: similarly: hence the expected velocity of the electron is equal to the current velocity defined in equation (57) and in the same direction: The standard deviation of the momentum operator in any direction is infinite due to the ideal discontinues form of the wave function, meaning that when measuring the electron velocity any value can be obtained with non vanishing probability. We notice, however, that even for a more realistic and smooth wave packet we expect a non vanishing standard deviation of the electron momenta due to the Heisenberg uncertainty relation: hence the measured velocity of the electron may differ significantly from the expected 219 current velocity. As pointed out earlier the relativistic engine effect will be null for an hydrogen atom in either the ground or excited state due to the symmetry of the proton potential. However, if the symmetry is broken the relativistic engine effect is recovered. We thus assume an additional proton located at a distance d from the hydrogen atom. This proton will break the cylindrical symmetry. In what follows we will consider system 1 as the additional proton and system 2 will be the electron. We will assume that the proton is at a distance d from the axis origin in the negative x direction. The potential generated by this additional proton locate at x 1 = −dx is according to equation (5): From now on we drop the index 2 and use standard spherical coordinates. Hence r = | x 2 | and x 2 = r sin θ cos ϕ, thus: the potential ϕ dependence clearly demonstrates the symmetry breaking of the above potential. We can now calculate the relativistic motor momentum defined in equation (7) by inserting the potential of equation (76) and the current obtained in equation (45) for an electron in an eigen state of the hydrogen atom: (77) in whichJ is given in equation (51), m is the magnetic quantum number (not the mass), andφ is given through equation (47). Finally: is the dimensionless state amplitude in which we suppress the quantum indices nlm. The above expression can be somewhat simplified as follows: in the above: Obviously the cylindrical symmetry is restored when B = 0. This happens for either small d or large d, that is if the proton is too close or too far from the electron: that sin θ ≥ 0 as θ ∈ [0, π], thus we are only interested in B ≥ 0. We also notice that the 223 maximal value of B is for θ = π 2 and for r = d for which B max = 1 as depicted in figure 2. A triple integral needs to be evaluated in order to calculate the momentum, this cannot be done using only analytic techniques, however, at least part of the integration can be done analytically. Consider the integrals: Integrating analytically we arrive at the following results: in the above Elliptic E and Elliptic K are the elliptic E and K functions respectively. The 225 function I y is an even function that is I y (B) = I y (−B) and is depicted in figure 3. We remark that since B ≥ 0 it follows that I y (B) ≤ 0, thus only the right hand 227 part of figure 3 is of interest. We also notice that I y (B) is monotonic decreasing function 228 of B hence the larger values of B which are around r = d, θ = π 2 will make the most 229 significant contribution (in absolute terms) to I y (B). For B = 1 this function is singular, 230 however, the singularity will be integrated away when integrating over θ. 231 We have obtained a counter intuitive result that the momentum is generated in the y direction despite the fact that proton which breaks the symmetry is located in the x direction, that is: Next we define the function F as follows: F will become larger if both A 2 and I y are peaked at the same r value, if the overlap is small so will be the relativistic momentum. We can thus write the momentum P in the form: Thus P is given in the form: P =PPŷ. in whichP is a dimensional constant independent of the quantum state of the electron: AndP is dimensionless, and depends on the quantum state.
in the above we have made explicit the dependence on the quantum numbers and have made use of the normalized r defined in equation (41) and also introduced a normalized d using a similar definition: We also notice that since the hydrogen eigen state is given as a function of r and also B being dimensionless can be written in terms of dimensionless quantity: it follows thatP nlm only depends on the quantum state and d .

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We shall now investigate the case d = 2a 0 . In such a case the Hamiltonian of the electron is modified significantly and thus will have different eigenstates from the ones that are obtained for an isolated Hydrogen atom. Hence the eigen states described in equation (38) can only be considered as a superposition of the true eigen states. Nevertheless, it will suffice to study the relativistic momentum generated by the Hydrogen eigen states in this preliminary study. In this case: The velocity associate withe this momentum is: in we have taken into account both the mass of the proton and the mass of the Hydrogen atom. Hence for an unloaded relativistic motor we will obtain a velocity of: We shall look at two cases of excited states.  It is seen that the function describes torii of equi-value. The function peaks at θ = π 2 237 at r = 2a 0 . A cross section of the same for θ = π 2 is described in figure 5: We are now able to evaluate numerically F 211 defined in equation (85) Obviously even if many such systems are accumulated the total mass will grow accord-241 ingly and the final achievable velocity will be rather slow. It is seen that the function describes torii of equi-value. The function peaks at θ = π 2 246 at r 12a 0 , hence the Hydrogen atom becomes much larger. A cross section of the same 247 for θ = π 2 is described in figure 8: We are now able to evaluate numerically F 433 defined in equation (85) which is 249 depicted in figure 9. Finally using F 433 we can calculate numericallyP 433 using equation (89) and obtain the value:P 433 −0.002.
this is surprisingly smaller than P 211 , but can be understood due to the small overlap between I y (B) and A 2 433 . Now according to equation (87) and equation (94) we obtain: Obviously even if many such systems are accumulated the total mass will grow accordingly and the final achievable velocity will be rather slow. The situation can be some what improved if the overlap between I y (B) and A 2 433 is better. This will happen if we choose d 12 a 0 , however, a detailed calculation gives in this case: this is better than before but worse than what is obtained in a low excited state 211. A similar analysis was attempted for a very high excited state n = 11, l = 10, m = 10, in this case which appears in Humphrey's series [29] A 2 11 10 10 peaks at 110 a 0 5.8 nano m, which thus justly deserves the name a nano relativistic motor. Choosing d to coincide with the above value we obtain, however, after a detailed calculation the values: P 11 10 10 −2.0 10 −32ŷ kg m/s v 11 10 10 = −5.9 10 −6ŷ m/s.
this does not qualify as an improvement.

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In a previous paper, we have shown that, in general, Newton's third law is not 255 compatible with the principles of special relativity and the total force on a two charged 256 body system is not zero. Still, momentum is conserved if one takes the field momentum 257 into account, and the same is true for energy.

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The dielectric strength's and current density limitations of macroscopic bodies led 259 us to consider the microscopic realm in which charge densities and current densities are 260 considerably higher.

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The main results of this paper are the possibility of implementing a relativistic 262 motor in the atomic and nano scales. It is shown that a Hydrogen atom whether in a 263 ground or excited state does not produce any momentum according to the relativistic 264 motor equation.

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Thus two configurations are studied, in one the Hydrogen electron is put in a wave 266 packet state rather than in an eigen state. In this approach one can reach high momentum 267 which means high velocity for both a loaded and an unloaded engine. However, for a 268 considerable momentum to be realized the wave packet must be highly localized to a 269 subatomic scale. To achieve even higher momentum, sub nuclear scales are required. In

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To conclude we remark that despite the possibility in principle to construct a 279 working relativistic motor, this is not a trivial task and involves the creation of a highly 280 localized wave packet. Thus in a study which is not a merely preliminary as this one, 281 the electromagnetic field needed to achieve this goal must be specified. And we notice 282 that the wave packet should not only be created but also maintained for an engine that 283 may become useful.

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Additional directions for future studies which are arise from this paper include:

3.
For the symmetry breaking approach, despite the fact that it shows little promise 293 for a useful relativistic motor, one should use true eigenstates (and not eigen-states of the Hydrogen atom which are clearly not appropriate). Perhaps for those 295 eigenstates the results will be more encouraging.