ENGINEERING A BRIDGE BETWEEN QUANTUM ELECTODYNAMICS AND QUANTUM GRAVITY – AN ENGINEERING MODEL
TODD J. DESIATO Statesville, NC 28625 [email protected]
Abstract For engineering purposes, it is proposed that gravitational fields may be interpreted as a reduction in the relative available driving power (Watts) of the Electromagnetic, Zero-Point Field. It is shown that variations in the relative power are covariant with variations in the coordinate speed of light as measured by a distant observer, outside the gravitational field. Gravitational time dilation and length contraction may then be interpreted as a loss of power from the interaction between matter and the zero-point field. It is hypothesized that the loss of power is due to increased radiative damping within matter due to Larmor radiation. The relative radiative damping factor affects the relative ground state energy of the quantum mechanical harmonic oscillator such that, the mean-square fluctuations of matter reproduce the behavior attributed to and resulting from the space-time metric of General Relativity. From this principle the phenomenon observed by a distant observer that are due to gravity, may be reproduced from the variable relative damping factor acting on the quantum harmonic oscillator. Utilizing the technics of Nuclear Magnetic Resonance to stimulate Larmor radiation, experiments may be conducted which may pave the way for the creation of breakthrough technologies, such as, Artificial gravity, Anti-gravity and Warp Drive.
Keywords: quantum engineering, quantum gravity, quantum electrodynamics, artificial gravity, anti-gravity, warp drive, electromagnetic zero-point field, general relativity, alternative models of gravity, breakthrough technology
ENGINEERING A BRIDGE Updated - 3/27/20 1 Nomenclature
gµν metric tensor where, µ and ν are indices in this context χ = (V-s) represents a scalar field of random magnetic flux quanta
K = g11 / −g00 the relative coordinate dependent refractive index of the vacuum
c0 = (m/s) the speed of light in vacuum as measured in a local inertial reference frame
cK = c0 / K = (m/s) the relative coordinate speed of light as measured from outside the gravitational field
Δx0 = (m) an interval along the x axis as measured in a local inertial reference frame
Δx = Δx0 / K = (m) an interval along the x axis as measured from outside the gravitational field
Δt0 = (s) an interval of time as measured in a local, inertial reference frame
Δt = Δt0 K = (s) an interval of time as measured from outside the gravitational field q2 = (C) squared magnitude of the electrical charge quantum e ! = (J-s/rad) the reduced Planck's constant, h / 2π
ε0 = (Farad/m) the dielectric permittivity of vacuum as measured in a local, inertial reference frame
µ0 = (Henry/m) the dielectric permeability of vacuum as measured in a local, inertial reference frame G = (N-m2/kg2) the gravitational constant
1. INTRODUCTION
Practically speaking, time is measured with a clock and space is measured with a ruler. Regardless of their unique construction, each is just a device used to compare with other similar devices at different sets of coordinates. An observer peering into the night sky uses his own devices to establish a coordinate system with which to compare his observations to identical devices at distant coordinates. He chooses for example, to observe the light emitted by distant supernovae and then compare it to the light emitted by other similar events at various locations, [1]. From this data and knowledge of the atomic reactions that generate these explosions, the distance to these events, and their motion relative to the observer is determined.
There are other ways to achieve this of course. Clocks, rulers or supernovae are examples given to illustrate the point, that measurements are made using physical tools of our choosing which are composed of some form of matter. To date, it appears that all matter is subjected to the physical effects of gravity in the same way. In other words; all objects fall at the same rate such that; there are no absolute rulers, or absolute clocks that are impervious to the physical effects of gravity.
ENGINEERING A BRIDGE Updated - 3/27/20 2
Working in the reference frame of a distant observer, space and time appear to be variables. General Relativity (GR) interprets these variables as space-time curvature when the local devices and the remote devices disagree. They disagree because matter experiences gravitational length contraction and time dilation in the presence of gravitational fields. This is not an illusion. Time dilation and length contraction are real, physical effects whose action can be described using elementary quantum mechanics, and the correct procedure to do so, which shall be shown here.
The model presented here-in uses the reference frame of the distant observer to approximate GR as a scalar field, because it allows all observations to be consistently scaled without the need of complicated tensor coordinate transformations, as this is all that is required to illustrate the engineering concepts to be conveyed. However, like a photograph displayed on a monitor appears to be a continuously smooth image when in fact it is pixelated at a scale that is below the resolution of our senses. Individual quantum oscillators behave in such a way that, in large numbers their averages reproduce the behavior of classical test particles on a curved space-time manifold, as directed by Einstein’s field equations of GR, [2].
Due to this quantum to classical correspondence, it is necessary to drop any expectation of using quantum field theory on a curved space-time manifold. In this model, space-time is considered to be perfectly flat background stage upon which the observations of curved space-time are an emergent property. As such, the simplified equations of physics done in flat space-time will be applied throughout.
Engineers are clever, but aside from the calibration of the Global Positioning Satellite network, engineers really don’t know what to do with space-time curvature as a means to manipulate gravity. The Gravitic Caliper is not yet a tool in our toolbox. Likewise, referring to gravitational fields as a variable refractive index, as is done in the Polarizable Vacuum (PV Model) Representation of GR [3, 4, 5, 6, 7] adds some intuitive, pedagogical value to understanding gravitational fields, but does not address the pressing issue of; “What to do to create or mimic gravity?” What engineers require is a more practical set of tools to work with when dealing with the effects of gravitational fields, so that they can acquire a deeper understanding of the “Nuts and Bolts” regarding how gravity and matter interact. These tools are precisely what this paper will address.
Space-time curvature is a useful mathematical description of the available data regarding gravity, but it is not the only useful interpretation of the data. The interpretation presented herein, describes gravitational time dilation and length contraction in the proximity of large gravitational bodies, as a physical effect acting on clocks and rulers at the quantum scale. This physical effect begins with a simple harmonic oscillator. Something most engineers should be familiar with. For the practical purposes of discussion, matter may be usefully approximated as being comprised of such oscillators, [8].
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If there is power dissipation (damping) occurring within the oscillator, eventually the oscillation will exponentially decay to its lowest energy state. In a passive electronic oscillator circuit for example, there may be a sinusoidal power supply (a.c. source) driving a resonant LC circuit, [9]. In the circuit there may be a resistance, R which dissipates power and damps the oscillation. Eventually, the source of power and the dissipation reach an equilibrium condition.
In the case of matter, when a system of particles decays to its lowest energy state, it is in the ground-state. Where, the minimum energy is not zero, [8, 10]. The minimum energy is the equilibrium between a uniform zero-point field, (ZPF) which drives the oscillators, and a variable damping function which damps them. The damping function is dependent on the presence of a local mass-energy density, which stimulates radiative damping as Larmor radiation, resulting in the observed behavior of oscillators in a gravitational field. Where, they have a lower ground state energy than they would in an unperturbed ZPF. In other words; the damping function lowers the relative ground state energy below that which the ZPF establishes as the natural ground state in a vacuum far from any gravitational bodies. In GR, this reduction of the ground state energy is interpreted as gravity possessing negative energy, [11].
In section 2, the physical effects of gravitation are derived from the space-time metric and associated with the variable refractive index of the PV Model for illustration. In section 3, the quantum vacuum processes that determine the ground state equilibrium condition between matter and vacuum are discussed, in addition to the co-variant relationship between relative power and the relative coordinate velocity of light.
In section 4, the relative radiative damping factor is derived and the connection to gravity is established. It is shown that the variable metric coefficients result from variations in the radiative damping factor that reduces the relative available power of the ZPF, making a test particle move in a way which may be interpreted as curved space-time.
In sections 5, Inflated Matter is introduced as an alternative interpretation of Exotic Matter. Exotic Matter is defined as negative energy density and is in violation of the strong and weak energy conditions of GR. It is a state in which something is below the minimum energy density of the surrounding vacuum. The idea if inflated matter being that since volume increases faster than energy content, matter which is inflated to a larger scale has a lower energy density than mater which has been contracted by gravity. Hence, inflated matter is an alternative way to look at exotic matter and anti-gravity. It is the opposite effect to gravitational length contraction.
In section 6, the idea of gravity as power loss is brought down to Earth, showing that the rate at which matter in free-fall loses power is, −g2 the square of the gravitational acceleration at the surface.
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In section 7, the notion of quantized magnetic flux is introduced. It is argued that a model of quantum gravity based on quantized scalar magnetic flux will have no effect on Maxwell’s equations. Although the flux might be physically real, it appears only as a simple gauge transformation of the Electromagnetic Potentials. It is shown how this process can affect the refractive index of the vacuum, defining a quantum mechanical process for gravitational lensing.
In section 8, it is demonstrated how a variable magnetic flux density, applied when conducting spectroscopy measurements using Nuclear Magnetic Resonance, (NMR) will lead to physical effects equivalent to gravity. The magnitude of the effect to be observed is dependent upon the amount of coupling and penetration of the applied fields into the volume of matter to be affected. Here, the rates of change in the applied magnetic field and the power requirements for such a test are considered.
2. THE PHYSICAL EFFECTS OF GRAVITATION
It shall be shown that a gravitational field may be interpreted as a variable refractive index that alters space-time and determines the relative scale of rulers and clocks in the altered, as measured by a distant observer in an unaltered region of space-time outside the gravitational field, [2, 3, 4, 5, 6, 7]. A brief introduction to the physical effects that engineers will encounter when working with modified space-time and matter, in the context of GR and the PV Model, will be presented in this section.
One obvious disadvantage of having the perspective of a local observer is that the speed of light remains constant in the local inertial reference frame. Observers in the local frame cannot measure light moving faster than c0 , the speed of light in vacuum. Nor can they measure light moving slower than c0 , while using rulers and clocks immersed in the same local vacuum. Therefore, it is advantageous for engineers to understand what to expect, what to look for and why there is a need to make observations from the perspective of a distant observer in an unaltered reference frame, outside of the gravitational effects to be measured.
In GR, the four-dimensional line element is given by the expression,
2 µ ν ds = gµν dx dx (1) where summation is assumed for repeated indices. In a flat space-time, the line element reduces to the more familiar expression,
2 2 2 2 2 2 ds = −c0 dt + dx + dy + dz (2)
ENGINEERING A BRIDGE Updated - 3/27/20 5 The reader does not need to be well versed in GR to follow most of what is presented here. Think of this as calculating the length of the hypotenuse of a right-triangle in two dimensions. In two 2 2 2 2 dimensions, ds = −c0 dt + dx . The metric coefficients from equation (1) are, g00 = −1, g11 = 1, 0 1 and gµν = 0 for µ ≠ ν . Where, dx = c0dt and dx = dx in Cartesian coordinates.
dx c For any light ray, the squared length, ds2 = 0 and may be solved to discover, c = = 0 is K dt K the relative coordinate velocity of light. The refractive index of this metric is then simply defined by, K ≡ c0 / cK . The absolute value brackets are necessary to indicate that the refractive index is positive. The implications of a negative refractive index are not considered here-in for brevity.
Similarly, Eq. (2) may be written in terms of variable metric coefficients g00 and g11 , in a curved space-time. Typically, they take on values that are determined by a solution of Einstein’s field equations of GR, such as the Schwarzschild solution.
For simplicity, in two dimensions the resulting line element becomes,
2 2 2 2 ds = g00c0 dt + g11dx (3)
The refractive index can now be read off as shown in equation (4).
K = g11 / −g00 (4)
In GR, the metric coefficients alter the scale of rulers and clocks in their region of influence, as compared to those of the observer in a distant unaltered region. For example; when −g00 < 1 and g11 > 1 , then −g00 dt < dt and, g11 dx > dx . Clocks in the altered region, as well as atomic oscillations there, appear to have slowed down; Δt = Δt0 / −g00 and rulers in the altered region, as well as atomic spacing, appear to have contracted; Δx = Δx0 / g11 , as compared to those rulers and clocks used by the distant observer, [3, 7]. This is simply gravitational time dilation and length contraction as described in GR.
The refractive index representation is accompanied by the same physical effects in the gravitationally altered region of space-time. In the special case where −1/ g00 = g11 = K , the physical effects of altering the refractive index can be simplified and tabulated for engineering purposes in terms of K , as shown in TABLE 1, [7].
ENGINEERING A BRIDGE Updated - 3/27/20 6 TABLE 1: Physical effects of space-time acting on matter in a gravitationally altered region, as measured by a distant observer in an unaltered region of space-time.
Anti- Gravity of Variable Gravity or a Massive Refractive Index FTL Star K = g / −g Effects 11 00 K > 1 K < 1 speed of speed of
cK = c0 / K light is light is slowed faster rulers rulers Δx = Δx / K 0 contract expand clocks run clocks run Δt = Δt K 0 slower faster velocity is velocity is v = Δx / Δt = v / K 0 slower faster acceleration acceleration a = a / K 3/2 0 is decreased is increased force is force is invariant invariant m = F / a = m K 3/2 0 mass is mass is increased decreased frequency is frequency ω = 2π / Δt = ω / K 0 decreased is increased ground state ground ΔE = !ω = ΔE0 / K energy state energy decreases increases
Power is measured in Watts. It is the change in energy per unit of time. Power, PK varies inversely with the refractive index and is therefore covariant with the relative coordinate velocity of light, cK . Referring to TABLE 1, it can be shown that,
PK = ΔE / Δt = P0 / K, W m (5) c = c / K, K 0 s
ENGINEERING A BRIDGE Updated - 3/27/20 7 Why this is true will become evident in section 4 where the effects of damping on the oscillator’s power will be derived from first principles.
From the perspective of the distant observer in an unaltered region of space-time it is observed that, in a gravitational field near a massive star for example; clocks run slow, rulers are shorter, and the speed of light has become slower. The conclusion drawn would be that, matter in the region is running low on power. There is not enough power to inflate matter to its "proper" size, as was presented in [7].
Alternatively, from the same perspective, in a region of space-time where matter is observed to be moving faster than light (FTL). As would be the case for an observer looking up from a gravity well, such as; hovering near the event horizon of a black hole. Matter is relatively inflated, clocks are running fast, rulers are expanded, and the speed of light has increased. The conclusion would then be drawn that; the scale of matter and the speed of light is regulated by the relative driving power available to do work to drive these physical processes, as will be shown in the following sections.
3. ZERO-POINT EQUILIBRIUM
Although a charged particle is constantly undergoing accelerated motion due to interactions with the electromagnetic ZPF, it does not appear to radiate, [8, 10]. The reason for this apparent lack of radiation is that the ground state of the particle is at steady-state equilibrium with the electromagnetic ZPF of the quantum vacuum, in which it is immersed.
In section 3.3 of The Quantum Vacuum, Milonni writes;
"The fact that an accelerating charge loses energy by radiating implies, according to classical ideas, that an electron should spiral into the nucleus and that atoms should not be stable. The balancing of the effects of radiation reaction and the vacuum field..., however, suggest that the stability of atoms might be attributable to the influence on the atom of the vacuum field.... We now know that the vacuum field is in fact formally necessary for the stability of atoms in quantum theory. As we saw..., radiation reaction will cause canonical commutators [x, px ] to decay to zero unless the fluctuating vacuum field is included, in which case commutators are consistently preserved." [8]
In an inertial reference frame where a charged particle is in bounded, steady-state motion, such as in a harmonic oscillator, an atom or matter in general. There is a non-zero ground state that is in equilibrium with the local ZPF. All fields, including the Dirac field (fermions) that makes up matter, and force carriers, (photons and gluons) have a ZPF where the ground state energy is,
Egs = !ω / 2 , per frequency mode.
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An intuitive way for engineers to look at this is that in the ground state, the mean power absorbed by the particle from the ZPF is equal in magnitude to the power radiated from the particle, due to instantaneous accelerations from collisions, stimulated and/or spontaneous emissions, i.e., random noise. This leads to a natural equilibrium symmetry,
rad abs Pa = Pzp (6)
The works of Milonni, [8, 12, 13] and also Puthoff, [10] illustrate this clearly. Power absorbed from the ZPF by a charged particle in bounded, steady-state motion is given by,
2 3 abs q !ω 0 Pzp = 3 , W (7) 12πε0m0c0
Where, ω 0 is the natural resonant frequency of oscillation. Power radiated from an accelerated charge to the vacuum is given by Larmor radiation, [14].
2 2 rad q a Pa = 3 , W (8) 6πε0c0
Where, a2 is the mean-square acceleration fluctuation given in Eq. (15). It may be inferred that when the symmetry of this equilibrium state is broken, the system accelerates.
For simplicity and its intuitive, classical interpretation, Puthoff, in [10] uses the Stochastic Electrodynamics (SED) Fourier composition of the ZPF’s electric field as the proposed driving function in Eq. (9). However, the end results are the same when adhering to strict Quantum Electrodynamics’ (QED) formalities, [8]. It is not necessary for engineers to learn or use Dirac’s Bra-ket notation, [15] in order to understand the information which follows. The objective of this article is to convey An Engineering Model in the most useful way possible, to facility understanding of the concepts to be used as tools of the trade.
The electric field is given by,
2 1/2 V E = Re d 3kεˆ ω / 8π 3ε eik⋅r−iωt+iθ(k,σ ), (9) ZP ∑∫ (! 0 ) σ =1 m
ENGINEERING A BRIDGE Updated - 3/27/20 9 The exponential term is the wave part of the equation. A similar equation can be expressed for the magnetic field by replacing E with H , the unit vector ˆ with ˆ ˆ and with . ZP ZP ε (k × ε ) ε0 µ0 However, this example is sufficient to describe the concepts to be conveyed.
The indefinite integral over the field modes, k in Eq. (9) may be tuned by utilizing matter in ways that alter the limits of the integration, as is done in the Casimir effect, [8, 16]. These limits are referred to as the cut-off modes of the field. The frequency limits of the ZPF are formally infinite but can be modified or limited by the presence of matter.
The electromagnetic ZPF is used to calculate the mean-square fluctuations in position, velocity and acceleration of a particle in bounded, steady-state motion. These fluctuations are determined by integrating over the modes and summing the various polarizations of EZP , as is typically done in quantum mechanics. The derived equations of motion for a particle with charge to mass ratio, q / m are as follows, [8, 10].
1/2 2 q 3 ⎛ !ω ⎞ ⎛ 1 ⎞ ik⋅r−iωt+iθ k,σ x = Re d k εˆ × xˆ e ( ), m ∑∫ ( )⎜ 3 ⎟ ⎜ ⎟ m σ =1 ⎝ 8π εo ⎠ ⎝ D⎠ 1/2 2 q 3 ⎛ !ω ⎞ ⎛ ω ⎞ ik⋅r−iωt+iθ k,σ m v = x! = Re d k εˆ × xˆ −i e ( ), (10) ∑∫ ( )⎜ 3 ⎟ ⎜ ⎟ m σ =1 ⎝ 8π εo ⎠ ⎝ D⎠ s 1/2 q 2 ⎛ !ω ⎞ ⎛ ω 2 ⎞ m a = !x! = Re d 3k εˆ × xˆ − eik⋅r−iωt+iθ(k,σ ), ∑∫ ( )⎜ 3 ⎟ ⎜ ⎟ 2 m σ =1 ⎝ 8π εo ⎠ ⎝ D ⎠ s
where the denominator, D expresses a resonance condition at the natural frequency, ω 0 .
2 2 3 D = −ω +ω 0 − i Γω (11)
The natural radiative damping function, Γ is expressed as,
q2 ⎛ 2α ⎞ ! Γ = 3 = ⎜ ⎟ 2 , s (12) 6πε0m0c0 ⎝ 3 ⎠ m0c0
Equations (8) and (12), come from the Larmor radiation power formula, [10, 14, 17]. For an electron in the reference frame of the local observer, the value is extremely small, −24 Γ0 = 6.336 ×10 s . In comparison to the Compton frequency of an electron, ω 0 , Γ0ω 0 = .005 .
4. PARTICLE FLUCTUATIONS AND GRAVITY
ENGINEERING A BRIDGE Updated - 3/27/20 10 From the equations of motion above, the mean-square fluctuations in the particle's motion are derived. For example, from equation (10), the mean-square position fluctuation is,
2 ∞ 3 2 q ! ω dω 2 x0 = 2 2 3 , m (13) ∫ 2 2 2 2 6 6π ε0m0 c0 0 ω −ω + Γ ω (( 0 ) 0 )
The derived integrand is almost precisely a Lorentzian line-shape, because, Γ0ω 0 ≪1 is small and the integrand in equation (13) is sharply peaked. Therefore, the standard resonance approximation for a harmonic oscillator may be used to high precision. Making the appropriate substitutions, simplifying and noting that the Lorentzian line-shape integral is unity, the resulting mean-square position fluctuation is simplified to,
∞ 2 / 2 2 ! 1 (Γ0ω 0 ) ! 2 x = dω = , m (14) 0 2m ω ∫ π 2 2 2 2m ω 0 0 −∞ (ω 0 −ω ) + (Γ0ω 0 / 2) 0 0
An easy to follow derivation of the mean-square fluctuations is found in, [10]. It will not be reproduced here. However, these results are the standard quantum mechanical values for the mean- square fluctuations in position, velocity and acceleration, that will be referred to in what follows.
2 ! 2 x0 = , m 2m0ω 0 2 2 !ω 0 m v0 = , 2 (15) 2m0 s 3 2 2 !ω 0 m a0 = , 4 2m0 s
The mean-square fluctuations in Eq. (15), provide the coupling to the oscillator and the gravitational effects. The mean-square power fluctuation of the oscillator is given by,
2 2 2 2 2 2 2 PM = m0 a0 v0 = (!ω 0 / 2) , W (16)
The smallness of Γ0ω 0 ≪1 implies that the oscillator is extremely underdamped, [7, 10, 14]. Meaning, once stimulated it will continue to oscillate for a long time. This fact greatly simplifies what follows.
In an underdamped oscillator, the relative damping factor, ζ may be defined in terms of the power lost from the mean power fluctuations at the natural frequency.
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2 2 2 !ωζ !ω 0 !(ω 0 ⋅ζ ) = − , W (17) 2 2 2
Where the power term on the left is the difference between the initial mean power of the oscillator and the power lost to the local environment. Meaning, the power was radiated away and absorbed back into the local EM field.
The value of ζ is a variable in the coordinate system of the distant observer. It may be thought of as a deficit in the equilibrium condition of equation (6), resulting in a shift in the equilibrium value. The losses are understood to be photons that were radiated away and not reabsorbed by the oscillator, i.e., lost to the environment.
In agreement with the underdamped oscillator, a naturally variable frequency (energy) arises that is dependent on the relative damping factor.
rad ω = ω 1−ζ 2 , (18) ζ 0 s
Increased damping, such that ζ > 0 will result in ωζ < ω 0 and thereby, reduce the ground state energy of the oscillator below its natural value.
E0 = !ω 0 , J (19)
2 Eζ = E0 1−ζ , J
Similarly, substituting the new resonant frequency, ωζ into equation (16) yields a reduced power fluctuation. !ω 2 P = 0 , W 0 2 (20)
2 Pζ = P0 ⋅(1−ζ ), W
The key new idea that permits this phenomenon to be interpreted as space-time curvature is as follows; Given equation (5) and the understanding that in the coordinates of the distant observer, the relative coordinate velocity of light varies with the relative available power. The coordinate velocity of light, cK may be expressed anew as,
ENGINEERING A BRIDGE Updated - 3/27/20 12
m c = c ⋅ 1−ζ 2 , (21) ζ 0 ( ) s
From this, wavelength and mass may be determined from the dispersion relationship in the usual way,
c c λ = 2π ζ = 2π 0 1−ζ 2 , m (22) ωζ ω 0
!ωζ !ω 0 m = 2 = 3 , kg (23) cζ 2 2 2 c0 (1−ζ )
In total, all of the references in TABLE 1 that apply to GR and the PV Model, can be reproduced by substituting the metric components with the local relative damping factor, as shown in TABLE 2. Where, 1 K ≡ = g / −g (24) (1−ζ 2 ) 11 00
TABLE 2: General Comparison of Relative Damping Factor vs the PV Refractive Index.
Variable Variable Description Refractive Relative Damping Index 1 = g / −g 1 2 11 00 K = g11 / −g00 ( −ζ ) Speed of c = c / K c = c ⋅ 1−ζ 2 K 0 ζ 0 ( ) light
2 Length Δx = Δx0 / K Δxζ = Δx0 ⋅ 1−ζ 1 Time t t Δt = Δt K Δ ζ = Δ 0 ⋅ 0 1−ζ 2
2 Velocity v = v0 / K vζ = v0 ⋅(1−ζ ) 3/2 3/2 2 Acceleration a = a0 / K aζ = a0 ⋅ (1−ζ )
ENGINEERING A BRIDGE Updated - 3/27/20 13 1 Mass 3/2 m = m ⋅ m = m K ζ 0 3/2 0 (1−ζ 2 )
2 Frequency ω = ω 0 / K ωζ = ω 0 ⋅ 1−ζ
E E / K 2 Energy Δ = Δ 0 ΔEζ = ΔE0 ⋅ 1−ζ
It may be inferred by inspection that for a spherical mass with a negligible NET charge, such as 24 6 the planet Earth with mass, M E = 5.972 ×10 kg and radius, RE = 6.371×10 m , the relative damping factor is,
2GM E −5 ζ = 2 = 3.731×10 (25) c0 RE
2GM The ratio, 2 is the familiar gravitational potential found in the Schwarzschild solution of c0 R Einstein’s field equations of GR, [2, 18]. Where,
1 1 = (26) 1−ζ 2 2GM ( ) 1− 2 c0 R
The normalized frequency shift due to the Earth’s gravity, as seen by the distant observer may then be expressed as,
ω −ω 0 = 1− 1−ζ 2 = 6.97 ×10−10 (27) ω 0
The normalized value for the frequency shift of the oscillator is very small, but it results in a m gravitational acceleration of g = 9.82 . Notice that it does not require much of a frequency shift s2 to generate significant results. However, determining this shift in energy for each particle, multiplied by the number of particles to be shifted, such as levitating a car for example, requires a significant amount of power. This is one of many engineering problems to be tackled.
The mean-square fluctuations from equation (15) may be restated in terms of the relative damping factor,
ENGINEERING A BRIDGE Updated - 3/27/20 14 2 ! 2 2 xζ = ⋅(1−ζ ), m 2m0ω 0 2 2 2 !ω 0 2 m vζ = ⋅(1−ζ ) , 2 (28) 2m0 s 3 2 3 2 !ω 0 2 m aζ = ⋅(1−ζ ) , 4 2m0 s
This illustrates how the mean-square particle fluctuations transform according to the relative damping factor. Note that this is identical to how such variables behave in a gravitational field, under GR.
As a result of increased relative damping, a particle’s frequency (clock) is slowed and its mean- square position fluctuation (length) contracts. There is less available power to inflate matter to its proper scale, as observed by a distant observer in an unaltered space-time.
There is less available power, because the power provided by the ZPF source is radiated away by the damping of the oscillation. This explains why gravity has negative energy density, i.e., an energy density less than the ZPF. Matter in a gravitational field is oscillating at energies less than the normal ground state energy, when observed from a region outside the range of significant gravitational field strength. The energy state is less than its ground state energy in a ZPF because the ZPF sets the baseline driving power (the a.c. source), and the radiative damping (power loss) reduces the available power to a value below that baseline. This energy deficit is why it is required to do work to climb uphill.
5. EXOTIC MATTER OR INFLATED MATTER?
It was illustrated that a value of K < 1 results in an increased relative driving power, Pζ . This is also associated with an increased relative ground state energy of matter, as shown in Eq. (18) and (19). Another important physical effect is that it also decreases the relative energy density of matter at equilibrium because, the mean-square position fluctuation and volume expand as the energy goes up. Here, the mean-square fluctuations of energy and volume are expressed in terms of the 2 equilibrium energy density, U0 and the square of the relative damping factor, ζ .
U J 0 U 1 2 , 2 ≈ 0 ( +ζ ) 3 (29) (1−ζ ) m
Exotic Matter is defined as negative energy density, it is not simply negative energy, or antimatter, as many people imagine it to be. The negative energy density contribution in Eq. (29) 2 occurs when the second term on the RHS, Uoζ < 0 . This negative contribution is what permits a
ENGINEERING A BRIDGE Updated - 3/27/20 15 relative value of K < 1. Referring to TABLE 2 illustrates why negative energy density is needed to achieve, cζ > c0 . However, it is not the addition of negative energy density that is required. It is the addition of real energy and real power to the system, to drive the expansion of the material that reduces the energy density of matter, [7].
Inflated matter is a means of storing energy. Like the LC resonant circuit described previously, matter stores energy as Reactive Power. Reactive Power uses imaginary numbers to describe the VA or Volt-Amps instead of Watts as the unit of measure. Given a real-world system, there are both Real and Reactive parts of the power, which represent the energy which does work and the stored energy, respectively. Similarly, a negative value of ζ 2 represents additional energy being stored as Reactive power within the resonant oscillator, resulting in inflated matter and reduced energy density.
Over time, exotic matter has come to mean different things to different people. Here, the term Inflated Matter shall be defined as;
"A reduction of the steady-state equilibrium energy density of matter, associated with an increased energy and volume of matter, due to an increase in the relative available driving
power of the locally damped ZPF. Pζ ."
This is what is required to achieve anti-gravity and FTL speeds. It does not entail extracting energy from the vacuum but rather, putting energy into matter and the ZPF to drive the expansion of the material.
The common example used to describe exotic matter is the Casimir effect, [7, 8, 16]. The Casimir effect excludes frequency modes in the ZPF by imposing a low frequency cut-off mode limit within the gap between the plates. Ultimately, this reduces the available power, Pζ as well as the energy contained within the ZPF spectrum that exists in the gap, resulting in a measurable attractive force.
Similarly, the attraction of gravity is also associated with a reduction in Pζ , and the minimum ground state energy. This occurs naturally, possibly as a result of matter imposing its own low and high frequency cut-off mode limits, and absorption spectrum onto the ZPF integrals given in Eq. (28). Gravity and the Casimir effect both demonstrate a reduction in the relative available power of the ZPF, resulting in a lower ground state energy and a reduced coordinate speed of light.
The Casimir effect demonstrates the existence of the ZPF, but what exists in the gap is not the exotic matter required for warp drive. The warp drive requires amplification of the ZPF with real power, to drive the inflation of real matter to higher ground state energies. In doing so, the relative speed limit for such matter is increased.
ENGINEERING A BRIDGE Updated - 3/27/20 16
In what follows, it will be shown that by using a magnetic scalar potential, the release of energy from inflated matter may be directed and used as a propulsion scheme with a preferred direction along the gradient, such that the deflating matter is put into free-fall. (Aka.; a Warp Drive.)
6. GRAVITY AS POWER LOSS
Take for example an electron. The electron is a Dirac particle. It is a point-like particle and has a minimum energy state that is in equilibrium with the local ZPF. The relative driving power required to inflate a point-like charge to a probability cloud the size of its Compton wavelength scale λe , and Compton frequency ω e , is calculated as follows;
2 2 m c2 !ω e ( e ⋅ 0 ) P = = = 5.058, MW (30) e 2 2 ⋅h
This power is inherently present in the equations of motion for the Dirac equation of QED, where the ZPF scatters against the point-like charge causing its Zitterbewegung motion. This is a natural jitter that results in the electron having a probability cloud of size, on the order of the Compton wavelength, [8].
Consider this jittering to be the source of the size of the electron’s probability spread. Gravity near the surface of the Earth steals from this power with losses that cause a contraction of the probability cloud (length contraction), and a reduction in the Compton frequency (time dilation). The rate of power loss for an electron in free-fall is simply;
dP W e = −m g2 = −8.76 ×10−29 (31) dt e s
In the Earth’s gravitational field, the loss of power per kilogram in free-fall is equal to;
W g2 = −96.43, (32) s⋅kg
This is the order of magnitude required to generate an artificial gravity field, or to reduce gravity with an applied anti-gravity field. Meaning; to negate the gradient in the potential with an appropriately applied spectral power excitation field used to reduce or replace those losses.
ENGINEERING A BRIDGE Updated - 3/27/20 17 Up until now, only underdamped cases where ζ ≪1 have been considered. The value near the surface of the Earth was calculated in Eq. (25) to be, ζ = 3.731×10−5 , which is underdamped. In the case where ζ = 1, the oscillator is said to be critically damped. From Eq. (26), this is understood to be the event horizon of a Schwarzschild Black Hole, [2, 15]. Falling past the event horizon to where ζ > 1, the oscillator is overdamped, and its oscillation will decay to zero exponentially. This description gives engineers an intuitive understanding for what it means when it is said that; “time stops”, at the event horizon of a black hole. Literally, it means that damping has increased to the point where the natural oscillations of matter can no longer exist. Matter which falls into a black hole will only add another bit to the surface area and more entropy to the event horizon, as predicted by Hawking and Bekenstein, and recently illustrated by Verlinde with regards to Quantum Information, [19, 20, 21].
7. QUANTIZED SCALAR MAGNETIC FLUX
So how does all this work? First, consider what the Quantum Vacuum really is. It is the summation of the minimum energy state of all quantum fields, including the Electromagnetic field. Let us only consider the Dirac field and the Electromagnetic field of QED, in order to describe the concepts to be conveyed. Electrons, and Quarks are particles of the Dirac field and as such, they obey the Dirac equation. Therefore, given the proper driving power, they will inflate to their proper size in the manner described above, leading to variations of Eq. (30). The motion of the Quarks determines the relative size of the Protons, and Neutrons.
The minimum energy state of the Electromagnetic field in QED is the EM ZPF. Quantization of the EM field has led to the discovery of photons. Similarly, magnetic fields are quantized by a minimum value of magnetic flux, given by, Φ0 , [22].
h Φ = = 2.068 ×10−15 , V⋅s (33) 0 2 ⋅q
Where, q is the electric charge quantum, e. This value was derived from experiments illustrating discrete magnetic flux vortices trapped inside superconductors, as illustrated in FIGURE 1.
FIGURE 1: Figure courtesy of web.mit.edu, The Physics of Topological Defects, 2014
ENGINEERING A BRIDGE Updated - 3/27/20 18
Likewise, a quantum of electrical resistance which leads to damping of electrical current flow is given by the same quantum of flux, as described by Ω0 .
h Ω = , Ohms (34) 0 2 ⋅q2
Therefore, ohmic resistance is determined by the magnetic flux in the vacuum intersecting the path of a charged particle. These flux quanta are the photons of the EM field, present everywhere in the ZPF and are naturally uniform across the entire EM spectrum. Note that the EM ZPF is Lorentz invariant, [8]. Meaning, its spectrum looks the same in all inertial reference frames. However, adding fixed cut-off limits to the spectrum, or fixed absorption lines will break that invariance in the local region.
Since the ZPF is random in nature, this results in a scalar field of random magnetic flux quanta,
χ0 , rather than a polarized magnetic flux density, B0 .
χ0 = ∑Φ0 = B0 dS, V⋅s (35) !∫∫Se
To illustrate this, B0 is taken to be the magnitude value of the instantaneous magnetic flux density at the location of the electron, and dS is its minimum element of cross-sectional area, Se . At any given instant within the ZPF, the magnetic flux density vector can point in any direction, and the ensemble, statistical average expectation value is zero.
ENGINEERING A BRIDGE Updated - 3/27/20 19
B0 = 0, T (36)
Scalar magnetic flux is just the magnitude of the field, with no vector direction. The presence of scalar magnetic flux has no observable effects on Maxwell’s equations, or the usual derived fields,
E and B . It enters only into the EM gauge potential, Aν and although the flux is physically real, the equation that governs it is identical to a simple gauge transformation of the EM potential. This is what should be expected because gravity has no effect on EM fields. The EM forces are invariant when transformed to a region with non-zero gravitational field. Therefore, if gravity can be attributed to scalar magnetic flux quanta, these quanta will have no effect on Maxwell’s equations.
The resulting EM field potential is modified such that;
V⋅s A′ = A + ∂ χ, (37) ν ν ν m
Where, (∂χ / dt) becomes the new baseline for zero voltage but does not affect the value of E , d∇χ dχ because the = ∇ . Similarly, the gradient, −∇χ becomes the new zero baseline for the dt dt magnetic vector potential but does not affect the value of B , because ∇ × ∇χ = 0 . Therefore, the
EM potentials, Aν′ are relatively shifted by gravity from Aν , but the change in the EM field values is locally unobservable, [22].
The effects of χ are observable when considering the refractive index, K . The presence of scalar magnetic flux increases the relative value of electric permittivity, ε0 and magnetic permeability, µ0 . Thus, value of K is transformed, and thereby the metric coefficients, by the scalar magnetic flux, such that;
⎛ χ ⎞ F Kε ∼ ε 0 , 0 0 ⎝⎜ χ ⎠⎟ m (38) ⎛ χ ⎞ H Kµ ∼ µ 0 , 0 0 ⎝⎜ χ ⎠⎟ m
Where, χ > 0 .
One of the tests of GR is that light is refracted when passing by massive objects or clusters of objects, [4]. Eq. (38) illustrates how scalar magnetic flux and gravity result in variability of the
ENGINEERING A BRIDGE Updated - 3/27/20 20 refractive index K , creating the many well-known Gravitational Lensing effects that have been observed.
8. MAGNETIC RESONANCE AND GRAVITY
A similar process occurs in Magnetic Resonance Imaging (MRI) spectroscopy, implemented using Electron Paramagnetic Resonance (EPR), or Nuclear Magnetic Resonance (NMR). As an example, consider the process whereby an applied magnetic field is used to polarize the spin axis of the subatomic particles within a tissue sample. This is possible because electrons, protons and neutrons (Fermions) have non-zero angular momentum. Their minimum angular momentum is, ±! / 2 . Although, with combinations of particles, spins are added together and can sometimes sum to zero. To analyze a sample using MRI, an RF or microwave frequency is applied to the sample and swept through a wide spectrum to find the resonance frequencies of the tissue sample. These frequencies are unique to certain atoms and molecules, which allows doctors to identify different tissues or other anomalies within the body.
At resonance, the electron flips its spin repeatedly between the two Spin states, up, Su = ! / 2 and down, Sd = −! / 2 . NMR is similar, but it is the nucleons that undergo these spin transitions. Classically, spin-flipping may be thought of as a precession of the particle around the axis of the applied, magnetic flux density vector, B . This is called Larmor precession which results in Larmor radiation. Where, in the case of MRI the Larmor radiation is then detected as the spectroscopy output signal. This type of spectroscopy allows for the determination of the internal shape and structure of materials. This is the same physics that Puthoff used to derive equations (6) through (8) in reference [10].
Since Larmor Radiation is what is shown in Eq. (8), to determine the radiated output power of matter interacting with the ZPF. A hypothesis may be established such that; the behavior of particles at resonance in EPR, NMR and Gravity are all related by Larmor Radiation. This hypothesis may then be tested by using such technology to mimic the natural process and demonstrate artificial gravity, anti-gravity or warp drive.
Likewise, there is a similarity in the behavior of the oscillation frequency to that of an oscillator in a gravitational field. Applying a magnetic field with a higher flux density will result in a higher oscillation frequency, as determined by the Gyromagnetic ratio. The Gyromagnetic ratio, γ e for the electron in its current environment allows the resonant frequency to be calculated.
fe = γ e ⋅ B , Hz (39)
Where, the gyromagnetic ratio of a free electron is given by;
ENGINEERING A BRIDGE Updated - 3/27/20 21 Hz γ = 2.802 ×1010 , (40) e T
Protons and neutrons, electrons and quarks, all have a different Gyromagnetic ratio, as do many other atomic and molecular structures. The ratios are proportional to the charge to mass ratios of these particles, as are the equations of motion used in Eq. (10). However, regardless of the exact values of for each particle, the precession and Larmor radiation formula remain the same. From equations (39) and (40), it can now be understood that;
When there is a gradient in the magnitude of the applied magnetic field, regardless of its direction of polarization or statistical lack thereof, the precession frequency will be shifted according to the instantaneous magnitude of B at each location. This is also the case for oscillators in a gravitational field, where B is unpolarized but whose magnitude varies relative to the gravitational center of mass.
To move the electron in the direction toward where the field is stronger requires work to be done to overcome the increase in frequency, i.e., it is moving uphill. Likewise, the decreasing precession frequency in the direction toward where the field is weaker, results in the electron falling freely in that direction, i.e., free-fall or geodesic motion.
Near the surface of the Earth, the field gradient required to nullify gravity for this electron can be calculated. Where, z is taken to be the coordinate in the “up” direction.
Ee (z) = h ⋅ fe (z), J
dE (z) d B e = h ⋅γ ⋅ 0 = m ⋅ g, N (41) dz e dz e
d B m ⋅ g T 0 = e = 4.811×10−7 , dz h ⋅γ e m
Similarly, there is a minimum scalar magnetic flux density within a narrow bandwidth centered 2 on the electron’s Compton frequency, fCe = mec0 / h . This field is enclosed by the cross-sectional area of the electron cloud, estimated roughly as the electron’s Compton wavelength squared.
2 ⎛ h ⎞ −24 2 (42) Se = ⎜ ⎟ = 5.887 ×10 , m ⎝ mec0 ⎠
ENGINEERING A BRIDGE Updated - 3/27/20 22 The value of the magnetic flux density necessary to inflate the electron to its proper size was found by equating Eq. (39) to the value found in Eq. (30);
2 2 2 (mec0 ) h ⋅ γ e ⋅ B0 _ e P = = ( ) = 5.058, MW (43) e 2 ⋅h 2
In order for this expression to hold true, the instantaneous flux density required is;
2 ⋅ Pe 9 B0 = 2 = 4.409 ×10 , T (44) h ⋅γ e
Given the minimum value for a flux quanta from Eq. (33), and the area previously calculated in Eq. (42). The number of flux quanta enclosed within the electron’s cloud at any given time, and with any given orientation in space, is precisely two times this value.
2 ⋅ B0 ⋅Se ne = = 25.109 ≈ 25 (45) Φ0
This calculated numeric value is only 0.4% from being a perfect integer number of flux lines, as expected.
9. DISCUSSION
In the engineering model presented here, matter is inflated to its maximum relative ground state energy by the ZPF. Its maximum natural state of inflation is when it is in a vacuum far from other matter, i.e., in the local frame of the distant observer the oscillator is undamped. The power required to inflate an electron to the its Compton wavelength and frequency was calculated for the electron to be in excess of 5 MW. This implies that in the frame of the distant observer, far from planets and other gravitational sources, the magnitude of the magnetic flux density is at its greatest value. As two objects are brought close together, their respective probability clouds and power spectrums overlap, such that; interference occurs that lowers the flux density, damps the resonant frequencies and lowers the energy state of both objects. It was shown that decreasing scalar magnetic flux density is associated with an increased radiative damping effect and thereby, lowers the relative energy (frequency) of the oscillators to energy states below their normal value, allowing them to attract each other gravitationally.
One might conjecture that this action is the electromagnetic manifestation of Quantum
Coherence. The undamped magnitude of B0 is the maximally coherent state of entanglement between particles which are linked by magnetic flux vortices. Whereas, the damped field near
ENGINEERING A BRIDGE Updated - 3/27/20 23 gravitational sources represents a reduced field, B ≤ B0 , which has less coherence and therefore less amplitude, because it has power losses given up to the environment. The process is strikingly similar and worthy of further investigation.
From a cosmological perspective here on Earth, if the relative damping in our solar system were increasing linearly with time, such that the length of 1 meter were contracting by just 6.935 nm/century relative to the most distant galaxies and supernova, it would appear from our perspective that the universe is expanding. Light arriving from the distant past is redshifted, because our local measuring devices have contracted over time more-so than those we observe in the distant past. When comparing the increasing velocity of expansion across great distances, far into the distant past to when matter was in a hotter, more inflated state, what is measured is interpreted as the present value of the Hubble constant, [23].
Consider atoms and nuclei which are cooling from a hotter, less dense state in the distant past. This contraction would be a natural thermodynamic process for matter to undergo. However, there would be no way to measure such a small effect locally, because there is nothing local that is unaffected by this. The only references to compare to are the distant stars. Therefore, the Hubble constant may be evidence that such a process is taking place here in our local solar system and that the thermodynamic energy of the universe is actually running down as it should be. As opposed to new energy being continuously created from nothing to fill the expanding space, as is the case in the Standard Model of Cosmology. The idea presented here allows re-evaluation of that assumption from the perspective of quantum mechanics.
On the other hand, if matter were inflating rather than contracting. In such a scenario, the universe would appear to be getting smaller. This is how a warp drive would be described and what may be seen from the bridge of a warp-ship ready for launch. The matter to be transported is inflated by a scalar magnetic warp field which makes the distance from point A to point B in the universe, literally shorter in scale, relative to the reference frame of the inflated ship. The warp- ship then moves in the desired direction by generating a gradient in the field along which the stored energy is discharged, moving the ship forward in free-fall as it deflates, [7].
Seth Lloyd, in [24], reproduced Einstein’s equations of GR simply by subtracting quantum areas the size of the Planck area, and proportional to the Heisenberg energy-time uncertainty relation. Where,
2ΔEΔt # ops = ! (46) π A ∼ # ops ℓ2 area 4 Planck
ENGINEERING A BRIDGE Updated - 3/27/20 24 From here, all that is required to recover Einstein’s equations is to recognize that ΔE is derived from the product of, T µν and the four-volume; and the area, “A” is proportional to the scalar curvature, which is the contracted Einstein tensor, G µν . For brevity, the reader is referred to the referenced paper [24], for his full derivation. It is an intuitive approach that illustrates that curvature is induced onto a surface by poking holes in that surface.
If the ZPF is interpreted to be a smooth space-time manifold of constant spectral energy density and from this, the power losses are subtracted. Then the spectral scalar curvature of the 3 spatial dimensions is, (47) R3 (ω ) = − 4πG(ρ0 (ω ) − ρζ (ω ))
Poking holes in a smooth surface is analogous to power loss from this smooth ZPF, as presented here-in. Therefore, with this one assumption, the procedure used in [24] permits the recovery of Einstein’s equations, directly from this model, in terms of the spectral energy content.
Note that this model rules out certain geometries in GR. For example, shift-only space-times where length contraction occurs without time dilation, or vis versa. Here, the clocks and rulers are linked by the same statistical processes that govern quantum mechanics and determine the mean- square fluctuations within matter.
Finally, regarding the 120 orders of magnitude problem, or vacuum catastrophe, [25] often cited as a criticism of the existence of the ZPF. In this model, the ZPF is not simply just another field resting on a background metric space-time. Where, its mass-energy contributes to the gravitational field of the universe. Instead, as the source of the cosmological constant in GR, the ZPF provides the minimum energy baseline governing the relative scale of material objects as well as the relative rate at which time passes. It is matter that is expanding and contracting and not the empty space in-between. The catastrophe is caused by wrong thinking, in assuming the ZPF should gravitate as matter does when in fact it plays a much more important part in the physics of gravitation itself. This problem is one of many examples where the geometrical interpretation of gravity has led us astray and made our job as engineers more difficult.
10. CONCLUSION
It was shown that an alternative interpretation exists for engineering purposes that equally describes the basic observations of gravitational length contraction and time dilation, as derived from GR. The observations of space-time curvature are indistinguishable from those made as a result of a relative radiative damping factor of the quantum mechanical harmonic oscillator. The relative frequency and energy of the oscillators vary as a function of the damping factor, identically in accordance with the metric coefficients of the Schwarzschild solution of Einstein’s field equation. It was shown that the damping factor can take the form of the gravitational potential, by
ENGINEERING A BRIDGE Updated - 3/27/20 25 matching it to the Schwarzschild solution and calculating the frequency shift and power lost from matter in free-fall near the surface of the Earth.
Variations in frequency, and therefore in the energy of the oscillators from which matter is composed, affects its mean-square fluctuations in length, velocity, acceleration and power. Resulting in what is perceived and interpreted (misinterpreted?) as space-time curvature in the classical theory.
Losing power results in a reduced Compton frequency and reduced spread of the particle’s probability cloud, which has been interpreted to be the observed gravitational time dilation and length contraction, originally derived in GR.
Here, it was proposed that a loss of power is the root cause for these uniquely gravitational effects. Therefore, a unified model of gravitational effects and quantum mechanics has been presented without introducing any unobserved new requirements, such as; higher dimensions, missing particles or strings. Only the recognition of an existing symmetry was introduced that requires a balance between the power into, and out of a quantum probability cloud. The model presented can be tested using NMR, which makes it practical and well suited to facilitate its understanding amongst engineers, and physicists alike.
This engineering model firmly establishes a viable solution to Quantum Gravity for Engineers within the Standard Model of particle physics, along with the proposed method to test and implement it using NMR and EPR spectroscopy equipment. It opens the door to new innovations that might permit breakthrough technologies, such as; artificial gravity, anti-gravity or warp drive propulsion. Technologies yet to be invented, through the use of increased or reduced radiative damping within materials, or by amplification of the resonant driving fields that inflate matter to higher ground state energies. Engineers now have a new set of old, familiar tools to work with when thinking about Gravity and Metric Engineering.
What was presented herein puts gravity in the hands of engineers, who could potentially advance such technologies as; warp drive, artificial gravity and anti-gravity, from pure speculation, to achievable endeavors in our lifetime. It is the author’s sincere hope that others will see the value in this engineering approach and continue this work toward these goals until plausible technology can be developed, to better the everyday lives of mankind.
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