the extraterrestrial casimir effect

8/3/2019 The Extraterrestrial Casimir Effect

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The Extraterrestrial Casimir Effect

Riccardo C. StortiDelta Group Engineering

Keywords: Casimir Effect, Electro-Gravi-Magnetics, EGM, Photons, Polarisable Vacuum, PV, Zero-Point-Field, ZPF

Abstract

Application of the Electro-Gravi-Magnetic (EGM) Photon radiation method to the Casimir Effect (CE),

suggests that the experimentally verified (terrestrially) neutrally charged Parallel-Plate configuration force, may differ

within extraterrestrial gravitational environments from the gravitationally independent formulation by Casimir.

Consequently, the derivation presented herein implies that a gravitationally dependent CE may become an important

design factor in nanotechnology for extraterrestrial applications (ignoring finite conductivity + temperature effects andevading the requirement for Casimir Force corrections due to surface roughness).

1 Introduction1.1 Manuscript Synopsis

Hypothesis to be tested: is the historically derived Casimir Force associated with a Parallel-Plate

configuration measurably different in extraterrestrial environments?

Substantial effort has been invested into research of the Casimir Effect (CE); with many variations upon the

central theme being rigorously investigated. However, precise physical measurement of the Casimir Force (CF) has only

occurred in terrestrial laboratory environments. This is experimentally significant as the base mathematical formulation

of the CF per unit area explicitly states that the CE is homogenous throughout the Universe, i.e. the base formulation

defined by [Eq. (1)] contains no gravitationally dependent terms. This implies that, within proximity of the event

horizon of a black-hole (i.e. in a curved space-time manifold), the CF associated with a Parallel-Plate experiment would

register a physically measured result, equal to an identical experiment executed in a flat space-time manifold; although

unverifiable, this is seemingly counter intuitive. Moreover, as the plate separation distance r tends to zero, the CF is

predicted to approach infinity. In addition, van-der-Waal forces may be applied to interpret the CE without reference tothe Zero-Point-Field (ZPF) or the virtual particles associated with quantum fields. However, in practical terms, the

Casimir and van-der-Waals forces are quite different; the van-der-Waals force is always attractive, whereas the sign of

the Casimir force is geometry dependent. For example, if a thin spherical conducting shell is cut in half, the two

hemispheres will experience a mutual repulsive force. [1]

FPP

h. c. APP

.

480 r4

.

(1)

Lamoreaux et. Al. stated that at least two corrections are required to be incorporated into a practical

measurement of the CF (i.e. finite temperature + conductivity). Mohideen et. Al. took the next logical step utilising

an atomic force microscope to increase the precision of the measurement. They determined that by incorporating

surface roughness contributions into their investigative regime, physical measurements of the CF to within 1(%)

of the theoretical prediction were observed [2]. CF measurements were further championed by Bressi et. Al., where theefforts of the experimentalists were specifically confined to a Parallel-Plate configuration (rather than a Plate-Sphere

apparatus), yielding observational agreement to within 15(%) of the historically predicted result [3]. Therefore, based

upon experimental confirmation that the theoretically predicted CF for a Parallel-Plate configuration is sufficiently

different from practical observation, one is forced to the wonder if additional parameters, not yet considered, might also

affect the measurement. Given the exponential growth of manufacturing capability in the modern era, and the

proliferation of communications technology beyond the terrestrial boundary, we explore the impact to the CF associated

with a Parallel-Plate configuration, of a heterogeneous rather than homogeneous ZPF. Herein, we propose that the ZPF

Spectral-Energy-Density distribution as presented by Haisch et. Al. in [Eq. (2): when is expressed in Hz] [4], is


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modified by the presence of a gravitational field ( i.e. the terrestrial laboratory environment being a curved field), such

that the idealised Parallel-Plate CF tends to zero in a vanishing gravitational acceleration field.

0 ( )2 h.

3.

c3

(2)

In 2006, [5] demonstrated that the minimum and maximum spectral limits of the ZPF may be computed byassuming that the constitution of spectral frequencies between these limits, obeys a Fourier distribution such that the

spectral energy contained locally within the ZPF is equal to the rest-mass-energy of the matter content present. The

computed spectral limits were subsequently utilised to formulate many observationally verified solutions to key Particle-

Physics and Cosmological problems ([6] and [7] respectively). Hence, outside the minimum and maximum spectral

limits computed by the authors, the ZPF cannot be said to exist. One of the fundamental benefits of such a construct is

that it evades the infinite energy in a vanishing volume problem of contemporary Quantum-Electro-Dynamics (QED);

as no more spectral energy exists in the ZPF surrounding an object, than the rest-mass-energy of the object itself ( i.e.

matter exists in equilibrium with the ZPF surrounding it).

The Electro-Gravi-Magnetic (EGM) construct was applied in [8] to derive a unique solution specifically for a

Parallel-Plate configuration of the CF from first principles [ i.e. for a plate separation of r = 1(mm)], demonstratingthat it differs depending upon ambient gravitational field strength; for example, the CF will be slightly different on Earth

than Jupiter or the Moon. The construct presented herein, develops this unique solution into a generalised representation

of the CF, valid for all experimentally practical values of r. Consequently, it is hypothesised that engineers may berequired to incorporate broader operational tolerances when designing and constructing nanotechnology for

extraterrestrial applications. The following table articulates the EGM predicted approximate average change in CF per

unit area [i.e. the Casimir Pressure (CP)] over the Parallel-Plate separation range 100(nm) to 900(nm) for a selection of

extraterrestrial environments of potentially practical significance (relative to a terrestrial experiment conducted at thesurface of the Earth);

Environment CP (%) Location [9]Lunar Surface +37

Mean Equatorial

Radius (RM, R, R)Martian Surface +12

Venusian Surface -0

Low Earth Orbit -5 180(km)

Altitu

de

Sun-Synchronous Orbit -15 705(km)

Mid Earth Orbit -35 2,000(km)

GPS Orbit -88 20,200(km)

High Earth Orbit -95 42,164(km)

Tab. (1),

The predictions listed in Tab. (1) have been generated based upon the proceeding set of assumptions:

1) System or environmental angular / spin-angular momentum factors, including frame-dragging, are negligible.2) The rest-mass-energy of an object is energetically equivalent to the spectral energy of the ZPF surrounding it.3) Influences of the experimental apparatus upon the ZPF are usefully negligible. Note: the reader is encouraged to refer to the table of Definitions and Nomenclature contained in Appendix

A, as and when required.

1.2 General RelativityMichelson and Morley disproved the existence of the mechanical luminiferous aether conceptualised in

Maxwells era, but it did little to arrest the emergence of a contemporary version. Einstein is, at least partially,

responsible for destroying the mechanical aether of old and replacing it with a new aether. Einsteins development of

Relativity and the notion of a new aether termed curved space-time evaporated the concept that gravity was a force

mediated by the ill-defined aether of Newtons time. Einsteins equations demonstrate that an objects motion in a

gravitational field is determined by its geodesic path. Einstein introduced this concept to describe gravitational

interactions between mass-objects, eliminating the necessity for action-at-a-distance. Curved space-time is a geometric

contrivance, but exactly what is being curved? And if the vacuum of space is indeed a formless void, then how may


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nothing have shape? General Relativity (GR) not only invokes, but requires the existence of a medium (i.e. manifold)

capable of conveying information indicating whether the space-time a mass-object transits is curved.

1.3 The Polarisable Vacuum Approach to General RelativityBernard Haisch and Alfonso Rueda introduced a model describing matter as being immersed-in and wholly

dependent upon the Quantum Vacuum (QV) for its existence. This fed an intuitively appealing interpretation of space-time curvature termed the Polarisable (PV) Approach to General Relativity [10]. The PV model is an optical

interpretation of gravity because it applies optical principles to define the topological features of space-time, otherwise

represented geometrically within GR. It attributes space-time with a variable Refractive Index KPV, not curvature.

The value of KPV is proportional to the energy density associated with a gravitational field. As light passes a mass-

object, it transits through regions of variable KPV and refracts in accordance with the experimentally verified results

within the GR construct (at the very least, within the weak field). The PV model ascribes a value of KPV to the QV

such that all matter generates a gradient in the energy density of the QV surrounding it. The gradient relates to a change

in KPV acting as a space-time lens causing light to bend. Hence, the PV model demonstrates that substituting the

metaphysical conceptualisation of space-time curvature with a physically meaningful optical construct yields a congruent

interpretation of gravity to that of GR.

The key difference between interpretations is that the PV model describes the physical manner by which space-

time is curved, GR does not. However, neither GR nor the PV model specifically addresses the precise mechanism by

which matter physically polarises space-time. Fortunately, the PV model is not required to do so because QED explainsthis mechanism based upon the premise that within a volume of space-time devoid of matter, a chaotic and equally

distributed mix of virtual Electron-Positron particle pairs is said to pop into and out of existence. The PV model asserts

that matterpolarises the QV (i.e. enforcing direction and order) into variable regions of energy density which, in turn,

generates regions of variable KPV. A well-developed precedent for the existence of vacuum polarisation exists, based

upon the generally accepted model of the Electron. The contemporary model of the Electron stems from QED [11],

modelling it as a negatively charged point core surrounded by a cloud of virtual-particle-pairs, constantly emerging from

and disappearing into the QV. According to QED and the relativistic Quantum-Field-Theory (QFT) of the interaction of

charged particles and Photons, an Electron may emit virtual Photons which, in turn, may become virtual Electron-

Positron pairs. The virtual Positrons are attracted to the bare Electron whilst the virtual Electrons are repelled from it.

The bare Electron is therefore screened due to polarisation. The presence of the negatively charged core attracts the

virtual positive charges and repels the virtual negative charges present in the vacuum, biasing the QV, resulting in a

vacuum gradient as it segregates clustered regions of virtual charges. In this state the vacuum is no longer uniform it

has beenpolarised. The effect of an Electron upon the QV is termed vacuum polarisation and the property ofchargeemerges due to a change in the Quantum-Vacuum-Energy (QVE) distribution of the surrounding space-time. Thus, if the

QV is effervescent with virtual-particle-pairs, we must consider its effect on all elementary particles, not just the

Electron.From the perspective of the PV model; matter polarises the QV, forming gravitational fields because its atomic

constituents are composed of large populations of elementary particles, all generating their own localised polarisations of

the vacuum such that the cumulative effect results in a large-scale, synergistic polarisation. Conceptualising the space-

time manifold in terms of vacuum polarisation yields an isomorphic representation of GR.

1.4 The Terrestrial Casimir EffectQFT models the vacuum of space as something quite different to the implied rigidity of the geometric

interpretation within GR. The Casimir Effect (CE) occurs when two neutrally charged conducting plates are placed in

close proximity and parallel to one another, establishing boundary conditions in the QV. In such a configuration, an

attractive force is observed between the plates, beyond that which may be attributed to gravitational attraction. The QVis comprised of ElectroMagnetic (EM) wavefunctions which may only exist between the plates if their lengths are equal

to or less than the plate separation distance r; any wave of longer length cannot exist within the gap. For example, ifr = 1(m), only the QV modes of wavelength less than or equal to 1(m) may physically exist within that space.Presently, the CE has only been experimentally confirmed to exist in gravitational fields, demonstrating that when small

distances separate two flat neutrally-charged metal plates, virtual Photons in the ambient gravitational field with

wavelengths larger than the plate separation distance are excluded from the spatial cavity, resulting in an attractive force

between the plates due to the bias in QVE across the system. The QVE density is lower within the cavity and higher

outside, resulting in a pressure imbalance pushing the plates together.


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1.5 Electro-Gravi-Magnetics (EGM)1.5.1 Section Synopsis

The PV model of gravity asserts that the metaphysical concept of space-time curvature may be replaced by an

optical representation of QV polarisation. Thus, it follows that the formation of gravitational fields are a result of QVE

displacement due to the presence of matter. Recognising that QVE is EM in composition, a fundamental relationshipbetween matter, EM energy and gravity is implied. This may described utilising a mathematical method termed Electro-

Gravi-Magnetics (EGM) [12], developed from the application of standard engineering principles, modelling the manner

in which matter equilibrates with, and is constrained by, the local QV as asystem. The initial premise in the development

of the EGM method is the assumption that gravity and ElectroMagnetism may be unified via Quantum Mechanics (QM)

in terms of the QV, utilising Buckingham Theory (BPT). BPT is a well established and widely used engineering principle developed by Edgar Buckingham in the early 1900s. BPT is applied to simplify complex systems and

determine which parameters are necessary (or unnecessary) to adequately represent it. The Greek letter denotes theformulation of dimensionless groups describing the system. BPT is utilised to model the behaviour of a whole system

without requiring precise interactional knowledge of all components simultaneously. BPT formulations are executed

within the structural framework of Dimensional Analysis Techniques (DATs), indicating that similar systems may be

described in like terms.

An important consideration involving DATs and BPT is the rule of similitude. In order to compare a

mathematical model to a physical system, certain criteria must be satisfied. The model must have dynamic, kinematic orgeometric similarity to the real-world system (any of, or all of these if applicable). Dynamic similarity relates forces,Kinematic similarity relates motion (i.e. synonymous with the time domain) and Geometric similarity relates shape

(e.g. the topology of space-time curvature within the context of GR). Once the design principles of similitude are

satisfied, the mathematical model is considered applicable to the real-world system. The EGM method commences by

mathematically representing mass as an equivalent localised density of wavefunction energy, contained by the QV

surrounding it. Properties of Fourier harmonics are utilised to mathematically decompile the mass-energy into a

spectrum of EM frequencies. This technique considers gravity to be the resultof an interaction between matter and the

space-time manifold; leading to the following precepts,

4) An object at rest polarises, and exists in equilibrium with, the QV surrounding it.5) The magnitude of QVE surrounding an object at rest is equivalent to E = mc2.6) The frequency distribution of the QVE surrounding an object at rest is cubic.

1.5.2 The QV SpectrumHistorically, the QV has been considered to comprise of a potentially infinite spectrum of randomly orientated

wavefunctions (i.e. in the form of virtual-particle-pairs), each of specific frequency and amplitude, analogous to the

static one observes on a dead television channel. However, the EGM construct disagrees with this historical conception

as it implies the existence of a potentially infinite quantity of energy in a vanishing volume; i.e. free space contains a

potentially infinite amount of energy because in Eq. (2) may equal an infinite quantity. EGM asserts that thelocalised QV surrounding an object is more appropriately described as a finite spectrum whose wavefunction population

(i.e. virtual-particle-pair population) is governed by the quantity of mass-energy influencing or occupying a specific

volume (i.e. free space contains a near zero amount of energy).

1.5.3 The ZPF SpectrumThe ZPF Spectrum (ZPFS) is defined by Eq. (2) and refers to the QV Spectrum (QVS) associated with

Minkowski space. The ZPFS is considered to be dispersed homogeneously throughout the Universe; consequently, the

spectral energy of the virtual-particle-pairs within it denotes the ground state of the QV. However, standard Quantum

Mechanics (QM) implies the existence of a potentially infinite quantity of energy in a vanishing volume, due to the

potential for high frequency virtual-particle-pair creation and annihilation. Fortunately, EGM resolves this conflict such

that a vanishingly small volume of flat space-time does notcontain an infinite amount of energy because although the

potential for such virtual-particle-pair creation and annihilation processes exists within the EGM construct, the

probability of high frequency virtual-particle-pair creation approaches zero in the absence of matter-energy (i.e. the

probability of low frequency virtual-particle-pair creation and annihilation approaches unity); the probability of low or


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high frequency virtual-particle-pair creation and annihilation is biased by the presence of matter-energy within a defined

region of Minkowski space (i.e. the greater the quantity of matter-energy present, the greater the probability of high

frequency virtual-particle-pair creation and annihilation). Mathematically within the EGM construct, this is achieved by

merging the continuous cubic frequency characteristic of the ZPF with a discrete and finite Fourier distribution such that

the highest frequency mode within the ZPFS tends to 0(Hz) in a vanishing gravitational acceleration field.

1.5.4 The EGM SpectrumThe energy spectrum associated with matter is termed the EGM Spectrum (EGMS). This is a harmonic

wavefunction representation of mass-energy obeying a Fourier distribution, in terms of conjugate wavefunction pairs,

such that the number of spectral frequency modes decreases as energy density increases (i.e. the number of modes is

inversely proportional to the energy density of the space-time manifold) [5], implying that the energy density of free-

space approaches zero, avoiding the infinite energy in a vanishing volume problem. The EGMS is based upon the Unit

Harmonic Operator (t); i.e. the number one (1) expressed as the summation of harmonic wavefunctions in the time

domain, obeying a Fourier Distribution. Within the EGM construct, (t) utilises a Fourier Distribution in Complex

form to operate upon a scalar function in order to harmonically quantise it over the Real and Imaginary planes. It is

important to recognise that for any harmonic decomposition of a constant function, unity in our case, only odd harmonics

are required to be summed, and the summation of Imaginary terms equals zero; (t) may be written in Complex form

as follows,

t( ) i

n 0

2

n 0.

e n 0

.

0.

t.

i.

.. 1

(3)

where,

7) The harmonic mode distribution is given by the odd sequence n0 = -N, 2-N ... N and N +; e.g. -21, -19,-17, -15 ... 21.

8) The maximum period is given by t0 = 1 / 0.9) 0 denotes the fundamental (i.e. minimum) spectral frequency; this is an arbitrary value. However, if one

applies the Hubble-Age as a physical limit (i.e. the age of the observable Universe), then 0 = 1 / Hubble-Age. Reference [7] derives the Hubble-Age to be 14.6 Billion Years, hence, the fundamental ( i.e.

minimum) Cosmological frequency becomes 0 = 1 / 14.6(Gyr) = 2.2 x10-18

(Hz).

Summing the first eleven (11) Real terms ( for illustrational purposes only) of

(t) yields a graphical representationconverging to unity as follows,

0.5

1

Time

1

Re t( )( )

t 0

2

t

Fig. (1): Unit Harmonic Operator,

Verifying by integration yields the appropriate results as follows,


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00

t 0

tRe t( )( ) d. 1=

(4)

00

t 0

tIm t( )( ) d.

0=

(5)

Therefore, utilising (t), mass-energy may be harmonically quantised according to the Massive Harmonic Operator

M(M,t) as follows,

M

M t,( ) M Re t( )( ).(6)

Similarly, the Gravitational Harmonic Operator g(r,M,t) may be written according to,

g

r M, t,( )G

r2

M

M t,( ).

(7)

Hence, the EGM mass-energy amplitude andfrequency spectra for (t), M(M,t) and g(r,M,t) may be described

according to,

Operator EGM Amplitude Spectrum EGM Frequency Spectrum

Unit: (t)Cn0

2

n0

.

(Eq. 8)

n0

n00

.

(Eq. 11)Massive: M(M,t) C

Mn

0M, M C

n

0.

(Eq. 9)

Gravitational: g(r,M,t)Cg n 0 r, M,

G

r2CM n 0 M,.

(Eq. 10)

Tab. (2),

1.5.5 The PV Spectrum1.5.5.1 Derivation Thereof

The energy spectrum associated with gravitational acceleration g is termed the PV Spectrum (PVS).

Consider the action of adding a stationary, non-rotating, neutrally charged point mass to an empty Universe. This action

superimposes the EGMS of the point mass onto the ZPFS of the Universe; doing so forms the PVS (i.e. a quantised

representation of the gravitational field in terms of g) surrounding the point mass. This modifies the KPV value of the

space-time manifold such that it changes at the same rate as g, radially outwards from the point mass. Merging the

EGM and ZPF spectra results in a cross-fertilisation of characteristics; the complete mathematical derivation is contained

in [5]. The EGM method produces a PVS such that the infinite energy dilemma of ZPF Theory (derived by

contemporary QM methods) is averted by assuming that the mass-energy density of an object is equal to the spectral

energy density of the gravitational field surrounding it. Therefore, when the EGM and ZPF spectra are merged, the

continuous ZPFS is equated to the Fourier distribution of the EGM spectrum such that the resulting PV spectral limits

may be determined. This process mathematically transforms the continuous ZPFS to a discrete and finite Fourier

distribution of equivalent energy.

The infinite energy in a vanishing volume problem is evaded within the EGM construct by determining the

finite limits of the PVS by application of theEquivalence Principle, which indicates that an accelerated reference frame

is equivalent to a uniform gravitational field. Reference [13] demonstrates that a generalised representation of

acceleration a may be derived utilising DATs and BPT, incorporating the ZPF spectral frequency distribution,

according to,


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a K0 r, E, B, X,( )

3r

2.

c

.

(12)

where, K0(,r,E,B,X) denotes a dimensionless constant related to KPV as follows,

K0 r, E, B, X,( )

1

KPV3

(13)

Assuming a represents gravitational acceleration and may be related to g(r,M,t) via the Equivalence Principle, it is

immediately apparent that a problem exists because g(r,M,t) contains two spectra (amplitude and frequency), whilst

a contains one spectrum (frequency). This difference may be reconciled by synchronising the frequency spectrum of

a, with the amplitude spectrum of g(r,M,t) at the 1st harmonic (i.e. when n0 = 1); by doing so, it is possible to

derive an expression for the common fundamental spectral frequency of both equations as follows,

Step 1: substitute Eq. (13) into Eq. (12) to yield a generalised expression for acceleration in terms of the

Refractive Index (i.e. KPV) within the PV approach to GR:

a1

KPV

3

3r2

.

c

.

(14)

Step 2: substitute n0 = 1 into Eq. (8,9,10) to determine the amplitude of the 1st harmonic of g(r,M,t):

C g 1 r, M,( )G

r2

C M 1 M,( ).

G M.

r2

C

1( ).2 G. M.

r2

. (15)

Step 3: equate the RHS of Eq. (14) to the RHS of Eq. (15) and solve for ; this synchronises thefrequency spectrum of a to the frequency spectrum of g(r,M,t) at the 1

st harmonic amplitude of

g(r,M,t). Hence, when = 0, the minimum spectral frequency 0 common to both representations ofacceleration may be written as follows,

0

1

r

3

2 c.

G.

M.

r..

KPV.

(16)

Consequently, since the PV amplitude and frequency spectra are theorised to obey a harmonic Fourier distribution, the

synchronised PV frequency spectrum PV as a function of the odd harmonic sequence described by n0 may beformulated according to,

PV

n0

n0

n00

.

(17)

By inspection of Eq. (17), the PV frequency spectrum may be considered to exist as a subset of the EGM frequency

spectrum [described in Tab. (2)] because 0 0 as r with respect to the PV amplitude spectrum [describedby Eq. (10)], whereas r has no component within the EGM amplitude spectrum [described by Eq. (9)]. At this

juncture, it is convenient to discriminate between spectra by the introduction of a PV subscript with respect to the odd

harmonic distribution nPV, such that nPV = n0. Hence, the PV amplitude and frequency spectra as a function of nPV,

r and M [i.e. CPV(nPV,r,M) and PV(nPV,r,M) respectively] are given by,

C PV n PV r, M,1

n PV

C g 1 r, M,( ).

1

n PV

2 G. M.

r2

.

.

(18)

PV

nPV

r, M, nPV

0

.

nPV

r

32 c. G. M.

r.

. KPV

.

(19)


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1.5.5.2 Derivation of LimitsEq. (19) may be applied to define the lower spectral limit of the ZPF encasing matter within the PV model of

gravity (i.e. when nPV = 1). The next requirement is to derive the upper spectral limit, but the distribution utilised to

derive the lower limit implies that the magnitude of the Nth

harmonic approaches infinity (i.e. nPV ).

Consequently, the upper spectral limit also approaches infinity [i.e. PV(,r,M) ] and the infinite energy in a

vanishing volume problem remains unresolved. Thus, if we are seeking to overcome this functional impasse, analternative approach is required. Fortunately, the EGM construct is capable of deriving the upper spectral limit in

accordance with the following solution algorithm,

Step 1:

Integrate Eq. (2) over the frequency domain:

2 h.

c3

3d.

1

2

h

c3

.

4.

(20)

Step 2:

Transform the continuous frequency spectrum of the ZPF represented in Eq. (20), into a discrete frequency

spectrum described by a harmonic Fourier distribution. To execute this, substitute Eq. (19) into Eq. (20)over one change in odd harmonic mode number (i.e. over the odd harmonic range |nPV| to |nPV|+2). This

action concentrates the spectral energy contained within the ZPF implied by Eq. (20), into a narrow

bandwidth described by a harmonic Fourier distribution; the solution takes the form of being a scalar multiple

of the fundamental spectral frequency PV(1,r,M). Hence, let U(nPV,r,M) denote the spectral energy of thisbandwidth according to,

U

n PV r, M,h

2 c3

.

PV n PV 2 r, M,4

PV n PV r, M,4

.

h

2 c3

.

PV 1 r, M,( ). n PV 2

4n PV

4.

(21)

Moreover, let:

U

r M,( )h

2 c3

.

PV 1 r, M,( ).

h 0.

2 c3

.

(22)

such that,

U

n PV r, M, U r M,( ) n PV 24

n PV4

.

(23)

Step 3:

Assume that the magnitude of mass-energy per unit volume associated with a material object [i.e. described by

Eq. (24)] is equal to the magnitude of the ZPF spectral energy per unit volume surrounding it. Thus, equating

Eq. (24) to Eq. (23) yields an expression with a single unknown (i.e. nPV) described by Eq. (25) as

follows,

U m r M,( )3 M. c

2.

4 . r3

.

(24)

Um

r M,( ) U

nPV

r, M,(25)

Step 4:

Derive the Harmonic Cut-Off Mode n(r,M): let n(r,M) denote the harmonic mode associated with the

upper spectral limit of the ZPF in the presence of a material object such that n(r,M) = |nPV|+2; the solution

algorithm for the derivation of Eq. (26,27) is contained in Appendix B [shown as Eq. (B.6,B.7)].

n

r M,( ) r M,( )

12

4

r M,( )1

(26)


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where,

r M,( )

3

108U m r M,( )

U

r M,( )

. 12 768 81U m r M,( )

U

r M,( )

2

..

(27)

Step 5:

Derive the Harmonic Cut-Off Frequency (r,M): let (r,M) denote the harmonic frequency associatedwith the upper spectral limit of the ZPF in the presence of a material object; the solution algorithm for the

derivation of Eq. (28) is contained in Appendix B [shown as Eq. (B.8)].

r M,( ) n

r M,( ) PV 1 r, M,( ).

(28)

Note: PV(1,r,M) = 0.Thus, as radial displacement r at a mathematical point from a mass-object increases,

10) Gravitational field strength decreases.11) Spectral energy density decreases.12) The number of harmonic spectral frequency modes increases.

Note: this is analogous to the number of standing waves that fit into a cube. As the cube grows insize, so do the number of standing waves capable of occupying the dimensions of the cube. If itgrows to the size of the observable Universe, the number of standing waves capable of fitting

within it tends to infinity. Conversely, if the cube diminishes in size, approaching the Planck

length, the number of standing waves capable of occupying it tends to unity.

13) Greater numbers of spectral frequency modes are required to be summed for mass-energy densityequivalence.

1.5.5.3 Spectral ComparisonThe EGM interpretation of Gravity is similar to Newtons thoughts of an optical model such that the aether was

presumed to be denser farther away. The gradient in aether density causes light and objects to follow trajectories

characteristic of GR. EGM demonstrates that the increasing density of Newtons aether is analogous to increases in

harmonic frequency mode population in the PV. Hence, the PV is an EM frequency spectrum obeying a Fourier

distribution at displacement r describing a mass M induced gravitational field such that,14) It denotes a polarised form of the ZPFS (i.e. mass pushes the ZPF surrounding it uphill, against the

natural flux of space-time manifold expansion).

15) The population of spectral frequency modes decreases as mass-energy density increases ( i.e. thespectral frequency mode bandwidth compresses); tending to unity for a case approaching the Planck

energy density limit.

16) Spectral frequency limits (lower and upper) increase as mass increases; converging to a discretespectrum tending to the Planck frequency, for a case approaching the Planck energy density limit.

Thus, utilising Eq. (28) a spectral comparison diagram may be constructed demonstrating the key distinctions between

the ZPF and its transformed form (i.e. the PV field) as follows,


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Fig. (2) (illustrational only - not to scale),

Increasing mass-energy density

where,

Region / Zone A B C D

Applicable Category Cosmology Astro-Physics Particle-Physics Planck Scale

Gravitational Model ZPF PV PV PV

Space-Time Geometry Flat Curved Flat Curved

Tab. (3),

On a Cosmological scale, the ZPF upper spectral limit is influenced by the average energy density of the presentUniverse. The spectral density of the ZPF remains cubic; however, the present upper spectral frequency limit is lower

than it was in the early Universe. Hence, the majority of Zero-Point-Energy (ZPE) is presently in the form of low-

frequency modes, each containing a relatively small amount of energy. The few high-frequency modes characterising the

early Universe have bifurcated into a very large bandwidth of lower-frequency modes as the Universe expanded to its

present form. The total energy of the Universe remains constant, but is spread out over a much greater volume as

Cosmological expansion continues. It is demonstrated by derivation in [14] that the majority proportion of gravitational

acceleration in a field biases the maximum frequency limit such that lower frequencies may be usefully neglected for

investigative purposes; the assertion of high spectral frequency bias is supported by [15]. By application of this

proportional spectral frequency characteristic, [16] demonstrates that a modal comparison diagram between the ZPF and

its transformed form (i.e. the PV field) may be constructed as follows,


11/35

Fig. (3) (illustrational only - not to scale),

Decreasing mass-energy density

where,

Region / Zone E F G H

Applicable Category Planck Scale Particle-Physics Astro-Physics Cosmology

Gravitational Model PV PV PV ZPF

Space-Time Geometry Curved Flat Curved Flat

Tab. (4),

Note: the 1 / nPV distribution depicted in Fig. (3) is analogous to the modal distribution of the amplitudespectrum described by Eq. (18) [i.e. CPV(nPV,r,M) for fixed values of r and M].

Further comparative characteristics between QV, ZPF, EGM and PV spectra may be tabulated as follows,

Spectrum Characteristics Bandwidth (Hz)

QV 17) Generalised reference to the QM nature of the vacuum18) Random and continuous19) Unspecified frequency distribution20) Within the EGM construct, in flat space-time geometries (i.e. Minkowski

Space), the QVS transforms to the ZPFS

21) Within the EGM construct, in curved space-time geometries (i.e.gravitational fields), the QVS transforms to the PVS

22)No significant governing equation/s


12/35

integer multiple of the lowest spectral frequency

31) Primary equations: Eq. (3-11)PV 32) Relates to gravitational acceleration

33) Formulated by relating the ZPF and EGM spectra34) Discrete and finite35) Obeys a Fourier Distribution; i.e. the maximum spectral frequency is an

integer multiple of the lowest spectral frequency36) Primary equations: Eq. (18,19,26-28)

hPV +h

Tab. (5),

1.5.6 PV TransformationsThe historical derivation of the PV model exhibits isomorphism to GR in weak field approximation. By

comparison, the similarities between EGM and the PV model demonstrate that EGM is also isomorphic to GR in the

weak field. However, differences exist between the two representations, primarily due to the introduction of a

superposition of fields, facilitating the formulation of engineering tools which may be utilised in practical applications.

Within the EGM construct, KPV is a function of 0 by wavefunction superposition at each point in a gravitationalfield. EGM supports the conjecture of the PV model such that measurements by rulers and clocks depend upon KPV

of the medium, by applying transformations to the ZPF. Hence, a PV transformation table for application to metric

engineering effects was articulated in [17] (see also: [18-21]) according to the proceeding table,

Important information with respect to Tab. (6):

37) The subscript relates to values, as would be measured or defined by a non-local observer, in a globally flatspace-time manifold (i.e. at infinity).

38) The non-subscripted parameters [e.g. c as a function of Refractive Index c(KPV)] relate to measurementsperformed by a local observer.

39) The non-subscripted parameters of , and Z do not refer to the classical representation of relativepermeability, permittivity and impedance (i.e. they are generalised references to the constants only).

40) The section Unit of Measure denotes PV transformations at the physical scale (i.e. rulers and clocks).41) The section Planck Measure denotes PV transformations at the Planck scale.42) The subscript h relates to Planck scale values, as would be measured or defined by a non-local observer, in

a globally flat space-time manifold (i.e. at infinity)

43) The section Relative Measure demonstrates the consistency of relative measures of the PV model (i.e. therelationship between the physical and Planck scales).

Physical Constant PV Representation of GR

Velocity of light c(KPV) = KPV-1c

Planck h(KPV) = h = h

Dirac (i.e. h / 2) (KPV) = = Gravitation G(KPV) = G = G

Permeability (KPV) = KPVPermittivity (KPV) = KPVImpedance Z(KPV) = Z = (/)

Unit of Measure

Mass (m)m(KPV) = KPV

3/2

mLength (r) r(KPV) = KPV-r

Time (t) t(KPV) = KPVt

Energy (E) E(KPV) = KPV-E

Planck Measure

Mass (mh) mh(KPV) = KPV-mh_

Length (h) h(KPV) = KPV3/2h_

Time (th) th(KPV) = KPV5/2th_

Energy (Eh) Eh(KPV) = KPV-5/2Eh_


13/35


14/35

46) Unconstrained Modelling refers to the ZPF across an elemental displacement (i.e. r) in agravitational acceleration field in the absence of Casimir boundaries (i.e. metal plates).

47) Quasi-Constrained Modelling refers to the geometric similarity requirements of ConstrainedModelling, expressed in terms of the maximum permissible wavelength (i.e. minimum frequency)

associated with an unconstrained model; i.e. the maximum permissible wavelength of a standing

wavefunction across r in a constrained model, is half the wavelength of an unconstrainedwavefunction transiting r.

48) Constrained Modelling refers to the ZPF across an elemental displacement (i.e. r) in agravitational acceleration field in the presence of Casimir boundaries (i.e. metal plates).

2.2 Derivation of the Casimir Force Utilising EGM Methodology2.2.1 Unique Solution2.2.1.1 Unconstrained Modelling

Reference [8] derives a unique solution for the CF associated with a Parallel-Plate experiment conducted at the

surface of the Earth, constructed upon foundation-work from [13] utilising the application of Buckingham (Pi)Theory (BPT). This lead to two General Modelling Equations (GMEs) formulated in [30], in-turn leading to a set ofGSEs developed in [14]. The authors commence their derivation by assuming that a superposition of applied EM

wavefunctions mimic a gravitational acceleration field. The GSEs were constructed from a previously defined

parameter in [30], termed the Critical Ratio KR; a ratio indicative of the similarity between a point in an ambient

gravitational acceleration field, and an applied EM field characterised by the superposition of EM wavefunctions. If the

applied EM field reproduces allof the scalar characteristics of the ambient gravitational acceleration field, then the ratio

of the applied EM field to the ambient gravitational acceleration field equals unity. Conceptually, this may be expressed

as follows,

Superposition_of_Applied_EM_Fieldsm

s2

.

Ambient_Gravitational_Acceleration_Fieldm

s2

.

Critical_Ratio 1

(34)

The leap from GMEs to GSEs is important because it represents the transition from a modelling technique

based upon dimensional similarity, to an initiating step in the formulation of a testable engineering outcome. The GMEs

were derived upon the presumption that the particulate constitution of the ZPF at the surface of the Earth is random,

however, the GSEs were derived post [5]. The significance of this being that the practical application of the GSEs

assumes that the ZPF in the presence of matter becomes structured in accordance with a Fourier distribution, i.e. it is nolonger a randomised vector field and may be described as a Polarised Vacuum field. Since [5] calculates the ZPF spectral

limits of a gravitational acceleration field, the transition from GMEs to GSEs signifies the transition from constructing

a generalised acceleration field via the superposition of EM wavefunctions, to the construction of an EM field utilising

the superposition of EM wavefunctions obeying a Fourier distribution, hypothesised by the authors to mimic a

gravitational acceleration field. Reference [8] applies GSE3 from [14] in their derivation workflow because it contains

the change in a parameter termed the Critical Factor KC [30]; a measure of the applied EM field intensity within an

experimental test volume. Hence, the change in KC ( specifically from zero) is termed the Change in Critical FactorKC, representing a proportional measure of the magnitude of the applied Poynting Vectors as the number of appliedharmonic EM field modes nA tends to infinity for the local observer as follows,

GSE r r, M,( )3

KC r M,( )

U PV r r, M,( )

0

0

.

(35)


15/35

where,

KC r M,( )1

KPV r M,( )2

n A

E A n A t,2

n A

B A n A t,2

..

(36)

U PV r

r,

M,

( ) U m r

r M,

( ) U m r M,

( ) (37)

Note: (i) for complete (i.e. ideal) similarity between the applied EM and ambient gravitational accelerationfields, GSE(r,r,M)3 = 1; (ii) all applied EM fields are coplanar with g; ( iii) KC is a constantfunction constructed from a superposition of EM fields at odd harmonics, obeying a distribution given by

nA = -N, 2-N ... N and N +; e.g. -21, -19, -17, -15 ... 21.

Once the authors established a Fourier based mathematical formalism for mimicking a local gravitational

acceleration field in [14], they applied an important pragmatic approximation in [22]; being that gravitational

acceleration may be usefully modelled as a one-dimensional 1-D phenomenon at the surface of the Earth ( i.e. g is

constant at the laboratory scale). This pragmatic approach facilitated the application of KR in a new manner, i.e. KR

is mathematically expressed as the sum of a family of wavefunctions obeying a Fourier distribution across r. Thisdiffers to the application of KR with respect to the formulation of GSE3 because KR as applied in [22] [expressed

as KR(r,r,M) therein], pertains to the harmonic structure of the ambient gravitational acceleration field across r,

rather than the applied EM wavefunctions implicit with respect to GSE3. Hence, the distinction between the forms of

KR applied in [14] and [22] may be characterised by the following attributes:

49) Reference [14] assumes that the particulate constitution of the ZPF is random within a gravitationalacceleration field, but may be mimicked by the application of EM fields obeying a Fourier distribution;

at conditions of ideal similarity between the ambient gravitational acceleration field and the applied

EM fields such that KR= 1.

50) Reference [22] conjectures that the particulate constitution of the ZPF is structured within agravitational acceleration field, such that for constant g approximations across r, g may bemodelled as a summation of harmonic wavefunctions obeying a Fourier distribution. Consequently, the

Critical Ratio associated with the value of g spanning r equals unity {expressed as KR(r,r,M) =1 in [22]} when all composite Fourier harmonics across r have been reproduced by the

superposition of artificial EM fields.The next step in the executed procedure was to develop a set of HSEs [22]; comprising of the Critical Ratio [of

the form KR(r,r,M)] divided by the appropriate GSE. The nomenclature of Harmonic Similarity stems from theharmonic constitution of KR(r,r,M) being divided by (i.e. compared to) the appropriate GSE; such that the number ofZPF harmonic frequency modes nPV tends to infinity where KR(r,r,M) denotes a constant function. Hence,KR(r,r,M) may be formulated utilising the Gravitational Harmonic Operator [i.e. Eq. (7)] applied across r asfollows,

Gravitational_Acceleration_across_ rm

s2

.

Gravitational_Acceleration_at_a_point_within_ rm

s2

.

Critical_Ratio 1

(38)

Gravitational_Acceleration_across_ rm

s2

.

Gravitational_Acceleration_at_a_point_within_ rm

s2

.

G M.

r2

i

n PV

2

n PV.

e n PV

. r 1 r, r, M,( )

. t. i....

G M.

r2

(39)


16/35

Thus, the Critical Ratio may be stated in the following form; equivalentto the Unit Harmonic Operator defined by Eq.

(3) according to,

KR r r, M,( ) i

n PV

2

n PV.

e n PV

. r 1 r, r, M,( )

. t. i...

(40)

where,

r 1 r, r, M,( ) PV 1 r r, M,( ) PV 1 r, M,( ) (41)

Note: (i) for complete (i.e. ideal) similarity between the constant g approximation across r and g ata point within r, KR(r,r,M) = 1; (ii) r(1,r,r,M) represents the magnitude of the change in thelower ZPF harmonic spectral frequency limit across r (i.e. the magnitude of the change in fundamentalfrequency); (iii) The harmonic distribution is given by the odd mode sequence nPV = -N, 2-N ... N and N

+; e.g. -21, -19, -17, -15 ... 21.

Hence, the similarity of an engineering solution utilising applied EM field harmonics, to a constant g approximation of

the local gravitational acceleration field is given by HSE3 as follows,

HSE r r, M,( )3

KR r r, M,( )

GSE r r, M,( )3 (42)

Substituting Eq. (35,36,40) into Eq. (42) yields,

HSE r r, M,( )3

i U PV r r, M,( ).

n PV

2

n PV.

e n PV

. r 1 r, r, M,( )

. t. i...

0

0

1

KPV r M,( )2

.

n A

E A n A t,2

n A

B A n A t,2..

(43)

Note: (i) for complete (i.e. ideal) similarity between the constant g approximation across r and theapplied EM fields, HSE(r,r,M)3 = GSE(r,r,M)3 = KR(r,r,M) = 1; (ii) all applied EM fields arecoplanar with g; (iii) the denominator is a constant function constructed from a superposition of EM

fields at odd harmonics, obeying a distribution given by nA = -N, 2-N ... N and N +; e.g. -21, -19, -

17, -15 ... 21.

The family of harmonic wavefunctions described by Eq. (43) may be decomposed into modal form by

discrete similarity; a piecewise representation of harmonic constitution according to,

HSE E A B A, n A, n PV, r, r, M, t,3

i U PV r r, M,( ).

2

n PV.

. e n PV

. r 1 r, r, M,( )

. t. i..

0

0

1

KPV

r M,( )

. E A n A t,. B A n A t,

.

(44)

Note: due to decomposition, the denominator of Eq. (43) is no longer constant and the time dependentconsequence of the executed procedure is incorporated into the Left-Hand-Side (LHS) of Eq. (44); via

the inclusion of EA, BA, nA and t as functional parameters.

However, Eq. (44) may be simplified by recognising that,

i e n PV

. r 1 r, r, M,( )

. t. i.. 1 (45)


17/35

Thus, the reduced form of Eq. (44) is given by,

HSE E A B A, n A, n PV, r, r, M, t,3 R

KPV r M,( ) U PV r r, M,( ).

E A n A t, B A n A t,.

0

0

. KR n PVH

.

(46)

where, KR(nPV)H denotes the harmonic amplitude spectrum of Eq. (40,46) according to,

KRnPV

H

2

nPV

.

(47)

Thus, harmonic similarity between an approximation of constant g across r and an applied EM field may becharacterised by the properties of the amplitude spectrum of the local ZPF. The significance of this being that ZPF

frequency information is implicitly embedded into Eq. (46) due the relationship between amplitude and frequency

within a Fourier spectrum; i.e. harmonic amplitude decreases asymptotically whilst harmonic frequency increases

linearly as nPV tends to infinity.

The engineering techniques utilised in [13] suggest that, for interactive physical modelling with the ZPF, it may

be possible to engineer the Polarizable Vacuum ( i.e. the ZPF in the presence of matter) by the selective manipulation of a

specific ZPF frequency. Moreover, [14] extends this proposition by demonstrating that the significant majority [ i.e.>>99.99(%)] of Polarisable Vacuum spectral energy (i.e. the ZPF spectral energy in the presence of matter) exists at

(or near) the upper limit of the local ZPF frequency spectrum. Consequently, Eq. (46) may be conveniently

reconstituted by stipulating and insisting that all applied EM field configurations for experimental purposes be executed

at conditions of maximal frequency. Hence we shall, in a manner of speaking, force the denominator of Eq. (46)

towards the characteristics of a constant function; offering a pragmatic engineering approach forward of the form shown

by Eq. (55).

Let St(r,r,M) be termed the Range Factor and the applied EM fields [i.e. EA(nA,t),BA(nA,t),Erms,Brms] be expressedin complex phasor form such that,

St

r r, M,( ) U PV r r, M,( ) 0

0

.

(48)

EA

nA

t, E0e

2 .

A n A.

t.

2i.

.

Eq. (49) B

AnA

t, B0e

2 .

A n A.

t.

2 i

.

.

Eq. (50)

Erms

E0

2 Eq. (51)

Brms

B0

2 Eq. (52)

Tab. (8),

Performing the appropriate substitution of Eq. (47-52) into Eq. (46) and evaluating utilising the MathCad 8

Professional symbolic processing engine, yields the following;

Note: the proceeding solution algorithm appears in standard product notation and form, and is a verbatimextraction of computational output.


18/35

Primary Sub-Routine:

KPV r M,( ) St r r, M,( ).

E A n A t, B A n A t,.

2

n PV.

.

substitute E A n A t, E 0 e

2 . A n A. t.

2i.

.,

substitute B A n A t, B 0 e

2 . A n A. t.

2 i.

.,

substitute B 0 2 B rms.

,

substitute E 0 2 E rms.

,

simplify

KPV r M,( )St r r, M,( )

n PV E rms B rms...

. exp i 4 . A n A. t. ..

(53)

Secondary Sub-Routine:

KPV r M,( )St r r, M,( )

n PV. E rms

. B rms.

. exp i 4 . A n A. t. .. simplify

1

KPV r M,( )

St r r, M,( )

n PV E rms B rms..

..

(54)

Hence, Eq. (46) may be reconstituted according to,

HSE 3 E rms B rms, n PV, r, r, M,3 R

KPV r M,( ) St r r, M,( ).

n PV. E rms

. B rms.

(55)

Eq. (55) provides a relative assessment of the similarity between a specific ZPF harmonic frequency mode

and an arbitrary experimental configuration. However, whilst the process of its formulation is mathematically consistent,

its implementation lacks the required pragmatism from an engineering perspective. Moreover, as stated in the preceding

dialogue, the applied EM fields mathematically act as a constant function. Thus, minimal advantage is gained by

comparing a sinusoidal function (i.e. the numerator) to a pseudo-constant function (i.e. the denominator). To overcome

this pragmatic impasse, [14] introduces the concept of spectral similarity via the development of SSEs; SSE3 being of

specific interest herein. SSEs define the similarity between a mode averaged ZPF across r, and the applied EMfields, providing a practical target for engineers. Reference [14] defines SSE3 as the average Harmonic Similarity per

mode according to,

SSE E rms B rms, r, r, M,3

1

n

r r, M,( )ZPF

1n

PV

HSE E rms B rms, n PV, r, r, M,3 R

.

(56)

where,

51) n(r,r,M)ZPF is termed the ZPF Beat Cut-Off Mode, and defined as an oddnumber approximating the upperZPF mode limit across r; thus, the harmonic distribution of the Polarisable Vacuum ( i.e. the ZPF in the

presence of matter) is given by the odd sequence as follows,

o nPV = -n(r,r,M)ZPF, 2-n(r,r,M)ZPF ... n(r,r,M)ZPF and n(r,r,M)ZPF +.o nPV is a double sided odd spectrum, symmetrical about the 0th mode; e.g. -21, -19, -17, -15 ... 21.

52) n(r,r,M)ZPF + 1 represents the number of harmonic modes across r for an odd harmonic distribution asdemonstrated by the following,

o e.g. n(r,r,M)ZPF = 9 yields the distribution -9, -7, -5, -3, -1, 1, 3, 5, 7, 9, possessing 10 terms.o Hence, the total number of modes across r is represented by n(r,r,M)ZPF + 1.o When n(r,r,M)ZPF >> 1, n(r,r,M)ZPF + 1 n(r,r,M)ZPF.

Eq. (56) demonstrates that two key units of information are required in order to artificially mimic the ZPF associated

with a gravitational acceleration field; i.e. n(r,r,M)ZPF and the sum of all harmonic modes across r. From apragmatic engineering perspective, any simplification is typically advantageous and consistent with the best practice

design philosophy of Keep-It-Simple (KIS). Subsequently, we may simplify Eq. (56) such that it is reliant upon one

key unit of information rather than two, according to the following construct.


19/35

Decomposing Eq. (56) into three distinct products yields,


1

n

r r, M,( )ZPF

1

KPV r M,( ) St r r, M,( ).

E rms. B rms

.

.

n PV

1

n PV

.

(57)

For solutions where nPV >> 1 and = Eulers Constant,

n PV

1

n PV

ln 2 n r r, M,( )ZPF

.

(58)

Note: (i) standard QED postulates that the upper modal limit of the local ZPF approaches infinity; (ii) theerror associated with Eq. (58) when n(r,r,M)ZPF = 10

6+1 is less than 6.6 x10

-6(%); (iii) 10

6+1 1.Thus, reconstituting Eq. (57) yields,


KPV r M,( ) St r r, M,( ).

E rms. B rms

. N X r r, M,( ).

(60)

Note: (i) for complete (i.e. ideal) similarity between the constant g approximation across r and theapplied EM fields, SSE(Erms,Brms,r,r,M)3 = HSE(r,r,M)3 = GSE(r,r,M)3 = KR(r,r,M) = 1; (ii) allapplied EM fields are coplanar with g; (iii) at complete (i.e. ideal) similarity, Erms and Brms are

representative of constant functions constructed from a superposition of EM fields at odd harmonics,obeying a distribution given by nA = -N, 2-N ... N and N +; e.g. -21, -19, -17, -15 ... 21; (iv) for

practical engineering investigations, Erms and Brms are time dependent functions [e.g. Eq. (49,50)]

representative of the maximum possible EM frequencies achievable with the resources at hand.

Eq. (59,60) clearly articulate that n(r,r,M)ZPF is an important parameter with respect to physicallymodelling the interaction between the ZPF and applied EM fields. It also demonstrates that the upper spectral limit of the

local ZPF is the dominating design factor for experimental investigations; consistent with the conclusions drawn by the

authors in [14]. However, in the absence of a precise numerical evaluation of n(r,r,M)ZPF, it has no practical valuefor engineering purposes. Reference [14] overcame this constraint by defining it in a manner analogous to (i.e. consistent

with) n(r,M) from Eq. (28) by transposition. Hence;

Let n(r,r,M)ZPF be mathematically defined as the ratio of the upper ZPF spectral frequency limit across r, termed

the ZPF Beat Cut-Off Frequency (r,r,M)ZPF, to the lower ZPF spectral frequency limit at r [ i.e. PV(1,r,M)]according to,

n

r r, M,( )ZPF

r r, M,( )ZPF

PV 1 r, M,( ) (61)

By inspection of Eq. (61), the determination of (r,r,M)ZPF presents a problem. Since the number of modes at r

differs significantly to the number of modes at r+r, a simple difference calculation is inadequate to negotiate the wayforward to a working solution. Moreover, the value of the upper ZPF spectral limit also varies significantly from r to


20/35

r+r; hence, a workaround to this problem is required. To achieve this, we shall leverage-off specific properties of themathematical formalism as constructed. All values of r possess a fundamental frequency, and all values of

fundamental frequency occur at nPV = 1. Consequently, it follows that the change in the lower ZPF spectral limit across

r [i.e. Eq. (41)] should represent the lower ZPF spectral boundary for engineering investigations. In addition,

recognising that (r,r,M)ZPF is only an approximation of the upper ZPF spectral limit across r (i.e. due to theconstant g approximation being the cornerstone of the construct), we shall assume that Eq. (20,21) are applicable in

the following form,

U PV r r, M,( )h

2 c3

.

r r, M,( )ZPF

4

r 1 r, r, M,( )4

.

(62)

Transposing yields,

r r, M,( )ZPF

4

2 c3

.

hU PV r r, M,( )

. r 1 r, r, M,( )

4

(63)

Therefore, Eq. (59,61,63) denote the key design parameters required to physically mimic the effects of an ambient

gravitational acceleration field (by useful approximation) utilising applied EM fields for engineering investigations

across r.

2.2.1.2 Quasi-Constrained ModellingQuasi-Constrained Modelling refers to the geometric similarity requirements of Constrained Modelling,

expressed in terms of the maximum permissible wavelength (i.e. minimum frequency) associated with an unconstrained

model; i.e. the maximum permissible wavelength of a standing wavefunction across r in a constrained model, is halfthe wavelength of an unconstrained wavefunction transiting r. The associated frequency is termed the CriticalFrequency C and defined by Eq. (64). Notably thus far, the engineering model constructed has been developedwithout the physical boundary plates required by a Casimir experiment, only the provision for physical boundaries by the

stipulation of phenomena spanning r. In preparation for the enforcement of physical boundary conditions (i.e. metalplates either side of r), let C be defined such that geometric similarity between an unconstrained system (i.e.without metal plates either side of r) and a bounded system (i.e. with metal plates either side of r) is preservedand given by Eq. (64), described as quasi-constrained according to,

C r( )c

2 r. (64)

Fig. (4): Quasi-Constrained Modelling (illustrational only - not to scale),

Hence, let the Critical Mode NC(r,r,M) of the quasi-constrained model be defined in a manner analogous to Eq.(28,61) as follows,

N C r r, M,( ) C r( )

PV 1 r, M,( ) (65)


21/35

Note: Eq. (64) has been formulated (as stated) to preserve geometric characteristics of an equivalentstanding wavefunction induced by metal plates either side of r.

2.2.1.3 Constrained ModellingConstrained Modelling refers to the ZPF across an elemental displacement (i.e. r) in a gravitational

acceleration field in the presence of a Casimir Parallel-Plate Experiment. Reference [8] advances their derivation by

conceptualising the transition from Quasi-Constrained to Constrained modelling by approximating the difference in

sum between NC(r,r,M) and NX(r,r,M); in the context of ZPF harmonic modes being constrained by the presenceof physical boundaries (i.e. Casimir plates) either side of r. Consequently, the difference in harmonic modalsummation is approximated by Eq. (66) according to,

ln 2 N C r r, M,( ). ln 2 N X r r, M,( )

. lnN C r r, M,( )

N X r r, M,( ) (66)

In this context, NC(r,r,M) represents a boundary condition of the end state achieved by the physical insertion ofCasimir Plates either side of r (by geometric similarity), whereas NX(r,r,M) represents the initial state of anunmodified system. Hence, the difference between the target state and initial state constitutes the effective change

induced by the insertion of Casimir Plates such that,

r lnN C r r, M,( )

N X r r, M,( ) (67)

Based upon the preceding premise, [8] proposes that the CF for a Parallel-Plate configuration (i.e. FPP) is proportional

to the following Similarity Indicators,

Eq. (68) Initial State Eq. (69) Target State

FPP

U PV r r, M,( )

N X r r, M,( )FPP

N C r r, M,( )

N X r r, M,( )

Tab. (9): Similarity Indicators,

Thus, by inspection and reinterpretation of Eq. (1), an EGM analogue to the CF for a Parallel-Plate configuration may

be formulated according to Eq. (70) as follows,

F PV r r, M,( ) A PP r( )U PV r r, M,( )

N X r r, M,( )

.

N C r r, M,( )

N X r r, M,( )

. lnN C r r, M,( )

N X r r, M,( )

4

.

(70)

where, APP(r) denotes Parallel-Plate area as a function of radial displacement according to the spherical surface area

relationship as follows,

A PP r( ) 4 . r

2.

(71)

Note: APP is expressed in terms of radial displacement r to ensure that a very large area is utilised forgeometric similarity between the EGM construct and the infinite plate area assumed by Casimir in theoriginal derivation.

2.2.1.4 Moving to InfinityEq. (70) demonstrates that, for an experiment ofunit plate area at a separation of r = 1(mm), the Casimir

Pressure induced by a planetary point mass increases with M and decreases with radial displacement r, tending to

zero as r tends to infinity [see: Fig. (5)]. However, this result should not be confused with the forthcoming prediction

that a physical measurement of the CF will be greater on the Moon than on Jupiter (refer to proceeding section). The


22/35

reason for this resides in the fact that a physical experiment at the surface of a planetary body can only be performed

subject to the fixed gravitational acceleration conditions at its surface (i.e. radius and mass are unique combinations

associated with specific planetary surface conditions); for the graphical illustration shown, r is a variable parameter.

The Moon

The EarthJupiter

Radial Displacement (m)

MagnitudeofCasimirPress.

(Pa)

5 RE. 10 RE

.

Fig. (5): Plot of Eq. (70): Magnitude of Casimir Pressure vs. Radial Displacement,

o Plate Separation = r = 1(mm).o The Lunar curve obeys the proportion FPV(r,r,MM) / APP(r).o The Terrestrial curve obeys the proportion FPV(r,r,ME) / APP(r).o The Jovian curve obeys the proportion FPV(r,r,MJ) / APP(r).o Y-Axis (log scale).

2.2.1.5 Engineering InfinityWith the formulation of Eq. (70), [8] consequently initiates a pragmatic determination of infinite Parallel-

Plate separation in the context of real-world engineering outcomes. Although the original derivation by Casimir

accommodates the computation of the force as Parallel-Plate separation approaches infinity (literally), it is a valueless

feature to the practical engineer. In order to maximise engineering advantage, it is highly desirable (where applicable) to

articulate Engineering Infinity. In the specific case of a Parallel-Plate configuration, [8] determined that Eq. (70)

usefully approximates Eq. (1) (by coincidence) at a Parallel-Plate separation of r = 1(mm) at the surface of theEarth; hence, it may be stated that Eq. (70) = Eq. (1) with trivial error [approx. -0.1122(%)] according to the

following computational output,

Relationship CP

FPP(r) / APP -1.3001 fPa(x10

-15Pa)FPV(RE,r,ME) / APP(RE) -1.2983

Legend:

r = Parallel-Plate Separation = 1(mm)APP = Parallel-Plate Area = 1(m

2)

RE = Radius of the Earth = 6.3781 x103(km)

ME = Mass of the Earth = 5.9722 x1024

(kg)

APP(RE) = 5.1121 x108(km

2)

Tab. (10): Casimir Pressure (i.e. Casimir Force per unit area),


23/35

2.2.1.6 Qualitative AnalysisUpon determining a convergence of forms between Eq. (1,70) at Engineering Infinity, [8] consequently

initiated a qualitative analysis of the CF for a Parallel-Plate configuration at the surface of other celestial bodies relative

to the Earth. Solving Eq. (70) with Lunar and Jovian parameters yields,

Relationship CPFPV(RM,r,MM) / APP(RM) -2.3480 fPa

(x10-15

Pa)FPV(RJ,r,MJ) / APP(RJ) -0.0742Legend:

RM = Radius of the Moon = 1.7381 x103(km)

MM = Mass of the Moon = 7.349 x1022

(kg)

RJ = Radius of Jupiter = 7.1492 x104(km)

MJ = Mass of Jupiter = 1.8986 x1027

(kg)

APP(RM) = 3.7963 x107(km

2)

APP(RJ) = 6.4228 x1010

(km2)

Tab. (11): Casimir Pressure (qualitative prediction only),

Hence, the CF differs depending upon ambient gravitational field strength; for example, the CF will be slightly different

on Earth than Jupiter or the Moon because the ZPFS, as described by Eq. (2), is compressed by the gravitational

acceleration field induced by the presence of matter. Thus, it may be stated that the ZPF exhibits the following

characteristics,

53) As gravitational acceleration field values approach zero, the ZPF modal bandwidth tends to infinity(i.e. it is described as being infinitely broad). This does not imply that the upper spectral limit

approaches infinity, as demonstrated by the following conceptualisation,

o Let (i) g(r,M) = 10-500(m/s2) represent the gravitational acceleration field value at r due tofield source M, of the local ZPF under consideration; ( ii) let the upper modal limit be

defined by n(r,M) = 10500

; (iii) let the lower spectral frequency limit be defined by

PV(1,r,M) = 10-500

(Hz).

o Obeying a Fourier distribution, the upper ZPF spectral frequency limit equals the upper modallimit multiplied by the lower frequency limit; 10

500x 10

-500(Hz) = 1(Hz). Therefore, for an

infinitely broad ZPF modalbandwidth, the lower and upper spectral frequency limits tend tozero; i.e. in the absence ofany form of field energy.

o . a Casimir Experiment conducted in free space will produce an extremely small force(tending to zero) due to the lack of initial background field pressure. Since the Casimir Force

arises from a pressure imbalance, the lack of significant ambient field pressure between the

plates prevents the formation of large Casimir Forces.

54) The ZPF modalbandwidth associated with the gravitational acceleration field at the surface of Jupiteris compressed relative to the Earth. Similarly, the ZPF modalbandwidth at the surface of the Earth is

compressed relative to the Moon; thus,

o . an Earth based equivalent Casimir experiment conducted on Jupiter will exclude fewerlow frequency modes preserving higher frequency modes that simply pass through the

plates, resulting in a smaller Casimir Force. By contrast, the same experiment conducted onthe Moon will produce a larger Casimir Force.

55) As gravitational acceleration field values approach their maximum permissible magnitude (i.e. in the presence of a particle of Planck energy density), the local ZPF modalbandwidth is compressed such

that the number of field modes tends to unity, and both limits ( i.e. lower and upper) of the ZPF

frequency bandwidth approach the Planck frequency.


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2.2.2 General Solution2.2.2.1 Derivation Thereof

By inspection of Eq. (67), it is immediately apparent that a virtual asymptote (i.e. discontinuity) is induced

within r when NC(r,r,M) = NX(r,r,M) [see: Fig. (6)]. Discontinuities are often not an issue in the engineeringdomain and in-fact occur more commonly than laypeople may realise. Possibly the most widespread application ofdiscontinuous functions relates to structural mechanics, where changes in loading situations demand discontinuous

representations; refer to any university entry level structural mechanics text for graphical examples relating to shear

force and bending moment diagrams.

The MoonThe Earth

Jupiter

Plate Separation (m)

MagnitudeofCasimirPress.

(Pa)

rE rJ

Fig. (6): Plot of Eq. (70): Magnitude of Casimir Pressure vs. Plate Separation,

o Maximum plate separation = r = 0.9(m) = 900(nm).o The Lunar curve obeys the proportion FPV(RM,r,MM) / APP(RM).o The Terrestrial curve obeys the proportion FPV(RE,r,ME) / APP(RE).o The Jovian curve obeys the proportion FPV(RJ,r,MJ) / APP(RJ).o Y-Axis (log scale).

To formulate a generalised application of the CF within the context of the EGM construct, we are compelled to refine the

unique solution presented in the preceding section. To achieve this, we shall compensate for the induced virtual

asymptote within r and calibrate the result to the terrestrial environment ( i.e. the surface of the Earth). Moreover, weshall articulate the solution algorithm in a series of steps, utilising computational resources where applicable. Thus;

Step 1: Calculate the position of the virtual asymptote (discontinuity) within the Casimir cavity utilising the

Given and Find commands in the MathCad 8 Professional computational environment,

Let

rM

,

rE and

r

J denote the position of the virtual asymptote within the Casimir cavity, from

the left-hand inner face at the surface of the Lunar, Terrestrial and Jovian environments respectively;

hence,

Given

ln

NCRM

rM

, MM

,

NX

RM

rM

, MM

,

0

ln

NCREr

E, M

E,

NX

REr

E, M

E,

0

ln

NCRJr

J, M

J,

NX

RJr

J, M

J,

0


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rM

rE

rJ

Find rM

rE

, rJ

,

rM

rE

rJ

131.9395

155.1574

342.3369

nm( )=

Note: maximum plate separation = 900(nm).Step 2: Formulate an expression for CP based upon Eq. (70), whilst compensating for the discontinuity of the

virtual asymptote by translating its position such that it coincides exactly with the left-hand inner face of theCasimir cavity,

Let the Asymptotic Correction Factor rX denote the position of the virtual asymptote in accordancewith the solution algorithm specified in the preceding step (e.g. rX = rM, rX = rE, rX = rJ).Hence, let the Casimir Pressure be given by Eq. (72) according to,

CP PV r r, rX, M, U PV r r, M,( )N C r r rX, M,

N X r r rX, M,2

. lnN C r r rX, M,

N X r r rX, M,

4

.

(72)

Note: UPV(r,r,M) must not be asymptotically corrected, as it is not an EGM derivedrelationship; inclusion of rX will re-define the plate separation distance (i.e. changing theexperimental configuration).

The Moon

The EarthJupiter


Magnitudeo

fCasimirPress.

(Pa)

Fig. (7): Plot of Eq. (72): Magnitude of Asymptotically Corrected Casimir Pressure vs. Plate Separation,

o Maximum plate separation = r = 0.9(m) = 900(nm).o The Lunar curve obeys the function CPPV(RM,r,rM,MM).o The Terrestrial curve obeys the function CPPV(RE,r,rE,ME).o The Jovian curve obeys the function CPPV(RJ,r,rJ,MJ).o X-Axis (log scale).


26/35

Step 3: Formulate a Terrestrial Casimir Calibration Factor CCF(r) utilising Eq. (1,72) as follows,

C CF r( )F PP r( )

A PP CP PV RE r, rE, M E,.

(73)

Note: FPP(r) must notbe asymptotically corrected, as it is not an EGM derived relationship; inclusion ofrX will re-define the plate separation distance (i.e. changing the experimental configuration).

The EarthPlate Separation (m)

TerrestrialCasimirCalibrationFactor

Fig. (8): Plot of Eq. (73): Terrestrial Casimir Calibration Factor vs. Plate Separation,

o Maximum plate separation = r = 900(nm).o APP = 1(m2).

Step 4: Formulate an expression for the Calibrated Terrestrial CP.

The Calibrated Terrestrial CP TCPPV(r,r,rX,M) may be formulated by multiplying the CasimirCalibration Factor Eq. (73) by the asymptotically corrected CP as defined by Eq. (72) such that,

TCP PV r r, rX, M, C CF r( ) CP PV r r, rX, M,.

(74)

Eq. (74)

(x10-3

Pa)

Eq. (73)

(x108)

Eq. (72)

(x10-12

Pa)r(nm)

-1.3001 x104

46.3569 -2.8046 x103

100

-812.5785 18.1271 -448.266 200

-160.5093 9.8861 -162.3579 300

-50.7862 6.2023 -81.8832 400

-20.802 4.2333 -49.139 500

-10.0318 3.0684 -32.694 600

-5.4149 2.3096 -23.4458 700

-3.1741 1.7971 -17.6623 800

-1.9816 1.4312 -13.8461 900

Tab. (12): Calibrated Terrestrial Casimir Pressure,

where, r = RE, rX = rE and M = ME.


27/35

The Moon

The EarthJupiter


Mag.ofCal.

Terr.

CasimirPress.

(Pa)

Fig. (9): Plot of Eq. (74): Magnitude of Calibrated Terrestrial Casimir Pressure vs. Plate Separation,

o Maximum plate separation = r = 900(nm).o The Lunar curve obeys the function TCPPV(RM,r,rM,MM).o The Terrestrial curve obeys the function TCPPV(RE,r,rE,ME).o The Jovian curve obeys the function TCPPV(RJ,r,rJ,MJ).o X-Axis (log scale).

Step 5: Formulate an expression for the Magnitude of the Average Calibrated CP over the experimental range

rmin to rmax.

Given by Eq. (75) according to,

TCPPV_av r rX, M,1

rmax rmin rmin

rmax

rTCPPV r r, rX, M, d.

(75)

Environment Eq. (75) (Pa) rX (nm) Mass M Location r [9] r (nm)Lunar Surface 0.7414 131.9395 MM Mean

Equatorial

Radius

(RM,R,RE,R) rmin = 100to

rmax = 900

Martian Surface 0.6041 146.3822 M

Earth Surface 0.541 155.1574 ME

Venusian Surface 0.5398 155.0831 M

Low Earth Orbit 0.5162 158.7907

ME

180(km)

AltitudeSun-Synchronous Orbit 0.4578 169.0331 705(km)

Mid Earth Orbit 0.3537 193.6958 2,000(km)

GPS Orbit 0.0664 493.3333 20,200(km)High Earth Orbit 0.0296 802.8734 42,164(km)

Tab. (13): Magnitude of the Average Calibrated Casimir Pressure over a range of plate separations r,

Step 6: Formulate an expression for the Magnitude of the Relative Average Calibrated CP over the

experimental range rmin to rmax.

Given by Eq. (76) as follows,


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TCPPV_Rel_av r rX, M,TCPPV_av r rX, M,

TCPPV_av RE rE, M E, (76)

Environment Eq. (76) rX (nm) Mass M Location r [9] r (nm)Lunar Surface 1.3705 131.9395 MM Mean Equatorial

Radius(RM,R,RE,R)

rmin = 100to

rmax = 900

Martian Surface 1.1167 146.3822 MVenusian Surface 0.9978 155.0831 M

Low Earth Orbit 0.9542 158.7907

ME

180(km)

AltitudeSun-Synchronous Orbit 0.8463 169.0331 705(km)

Mid Earth Orbit 0.6538 193.6958 2,000(km)

GPS Orbit 0.1228 493.3333 20,200(km)

High Earth Orbit 0.0548 802.8734 42,164(km)

Tab. (14): Magnitude of the Relative Average Calibrated Casimir Pressure over a range of plate separations,

Step 7: Formulate an expression for the Magnitude of Proportional Change in the Relative Average Calibrated

CP over the experimental range rmin to rmax.

Given by Eq. (77) according to,

CP r rX, M, TCP PV_Rel_av r rX, M, 1 (77)

Environment Eq. (77) (%) rX (nm) Mass M Location r [9] r (nm)Lunar Surface +37 131.9395 MM Mean Equatorial

Radius

(RM,R,R)rmin = 100

to

rmax = 900

Martian Surface +12 146.3822 M

Venusian Surface -0 155.0831 M

Low Earth Orbit -5 158.7907

ME

180(km)

Altitude

Sun-Synchronous Orbit -15 169.0331 705(km)

Mid Earth Orbit -35 193.6958 2,000(km)

GPS Orbit -88 493.3333 20,200(km)

High Earth Orbit -95 802.8734 42,164(km)

Tab. (15): Mag. of Prop. Change in the Relative Average Calibrated CP over a range of plate separations,

2.2.2.2 EGM ToleranceApplication of the EGM construct to the CP associated with a Parallel-Plate configuration is an approximation

due to the initial simplification of the system during the formulation of the solution. It was assumed that the gravitational

acceleration field across all practical laboratory dimensions is uniform (i.e. g is constant at the human scale).

Consequently, this simplification introduced errors into the solution and it is important to understand the limitations of

the EGM construct as applied, in context; such that the tolerances associated with two key modelling parameters (plateseparation and radial displacement) may be computed by determining the similarity between a change in gravitational

acceleration field energy density [as described by Eq. (37)], and the constant gravitational acceleration field

characterised by the derivation of Eq. (74). Thus, as the change in gravitational acceleration field energy density

approaches zero due to a decrease in Parallel-Plate separation, the application of Eq. (74) approaches its limit.

Therefore, application tolerances associated with the EGM construct may be formulated utilising Eq. (37,74) to formEq. (78) as follows,

KU r r, rX, M,

U PV r r, M,( )

TCP PV r r, rX, M, (78)

Hence, the approximate EGM application tolerance with respect to plate separation limit r = rlim is given by thefollowing table of results; notably, the EGM construct appears to remain applicable to within X-Ray resolution ( i.e.

substantially exceeding the physical limitations of executed experiments).


29/35

Environment Eq. (78) (%) rX (nm) Mass M Location r [9] rlim (nm)Lunar Surface 3.4736 x10

-3131.9395 MM Mean

Equatorial

Radius

(RM,R,RE,R) 0.5

Martian Surface 2.3018 x10-3

146.3822 M

Earth Surface 9.2686 x10-4

155.1574 ME

Venusian Surface 2.0313 x10-3

155.0831 M

Low Earth Orbit 1.5529 x10

-3

158.7907

ME

180(km)

AltitudeSun-Synchronous Orbit 1.1129 x10

-169.0331 705(km)

Mid Earth Orbit 1.2665 x10-3

193.6958 2,000(km)

GPS Orbit 0.0187 493.3333 20,200(km) 1.9

High Earth Orbit 0.0236 802.8734 42,164(km) 3.8

Tab. (16): Approximate EGM application tolerance with respect to plate separation limit r = rlim,

Similarly, the EGM application limit (with respect to radial displacement) may be approximated by solving Eq. (78)

for values of r such that KU(r,r,rX,M) tends to zero according to,

Gravitational Source Eq. (78) (%) r r (nm) rX (nm) Mass MThe Moon 6.8913 x10

-12.1831 RM

100 900

MM

Mars 2.125 x10-5

10.7204 R M

The Earth 1.0032 x10-

9.9815 RE ME

Venus 1.5133 x10-4 9.9888 R M

Tab. (17): Approximate EGM application tolerance with respect to radial displacement r,

Note: this table of results assumes that the range of plate separation values utilised in the preceding sectionsof this manuscript [100(nm) to 900(nm)] govern application limit criteria.

3 ConclusionThe original derivation of the relationship now termed the CF was formulated based upon a series of non-

physical assumptions, such as infinitely large plate area at absolute zero temperature. Due to this specific assumption, we

have re-interpreted Eq. (1) as representing an average Cosmological quantity. This re-interpretation is qualitatively

supported by the fact that, without substantial relaxation (i.e. from thermal influences etc.), the historically derived form

does not constitute a definitive outcome by experimental validation. To date, precise measurements of the CF have been

executed incorporating substantial geometric departures from the ideal configuration (i.e. plate-sphere rather than plate-plate). Moreover, the experimental error associated with a plate-plate configuration is as high as 15(%). Perhaps more

harshly, from an engineering perspective, it is tempting to describe Eq. (1) as being poorly formulated due to its

inability to predict a verifiable result without significant deviation or correction. The core concepts formulated by re-

interpreting the historically defined CF are:

o It denotes an Average Cosmological Value (ACV).o Terrestrial measurements only approach the ACV after substantial correction; e.g. temperature etc.o An experimental tolerance of 15(%) envelopes sufficient latitude to warrant the search for physical

influences beyond the traditional; e.g. gravitational acceleration.

The key conclusions derived herein for Parallel-Plate configurations of 100 r(nm) 900 are:o The local value of g affects the CF.o The minimum permissible value of r is approximately 4(nm).o The maximum permissible value of r is approximately 10 Earth Radii.

Therefore, application of the EGM Photon radiation method to the CE, suggests that the experimentally verified

(terrestrially) neutrally charged Parallel-Plate configuration force, may differ within extraterrestrial gravitationalenvironments from the gravitationally independent formulation by Casimir. Consequently, the derivation presented

herein implies that a gravitationally dependent CE may become an important design factor in nanotechnology for

extraterrestrial applications (ignoring finite conductivity + temperature effects and evading the requirement for Casimir

Force corrections due to surface roughness).


30/35

Appendix A

Definitions and NomenclatureSymbol Description Definition or Units

a Acceleration (m/s2)

ACV Average Cosmological Value N/A

APP Parallel-Plate Area The projected area of a Parallel-Plate Casimir experimentBPT Buckingham Theory N/Ac,c Speed of light in a vacuum Definition: http://physics.nist.gov/cuu/Constants/

CCF(r) Terr. Casimir Calib. Factor DimensionlessCE, CF, CP Casimir Effect, Casimir

Force, Casimir Pressure

The effect associated with the quantum force / pressure of

attraction between two neutrally charged parallel

conducting plates

Cg(1,r,M) EGM Amplitude The amplitude of the 1st Harmonic associated with

g(r,M,t)

Cg(n0,r,M) EGM Amplitude Spectrum The amplitude spectrum associated with g(r,M,t)

CM(1,M) EGM Amplitude The amplitude of the 1st Harmonic associated with

M(M,t)

CM(n0,M) EGM Amplitude Spectrum The amplitude spectrum associated with M(M,t)CPPV(r,r,rX,M) Parallel-Plate CP (Pa)

CPV(nPV,r,M) PV Amplitude The amplitude spectrum associated with the PV

C(n0) EGM Amplitude Spectrum The amplitude spectrum associated with (t)

C(n0) EGM Amplitude The amplitude of the 1st Harmonic associated with

(t)

DATs Dim. Analysis Techniques N/A

E,E,Eh,Eh Energy, Planck Energy (J)

E0,B0 EM Amplitude Amplitude of the applied Electric and Magnetic forcing

functions respectively

EA,BA Applied EM Fields Applied electric and magnetic fields respectively

EGM Electro-Gravi-Magnetics A theoretical relationship between EM fields and

gravitational acceleration g

EGMS EGM Spectrum The discrete and harmonically continuous energy spectrum

associated with matter, obeying a Fourier distribution

EM ElectroMagnetic N/A

Erms,Brms EM Root-Mean-Square Root-Mean-Square values of E0,B0

FPPParallel-Plate CF (N)

FPV(r,r,M)g Gravitational Acceleration (m/s

2)

G,G Newtonian Grav. Constant 6.67428 x10-

(m kg-

s-

)

GMEs General Modelling Equations (m/s2)

GR General Relativity Isomorphic to the PV Model of Gravity

GSEs General Similarity Equations Dimensionless

GSE3 GSE 3 The 3rd

member from the family of GSEs

Gyr Giga-Year 10 Years = 1 Billion Yearsg(r,M,t) Grav. Harmonic Operator Scales M(M,t) by G/r

2

h,h Plancks Constant 6.62606896 x10-34

(Js)

HSEs Harmonic Similarity Eqn.s N/A

HSE3 HSE 3 The 3rd

member from the family of HSEs

HSE3 R Reduced form of HSE 3 The reduced form of the 3rd

member from the family of

HSEs

K0(,r,E,B,X) Constant DimensionlessKC(r,M) Critical Factor (Pa)


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KPV Refractive Index The Refractive Index at a point in the PV

KR(nPV)H Harmonic Amp. Spectrum The harmonic amplitude spectrum associated with

KR(r,r,M)KR(r,r,M) Critical Ratio Dimensionless

KU(r,r,rX,M) EGM Application Tolerance (%)M,m,mh,mh Mass, Planck Mass (kg)

M(M,t) Massive Harmonic Operator Scales (t) by mass M

N Odd Harmonic Limit Nth

Harmonic Limit of the n0,nA,nPV Odd Harmonic

Distribution;

Note: n0,nA,nPV = -21 is a shorthand referenceto the Odd Harmonic Limit N = +21 of the

n0,nPV Odd Harmonic Distribution (example

only)

n0,nA,nPV Odd Harmonic Distribution n0,nA,nPV

the extraterrestrial casimir effect

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