Embed Size (px)
Citation preview
i
SuperconductivityThe Structure Scale of the Universe
Seventh Edition
August 01, 2005
Richard D. Saam P.E.
Proteus Systems Inc.
525 Louisiana Ave
Corpus Christi, Texas 78404 USA
e-mail: [email protected]
iiABSTRACT
A theoretical framework supported by experimental evidence is presented which indicatesthat superconductivity is a relativistic phenomenon and congruent with the concept of aBose Einstein Condensate and Charge Conjugation, Parity Change and Time Reversal(CPT) theorem. A momentum and energy conserving (elastic) CPT lattice and associatedsuperconducting theory is postulated whereby electromagnetic and gravitational forces aremediated by a particle of relativistic velocity transformed mass (mt) (110.123 x electronmass) and unit charge congruent with Heisenberg Uncertainty, such that the establishedelectron/proton mass is maintained, electron and proton charge is maintained and theuniverse radius is 2.25E28 cm, the universe mass is 3.02E56 gram, the universe density is6.38E-30 g/cm3or 2/3 the critical density. The universe time or age is 1.37E10 years and theuniverse Hubble constant is 2.31E-18/sec (71.2 km/sec-million parsec). The calculateduniverse mass and density are based on an isotropic homogeneous media filling the vacuumof space analogous to the 'aether' referred to in the 19th century (but still in conformancewith Einstein's relativity theory) and could be considered a candidate for the 'darkmatter/energy' in present universe theories. Each particle of mass mt in the proposed darkmatter has a volume of 15733 cm3. In this context the universe cosmic microwavebackground radiation (CMBR) black body temperature is linked to universe darkmatter/energy superconducting temperature. The model predicts an acceleration value withobserved universe expansion and consistent with observed Pioneer 10 &11 deep spacetranslational and rotational deceleration. Also, a reasonable value for the cosmologicalconstant is derived having dimensions of the known universe. Also, a YBCOsuperconductor should loose .04% of its weight while 100% in superconducting modewhich is close to reported results by Podkletnov and Nieminen (.05%). All of this is placedin context of the virial theorem. Also this trisine model predicts 1, 2 and 3 dimensionalsuperconductors which has been verified by observed magnesium diboride (MgB2)superconductor critical data. Also dimensional guidelines are provided for design of roomtemperature superconductors and beyond and which are related to practical goals such asfabricating a superconductor with the energy content equivalent to the combustion energyof gasoline. These dimensional guidelines have been experimentally verified by Homes’Law and generally fit the parameters of a superconductor operating in the “dirty” limit.
iii
PREVIOUS PUBLICATION
First Edition October 15, 1996http://xxx.lanl.gov/abs/physics/9705007Title: Superconductivity, the Structure Scale of the UniverseAuthor: Richard D. SaamComments: 61 pages, 27 references, 11 figures(.pdf available on request) from [email protected]: General Physics
Second Edition May 1, 1999http://xxx.lanl.gov/abs/physics/9905007, version 6Title: Superconductivity, the Structure Scale of the UniverseAuthor: Richard D. SaamComments: 61 pages, 27 references, 16 tables, 16 figures,Subj-class: General Physics
Third Edition February 23, 2002http://xxx.lanl.gov/abs/physics/9905007 , version 7Title: Superconductivity, the Structure Scale of the UniverseAuthor: Richard D. SaamComments: 66 pages, 50 references, 27 tables, 17 figuresSubj-class: General Physics
Fourth Edition February 23, 2005http://xxx.lanl.gov/abs/physics/9905007 , version 9Title: Superconductivity, the Structure Scale of the UniverseAuthor: Richard D. SaamComments: 46 columnated pages, 21 figures, 17 tables, 192 equations, 58 referencesSubj-class: General Physics
Fifth Edition April 15, 2005http://xxx.lanl.gov/abs/physics/9905007 , version 11Title: Superconductivity, the Structure Scale of the UniverseAuthor: Richard D. SaamComments: 46 columnated pages, 21 figures, 17 tables, 192 equations, 58 referencesSubj-class: General Physics
Sixth Edition June 01, 2005http://xxx.lanl.gov/abs/physics/9905007 , version 13Title: Superconductivity, the Structure Scale of the UniverseAuthor: Richard D. SaamComments: 47 columnated pages, 21 figures, 17 tables, 192 equations, 63 referencesSubj-class: General Physics
Copyright © 1996, 1999, 2002, 2005 Proteus Systems Inc.
All rights reserved.
Reference of this material is encouraged.
iv
TABLE OF CONTENTS
1. INTRODUCTION
2. TRISINE MODEL DEVELOPMENT
2.1 DEFINING MODEL RELATIONSHIPS
2.2 TRISINE GEOMETRY
2.3 TRISINE CHARACTERISTIC WAVE VECTORS
2.4 TRISINE CHARACTERISTIC DE BROGLIE VELOCITIES AND SAGNAC RELATIONSHIP
2.5 SUPERCONDUCTOR DIELECTRIC CONSTANT AND MAGNETIC PERMEABILITY
2.6 FLUXOID AND CRITICAL FIELDS
2.7 SUPERCONDUCTOR INTERNAL PRESSURE, CASIMIR FORCE AND DEEP SPACE DRAG FORCE(PIONEER 10 & 11
2.8 SUPERCONDUCTING CURRENT, VOLTAGE AND CONDUCTANCE (AND HOMES’ LAW)
2.9 SUPERCONDUCTOR WEIGHT REDUCTION IN A GRAVITATIONAL FIELD
2.10 BCS VERIFYING CONSTANTS
2.11 SUPERCONDUCTING COSMOLOGICAL CONSTANT
2.12 SUPERCONDUCTING VARIANCE AT T/Tc
2.13 SUPERCONDUCTOR ENERGY CONTENT
2.14 SUPERCONDUCTOR GRAVITATIONAL ENERGY
3. DISCUSSION
4. CONCLUSION
5. VARIABLE AND CONSTANT DEFINITIONS
APPENDICES
A. TRISINE NUMBER AND B/A RATIO DERIVATION
B. DEBYE MODEL NORMAL AND TRISINE RECIPROCAL LATTICE WAVE VECTORS
C. ONE DIMENSION AND TRISINE DENSITY OF STATES
D. GINZBURG-LANDAU EQUATION AND TRISINE STRUCTURE RELATIONSHIP
E. LORENTZ EINSTEIN TRANSFORMATION CONSEQUENCE
6. REFERENCES
1
1. INTRODUCTION
Trisine structure is a model for superconductivity based onGaussian surfaces surrounding superconducting Cooper CPTCharge conjugated pairs. This Gaussian surface has the sameconfiguration as a particular matrix geometry as defined inreference [1], and essentially consists of mirror image nonparallel surface pairs. The main purpose of the lattice has andis to control the flow of particles suspended in a fluid streamand generally has been fabricated on a macroscopic scale. Inthis particular configuration, a lattice is described whereinthere is a perfect elasticity to fluid flow. In other words thereis 100 percent conservation of energy and momentum whichby definition we will assume to describe the phenomenon ofsuperconductivity.
The superconducting model presented herein is a logicaltranslation of this geometry in terms of classical and quantumtheory to provide a basis for explaining aspects of thesuperconducting phenomenon and integration of thisphenomenon with gravitational forces within the context ofreferences[1-63].
This approach is an attempt to articulate a geometrical modelfor superconductivity in order to anticipate dimensionalregimes in which one could look for higher performancematerials. This approach does not address specific particlessuch as polarons, exitons, magnons, plasmons which may benecessary for the expression of superconductivity in a givenmaterial. Indeed, the Trisine structure or lattice as presentedin this report may be expressed in terms of something moreprofound and elementary and that is the Charge ConjugateParity Change Time Reversal (CPT) Theorem.
Superconductivity is keyed to a critical temperature Tc( )which is representative of the energy at which thesuperconductive property takes place in a material. Thematerial that is characterized as a superconductor has thisproperty at all temperatures below the critical temperature.In terms of the trisine model, any critical temperature can beinserted, but for the purposes of this report, six criticaltemperatures are selected as follows:
1.1 Tc 8.11E-16 o K Tb 2.729 o K
The background temperature of the Universerepresented by the black body cosmic microwavebackground radiation (CMBR) of 2.728 ± .004 o Kas detected in all directions of space in 60 - 630 GHzfrequency range by the FIRAS instrument in the Cobesatellite and 2.730 ± .014 o K as measured at 10.7GHz from a balloon platform [26]. The most recent
NASA estimation of CMBR temperature is 2.725 ±.002 K. The particular temperature of 2.729 o Kindicated above which is essentially equal to theCMBR temperature was established by the trisinemodel wherein the superconducting Cooper CPTCharge conjugated pair density (see table 2.9.1)(6.38E-30 g/cm3) equals the density of the universe(see equation 2.11.5).
1.2 Tc 93 o K Tb 7.05E8 o K
The critical temperature of the YBa Cu O x2 3 7−
superconductor as discovered by Paul Chu at theUniversity of Houston and which was the basis forthe gravitational shielding effect of .05% as observedby Podkletnov and Nieminen(see Table 2.9.1).
1.3 Tc 447 o K Tb 1.55E9 o K
The critical temperature of an anticipated roomtemperature superconductor which is calculated to be1.5 x 298 degree Kelvin such that thesuperconducting material may have substantialsuperconducting properties at an operatingtemperature of 298 degree Kelvin.
1.4 Tc 1190 o K Tb 2.52E9 o K
The critical temperature of an anticipated materialwhich would have an energy content equivalent to thecombustion energy content of gasoline(see Table2.13.1).
1.5 Tc 2,135 o K Tb 3.23E9 o K
The critical temperature of an anticipatedsuperconducting material with the assumed density(6.39 g/cm3 ) [17] of YBa Cu O x2 3 7− which wouldgenerate its own gravity to the extent that it would beweightless in earth's gravitational field (See Table2.9.1).
1.6 Tc 3,100,000 o K Tb 1.29E11 o K
The upper bounds of the superconductivephenomenon as dictated by relativistic effectswherein material dielectric equals one (See Table2.5.1).
As a general comment, a superconductor should be operated atabout 2/3 of Tc( ) for maximum effect of most propertieswhich are usually expressed as when extrapolated to T = 0
2(see section 2.12). In this light, critical temperatures Tc( ) initems 1.4 & 1.5 could be operated up to 2/3 of the indicatedtemperature (793 and 1300 degree Kelvin respectively) andstill retain superconducting properties characteristic of theindicated Tc( ) . Also, the present development is for onedimensional superconductivity. Consideration should begiven to 2 and 3 dimensional superconductivity by 2 or 3multiplier to Tc ‘s contained herein interpret experimentalresults.
2. TRISINE MODEL DEVELOPMENT
In this report, mathematical relationships which link trisinegeometry and superconducting theory are developed and thennumerical values are presented in accordance with theserelationships. Centimeter, gram and second(cgs) as well asKelvin temperature units are used unless otherwise specified.The actual model is developed in spread sheet format with allmodel equations computed simultaneously and interactively asrequired.
2.1 D E F I N I N G M O D E LRELATIONSHIPS
The four defining model equations are presented as 2.1.1 -2.1.4 below:
d Area Cooper e( ) = ( )∫∫ ±4π (2.1.1)
2
12
m k T KKt b ctrisine
Eule
= ( )−
=
e
e rr
B
KK
trisine
K
π( )
( )=
sinh
2
(2.1.2)
trisineD t
=∈( ) ⋅1
V(2.1.3)
f
m
m
e
m vcavity Ht
e ec
⎛⎝⎜
⎞⎠⎟
=±
2 ε
(2.1.4)
Equation 2.1.1 is a representation of Gauss's law with a
Cooper CPT Charge conjugated pair charge contained withina bounding surface area. Equation 2.1.2 and 2.1.3 define thesuperconducting model as developed by Bardeen, Cooper andSchrieffer in references [2] and Kittel in in references [4].
The objective is then to select a particular
wave vector or KK( ) ( )2 ω based on trisine geometry (see
figures 2.2.1-2.2.5) that fits the model relationship asindicated in equation 2.1.2 with KB
2 being defined as thet r i s ine wave vector or KK( ) ( )2
associa ted wi thsuperconductivity of Cooper CPT Charge conjugated pairsthrough the lattice. This procedure establishes the trisinegeometrical dimensions, then equation 2.1.4 is used toestablish an effective mass function f m mt e( ) of the particlesin order for particles to flip in spin when going from trisinecavity to cavity while in thermodynamic equilibrium withcritical field Hc . As a check on this method, conservation ofmomentum p and energyΕ are assumed such that:
∆ ∆ =
+ +
=∑ p x
K K K K
n nn 1 2 3 4
1 2 3 4
0, , ,
(conservation of momentum)
(2.1.5)
∆ ∆ =
+ +
=∑ Εn n
n
t
K K K K K K K K
1 2 3 4
1 1 2 2 3 3 4 4
0, , ,
(consservation of energy)
(2.1.6)
These conditions of conservation of momentum and energyprovide the necessary condition of perfect elasticity whichmust exist in the superconducting state. In addition, thesuperconducting state may be viewed as boiling of states∆ ∆ ∆ ∆Εn n n nt p x, , , on top of the zero point state in acoordinated manner.
The trisine symmetry (as visually seen in figures 2.2.1-2.2.5)allows movement of Cooper CPT Charge conjugated pairswith center of mass wave vector equal to zero as required bysuperconductor theory as presented in references [2, 4].
Wave vectors K( ) as applied to trisine are assumed to be freeparticles (ones moving in the absence of any potential field)and are solutions to the one dimensional Schrödinger equation[3]:
d
ds
mEnergyt
2
2 2
20
ψ ψ+ = (2.1.7)
3The well known solution to the one dimensional Schrödingerequation is in terms of a wave function ψ( ) is:
ψ = ⎛⎝⎜
⎞⎠⎟
⎛
⎝⎜⎜
⎞
⎠⎟⎟
Amplitudem
Energy st sin2
2
1
2
= ⋅( )Amplitude K ssin
(2.1.8)
Wave vector solutions KK( ) are constrained to fixed trisinecell boundaries such that energy eigen values are establishedby the condition s S= = and ψ 0 such that
22
1
2mEnergy S K S nt⎛
⎝⎜⎞⎠⎟
= = π(2.1.9)
Energy eigen values are then described in terms of what isgenerally called the particle in the box relationship as follows:
Energy
n
m St
= ⎛⎝⎜
⎞⎠⎟
2 2 2
2
π (2.1.10)
For our model development, we assume the quantum numbern = 2 and this quantum number is incorporated into the wavevectors KK( ) as presented as follows:
Energym S
m S
t
t
= ⎛⎝⎜
⎞⎠⎟
=
2
2
2
2
2 2 2
2
π
π
⎛⎛⎝⎜
⎞⎠⎟
= ( )
2
2
2
mKK
t
(2.1.11)
Trisine geometry is described in Figure 1-5 is in generalcharacterized by relativistic dimensions A and B with acharacteristic ratio B/A of 2.37933 and correspondingcharacteristic angle θ( ) where:
θ = ⎛⎝⎜
⎞⎠⎟
= ⎛⎝⎜
⎞⎠⎟
=− −tan tan .1 1 1 2
322 80
A
Bo
e(2.1.12)
The superconducting model is described with equations 2.1.13- 2.1.16, with variable definitions described in the remainingequations. The model essentially falls within the concept of aBose Einstein Condensate (BEC). The model convergesaround a particular relativistic velocity transformed mass
mt( ) (110.123 x electron mass me( ) ) and dimensionalratio B A/( ) of 2.37855.
k T
K
m
K K
m
K
b c
B
t
C Ds
ttrisine
C
=
+( )−
2 2
2 2 2
2
2
2
1
1e
22 2
2
2
+( )( )
+
K
m trisine
m m
m m
Ds
t
Euler
e t
e t
e
π sinh
22
1
8
2 2
2
Bm m
m mA
g
cavityDE
t p
t p
s
( ) ++
( )
⎛⎝⎜
⎞⎠⎟
π
11
2
3
2
g
chain
cavity
e
B
time
s
±
⎧
⎨
⎪⎪⎪
ε
π
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪
⎫
⎬
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪
⎭
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪
= =
m vt dx
2
2
11
2ω
(2.1.13)
BCS theory is adhered to as verified by equation 2.1.13, arrayelement 3 in conjunction with equations A.3, A.4, and A.5 asderived in Appendix A. Also, Bose Einstein Condensatecriteria is satisfied as:
cavityh
m k Tt b
1
3
2=
π
(2.1.13a)
trisineK K
K K
D
B P
P B
=+
=∈( )
2 2
1
V
e= − −⎛⎝⎜
⎞⎠⎟
ln2
1π
Euler
(2.1.14)
4
density of states Dcavity m K
Tt C
∈( ) =3
4 3 2π
=1
k Tb c
(2.1.15)
The attractive Cooper CPT Charge conjugated pair energy V( )is:
VK K
K Kk TP B
B Pb c=
+2 2 (2.1.16)
Table 2.1.1 with Density of States (/erg) Plot
T Kc
0( ) 3,100,000 2,135 1,190 447 93 8.11E-16
T Kb
0( ) 1.29E+11 3.38E+09 2.52E+09 1.55E+09 7.05E+08 2.729
V(erg)
2.12E-10 1.46E-13 8.13E-14 3.05E-14 6.35E-15 5.54E-32
D ∈( )(/erg)
2.34E+09 3.39E+12 6.09E+12 1.62E+13 7.79E+13 8.93E+30
1E+11
1E+12
1E+13
1E+14
1E+15
1E+16
1E+17
1 10 100 1000 10000
Critical Temperature Tc (K)
Den
sity
of
Stat
es (
/erg
)
The equation 2.1.13 array element 2 and 3 along with
equations 2.1.14, 2.1.15, 2.1.16, 2.1.17 and 2.1.18 are
consistent with the BCS weak link relationship:
k T
e
mK K eb c
Euler
tC Ds
D V= +( )−∈( )2
2
22 2
1
π(2.1.17)
k T
g
chain
cavity m
Kb c
s t
A=⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
1
2
2 2 ⎞⎞⎠⎟
−∈( )eD V
1(2.1.18)
The relationship in equation 2.1.19 is based on the Bohrmagneton e m ve2 ε (which has dimensions of magnetic fieldx volume) with a material dielectric ε( ) modified speed oflight vε( ) and establishes the spin flip/flop as Cooper CPTCharge conjugated pairs go from one cavity to the next(seefigure 2.3.1). The 1/2 spin factor and electron gyromagneticfactor ge( ) and superconducting gyro factor gs( ) arem u l t i p l i e r s o f t h e B o h r m a g n e t o nwhere g m m m m m ms t t e p p e= −( )( ) −( )( ) as related to the
Dirac Number by (3/4)(gs-1)=2 (g -1)dπ . Also becauseparticles must advance in pairs, a symmetry is created forsuperconducting current to advance as bosons.
1
2 2g g
m
m
e
m vcavity He s
t
e ec
± =ε
(2.1.19)
Proton to electron mass ratio m mp e( ) is verified through theWerner Heisenberg Uncertainty Principle volume∆ ∆ ∆x y z⋅ ⋅( ) as follows:
∆
∆
∆
∆
∆
∆
x
y
z
p
p
p
x
y
z
⎧
⎨
⎪⎪⎪
⎩
⎪⎪⎪
⎫
⎬
⎪⎪⎪
⎭
⎪⎪⎪
=
2
1
2
1
2
1
⎧⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
⎫
⎬
⎪⎪⎪⎪⎪
⎭
⎪⎪⎪⎪⎪
=
( )2
1
2
1
∆
∆
K
K
B
PP
AK
B
( )
( )
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
⎫
⎬
⎪⎪⎪⎪⎪
⎭
⎪⎪⎪⎪⎪2
1
2
1
∆
∆ππ
π
π
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪∆
∆
3
2 3
1
2
2
1
2
B
A
⎪⎪⎪⎪⎪
⎫
⎬
⎪⎪⎪⎪⎪
⎭
⎪⎪⎪⎪⎪
(2.1.20)
∆ ∆ ∆x y zK K KB P A
⋅ ⋅( ) = ( )( )( )1
2 2 2(2.1.21)
m
m
m m
m m
cavity
x y z
m m
mp
e
t e
t e
t e
e
=−+ ( ) −
+∆ ∆ ∆
2 (2.1.22)
Equations 2.1.20 - 2.1.22 provides us with the confidence( ∆ ∆ ∆x y z⋅ ⋅ is well contained within the cavity ) that we canproceed in a semi-classical manner outside the envelop of theuncertainty principle with the relativistic velocity transformedmass mt( ) and De Broglie velocities vd( ) used in this trisinesuperconductor model or in other words equation 2.1.23 holds.
22
2
1
2
KK
mm v
tt d
( )⎛⎝⎜
⎞⎠⎟
= ⎛⎝⎜
⎞⎠⎟
=( )m v
mt d
t
2
(2.1.23)
The Meissner condition as defined by total magnetic fieldexclusion from trisine chain as indicated by diamagneticsusceptibility Χ( ) = −1 4π , is adhered to by the followingequation:
Χ = − = −
k T
chain Hb c
c2
1
4π(2.1.24)
Equation 2.1.25 provides a representation of the relativistic
5velocity transformed mass mt( )
m v
c v
c
mv
c
t d
d
r
d
2
2 2
2
2
2
2
1
1 1
1
1 1− −=
− −
= mm
where v ct
d <<
(2.1.25)
Table 2.1.2 with Wave Length (cm) Plot
T Kc
0( ) 3,100,000 2,135 1,190 447 93 8.11E-16
T Kb
0( ) 1.29E+11 3.38E+09 2.52E+09 1.55E+09 7.05E+08 2.729
mt (g) 1.00E-25 1.00E-25 1.00E-25 1.00E-25 1.00E-25 1.00E-25
mr (g) 4.76E-31 3.28E-34 1.83E-34 6.87E-35 1.43E-35 1.25E-52
Frequency(Hz)
8.12E+17 5.59E+14 3.12E+14 1.17E+14 2.44E+13 2.12E-04
Wavelength(cm)
2.32E-07 3.37E-04 6.05E-04 1.61E-03 7.74E-03 8.87E+14
1E-05
1E-04
1E-03
1E-02
1E-01
1E+00
1 10 100 1000 10000
Critical Temperature Tc (K)
Wav
e L
engt
h (c
m)
Equation 2.1.25 is in conformance with the conventionallyheld principle that the photon rest mass equals zero.
The relativistic nature of the superconducting phenomenon isagain demonstrated by equation 2.1.26 which relates theclassical electron radius Rce( ) (2.818E-13 cm) to a relativisticlength we shall call B. The velocity vε( ) is a speed of lightdependent on the permittivity and permeability properties ofthe particular medium studied.
B Rm
m v
c
e
m c
m
m
cet
e
e
t
e
= ⋅ ⋅ ⋅− −
= ⋅ ⋅ ⋅−
61
1 1
61
1
2
2
2
2
ε
112
2−
<<
v
cwhere v c
ε
ε
(2.1.26)
Also a relativistic CPT time± is defined in terms of the
classical electron radius Rce( ) divided by the speed of lightc( ) or time± interval of 9.400E-24 sec.
timeR
c v
c
e
m c c
e
d
e
= ⋅ ⋅− −
= ⋅ ⋅−
±
41
1 1
41 1
1
2
2
2
2
112
2−
<<
v
cwhere v c
d
d
(2.1.27)
Where vd( ) is the De Broglie velocity as in equation 2.1.28and also related to relativistic CPT time± by equation 2.1.29.
v
m Bdt
=π (2.1.28)
timeB
vd
=⋅2 (2.1.29)
Also a relationship among zero point base energy ω 2( ) , DeBroglie velocity, relativistic CPT time±, speed of light andrelativistic mass is presented in equation 2.1.30.
energy mv dv
v
c
m cv
ctd d
d
td=
−= − −
⎛
⎝⎜
⎞
⎠⎟∫
1
1 12
2
22
2
=
m cv
c
ti
t
d
2
2
2
1
2
1
1 1
1
2
2
⋅ ⋅ ⋅− −
= ⋅ ⋅
ω
πmme v
c
m vv
c
d
t d
d
⋅− −
= ⋅ ⋅ ⋅− −
1
1 1
1
2
1
1 1
2
2
2
2
2
where v cd <<
(2.1.30)
Conceptually, it is though the electron radius and time± werespread out or dilated to the trisine dimensions. LorentzEinstein transformations are in accordance with Appendix E.
The relationship between vdx( ) and v xε( ) is per thegroup/phase velocity relationship presented in equation 2.1.31.
6( ) ( )
( )
group velocity phase velocity c
vdx
⋅ =
⋅
2
1
ππε εv
vc v
B
A
v
vcx
dxdx
x
dx
2
2 1
4
2
2
1
3
⋅⎛⎝⎜
⎞⎠⎟
= ⋅ ⋅
⎛
⎝
( )⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟
= c2
(2.1.31)
2.2 TRISINE GEOMETRY
The characteristic trisine relativistic dimensions A and B areindicated as a function of temperature in table 2.2.1 ascomputed from the equation 2.2.1
k T
K
m
K
m
B
Ab cB
t
A
t
= = ⎛⎝⎜
⎞⎠⎟
2 2 2 2 2
2 2(2.2.1)
Trisine Geometry is subject to Charge conjugation Paritychange Time reversal (CPT) -symmetry which is afundamental symmetry of physical laws under transformationsthat involve the inversions of charge , parity and timesimultaneously.
Table 2.2.1 with B vs Tc plot
T Kc
0( ) 3,100,000 2,135 1,190 447 93 8.11E-16
T Kb
0( ) 1.29E+11 3.38E+09 2.52E+09 1.55E+09 7.05E+08 2.729
A (cm) 1.50E-10 5.73E-09 7.67E-09 1.25E-08 2.74E-08 9.29
B (cm) 3.58E-10 1.36E-08 1.82E-08 2.98E-08 6.53E-08 22.11
1E-09
1E-08
1E-07
1E-06
1 10 100 1000 10000
Critical Temperature Tc (K)
Len
gth
B (
cm)
Ae
m c v
c
e
time
Euler
e y
=
− −
=⋅
±
±
±
2 3 1
1 1
2
2 2
2
e
πε
HH v
m
mc x
t
e⋅⎛⎝⎜
⎞⎠⎟ε
Biot-Savart
eqn 2.6.3
(2.2.2)
B ge
m c v
c
se x
=− −
±6 1
1 1
2
2 2
2
ε
(2.2.3)
Figure 2.2.1 Trisine Steric (Mirror Image) Parity Forms
Rotation
0 Degree
120 Degree
240 DegreeA
2A
2B3
2B3
A2 + B
2 = C2
B
CA
Steric FormOne
Steric FormTwo
z
Trisine characteristic volumes with variable names cavity andchain as well as characteristic areas with variable namessection, approach and side are defined in equations 2.2.2 -2.2.6. See figures 2.2.1 - 2.2.4 for a visual description of theseparameters. The mirror image forms in figure 2.2.1 are inconformance with parity requirement in Charge conjugationParity change Time reversal (CPT) theorem as established bydeterminant identity in equation 2.2.3a.
−
−
−
= − −
−
B
B B
A A
B
B B
A A
0 0
03 3
20
2
0 0
03 3
20
2
(2.2.3a)
cavity AB= 2 3 2 (2.2.4)
chain cavity=2
3(2.2.5)
The section is the trisine cell cavity projection on to the x, yplane as indicated in figure 2.2.2.
section B= 2 3 2 (2.2.6)
7
Figure 2.2.2 Trisine Cavity and Chain Geometry fromSteric (Mirror Image) Forms
2B3
A2 + B
2 = C2
B
CA
y2B
x
Indicates Trisine Peak
Cell Section2B
3
B
The approach is the trisine cell cavity projection on to the y, zplane as indicated in figure 2.2.3.
approachB
A AB= =1
2
3
32 3 (2.2.7)
Figure 2.2.3 Trisine approach from both cavityapproaches
The side is the trisine cell cavity projection on to the x, z planeas indicated in figure 2.2.4.
side A B AB= =1
22 2 2 (2.2.8)
Figure 2.2.4 Trisine side view from both cavity sides
Table 2.2.2 with cavity and section plots
T Kc
0( ) 3,100,000 2,135 1,190 447 93 8.11E-16
T Kb
0( ) 1.29E+11 3.38E+09 2.52E+09 1.55E+09 7.05E+08 2.729
Cavity
(cm3)6.65E-29 3.68E-24 8.85E-24 3.84E-23 4.05E-22 15733
chain
(cm3)4.44E-29 2.45E-24 5.90E-24 2.56E-23 2.70E-22 10489
section
(cm2)4.43E-19 6.43E-16 1.15E-15 3.07E-15 1.48E-14 1693
approach
(cm2)9.31E-20 1.35E-16 2.42E-16 6.45E-16 3.10E-15 356
side
(cm2)1.07E-19 1.56E-16 2.80E-16 7.45E-16 3.58E-15 411
1E-251E-241E-23
1E-221E-211E-201E-191E-18
1 10 100 1000 10000
Critical Temperature Tc (K)
Cav
ity (
cm^3
)
1E-16
1E-15
1E-14
1E-13
1E-12
1E-11
1 10 100 1000 10000
Critical Temperature Tc (K)
Sect
ion
(cm
^2)
8
Figure 2.2.5 Trisine Geometry
2B3
2B3
2B
3
B2B
Indicates Trisine Peak Each Peak is raised by height A
B
C
A yx A
2 + B2 = C
2
2.3 TRISINE CHARACTERISTIC WAVE VECTORS
Superconducting model trisine characteristic wave vectorsK K KDn Ds C, and are defined in equations 2.3.1 - 2.3.3 below.
KcavityDn =
⎛⎝⎜
⎞⎠⎟
6 21
3π (2.3.1)
The relationship in equation 2.3.1 represents the normalDebye wave vector KDn( ) assuming the cavity volumeconforming to a sphere in K space and also defined in terms ofK K KA P B, and as defined in equations 2.3.4, 2.3.5 and 2.3.6.
See Appendix B for the derivation of equation 2.3.1.
Kcavity
K K KDs A B P=⎛⎝⎜
⎞⎠⎟
= ( )8 31
3 1
3π
(2.3.2)
The relationship in equation 2.3.2 represents the normalDebye wave vector KDs( ) assuming the cavity volumeconforming to a characteristic trisine cell in K space.
See Appendix B for the derivation of equation 2.3.2.
KA
C =4
3 3
π (2.3.3)
The relationship in equation 2.3.3 represents the result ofequating one dimensional and trisine density of states.
See Appendix C for the derivation of equation 2.3.3.
Equations 2.3.4, 2.3.5, and 2.3.6 translate the trisine wavevectors KC , KDs , and KDn into x, y, z Cartesian coordinates asrepresented by KB , KP , and KA respectively. Thesuperconducting current is in the x direction.
Kg
K K KBB
sDn Ds C=
⎛⎝⎜
⎞⎠⎟
+( ) − =1 2
2
π (2.3.4)
K g K K KBP s Dn Ds C= −( ) + =
2 3
3
π (2.3.5)
K g K K KAA s Ds Dn C= ( ) + + =4 2π (2.3.6)
Note that the energy relationships of wave vectors are asfollows:
K K K K KA C Ds P B2 2 2 2 2> > > > (2.3.7)
Equation 2.3.8 relates addition of wave function amplitudesB P B3 2 2( ) ( ) ( ), , in terms of superconducting
energy k Tb c( ) .
k Tcavity
chain time
K
B
B
K
P
Pb c
B
P=
+ ⎛⎝⎜
⎞⎠⎟
+±
2 3
2 2
2
⎛⎛⎝⎜
⎞⎠⎟
+ ⎛⎝⎜
⎞⎠⎟
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
2
2
2 2
K
A
AA
(2.3.8)
The wave vector KB , being the lowest energy, is the carrier ofthe superconducting energy and in accordance with derivationin appendix A. All of the other wave vectors are containedwithin the cell cavity.
m m
m m K
m m
m m Kg m
Ke t
e t B
t p
t p As t
C++
+=
1 1 1
6
12 2
22
(2.3.9)
Kg A B
Cs
=+
1 22 2
π (2.3.10)
9The conservation of momentum and energy relationships in2.3.11 and 2.3.12 are dictated and verified by the B/A ratio of2.37933.
∆ ∆ =
+( ) +
=∑ p x
g K K K K
n nn B C Ds Dn
s B c Ds Dn
, , ,
0
(conservation of momentum)
(2.3.11)
∆ ∆ =
+ +( )
=∑ E t
K K g K K
n nn B C Ds Dn
B C s Ds Dn
, , ,
0
2 2 2 2
(conservation of energy)
(2.3.12)
Figure 2.3.1 Trisine Steric Charge Conjugate PairChange Time Reversal CPT Geometry in theSuperconducting (Bose Einstein Condensate) Mode
Figure 2.3.1 in conjunction with mirror image parity images inFigure 2.2.1 clearly depict the trisine symmetry congruent
with the Charge conjugation Parity change Time reversal(CPT) theorem. Negative and positive charge reversal as wellas time reversal take place in mirror image parity pairs.
Table 2.3.1 with KB Plot
T Kc
0( ) 3,100,000 2,135 1,190 447 93 8.11E-16
T Kb
0( ) 1.29E+11 3.38E+09 2.52E+09 1.55E+09 7.05E+08 2.729
KB (/cm) 8.79E+09 2.31E+08 1.72E+08 1.06E+08 4.81E+07 1.42E-01
KDn (/cm) 9.62E+09 2.52E+08 1.88E+08 1.16E+08 5.27E+07 1.56E-01
KDs (/cm) 1.55E+10 4.07E+08 3.04E+08 1.86E+08 8.49E+07 2.51E-01
KC (/cm) 1.61E+10 4.22E+08 3.15E+08 1.93E+08 8.82E+07 2.60E-01
KP (/cm) 1.02E+10 2.67E+08 1.99E+08 1.22E+08 5.56E+07 1.64E-01
KA (/cm) 4.18E+10 1.10E+09 8.19E+08 5.02E+08 2.29E+08 6.76E-01
1E+06
1E+07
1E+08
1E+09
1 10 100 1000 10000
Critical Temperature Tc (K)
KB
(cm
^-1)
Figure 2.3.2 Trisine Steric CPT Geometry in theSuperconducting Mode showing the relationship betweenlayered superconducting planes and wave vectors KB .Layers are so constructed in x, y and z directions makingup a lattice of arbitrary lattice or Gaussian surfacevolume xyz.
2.4 TRISINE CHARACTERISTIC DE BROGLIEVELOCITIES
The Cartesian De Broglie velocities vdx( ) , vdy( ) , and vdz( ) aswell as vdC( ) are computed with the trisine Cooper CPTCharge conjugated pair residence relativistic CPT time± and
10characteristic frequency ω( ) in phase with trisinesuperconducting dimension 2B( ) in equations 2.4.1-2.4.5.
v
K
m
B
timeB
eH
m vg Bdx
B
t
c
es= = = =
±
2
22 6ω
π ε
(2.4.1)
In equation 2.4.1, note that m v eHe cε is the electron spin axisprecession rate in the critical magnetic field Hc( ) . Thiselectron spin rate is in tune with the electron moving with DeBroglie velocity vdx( ) with a CPT residence time± in eachcavity. In other words the Cooper CPT Charge conjugatedpair flip spin twice per cavity and because each Cooper CPTCharge conjugated pair flips simultaneously, the quantum isan 2 1 2( ) or integer which corresponds to a boson.
v
K
mdyP
t
= (2.4.2)
v
K
mdzA
t
= (2.4.3)
v
K
mdCC
t
= (2.4.4)
The vector sum of the x, y, z and C De Broglie velocitycomponents are used to compute a three dimensional helicalor tangential De Broglie velocity vdT( ) as follows:
v v v v
mK K K
dT dx dy dz
tA B P
= + +
= + +
2 2 2
2 2 2
== ( )3
gv
sdCcos θ
(2.4.5)
Table 2.4.1 Listing of De Broglie velocities, trisine cell pairresidence relativistic CPT time± and characteristicfrequency ω( ) as a function of selected criticaltemperature Tc( ) and Debye black body temperature Tb( )along with a time± plot.
T Kc
0( ) 3,100,000 2,135 1,190 447 93 8.11E-16
T Kb
0( ) 1.29E+11 3.38E+09 2.52E+09 1.55E+09 7.05E+08 2.729
vdx (cm/sec) 9.24E+07 2.42E+06 1.81E+06 1.11E+06 5.06E+05 1.49E-03
vdy (cm/sec) 1.07E+08 2.80E+06 2.09E+06 1.28E+06 5.84E+05 1.72E-03
vdz (cm/sec) 4.40E+08 1.15E+07 8.61E+06 5.28E+06 2.41E+06 7.11E-03
vdC (cm/sec) 1.69E+08 4.44E+06 3.31E+06 2.03E+06 9.27E+05 2.74E-03
vdT (cm/sec) 4.62E+08 1.21E+07 9.05E+06 5.54E+06 2.53E+06 7.47E-03
Time± (sec) 7.74E-18 1.12E-14 2.02E-14 5.37E-14 2.58E-13 2.96E+04
ω (/sec) 8.12E+17 5.59E+14 3.12E+14 1.17E+14 2.44E+13 2.12E-04
1E-15
1E-14
1E-13
1E-12
1E-11
1E-10
1 10 100 1000 10000
Critical Temperature Tc (K)
Tim
e (
sec)
Equation 2.4.6 represents a check on the trisine cell residence
relativistic CPT time± as computed from the precession of
each electron in a Cooper CPT Charge conjugated pair underthe influence of the perpendicular critical field Hc( ) . Note
that the precession is based on the electron mass me( ) and not
trisine relativistic velocity transformed mass mt( ) .
timeg
m v
e H Bs
e
c±
±
=1 1
6ε (2.4.6)
Another check on the trisine cell residence relativistic CPTtime± is computed on the position of the Cooper CPT Chargeconjugated pair e( ) particles under the charge influence ofeach other within the dielectric ε( ) as they travel in the ydirection as expressed in equation 2.4.7.
md y
dt
e
yta
2
2
2
2
1= ( )
±Με θcos
(2.4.7)
From figure 2.3.1 it is seen that the Cooper CPT Chargeconjugated pairs travel over distance 3 3 2 3B B−( ) in
11each quarter cycle or relativistic CPT time±/4.
timem
e
B Bt
a±
±
= ( )⎛⎝⎜
⎞⎠⎟
−⎛⎝⎜
⎞4
8
81
3
3
2
32
1
2ε θcos
Μ ⎠⎠⎟
3
2(2.4.8)
where the Madelung constant M y( ) in the y direction iscalculated as:
Mn ny
n
n
=+( ) −
−+( ) +
⎛⎝⎜
⎞⎠⎟=
= ∞
∑31
3 1 2 1
1
3 1 2 10
0.523598679=
(2.4.9)
And the Thomas scattering formula[43] holds in terms of thefollowing equation:
SideK
K
e
M m vB
Ds y t dy
=⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
⎛
⎝⎜⎞
⎠⎟±8
3
2
2
πε
22
(2.4.10)
A sense of the rotational character of the De Broglie vdT( )velocity can be attained by the following energy equation:
1
2
1 1
222 2 2m v
g
chain
cavitym Bt dT
st= ( ) ( )( ) cos θ ω
+1 1
2g
chain
cavitym A
stcos θ( ) ( ))( )2 2ω
(2.4.11)
Equation 2.4.12 provides an expression for the Sagnac time.
timeA B
B time
section
vdx±
±
=+2 2
2
2
6
1π (2.4.12)
2.5 SUPERCONDUCTOR DIELECTRIC CONSTANTAND MAGNETIC PERMEABILITY
The material dielectric is computed by determining adisplacement D( ) (equation 2.5.1) and electric field E( )(equat ion 2 .5 .2) and which es tabl ishes adielectric ε or D E/( ) (equation 2.5.3) and modified speed oflight v cε ε or /( ) (equation 2.5.4). Then thesuperconducting fluxoid Φε( ) (equation 2.5.5) can becalculated.
The electric field(E) is calculated by taking the translationalenergy m vt dx
2 2( ) (which is equivalent to k Tb c ) over a distancewhich is the trisine cell volume to surface ratio. This concept
of the surface to volume ratio is used extensively in fluidmechanics for computing energies of fluids passing throughvarious geometric forms and is called the hydraulic ratio.
Eg
m v section
s
t dx=⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟ ( )
⎛⎝
1
2
22
2
cos θ⎜⎜⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
=
±
1
4 251
2
2
e cavity
m v
et dx.
±±
A
(2.5.1)
Secondarily, the electric field(E) is calculated in terms of theforces m v timet d /( ) exerted by the wave vectors K( ) .
E
gm v
time e
g
st dx
=
( )⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟± ±
101
91
2 cos θ
ss
t dy
s
t
m v
time e
g
m
2
1
21 1
± ±
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
( )
cos θvv
time e
m v
time
dz
t dC
± ±
±
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
( )
1
6
cos θ⎛⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
⎧
⎨
⎪⎪⎪⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪⎪⎪⎪
⎫
⎬
⎪⎪⎪⎪
±
1
e
⎪⎪⎪⎪⎪
⎭
⎪⎪⎪⎪⎪⎪⎪⎪
(2.5.2)
The trisine cell surface is section cos θ( )( ) and the volume isexpressed as cavity. Note that the trisine cavity, althoughbounded by 2 ⋅ ( )( )section / cos θ , still has the passageways forCooper CPT Charge conjugated pairs e( ) to move from cellcavity to cell cavity.
The displacement D( ) as a measure of Gaussian surface
containing free charges e±( ) is computed by taking the
4π( ) solid angle divided by the characteristic trisine area
section cos θ( )( ) with the two(2) charges e±( ) contained
therein and in accordance with Gauss's law as expressed in
equation 2.1.1.
D e
section= ( )
±42
πθ
cos (2.5.3)
Now a dielectric coefficient can be calculated from the electricfield E( ) and displacement field D( ) [61].
ε =D
E(2.5.4)
12Assuming the trisine geometry has the relative magneticpermeability of a vacuum km =( )1 then a modified velocity oflight vε( ) can be computed from the dielectric coefficient ε( )and the speed of light c( ) where ε0( ) = 1.
vc
k
c
m
ε εε
ε= =
0
(2.5.5)
Trisine incident/reflective angle (Figure 2.3.1) of 30 degrees isless than Brewster angle of tan− ( ) =1 1 2 045ε ε assuring totalreflectivity as particles travel from trisine lattice cell to trisinelattice cell.
Now the fluxoid Φε( ) can be computed quantized according to
Cooper e ( ) as experimentally observed in superconductors.
Φε
επ=
±
2 v
e (2.5.6)
Table 2.5.1 with dielectric plot
T Kc
0( ) 3,100,000 2,135 1,190 447 93 8.11E-16
T Kb
0( ) 1.29E+11 3.38E+09 2.52E+09 1.55E+09 7.05E+08 2.729
D (erg/e/cm) 1.26E+10 8.65E+06 4.82E+06 1.81E+06 3.77E+05 3.29E-12
E (erg/e/cm) 1.26E+10 2.28E+05 9.49E+04 2.18E+04 2.07E+03 5.34E-23
E (volt/cm) 3.79E+12 6.84E+07 2.85E+07 6.55E+06 6.22E+05 1.60E-20
ε 1.00 37.95 50.83 82.93 181.82 6.16E+10
vε (cm/sec) 3.00E+10 4.87E+09 4.21E+09 3.29E+09 2.22E+09 1.21E+05
Φε (gauss cm2) 2.07E-07 3.36E-08 2.90E-08 2.27E-08 1.53E-08 8.33E-13
Using the computed dielectric, the energy associated withsuperconductivity can be calculated in terms of the standardCoulomb's law electrostatic relationship e B2 ε ( ) as presentedin equation 2.5.7.
k T
chain
cavity
e
B
gchain
cavitb c s y= ( )
±2
1
ε
θ
cosΜ
yy
e
B
g
chain
cavity
e
As
±
±( )
⎧
⎨
⎪⎪⎪⎪⎪
⎩
2
2
3
1
ε
θε
tan
⎪⎪⎪⎪⎪⎪
⎫
⎬
⎪⎪⎪⎪⎪
⎭
⎪⎪⎪⎪⎪
(2.5.7)
where Μa is the Madelung constant in y direction ascomputed in equation 2.4.9.
A conversion between superconducting temperature and black
body temperature is calculated in equation 2.5.9. Twoprimary energy related factors are involved, the first being thesuperconducting velocity vdx
2( ) to light velocity vε2( ) and the
second is normal Debye wave vector KDn2( ) to
superconducting Debye wave vector KDs2( ) all of this followed
by a minor rotational factor for each factor involving me( )and mp( ) . For reference, a value of 2.71 degrees Kelvin isused for the universe black body temperature Tb( ) as indicatedby experimentally observed microwave radiation by theCosmic Background Explorer (COBE) and later satellites.Although the observed minor fluctuations (1 part in 100,000)in this universe background radiation indicative of clumps ofmatter forming shortly after the big bang, for the purposes ofthis report we will assume that the experimentally observeduniform radiation is indicative of present universe that isisotropic and homogeneous.
Verification of equation 2.5.6 is indicated by the calculationthat superconducting density m cavityt( ) (table 2.9.1) andpresent universe density (equation 2.11.4) are equal at thisDebye black body temperature Tb( ) of 2.71 o K .
T
v
vg Tb
dxs c=
⎛⎝⎜
⎞⎠⎟
12
3ε (2.5.8)
k T
cb b
b
= 2πλ
(2.5.9)
The black body temperatures Tb( ) appear to be high relativeto superconducting Tc( ) , but when the corresponding wavelength λb( ) is calculated in accordance with equation 2.5.7, itis nearly the same and just within the Heisenberg ∆z( )parameter for all Tc and Tb as calculated from equation 2.1.18and presented in table 2.5.2. It is suggested that a black bodyoscillator exists within such a volume as defined by ∆ ∆ ∆x y z( )and is the source of the microwave radiation at 2.71 o K . It isapparent that the high black body temperatures indicated forthe other higher critical superconducting temperatures are notexpressed external to the superconducting material. Could itbe that the reason we observe the 2.71 o K radiation isbecause we as observers are inside the universe as asuperconductor?
13
Table 2.5.2 based on equations 2.1.20 and 2.5.9
T Kc
0( ) 3,100,000 2,135 1,190 447 93 8.11E-16
T Kb
0( ) 1.29E+11 3.38E+09 2.52E+09 1.55E+09 7.05E+08 2.729
∆x (cm) 5.69E-11 2.17E-09 2.90E-09 4.74E-09 1.04E-08 3.52
∆y (cm) 4.93E-11 1.88E-09 2.52E-09 4.10E-09 9.00E-09 3.05
∆z (cm) 1.20E-11 4.56E-10 6.10E-10 9.96E-10 2.18E-09 0.74
λb (cm) 8.58E-12 3.27E-10 4.38E-10 7.15E-10 1.57E-09 0.53
Based on the same approach as presented in equations 2.5.1,2.5.2 and 2.5.3, Cartesian x, y, and z values for electric anddisplacement fields are presented in equations 2.5.10 and2.5.11.
E
E
E
F
e
F
e
F
e
x
y
z
x
y
z
⎫
⎬
⎪⎪⎪
⎭
⎪⎪⎪
=
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪
±
±
±
⎪⎪⎪⎪⎪
⎫
⎬
⎪⎪⎪⎪⎪
⎭
⎪⎪⎪⎪⎪
=
± ±
m v
time e
m v
time
t dx
t dy
1
±± ±
± ±
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
1
1
e
m v
time et dz
(2.5.10)
D
D
D
approache
sidee
s
x
y
z
⎫
⎬
⎪⎪⎪
⎭
⎪⎪⎪
=
±
±
4
4
4
π
π
π
eectione ±
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
(2.5.11)
Now assume that the superconductor material magneticpermeability km( ) is defined as per equation 2.5.12 notingthat k k k km mx my mz= = = .
k
k
k
k
D v
E
E
D
m
mx
my
mz
x x
x
x
x
=
⎧
⎨
⎪⎪⎪
⎩
⎪⎪⎪
⎫
⎬
⎪⎪⎪
⎭
⎪⎪⎪
=
ε2
vv
D v
E
E
D v
D v
E
E
D v
dx
y y
y
y
y dy
z z
z
z
z dz
2
2
2
2
2
ε
ε
⎧
⎨
⎪⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
⎫
⎬
⎪⎪⎪⎪⎪
⎭
⎪⎪⎪⎪⎪
=
v
v
v
v
v
v
x
dx
y
dy
z
ε
ε
ε
2
2
2
2
2
ddz2
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
⎫
⎬
⎪⎪⎪⎪⎪
⎭
⎪⎪⎪⎪⎪
(2.5.12)
Also note that the dielectric ε( ) and permeability km( ) are
related as follows:
km
mmt
e
=⎛⎝⎜
⎞⎠⎟
( )ε θ
3
22cos
(2.5.12a)
Then the De Broglie velocities are defined in terms of thespeed of light then v v vdx dy dz, and as per equations 2.4.1,2.4.2, 2.4.3 can be considered a modified speeds of lightinternal to the superconductor material as per equation 2.5.13.These relativistic De Broglie velocities are presented inequation 2.5.12 with results also presented in terms of wavevectors KA , KB and KP .
v
v
v
c
k
c
k
c
k
dx
dy
dz
m x
m y
m z
⎫
⎬
⎪⎪⎪
⎭
⎪⎪⎪
=
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪
ε
ε
ε
⎪⎪⎪⎪⎪
⎫
⎬
⎪⎪⎪⎪⎪
⎭
⎪⎪⎪⎪⎪
=
⎧
⎨
⎪c
kD
E
c
kD
E
c
kD
E
mx
x
my
y
mz
z
⎪⎪⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪⎪⎪
(2.5.13a)
and
v
v
v
c
kB
A
e m
K
c
kB
A
dx
dy
dz
mt
B
m
⎫
⎬
⎪⎪⎪
⎭
⎪⎪⎪
=
±83
4
2 2
2
2 2
2
2 2
24
3
e m
K
c
ke m
K
t
P
mt
A
±
±
⎧
⎨
⎪⎪⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪⎪⎪⎪
⎫
⎬
⎪⎪⎪⎪⎪⎪⎪
⎭
⎪⎪⎪⎪⎪⎪⎪
=
vB
Ac
K
K
vB
ex
B
P
ey
2
14
2
14
3
3AA
c
K
K
vB
Ac
B
A
ez2
143
⎧
⎨
⎪⎪⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪⎪⎪
(2.5.13b)
Combining equations 2.5.12 and 2.5.13, the Cartesiandielectric velocities can be computed and are presented inTable 2.5.2.
14
Table 2.5.2
T Kc
0( )3,100,000 2,135 1,190 447 93 8.11E-15
T Kb
0( )1.3E+11 3.4E+09 2.5E+09 1.6E+09 7.1E+08 2.729
km 1.0E+03 3.9E+04 5.2E+04 8.4E+04 1.9E+05 6.3E+13
v xε (cm/sec) 2.9E+09 4.8E+08 4.1E+08 3.2E+08 2.2E+08 1.2E+04
v yε (cm/sec) 3.4E+09 5.5E+08 4.8E+08 3.7E+08 2.5E+08 1.4E+04
v zε (cm/sec) 1.4E+10 2.3E+09 2.0E+09 1.5E+09 1.0E+09 5.6E+04
We note with special interest that the Lorentz-Einsteinrelativistic relationship expressed in equation 2.5.14 equals 2which we define as Cooper for allTc .
1
1
2
1
1 1
1
1 1
2
2
2
2
2
−= =
− −
− −
v
v
v
c
v
c
dx
dy
ex
ey
and
22
2
2
1
1 1− −
⎧
⎨
⎪⎪⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪⎪⎪
⎫
⎬
⎪⎪⎪⎪⎪⎪⎪
⎭
⎪⎪⎪
v
cez
⎪⎪⎪⎪⎪
=
− −
− −
1
1 1 3
1
1 1 3
2
2
2
2
2
2
2
2
B
A
v
v
K
K
B
A
v
dx
ex
P
B
dx22
2
2
2
2
2
2
2
1
1 1 3
v
K
K
B
A
v
v
ey
A
B
dx
ez
− −
⎧
⎨
⎪⎪⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪⎪⎪⎪
⎫
⎬
⎪⎪⎪⎪⎪⎪⎪
⎭
⎪⎪⎪⎪⎪⎪⎪
(2.5.14)
As a check on these dielectric Cartesian velocities, note thatthey are vectorially related to vε from equation 2.5.5 asindicated in equation 2.5.14. As indicated, the factor '2' isrelated to the ratio of Cartesian surfaces approach, section andside to trisine area cos θ( ) section and cavity/chain.
approachsection
chain
cavity( ) ( )⎛
⎝⎜⎞⎠⎟
⎛⎝⎜
cos θ ⎞⎞⎠⎟
+ ( ) ( )⎛⎝⎜
⎞⎠⎟
sectionsection
chain
cavi
cos θtty
sidesection
ch
⎛⎝⎜
⎞⎠⎟
+ ( ) ( )⎛⎝⎜
⎞⎠⎟
cos θ aain
cavity
v v v vx y z
⎛⎝⎜
⎞⎠⎟
=
= + +
2
2 2 2 2ε ε ε ε
(2.5.15)
Based on the Cartesian dielectric velocities, correspondingCartesian fluxoids can be computed as per equation 2.5.16.
Φ
Φ
Φ
ε
ε
ε
επ
x
y
z
xv
Cooper e⎧
⎨
⎪⎪⎪
⎩
⎪⎪⎪
⎫
⎬
⎪⎪⎪
⎭
⎪⎪⎪
=
±
2
22
2
π
π
ε
ε
v
Cooper e
v
Cooper e
y
z
±
±
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪⎪
⎫
⎬
⎪⎪⎪⎪⎪
⎭
⎪⎪⎪⎪⎪
(2.5.16)
The trisine residence relativistic CPT time± is confirmed interms of conventional capacitance C( ) and inductance L( )resonance circuit relationships in x, y and z as well as trisinedimensions.
15
time
L C
L C
L C
L C
x x
y y
z z
± =
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
⎫
⎬
⎪⎪⎪⎪⎪
⎭
⎪⎪⎪⎪⎪⎪
(2..5.17)
=
±
Φx
x
xtime
e v
approach
ε πε
4 22
4
3
3
B
time
e v
side
B
time
e v
y
y
y
z
Φ
Φ
±
±
ε
ε
πε
zz
z
s
section
A
time
e v g
section
4
1
42
πε
πε
Φ
± cos θθεθ( ) ( )
⎧
⎨
⎪⎪⎪⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
2
section
cavity cos
⎪⎪⎪⎪⎪
⎫
⎬
⎪⎪⎪⎪⎪⎪⎪⎪
⎭
⎪⎪⎪⎪⎪⎪⎪⎪
Where relationships between Cartesian and trisine capacitanceand inductance is as follows:
2 C C C Cx y z= = = (2.5.18)
L L L Lx y z= = =2 2 2 (2.5.19)
L L Lx y z= = (2.5.20)
C C Cx y z= = (2.5.21)
Table 2.5.3
T Kc
0( ) 3,100,000 2,135 1,190 447 93 8.1E-16
T Kb
0( ) 1.3E+11 3.4E+09 2.5E+09 1.6E+09 7.1E+08 2.729
C /v (fd/cm )3 1.4E+06 3.8E+04 2.8E+04 1.7E+04 7.9E+03 2.3E-05
L /v (h/cm )3 9.0E+15 2.4E+14 1.8E+14 1.1E+14 4.9E+13 1.5E+05
In terms of an extended Thompson cross section σT( ) , it is
noted that the 1 2/ R( ) factor [62] is analogous to a dielectric ε( )equation 2.5.4. Dimensionally the dielectric ε( ) isproportional to trisine cell length dimension.
The extended Thompson scattering cross section σT( ) [43
equation 78.5, 62 equation 33] then becomes as in equation2.5.22.
σ πεT
B
Ds t dy
K
K
e
m v=
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
⎛
⎝⎜⎞
⎠⎟≈±
2 2
2
2
8
3 sside
(2.5.22)
2.6 FLUXOID AND CRITICAL FIELDS
Based on the material fluxoid Φε( ) (equation 2.5.5), thecritical fields Hc1( ) (equation 2.6.1), Hc2( ) (equation 2.6.2) &
Hc( ) (equation 2.6.3) as well as penetration depth λ( )(equation 2.6.5) and Ginzburg-Landau coherence length ξ( )(equation 2.6.6) are computed. Also the critical field Hc( ) isalternately computed from a variation on the Biot-Savart law.
Hct
1 2=
Φε
πλ(2.6.1)
Hsection
g B
c
s
2
4
1 4
2
=
=⎛⎝⎜
⎞⎠⎟ ⎛
⎝⎜⎞⎠
π
π
π
ε
Φ
⎟⎟
( )2 Φε θ cos
(2.6.2)
H H He
v A time
m
mc c cx
t
e
= =⋅ ⋅
±1 2
ε
(2.6.3)
n
cavityc = (2.6.4)
Also it interesting to note that the following relationship holds
22 2
2 2π e
A
e
Bm v m ce dx e
± ± = (2.6.4a)
16
Figure 2.6.1 Experimental (Harshman Data) and TrisineModel Cooper CPT Charge conjugated Pair ThreeDimensional Concentration as per equation 2.6.4
1E+12
1E+13
1E+14
1E+15
1E+16
0 20 40 60 80 100 120 140
Critical Temperature Tc
Coo
per P
air A
rea
Con
cent
ratio
n (/
cm2)
Trisine ModelHarshman Data
Figure 2.6.2 Experimental (Harshman Data) and TrisineModel Cooper CPT Charge conjugated Pair TwoDimensional Area Concentration or section
1E+12
1E+13
1E+14
1E+15
1E+16
0 20 40 60 80 100 120 140
Critical Temperature Tc
Coo
per P
air A
rea
Con
cent
ratio
n (/c
m2)
Trisine ModelHarshman Data
λ ε22
2
2
1
2
1 1=
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟ ( )
=
g
m v
n es
t
c
11 1
2
1 12
2
g
m c
n es
t
c
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟ ( )±
ε 22
(2.6.5)
From equation 2.6.5, the trisine penetration depth λ iscalculated. This is plotted in the above figure along withHarshman[17] and Homes[51, 59] penetration data. TheHomes data is multiplied by a factor of ( )n chain Coopere
12
to get nc in equation 2.6.5. This is consistent with λ2 1∝ nc
in equation 2.6.5. It appears to be a good fit and better thanthe fit with data compiled by Harshman[17].
Penetration depth and gap data for magnesium diboride MgB2
[55] which has a critical temperature Tc of 39 K indicates a fitto the trisine model when a critical temperature of 39/3 or 13
K is assumed (see Table 2.6.1). This is interpreted as anidication that MgB2 is a three dimensional superconductor.
Table 2.6.1 Experimental ( MgB2 Data[55]) and TrisineModel Prediction
Trisine Data atTc
39/3 K or 13 K
Observed Data[55]
PenetrationDepth (nm)
276 260 ±20
Gap (meV) 1.12 3.3/3 or 1.1 ±.3
As indicated in equation 2.5.5, the material dielectric modifiedspeed of light vε is a function of material dielectric ε( )constant which is calculated in general and specifically fortrisine by ε = D E
(equation 2.5.4). The fact that chain
rather than cavity (related by 2/3 factor) must be used inobtaining a trisine model fit to Homes’ data indicates that thestripe concept may be valid in as much as chains in the trisinelattice visually form a striped pattern as in Figure 2.3.1.
Figure 2.6.3 Experimental (Homer and Harshmann) andTrisine Model Penetration Depth
1.0E-06
1.0E-05
1.0E-04
1.0E-03
0 20 40 60 80 100 120 140
Critical Temperature Tc (K)
Pene
trat
ion
Dep
th (
cm)
Harshman's DataTrisine TheoryHomes' Data
See derivation of Ginzberg-Landau coherence lengthξ inequation 2.6.6 in Appendix D. It is interesting to note thatt B ξ = ≈2 73. e for all Tc which makes the trisinesuperconductor mode fit the conventional description ofoperating in the “dirty” limit. [54]
ξ =m
m K
chain
cavityt
e B
12
(2.6.6)
17
Figure 2.6.4 Trisine Geometry in the Fluxoid Mode(Interference Pattern Ψ2 )
The geometry presented in Figure 2.6.4 is produced by threeintersecting coherent polarized standing waves which can becalled the trisine wave function. It is interesting to note that asingularity exists at the circular portion. This is similar to thesingularity at the Schwarzschild radius as derived in thegeneral theory of relativity. This can be compared to figure2.6.5 which presents the dimensional trisine geometry.
Figure 2.6.5 Trisine Geometry in the Fluxoid Mode
B2B
Indicates Trisine Peak Each Peak is raised by height A
Cell Cavity
A2 + B
2 = C2
B
C
A yx
2B3
2B3
2B3
2B3
B2
B2
Equation 2.6.7 represents the trisine wave function Ψ( ) . Thiswave function is the addition of three sin functions that are120 degrees from each other in the x y plane and form anangle 22.8 degrees with this same x y plane. Also note that22.8 degrees is related to the B A( ) ratioby tan .90 22 80 0−( ) = B A .
18
Ψ =+ +⎛
⎝⎜⎞⎠⎟
⎛
⎝⎜⎜
⎞
⎠⎟⎟
+
e
sin 2 1 0 1 0 1 0a x b y c zx x x
ee
e
sin
s
2 2 0 2 0 2 0a x b y c zx x x+ +⎛⎝⎜
⎞⎠⎟
⎛
⎝⎜⎜
⎞
⎠⎟⎟
+iin 2 3 0 3 0 3 0a x b y c zx x x
ax
+ +⎛⎝⎜
⎞⎠⎟
⎛
⎝⎜⎜
⎞
⎠⎟⎟
Where:
110 0
10
030 22 8
030 22
= ( ) ( )= ( )
cos cos .
sin cos .
bx 88
030
150 22 8
0
10
20 0
( )= − ( )= ( ) (
c
a
x
x
sin
cos cos . ))= ( ) ( )= − ( )
b
c
x
x
20 0
20
150 22 8
150
sin cos .
sin
aa
b
x
x
30 0
30
270 22 8
270 2
= ( ) ( )= ( )
cos cos .
sin cos
22 8
270
0
30
.
sin
( )= − ( )
cx
(2.6.7)
In table 2.6.2, note that the ratio of London penetrationlength λ( ) to Ginzburg-Landau correlation distance ξ( ) is theconstant κ( ) of 58. The trisine values compares favorablywith London penetration length λ( ) and constant κ( ) of1155(3) Å and 68 respectively as reported in reference [11,17] based on experimental data.
Table 2.6.2
T Kc
0( )3,100,000 2,135 1,190 447 93 8.1E-16
T Kb
0( ) 1.3E+11 3.4E+09 2.5E+09 1.6E+09 7.1E+08 2.729
Hc (gauss) 1.6E+10 1.7E+06 8.4E+05 2.5E+05 3.5E+04 1.6E-17
Hc1 (gauss) 2.1E+07 2.3E+03 1.1E+03 3.3E+02 4.6E+01 2.2E-20
Hc2 (gauss) 1.2E+13 1.3E+09 6.3E+08 1.9E+08 2.6E+07 1.2E-14
nc (/cm3) 3.0E+28 5.4E+23 2.3E+23 5.2E+22 4.9E+21 1.3E-04
λ (cm) 5.7E-08 2.2E-06 2.9E-06 4.7E-06 1.0E-05 3500
ξ (cm) 9.8E-10 3.7E-08 5.0E-08 8.1E-08 1.8E-07 60.30
Figure 2.6.5 Critical Magnetic Field Data (gauss)compared to Trisine Model as a function of Tc
.
0.001
0.01
0.1
1
1 0
100
1000
10000
100000
1000000
10000000
100000000
0.1 1 1 0 100 1000 10000
Critical Temperature Tc
Cri
tical
Fie
lds
Hc1
, Hc,
and
Hc2
Hc1 Data
Hc2 Data
Hc Data
Hc1 Trisine
Hc2 Trisine
Hc Trisine
2.7 SUPERCONDUCTOR INTERNALPRESSURE
The following pressure values in Table 2.71 as based onequations 2.7.1 indicate rather high values as superconductingTc( ) increases. This is an indication of the rather large forces
that must be contained in these anticipated materials.
pressure
K
time approach
K
time side
K
B
P=
±
±
AA
time section±
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
⎫
⎬
⎪⎪⎪⎪⎪
⎭
⎪⎪⎪⎪⎪
=
FF
approach
F
side
F
section
x
y
z
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
⎫
⎬
⎪⎪⎪⎪⎪⎪
⎭
⎪⎪⎪⎪⎪
(2.7.1a)
19
pressure
m v
time approach
m v
time side
t dx
t dy=
mm v
time sectiont dz
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
⎫
⎬
⎪⎪⎪⎪⎪
⎭
⎪⎪⎪⎪⎪⎪
=
Cooper k T
cavityb c
=+A B
B
cavity
chain
v
Adx
2 2 2
4240
π
Casimiir pressure( )
(2.7.1b)
To put the calculated pressures in perspective, the C-C bondhas a reported energy of 88 kcal/mole and a bond length of1.54 Å [24]. Given these parameters, the internal chemicalbond pressure CBP( ) for this bond is estimated to be:
CBPkcal
molex
erg
kcal= ⋅88 4 18 1010.
⋅1
6 02 1023. x
mole
bond
⋅( )−
1
1 54 10 8 3 3. x
bond
cm
= 1 67 10123
. xerg
cm
Within the context of the superconductor internal pressurenumbers in table 2.7.1, the internal pressure requirement forthe superconductor is less than the C-C chemical bondpressure at the order of design Tc of a few thousand degreesKelvin indicating that there is the possibility of usingmaterials with the bond strength of carbon to chemicallyengineer high performance superconducting materials.
In the context of interstellar space vacuum, the total pressurem c cavityt
2 may be available and will be of such a magnitudeas to provide for universe expansion as astronomicallyobserved. (see equation 2.11.21)
Table 2.7.1 with pressure plot
T Kc
0( ) 3,100,000 2,135 1,190 447 93 8.1E-16
T Kb
0( ) 1.3E+11 3.4E+09 2.5E+09 1.6E+09 7.1E+08 2.729
pressureerg/cm3 1.3E+19 1.6E+11 3.7E+10 3.2E+09 6.3E+07 1.4E-35
pressure( )psi
1.9E+14 2.3E+06 5.4E+05 4.7E+04 9.2E+02 2.1E-40
pressure( )pascal
1.3E+18 1.6E+10 3.7E+09 3.2E+08 6.3E+06 1.4E-36
Totalpressureerg/cm3
1.36E+24 2.45E+19 1.02E+19 2.35E+18 2.23E+17 5.73E-09
1E+00
1E+02
1E+04
1E+06
1E+08
1E+10
1E+12
1 10 100 1000 10000
Critical Temperature Tc (K)Pr
essu
re (
pasc
al)
Assuming the superconducting current moves as a
longitudinal wave with an adiabatic character, then equation2.7.2 for current or De Broglie velocity vdx( ) holds where the
ratio of heat capacity at constant pressure to the heat capacityat constant volume δ( ) equals 1.
v pressurecavity
mdxt
= δ (2.7.2)
The Casimir force is related to the trisine geometry byequation 2.7.3.
Casimir Force g
section v
Asdz = 3
2
43 240
π
(2.7.3)
In a recent report[53] by Jet Propulsion Laboratory, anunmodeled deceleration was observed in regards to Pioneer 10and 11 spacecraft as they exited the solar system. This dragcan be related to the following expression where m cavityt( )is the mass equivalent energy density value atT x Kc = − 08 1 10 16. representing conditions in space which is100 percent transferred to an object passing through it.
Drag Force AREAm
cavityct = − 2 (2.7.4)
20
Note that equation 2.7.4 is similar to the conventional dragequation as used in design of aircraft with v2 replaced with c2 .Of course Drag Force Ma = , so the Pioneer spacecraft M AREA/ , is an important factor in establishingdeceleration which was observed to be on the order of 8.74E-8cm2/sec.
The drag equation is repeated below in terms of Pioneerspacecraft and the assumption that space density is essentiallywhat has been empirically determined to be dark matter:
F Ma C Av
where
CR R
d c
de e
= =
= ++
+
ρ2
2
24 6
14
:
. [60] empirical
(2.7.4)
where
F
M
a
:
force
Pioneer 10 & 11 Mass
=== aacceleration
Pioneer 10 & 11 cross sectAc = iion
space fluid density (6.38x10 g/c-30ρ = mm
consistent with dark matter est
3)
iimates
object velocity
absolute fl
v ==µ uuid viscosity (1.21E-16 g/(cm sec)
kinν = eematic fluid viscosity (1.90x10 cm13µ ρ/ 22 /sec)
Reynolds' number
= iner
Re =ttial force
viscous forcedimensionless numb= eer
=vD
=(fluid density)(object velociρ
µtty)(object diameter)
(fluid absolute viscosiity)When the Reynolds’ number is low (laminar flow condition),then C Rd e= ( )24 and the drag equation reduces to the
Stokes’ equation F vD= 3πµ . [60]
In other parts of this report, a model is developed as to whatthis dark matter actually is. The model dark matter is keyed toa 6.38E-30 g/cc value in trisine model (NASA observed valueof 6E-30 g/cc). The proposed dark matter consists of massunits of mt (110 x electron mass) per volume (cavity- 15733cm3). These mass units are virtual particles that exist in acoordinated lattice at base energy ω / 2 under the conditionthat momentum and energy are conserved - in other wordscomplete elasticity. This dark matter lattice would haveinternal pressure (table 2.7.1) to withstand collapse intogravitational clumps.
Under these circumstances, the traditional drag equation formis valid but velocity (v) should be replaced by speed of light(c). Correspondingly, the space viscosity is computed asmomentum/area or m c xt ( )2 2∆( ) where ∆x is uncertaintydimension in table 2.5.1 at Tc = 8.1E-16 K and as perequation 2.1.20. This is in general agreement with gaseouskinetic theory [19].
where =K
Fdx mv
v
c
dv
m
B
t
v
t
dx
=−
∫ ∫0
2
2
0 1
BB
dx dx
dx
v=
v
and =v
=
2
π
π
B v x
mv B
t
=2
1
∆ ∆∆
∆∆∆ ∆
x
v v and x B
F B m cv
dx
t
and
then
= → = →
= − −
0 0
12 ddx
t dx
c
m c v c
2
2
2 at = − <<
(2.7.5)
It is understood that the above development assumes thatdv dt/ ≠ 0 , an approach which is not covered in standardtexts[61], but is assumed to be valid here because ofdimensional similarity to the Heisenberg Uncertainty principleas reflected in equation 2.1.20 and also applicability in acollision context.
In terms of an object passing through the trisine elastic spaceCPT lattice at some velocity v v x Bdx= , the force (F) is nolonger dependent of object velocity (v) at v v cdx < <<
but is a
constant relative to c2 which reflects the fact that the speed oflight is a constant in the universe and in consideration of thefollowing established relationship:
F Cm c
xdt= −
2 (2.7.6)
Where Cd is the standard fluid mechanical drag coefficient.This model was used to analyze the Pioneer UnmodeledDeceleration data 8.74E-08 cm/sec2 and there appeared to be agood fit with the observed space density 6E-30 g/cm3 (trisinemodel 6.38E-30 g/cm3). A drag coefficient of 59.67 indicatesa laminar flow condition. An absolute space viscosity of1.21E-16 g/(cm sec) is established within trisine model andused.
Pioneer (P) Translational Calculations
P mass 241,000 gram P diameter 274 cm (effective) P cross section 58,965 cm2 (effective)
21 P area/mass 0.24 cm2/g P velocity 1,117,600 cm/sec space kinematic viscosity 1.90E+13 cm2/sec space density 6.38E-30 g/cm3
Pioneer Reynold's number 4.31E-01 unitless drag coefficient 59.67 P deceleration 8.37E-08 cm/sec2
one year drag distance 258.51 mile laminar drag force 9.40E-03 dyne Time of object to stop 1.34E+13 sec
JPL, NASA raised a question concerning the universalapplication of the observed deceleration on Pioneer 10 & 11.In other words, why do not the planets and their satellitesexperience such an deceleration? The answer is in theA Mc ratio in the modified drag relationship.
a Cm c
xC
A
Mc C
A
M
m
cavitycd
td
cd
c t= − = − = −2
2 2ρ (2.7.6)
Using the earth as an example, the earth differentialmovement per year due to modeled deceleration of 4.94E-19cm sec2 would be 0.000246 centimeters (calculated as at 2 2 ,an unobservable distance amount). Also assuming the trisinesuperconductor model, the reported 6 nanoTesla (nT) (6E-5gauss) interplanetary magnetic field (IMF) in the vicinity ofearth, will not allow the formation of a superconductor CPTlattice due to the well known properties of a superconductor inthat magnetic fields above critical fields will destroy it. In thiscase, the space superconductor at Tc = 8.1E-16 K will bedestroyed by a critical magnetic field above Hc2 = 2.14E-14gauss as in table 2.6.1. It is conceivable that the IMFdecreases by some power law with distance from the sun, andperhaps at some distance the IMF diminishes to an extentwherein the space superconductor CPT lattice is allowed toform. This may be an explanation for the observed “kick in”of the Pioneer spacecraft deceleration phenomenon at 10Astronomical Units (AU).
Earth Translational Calculations
Earth mass 5.98E+27 gram Earth diameter 1.27E+09 cm Earth cross section 1.28E+18 cm2
Earth area/mass 2.13E-10 cm2/g Earth velocity 2,980,010 cm/sec space kinematic viscosity 1.90E+13 cm2/sec space density 6.38E-30 g/cm3
Earth Reynold's number 2.01E+06 unitless drag coefficient 0.40 unitless drag force (with drag coefficient) 2.954E+09 dyne Earth deceleration 4.94E-19 cm/sec2
One year drag distance 0.000246 cm
laminar drag force 4.61E-01 dyne Time for Earth to stop rotating 6.03E+24 sec
Now to test the universal applicability of the unmodeledPioneer 10 & 11 decelerations for other man made satellites,one can use the dimensions and mass of the Hubble spacetelescope. One arrives at a smaller deceleration of 6.79E-09cm/sec2 because of its smaller area/mass ratio. This lowdeceleration is swamped by other decelerations in the vicinityof earth, which would be multiples of those well detailed inreference 1, Table II Pioneer Deceleration Budget. Also thetrisine elastic space CPT lattice probably does not exist in theimmediate vicinity of the earth because of the earth’smilligaus magnetic field which would destroy the coherenceof the trisine elastic space CPT lattice.
Hubble Satellite Translational Calculations
Hubble mass 1.11E+07 gram Hubble diameter 7.45E+02 centimeter Hubble cross section 5.54E+05 cm2
Hubble area/mass 0.0499 cm2/g Hubble velocity 790,183 cm/sec space kinematic viscosity 1.90E+13 cm2/sec space density 6.38E-30 g/cm3
Hubble Reynolds' number 1.17E+00 unitless drag coefficient 23.76 drag force (with drag coefficient) 7.549E-02 dyne Hubble deceleration 6.79E-09 cm/sec2
One year drag distance 3,378,634 cm laminar drag force 7.15E-08 dyne
Time for Hubble to stop rotating 1.16E+14 sec
The fact that the unmodeled Pioneer 10 & 11 deceleration dataare statistically equal to each other and the two space craftexited the solar system essentially 190 degrees from eachother implies that the supporting fluid space density throughwhich they are traveling is co-moving with the solar system.But this may not be the case. Assuming that the supportingfluid density is actually related to the cosmic backgroundmicrowave radiation (CMBR), it is known that the solarsystem is moving at 500 km/sec relative to CMBR.Equivalent decelerations in irrespective to spacecraft headingrelative to the CMBR would further support the hypothesisthat deceleration is independent of spacecraft velocity.
22
Figure 2.7.1 Pioneer Trajectories (NASA JPL [53])with delineated solar satellite radial component r.
Executing a web based [58] computerized radial rate dr/dt(where r is distance of spacecraft to sun) of Pioneer 10 & 11Figure 2.7.1), it is apparent that the solar systems exitingvelocities are not equal. Based on this graphical method, thespacecraft’s velocities were in a range of about 9% (28,600 –27,500 mph for Pioneer 10 “3Jan87 - 22July88” and 26,145 –26,339 mph for Pioneer 11 ”5Jan87 - 01Oct90”). Thisreaffirms the basic drag equation used for computation in that,velocity magnitude of the spacecraft is not a factor, only itsdirection with the drag force opposite to that direction. Theobserved equal deceleration at different velocities wouldappear to rule out deceleration due to Kuiper belt dust as asource of the deceleration for a variation at 1.092 or 1.19would be expected.
Also it is observed that the Pioneer spacecraft spin is slowingdown from 7.32 rpm in 1987 to 7.23 rpm in 1991. Thisequates to an angular deceleration of .0225 rpm/year or 1.19E-11 rotation/sec^2. JPL[57] has acknowledged somesystematic forces that may contribute to this deceleration rateand suggests an unmodeled deceleration rate of .0067rpm/year or 3.54E-12 rotations/sec. [57]
Now using the standard torque formula:
Γ = Iω (2.7.7)
The deceleration value of 3.54E-12 rotation/sec2 can bereplicated assuming a Pioneer Moment of Inertia about spinaxis(I) of 5.88E+09 g cm2 and a 'paddle' cross section area of3,874 cm2 which is 6.5 % spacecraft 'frontal" cross sectionwhich seems reasonable. The gross spin deceleration rate of.0225 rpm/year or 1.19E-11 rotation/sec^2 results in a pseudo
paddle area of 13,000 cm2 or 22.1% of spacecraft 'frontal"cross section which again seems reasonable.
Also it is important to note that the angular deceleration ratefor each spacecraft (Pioneer 10, 11) is the same even thoughthey are spinning at the two angular rates 4 and 7 rpmrespectfully. This would further confirm that velocity is not afactor in measuring the space mass or energy density. Thecalculations below are based on NASA JPL problem setvalues[57]. Also, the sun is moving through the CMBR at600 km/sec. According to this theory, this velocity would notbe a factor in the spacecraft deceleration.
Pioneer (P) Rotational Calculations
P mass 241,000 g P moment of inertia 5.88E+09 g cm2
P diameter 274 cm P translational cross section 58,965 cm2
P radius of gyration k 99 cm P radius r 137 cm paddle cross section 3,874 cm2
paddle area/mass 0.02 cm2/g P rotation speed at k 4,517 cm/sec P rotation rate change 0.0067 rpm/year P rotation rate change 3.54E-12 rotation/sec2
P rotation rate change 2.22E-11 radian/sec2
P rotation deceleration at k 2.20E-09 cm/sec2
P force slowing it down 1.32E-03 dyne space kinematic viscosity 1.90E+13 cm2/sec space density 6.38E-30 g/cm3
P Reynolds' number 4.31E-01 unitless drag coefficient 59.67 drag force 1.32E-03 dyne
P Paddle rotationaldeceleration at k
2.23E-11 radian/sec2
P Paddle rotationaldeceleration at k
2.20E-09 cm/sec2
One year rotational dragdistance
1,093,344 cm
P rotational laminar drag force 7.32E-23 dyne Time for P to stop rotating 2.05E+12 sec
These calculations suggest a future spacecraft with generaldesign below to test this hypothesis. This design incorporatesgeneral features to measure translational and rotational drag.A wide spectrum of variations on this theme are envisioned.General Space Craft dimensions should be on the order ofTrisine geometry B at T K T E Kb c= = −2 711 8 12 16. . or 22cm (Table 2.2.1) or larger. The Space Craft Reynold’snumber would be nearly 1 at this dimension and the resultingdrag the greatest. The coefficient of drag would decreasetowards one with larger Space Craft dimensions. It is difficultto foresee what will happen with Space Craft dimensions
23below 22 cm.
Figure 2.7.2 Space craft general design to quantifytranslational and rotary drag effect. Brown isparabolic antenna, green is translational captureelement and blue is rotational capture element.
The spacecraft (Figure 2.7.2) would rotate on green bar axiswhile traveling in the direction of the green bar. The redpaddles would interact with the space matter as mc2 slowingthe rotation with time in conjunction with the overallspacecraft slowing decelerating.
A basic experiment cries out here to be executed and that is tolaunch a series of spacecraft of varying A Mc ratios andexiting the solar system at various angles while measuringtheir rotational and translational decelerations. This wouldgive a more precise measure of the spatial orientation of darkmatter in the immediate vicinity of our solar system. An equalcalculation of space energy density by translational androtational observation would add credibility to accumulateddata.
The assumption that the Pioneer spacecraft will continue onforever and eventually reach the next star constellation may bewrong. Calculations indicate the Pioneers will come to a stoprelative to space fluid in about 400,000 years. We may bemore isolated in our celestial position than previously thought.
When this drag model is applied to a generalized design for aSolar Sail [63], a deceleration of 3.49E-05 cm/sec2 iscalculated which is 3 orders higher than experienced byPioneer spacecraft. Consideration should be given to spinpaddles for these solar sails in order to accumulate spin
deceleration data also. In general, a solar sail with itscharacteristic high A Mc ratio such as this example should bevery sensitive to the trisine elastic space CPT lattice. Theintegrity of the sail should be maintained beyond its projectedsolar wind usefulness as the space craft travels into spacebeyond Jupiter.
Pioneer (P) Rotational Calculations
Sail mass 300,000 gram Sail diameter 40,000 centimeter Sail cross section 1,256,637,000 cm2
Sail area/mass 4,189 cm2/g Sail velocity 1,117,600 cm/sec space kinematic viscosity 1.90E+13 cm2/sec space density 6.38E-30 g/cm3
Sail Reynold's number 6.30E+01 unitless drag coefficient 1.45 drag force 10 dyne Sail acceleration 3.49E-05 cm/sec2
one year drag distance 1.73E+10 cm laminar drag force 1.37E+00 dyne Time of Sail to stop 1015 year
The following calculation establishes a distance from the sunat which the elastic space CPT lattice kicks in. There is aenergy density competition between the solar wind and theelastic space CPT lattice. Assuming the charged particledensity at earth is 5 / cm3 [57] and varying as 1/R2 andcomparing with elastic space density of kTc/cavity, then anequivalent energy density is achieved at 3.64 AU. This ofcourse is a dynamic situation with solar wind changing withtime which would result in the boundary of the elastic spaceCPT lattice changing with time. This is monitored by a spacecraft (Advanced Composition Explorer (ACE)) at the L1position (1% distance to sun) and can be viewed athttp://space.rice.edu/ISTP/dials.html. A more descriptive termwould be making the elastic space CPT lattice decoherent orcoherent. It is interesting to note that the calculated boundaryis close to that occupied by the Astroids and is considered theAstroid Belt.
Elastic Space CPT lattice intersectingwith solar wind
AU Mercury 0.39 Venus 0.72 Earth 1.00 Mars 1.53
24 Jupiter 5.22 Saturn 9.57 Uranus 19.26 Neptune 30.17 Pluto 39.60 AU 3.64 n at AU 0.38 cm-3
r 1/n1/3 1.38 cm coordination # 6 unitless volume 0.44 cm3
e
r volume
T
Tc
b
2 1
energy
volume 1.13E-34 erg / cm3
Trisine
k T
cavityb c
energy
volume 1.13E-34 erg / cm3
Now assuming the basic conservation of total kinetic andpotential energy for an object in circular orbit of the sun withvelocity v0( ) and at distance R0( ) equation 2.7.8 provides anindication of the time required to perturb the object out of thecircular orbit in the direction of the sun.
vGM
Rv v at
vGM
Rv v C
Aread
0
0
2
0
20
2
2
0
20
+ = + −( )
+ = + −MM
c t
where v v
ρ
δ
22
0
⎛⎝⎜
⎞⎠⎟
=
(2.7.8)
Assuming the sun orbital velocity perturbing factor δ to be .9and the time (t) at the estimated age of the solar system(5,000,000,000 years), then the smallest size objects that couldremain in sun orbit are listed below as a function of distance(AU) from the sun.
Space object uppersize
AU diameter (meter) Mercury 0.39 9.04 Venus 0.72 10.67 Earth 1.00 11.63 Mars 1.53 13.04 Jupiter 5.22 18.20 Saturn 9.57 21.50 Uranus 19.26 26.10 Neptune 30.17 29.60 Pluto 39.60 31.96 Kuiper Belt 50 35.95 60 37.57 70 39.03 Oort Cloud? 80 40.36 100 41.60
150 46.75 200 50.82 300 57.20 400 62.24
This calculation would indicate that there is not small particledust remaining in the solar system, which would include theKuiper Belt and hypothesized Oort Cloud. Also, thecalculated small object size would seem to be consistent withAsteroid Belt objects, which exist at thecoherence/decoherence boundary for the elastic space CPTlattice.
This analysis has implications for large-scale galacticstructures as well. It is conceivable that energy densities varyspatially within a galaxy with generally increased energydensity towards the center. This would imply that a generallyradial boundary condition would exist within a galaxy andmay be responsible for observed anomalous dark mattergalactic rotation. Apparent clumping of galactic matter maybe due to localities where elastic space CPT lattice does notexist due to decoherent effects of various energy sourcesexisting at that particular locality. Dark energy and Darkmatter may be one and the same.
2.8 SUPERCONDUCTING CURRENT, VOLTAGEAND CONDUCTANCE
The superconducting electrical current Ie( ) is expressed as interms of a volume chain containing a Cooper CPT Chargeconjugated pair moving through the trisine CPT lattice at a DeBroglie velocity vdx( ) .
I
area
Cooper e v
chaine dx= ± (2.8.1)
The superconducting mass current Im( ) is expressed as interms of a trisine relativistic velocity transformed mass mt( )per cavity containing a Cooper CPT Charge conjugated pairand the cavity Cooper CPT Charge conjugated pair residencerelativistic CPT time±.
Im
cavity timemt=
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
1
(2.8.2)
It is recognized that under strict CPT symmetry, there wouldbe zero current flow. It must be assumed that pinning of oneside of CPT will result in observed current flow. Pinning
25(beyond that which naturally occurs) of course has beenobserved in real superconductors to enhance actualsupercurrents.
Secondarily, this reasoning would provide reasoning why anet charge or current is not observed in a vacuum such astrisine model extended to critical space density in section 2.11.
Table 2.8.1 with super current amp cm/ 2( ) plot.
T Kc
0( ) 3,100,000 2,135 1,190 447 93 8.1E-16
T Kb
0( ) 1.3E+11 3.4E+09 2.5E+09 1.6E+09 7.1E+08 2.729
Ie amp cm/ 2( ) 6.7E+17 3.2E+11 9.8E+10 1.4E+10 6.0E+08 4.6E-26
Im g cm/ sec3( ) 1.95E+20 2.42E+12 5.62E+11 4.86E+10 9.60E+08 2.2E-34
1E+04
1E+06
1E+08
1E+10
1E+12
1E+14
1 10 100 1000 10000
Critical Temperature Tc (K)
Supe
rcur
rent
(am
p/cm
^2)
A standard Ohm's law can be expressed as follows with the
resistance expressed as the Hall resistance with a value of
25,815.62 ohms.
voltage current resistanc = ( ) ee
k T
e
e
chainapproach vb c
dx
( )
= ⎛⎝⎜
⎞⎠⎟±
±
22
πe±( )
⎛
⎝⎜
⎞
⎠⎟ (2.8.3)
Equation 2.8.4 represents the Poynting vector relationship orin other words the power per area per relativistic CPT time±
being transmitted from each superconducting cavity:
k Tm
mE H s time vb c
e
tx c= ( ) ±
1
22 ide ε
(2.8.4)
Equation 2.8.4 can be rearranged in terms of a relationshipbetween the De Broglie velocity vdx( ) and modified speed oflight velocity v xε( ) as expressed in equation 2.8.5.
vm
m
e B
Av
B
xt
edxε π
22
2 2
1
4
3
48
3
=⎛⎝⎜
⎞⎠⎟
( )
=
±
AA
c v
c v
dx
dx
⋅ ⋅
= ⋅ ⋅ π
(2.8.5)
This leads to the group/phase velocity relationship in equation
2.8.6
( ) ( )
( )
group velocity phase velocity c
vdx
⋅ = 2
or
⋅⋅ ⋅⎛⎝⎜
⎞⎠⎟
=1 2
22
πεv
vc cx
dx
(2.8.6)
In the Nature reference [51] and in [59], Homes’ reportsexperimental data conforming to the empirical relationshipHomes’ Law relating superconducting density ρs at justaboveTc , superconductor conductivity σ dc andTc .
ρ σs dc cT∝ Homes' Law ref: 51 (2.8.7)
The Trisine model is developed forTc , but using the Tanner’slaw observation that n = n 4S N (ref 52), we can assumethat E = E 4fN fS . This is verified by equation 2.5.1. Nowsubstituting E IfN e for superconductor conductivityσ dc , andusing cavity for super current density ρs , the followingrelationship is obtained and is given the name Homers’constant.
Homes' constant
=σ
ρdc b c
s
k T
= ±3
2
2 2
2
π e
m
A
Bt
= 191 5375
3,
sec
cm
(2.8.8)
The results for both c axis and a-b plane (ref 51) aregraphically presented in the following figure 2.8.1. Onaverage the Homes’ constant calculated from data exceeds theHomes’ constant derived from Trisine geometry by 1.72.There may be some degrees of freedomconsiderations1 2⋅ k Tb c / , 2 2⋅ k Tb c / , 3 2⋅ k Tb c / that warrantlooking into that may account for this divergence althoughconformity is noteworthy. Also 1.72 is close to sqrt(3), avalue that is inherent to Trisine geometry. I will have to givethis a closer look. In general, the conventional and hightemperature superconductor data falls in the same pattern. I
26would conclude that Homes’ Law can be extrapolated to anycritical temperatureTc .
Figure 2.8.1 Homes’ data compared to Trisine ModelHomes’ Constant
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
0 10 20 30 40 50 60 70 80 90 100Superconductor Critical Temperature Tc
Hom
es' D
ata/
Hom
es'
Con
stan
t
Homes' DataHomes' Constant
2.9 SUPERCONDUCTOR WEIGHT REDUCTION INA GRAVITATIONAL FIELD
Based on the superconducting helical or tangentialvelocity vdT( ) , the superconducting gravitational shielding
effect Θ( ) is computed based on Newton's gravitational law
and the geometry as presented in Figure 2.9.1. With amaterial density ρ( ) of 6.39 g/cc [17], the gravitational
shielding effect Θ( ) is .05% as observed by Podkletnov and
Nieminen (see Table 2.9.1).
ρCt tm
cavity
n m= =
2
(2.9.1)
F mv
R
Gm M
Rtdx t= =2
2(2.9.2)
Θx
Superconducting Force
Earth's Gravitation=
aal Force
Earth Radius
Earth Mass
vC dx =ρρ
2
GG
(2.9.3)
ΘT
Superconducting Force
Earth's Gravitation=
aal Force
Earth Radius
Earth Mass
vC dT =ρρ
22
G
(2.9.4)
Figure 2.9.1 Superconductor Gravitational Model Basedon Super current mass flow with Mass Velocities vdx( )and vdT( ) .
R
M
F
Earth
mtv
dT
mtv
dx
mtv
dy
mtv
dx
mtv
dz
Table 2.9.1
T Kc
0( ) 3,100,000 2,135 1,190 447 93 8.1E-16
T Kb
0( ) 1.3E+11 3.4E+09 2.5E+09 1.6E+09 7.1E+08 2.729
ρC (g/cm3) 1.5E+03 2.7E-02 1.1E-02 2.6E-03 2.5E-04 6.4E-30
Θx3.2E+06 4.0E-02 9.3E-03 8.0E-04 1.6E-05 3.5E-48
ΘT 8.0E+07 1.00 2.3E-01 2.0E-02 4.0E-04 8.9E-47
Experiments have been conducted to see if a rotating body'sweight is changed from that of a non-rotating body [21, 22,23]. The results of these experiments were negative as theyclearly should be because the center of rotation equaled thecenter mass. A variety of gyroscopes turned at variousangular velocities were tested. Typically a gyroscope witheffective radius 1.5 cm was turned at a maximum angularfrequency of 22,000 rpm. This equates to a tangential velocityof 3,502 cm/sec. At the surface of the earth, the tangentialvelocity is calculated to be gRE or 791,310 cm/sec. Theexpected weight reduction if center of mass was different thencenter of rotation r( ) according to equations 2.9.2 and 2.9.3would be 3 502 791 310
2, ,( ) or 1.964E-5 of the gyroscope
weight. The experiments were done with gyroscopes withweights of about 142 grams. This would indicate a requiredbalance sensitivity of 1.964E-5 x 142 g or 2.8 mg, whichapproaches the sensitivity of laboratory balances. Bothpositive[20] and negative results [22, 23] are experimentallyobserved in this area although it has been generally acceptedthat the positive results were spurious.
27Under the geometric scenario for superconducting materialspresented herein, the Cooper CPT Charge conjugated paircenters of mass and centers of rotation r( ) are differentmaking the numerical results in table 2.9.1 valid. Under thisscenario, it is the superconducting material that looses weight.The experimentally observed superconducting shielding effectcannot be explained in these terms.
2.10 BCS VERIFYING CONSTANTS
Equations 2.9.1 - 2.9.8 represent constants as derived fromBCS theory [2] and considered by Little [4] to be primaryconstraints on any model depicting the superconductingphenomenon. It can be seen that the trisine model constantscompare very favorably with BCS constants in brackets {}.
C D k T
cavityk
cavityD T b c b= ∈( ) =10 2 10 22. . (2.10.1)
γ π ε π
= ( ) =2
3
2
32 2
2
D kcavity
k
T cavityT bb
c
(2.10.2)
1178
18170
2
2
γ πT
Horc
c
= ⎧⎨⎩
⎫⎬⎭
. . (2.10.3)
C
T
jumpD
cγ π=
⋅ { } = { }3
21 52 1 55 1 52
2. . . (2.10.4)
22
2
210 2
KK
m k Tjump
ADn
t b c
−⎛⎝⎜
⎞⎠⎟
= { }.
= { }10 14 10 2. .
(2.10.5)
T cavitym
B
Ak
c
tt
b
3
23 3
12
32
32
3
6=
⎛⎝⎜
⎞⎠⎟
π
(Saam's Constan= × −3 6196 10 19. tt)
(2.10.6)
This equation basically assumes the ideal gas law forsuperconducting charge carriers. When the 1 / ρs data fromreference [51, 59] is used as a measure of cavity volume in theequation 2.10.6 , the following plot is achieved. It is a goodfit for some of the data and not for others. One can onlyconclude that the number of charge carriers in the cavityvolume varies considerably from one superconductor to
another with perhaps only the Cooper CPT Charge conjugatedpair being a small fraction of the total. Also, degrees offreedom may be a factor as in the kinetic theory of gas orother words – is a particular superconductor one, two or threedimensional ( / , / , / )1 2 2 2 3 2⋅ ⋅ ⋅k T k T k Tb c b c b c as Table2.6.1 data for MgB2 would imply? Also it is important toremember that Homes’ data is from just above Tc and not ator belowTc .
Figure 2.10.1 Homes’ Data Compared to Saam’sConstant
1.00E-22
1.00E-21
1.00E-20
1.00E-19
1.00E-18
1.00E-17
1.00E-16
1.00E-15
1.00E-14
0 10 20 30 40 50 60 70 80 90 100
Superconductor Critical Temperature Tc
Hom
es' D
ata
& S
aam
's C
onst
ant
Homes' DataSaam's Constant
γ πT
k
cavityc
b
=⎧⎨⎩
⎫⎬⎭
2
3
2 (2.10.7)
22 3 527754
∆oBCS
b c
Euler
k Te= { }−π . (2.10.8)
Table 2.10.1 Electronic Heat Jump k cavityb at Tc
T Kc
0( ) 3,100,000 2,135 1,190 447 93 8.11E-16
T Kb
0( ) 1.29E+11 3.38E+09 2.52E+09 1.55E+09 7.05E+08 2.729
k
cavity
erg
cmb
3
⎛⎝⎜
⎞⎠⎟ 4.20E+13 7.59E+08 3.16E+08 7.28E+07 6.90E+06 1.78E-19
Harshman [17] has compiled and reported an extended listingof electronic specific heat jump at Tc data for a number ofsuperconducting materials. This data is reported in units ofmJ/mole K 2 . With the volume per formula weight data for thesuperconductors also reported by Harshman and aftermultiplying byTc
2 , the specific heat jump at Tc is expressed interms of erg/cm
3 and graphed in figure 2.10.2.
28
Figure 2.10.2 Electronic Specific Heat Jump Cd( ) at Tc
comparing Harshman compiled experimental data andpredicted trisine values based on equation 2.10.1
1.00E+00
1.00E+01
1.00E+02
1.00E+03
1.00E+04
1.00E+05
1.00E+06
1.00E+07
1.00E+08
0 20 40 60 80 100 120 140
Critical Temperature Tc
Ele
ctro
nic
Spec
ific
Hea
t Jum
p at
Tc
Experimental DataTrisine Model
2.11 SUPERCONDUCTING COSMOLOGICALCONSTANT
The following relations of r a d i u s RU( ) , mass MU( ) ,density ρU( ) and time TU( ) in terms of the superconductingcosmological constant are in general agreement with knowndata. This could be significant. The cosmological constant aspresented in equation 2.11.1 is based on energy/area oruniverse surface tension σU( ) , as well as energy/volume oruniverse pressure pU( ) . The universe surface tension σU( ) isindependent of Tc( ) while the universe pressure pU( ) is thevalue at current CMBR temperature of 2.71 o K as presentedin table 2.7.1 and observed by COBE [26] and later satellites.The cosmological constant also reduces to an expressionrelating trisine relativistic velocity transformed mass mt( ) thegravitational constant G( ) and Planck's constant ( ) . Also areasonable value of 71.2km/sec-million parsec for the Hubbleconstant HU( ) from equation 2.11.13 is achieved based on auniverse absolute viscosity µU( ) and energy dissipationrate PU( ) .
cosmological constant
v
H
c
m G
dxU
U
t
=
16 Gπ σ
π
4
3
3
4
3 2
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
⎫
⎬
⎪⎪⎪⎪⎪
⎭
⎪⎪⎪⎪⎪
(2.11.1)
σU
b ck T
section=
(2.11.2)
Rcosmological constantU =
1 (2.11.3)
RU = 2.25E+28 cm
The mass of the universe MU( ) is established by assuming aspherical isotropic universe (center of mass in the universecenter) of radius RU( ) such that a particle with the speed oflight cannot escape from it as expressed in equation 2.11.4 andgraphically described in figure 2.11.1.
Figure 2.11.1 Closed Universe with Mass MU andRadius RU
M Rc
G
c
m G
U U
t
= ⋅
=
=
2
2 2
3 2
3
4
3.02E+56
π
g
(2.11.4)
29
ρπ
π
U UU
t
t
T
MR
m c G
m
cavity
= ⋅
=
=
3
4
1
4
3
3
6 2
4
cc
b
ET
E
= −=
= −
8 11 162 729
6 38 30
.
.
. g/cm3
(2.11.5)
This value for Universe density is much greater than thedensity of the universe calculated from observed celestialobjects. As reported in reference [27], standard cosmologymodel allows accurate determination of the universe baryondensity of between 1.7E-31 to 4.1E-31 g/cm3. These valuesare 2.7 - 6.4% of the universe density reported herein basedthe background universe as superconductor.
Universe time TR
cU
U( ) =3
=π4
2
3m c Gt
sec
= +4 32 17. E
or .37 years1 10E +
(2.11.6)
Figure 2.11.2 Shear forces along parallel planes of anelemental volume of fluid
As illustrated in Figure 2.11.2 equation 2.11.13 for theHubble constant HU( ) is derived by equating the forces actingon a cube of fluid in shear in one direction to those in theopposite direction, or
p y zd
dzz x y
x y pdp
U
UU
∆ ∆ ∆ ∆ ∆
∆ ∆
+ +⎡⎣⎢
⎤⎦⎥
= + +
τ τ
τddx
x y z∆ ∆ ∆⎡⎣⎢
⎤⎦⎥
(2.11.7)
Here pU is the universe pressure intensity, τ the shearintensity, and ∆x , ∆y , and ∆z are the dimensions of the cube.It follows that:
d
dz
dp
dxUτ
= (2.11.8)
The power expended, or the rate at which work is done by thecouple τ ∆ ∆x y( ) , equals the torque τ ∆ ∆ ∆x y z( ) times theangular velocitydv dz . Hence the universe power PU( )consumption per universe volume VU( ) of fluid is
P V x y z dv dz x y zU U = ( )⎡⎣ ⎤⎦ ( ) = dv dzτ τ∆ ∆ ∆ ∆ ∆ ∆ (( ) .
Defining τ µ= U dv dz
and H dv dzU =
then P V dv dz HU U U U U= ( ) = µ µ2 2
or H P VU U U U2 = ( )µ
The universe absolute viscosity Uµ( ) is based onmomentum/surface area as an outcome of classical kinetictheory of gas viscosity [19]. In terms of our development, themomentum changes over sectional area in each cavity suchthat:
µUt dxm v
section= (2.11.9)
Derivation of the kinematic viscosity υU( ) in terms ofabsolute viscosity as typically presented in fluid mechanics[18], a constant universe kinematic viscosity υU( ) is achievedin terms of Planck's constant ( ) and Cooper CPT Chargeconjugated pair relativistic velocity transformed mass mt( ) .
υ µ
π
Ut
U
t
cavity
m
m
A
B
E
=
=
= −
cm
s
2
1 39 02.eec
(2.11.10)
Derivation of the Universe Energy Dissipation Rate PU( )assumes that all of the universe mass MU( ) is in thesuperconducting state. In other words, our visible universe
30contributes an insignificant amount of mass to the universemass. This assumption appears to be valid due to thesubsequent calculation of the Hubble constant HU( ) of71.2km/sec-million parsec (equation 2.11.13) which fallswithin the experimentally observed value[20] on the order of70-80 km/sec-million parsec and the recently measured 70km/sec-million parsec within 10% as measured by a NASAHubble Space Telescope Key Project Team on May 25, 1999after analysis of 800 Cepheid stars, in 18 galaxies, as far awayas 65 million light years.
PM v v
RUU dx dx
U
= 2
2(2.11.11)
Universe volume V RU U
( ) =4
33π
=3
16
4 6
9 3
πm Gt
= +4 74 85 3. E cm
(2.11.12)
HP
V
m Gc
E
UU
U U
t
=
=
= −
µ
π
4
2 31 18
3
2
. / sec
(2.11.13)
or in more conventionally reported units:
HU = 71 231
.km
sec million parsec
Now from equation 2.11.1 and 2.11.13 the cosmologicalconstant is defined in terms of the Hubble constant HU( ) inequation 2.11.14.
Cosmological constantH
cU=3
=1
RU
4.45E-29 = −cm 1
(2.11.14)
Now from equation 2.11.4 and 2.11.13 the universe massMU( ) is defined in terms of the Hubble constant HU( ) in
equation 2.11.15.
Mc
H G
c
m G
R c
G
UU
t
U
=
=
=
3
3
4
3
2 2
3 2
2
π
= 3 02 56. E g
(2.11.15)
Now from equation 2.11.13 and 2.11.5 the universe densityρU( ) is defined in terms of the Hubble constant HU( ) in
equation 2.11.16. This differs from the universe criticaldensity equation in reference [27] by a factor of 2/3 whichindicates an inflationary universe.
ρπ πU
U UH
G
chain
cavity
H
G
E
= =
= −
2
8
3
8
6 38 30
2 2
.gg
cm3
(2.11.16)
Now from equation 2.11.13 and 2.11.6 the universe time TU( )is defined in terms of the Hubble constant HU( ) in equation2.11.17.
TH
E
UU
=
=
1
1 37 10
4.32E+17
or years.
(2.11.17)
Universe mass creation rate M
P
cUU ( ) =2
(2.11.18)
Table 2.11.1 Vacuum parameters
T Kc
0( ) 3,100,000 2,135 1,190 447 93 8.1E-16
T Kb
0( ) 1.3E+11 3.4E+09 2.5E+09 1.6E+09 7.1E+08 2.729
µU (g/cm/sec) 2.09E+01 3.78E-04 1.57E-04 3.62E-05 3.44E-06 8.85E-32
σU (erg/cm2) 5.02E+11 2.38E+05 7.40E+04 1.04E+04 4.52E+02 3.43E-32
PU (erg/sec) 5.31E+51 9.59E+46 3.99E+46 9.19E+45 8.72E+44 2.24E+19
MU (g/sec) 5.91E+30 1.07E+26 4.44E+25 1.02E+25 9.70E+23 2.50E-02
This development would allow the galactic collision rate
calculation in terms of the Hubble constant HU( ) (which is
considered a shear rate in our fluid mechanical concept of the
universe) in accordance with the Smolukowski relationship
[12, 13, 14] where JU represents the collision rate of galaxies
of two number concentrations n ni j,( ) and diameters d di j,( ) :
31
J n n H d dU i j U i j= +( )1
6
3 (2.11.19)
The superconducting Reynolds' number Re( ) , which isdefined as the ratio of inertial forces to viscous forces ispresented in equation 2.11.20 and has the value of one(1). Interms of conventional fluid mechanics this would indicate acondition of laminar flow [18].
Rm
cavityv cavity
section
m v
et
dx
t dx
=
=
1 1
µ
µµ
section
= 1
(2.11.20)
av
PUEuler
T ET
dy
c
b
=⎛
⎝⎜⎞
⎠⎟
=
−
= −=
πe2
8 11 162 729
2 ..
ccHP c
V
xcm
UU
U U
=
−
2
8210
µ
=6.93
sec
(2.11.21)
The numerical value of the universal deceleration aU( )expressed in equation 2.11.21 is nearly equal to that observedwith Pioneer 10 & 11 [53], but this near equality is viewed ascoincidence only. This acceleration component of theuniversal dark matter/energy may provide the mechanism forobserved universe expansion and primarily at the boundary ofthe elastic space CPT lattice and Baryonic matter (starsystems, galaxies, etc).
2.12 SUPERCONDUCTING VARIANCE AT T/Tc
Establish Gap and Critical Fields as a Function of T Tc
Using Statistical Mechanics [19]
z e es
K
m k T
chain
cavityB
e b c= =−
−2 2
22
3(2.12.1)
cavity
mB
Ak Tt
tb c
=⎛⎝⎜
⎞⎠⎟
3
6
3 3
1
2
3
2
3
2
3
2
π (2.12.2)
z
m k T cavity T
Ttt b c
c
=( )
≈⎛⎝⎜
⎞⎠⎟
2
8
3
2
3 3
1 3
3
2ππ
. % (2.12.3)
kz
ze
T
Te e
T
Ttt
s
k T
k T
c
T
T
c
b c
b
c
= =⎛⎝⎜
⎞⎠⎟
=⎛⎝⎜
⎞−
3
2 2
3
⎠⎠⎟−⎛
⎝⎜⎞⎠⎟
3
2 2
3eT
Tc (2.12.4)
k kchain
cavityks t t= − = −
2
3(2.12.5)
∆Tc
t
c t
T
T
k
T
T k= −
⎛⎝⎜
⎞⎠⎟
= + ( )11
3
1
11
1
3
1
lnln
(2.12.6)
H HT
e TT o Eulerc
12 2 2
2
12
3= −
⎛⎝⎜
⎞⎠⎟
⎡
⎣⎢⎢
⎤
⎦⎥⎥−π
π
at T
Tc
near 0
(2.12.7)
H H kT o s22 2= at
T
Tc
near 1 (2.12.8)
AvgHH H
H HcT
cT cT
cT c
2 12
22
12
=>if
then else TT 22
⎡
⎣⎢
⎤
⎦⎥
(2.12.9)
2.13 SUPERCONDUCTOR ENERGY CONTENT
Table 2.13.1 presents predictions of superconductivity energycontent in various units that may be useful in anticipating theuses of these materials.
Table 2.13.1 Superconductor Energy in Various Units.
T Kc
0( ) 3,100,000 2,135 1,190 447 93 8.1E-16
T Kb
0( ) 1.3E+11 3.4E+09 2.5E+09 1.6E+09 7.1E+08 2.729
erg/cm3 1.61E+20 2.00E+12 4.64E+11 4.01E+10 7.92E+08 1.78E-34
joule/liter 1.61E+16 2.00E+08 4.64E+07 4.01E+06 7.92E+04 1.78E-38
BTU/ft3 4.32E+14 5.37E+06 1.25E+06 1.08E+05 2.13E+03 4.77E-40
kwhr
ft3 1.26E+11 1.57E+03 3.65E+02 3.16E+01 6.23E-01 1.40E-43
kwhr
gallon2.27E+10 2.82E+02 6.54E+01 5.66E+00 1.12E-01 2.51E-44
gasolineequivalent
3.46E+08 4.30E+00 1.00 8.63E-02 1.70E-03 3.82E-46
Table 2.13.2 represents energy level of trisine energy statesand would be key indicators material superconductor
32characteristics at a particular critical temperature. The BCSgap is presented as a reference energy.
Table 2.13.2 Energy (ev) associated with Various TrisineWave Vectors. The energy is computed by
2 22m Kt( ) andexpressing as electron volts(ev). The BCS gap is computedas π exp Euler m Kt B( )( )( )2 22 .
T Kc
0( ) 3,100,000 2,135 1,190 447 93 8.1E-16
T Kb
0( ) 1.3E+11 3.4E+09 2.5E+09 1.6E+09 7.1E+08 2.729
KB2 2.67E+02 1.84E-01 1.03E-01 3.85E-02 8.01E-03 6.99E-20
KC2 8.96E+02 6.17E-01 3.44E-01 1.29E-01 2.69E-02 2.34E-19
KDs2 8.32E+02 5.73E-01 3.19E-01 1.20E-01 2.50E-02 2.18E-19
KDn2 3.20E+02 2.20E-01 1.23E-01 4.62E-02 9.60E-03 8.37E-20
KP2 3.56E+02 2.45E-01 1.37E-01 5.14E-02 1.07E-02 9.31E-20
K KDs B2 2+ 1.10E+03 7.57E-01 4.22E-01 1.58E-01 3.30E-02 2.87E-19
K KDn B2 2+ 5.87E+02 4.04E-01 2.25E-01 8.47E-02 1.76E-02 1.54E-19
K KDs C2 2+ 1.73E+03 1.19E+00 6.63E-01 2.49E-01 5.18E-02 4.52E-19
K KDn C2 2+ 1.22E+03 8.38E-01 4.67E-01 1.75E-01 3.65E-02 3.18E-19
K KDs P2 2+ 1.19E+03 8.18E-01 4.56E-01 1.71E-01 3.56E-02 3.11E-19
K KDn P2 2+ 6.76E+02 4.66E-01 2.60E-01 9.75E-02 2.03E-02 1.77E-19
KA2 6.05E+03 4.17E+00 2.32E+00 8.72E-01 1.81E-01 1.58E-18
BCS gap 4.71E+02 3.25E-01 1.81E-01 6.79E-02 1.41E-02 1.23E-19
2.14 SUPERCONDUCTOR GRAVITATIONALENERGY
As per reference[25], a spinning nonaxisymmetric objectrotating about its minor axis with angular velocity ω willradiate gravitational energy in accordance with equation2.14.1
EG
cIg
.
=32
5 5 32 2 6ς ω (2.14.1)
This expression has been derived based on weak field theoryassuming that the object has three principal moments of inertiaI1, I2, I3, respectively, about three principal axesa b c> > and ς is the ellipticity in the equatorial plane.
ς =−a b
ab(2.14.2)
Equations 2.14.1 with 2.14.2 were developed for predicting
gravitational energy emitted from pulsars but we use them tocalculate the gravitational energy emitted from rotatingCooper CPT Charge conjugated pairs in the trisinesuperconducting mode.
Assuming the three primary trisine axes C>B>Acorresponding to a>b>c in reference [25] as presented above,then the trisine ellipticity would be:
ς =−
=C B
CB1 (2.14.3)
Assuming a mass ms( ) with density ρ( ) containingsuperconducting pairs with density ρC( ) rotating around anonaxisymmetric axis of relativistic radius B to establishmoment of inertia I3( ) then equation 2.14.1 becomes:
EG
cm Bg s
c.
=⎛⎝⎜
⎞⎠⎟
32
5 52
2
6ρρ
ω(2.14.4)
Then integrating with respect to time whereω π= 2 time , andreplacing speed of light c( ) with De Broglie speed oflight vdx( ) , then equation 2.14.4 becomes:
EG
vm B
timegdx
sc= −
⎛⎝⎜
⎞⎠⎟
( )
=
32
5
6 25
2
2 6
5
ρρ
π
−−384
5
6 2
2
2π ρρ
Gm
Bc s
(2.14.5)
In the Podkletnov experiment [7], 5.47834 gram silicondioxide disc was weighed over a rotating superconductingdisk 145 mm in diameter and 6 mm in thickness with amass m( ) of 633 grams based on a YBa Cu O2 2 6 95. density of6.33 g/cm3. Based on equation 2.14.5 the gravitation energyassociated with this superconducting material would be -46.1ergs as indicated in table 2.14.1. Assuming this energy isreflected in gravitational potential energy in regards to theweighed silicon dioxide mass, then it can be reflected in aweight loss of .05% at 17 cm or in other words therelationship in equation 2.14.6 holds.
− = − ⋅ ⋅ ⋅mgh gcm
cm. .sec
0005 5 48 980 172
= −45 5. erg (2.14.6)
In the Podkletnov experiment [7], the .05% weight loss wasobserved with the silicon dioxide weight at 1.5 cm above thesuperconducting mass ms( ) . The size of the silicon dioxide
33mass is not given by Podkletnov but with a weight of 5.47834grams, its projected area onto the superconducting mass mustbe only a few square centimeters compared to the area of thesuperconducting disc of π 14 52. or 660 cm2. These relativearea's may have something to do with the less observed weightloss relative to what is predicted based on relative massesalone.
Table 2.14.1
T Kc
0( ) 3,100,000 2,135 1,190 447 93 8.1E-16
T Kb
0( ) 1.3E+11 3.4E+09 2.5E+09 1.6E+09 7.1E+08 2.729
−Eg
(-erg)-3.1E+17 -2.6E+06 -3.4E+05 -1.1E+04 -4.6E+01 -8.9E-59
All of this is consistent with the Virial Theorem for a systemof particles with total mass M.
Gravitational Potential Energy = Kinetic Energy
1
2
3 1
22
8 11 162 729
2Gmx y z
m vtT ET
r dxc
b
∆ ∆ ∆+ +=
= −=
.
.TT ET
c
b
= −=
8 11 162 729..
(2.14.7)
Also:
1
2
1
2
8 11 162 729
ε x mt dx
T ET
km v
chain
cavity
c
b
= −=
.
.
22
1 2
8 11 162 729
ε εx m T ET
k
e
B c
b
= −=
⎧
⎨
⎪⎪⎪
⎩
⎪⎪⎪
⎫
⎬
⎪⎪
.
.
⎪⎪
⎭
⎪⎪⎪
== −=
1
2
2
2
8 11 162 729
m v
where
r dxT ET
x
c
b
.
.
:
ε kkc
v v
c
mdx dx
T ET
c
b
= =− −
= −=
2 1
1 1
2
2 2
28 11 162 729..
(2.14.7a)
Where mt and mr are related as in Equation 2.1.25 and∆x , ∆y , ∆z are Heisenberg Uncertainties as expressed inequation 2.1.20.In the case of a superconductor, the radius R is the radius ofcurvature for pseudo particles described in this report as thevariable B. All of the superconductor pseudo particles movein asymmetric unison, making this virial equation applicableto this particular engineered superconducting situation. Thisis consistent with Misner Gravitation [35] page 978 where it is
indicated that gravitational power output is related to “powerflowing from one side of a system to the other” carefullyincorporating “only those internal power flows with a time-changing quadrupole moment”. The assumption in this reportis that a super current flowing through a trisinesuperconducting CPT lattice has the appropriate quadrupolemoment at the dimensional level of B. The key maybe alinear back and forth trisine system. Circular superconductingsystems such as represented by Gravity Probe B gyroscopicelements indicate the non gravitational radiation characteristicof such circular systems reinforcing the null results ofmechanical gyroscopes [21, 22, 23].
3. DISCUSSION
This effort is towards formulating of a theoretical frame worktowards explaining the Podkletnov's and Nieminen's resultsbut in the process, relationships between electron/proton mass,charge and gravitational forces were established which appearto be unified under this theory with links to the cosmologicalconstant, the dark matter of the universe, and the universeblack body radiation as a relic of its big bang origins.
The overall theory is an extension to what Sir ArthurEddington proposed during the early part of this century. Heattempted to relate proton and electron mass to the number ofparticles in the universe and the universe radius. Central tohis approach was the fine structure constant c e2( ) , which isapproximately equal to 137. This relationship is also found asa consequence of our development in equation 2.1.16, thedifference being that a material dielectric ε( ) modifiedvelocity of light vε( ) is used. To determine this dielectric vε( ) ,displacement D( ) and electric fields E( ) are determined forCooper CPT Charge conjugated pairs moving through thetrisine CPT lattice under conditions of the trisine CPT latticediamagnetism equal to −1 4π (the Meissner effect). Standardprinciples involving Gaussian surfaces are employed and interms of this development the following relationship isdeveloped based on equation 2.1.17 containing the constant
c e2( ) and relating electron mass me( ) , trisine relativistic
velocity transformed mass mt( ) and the characteristic trisinegeometric ratio B A( ) .
1
24 3 7 12
2
2
1
42
2
2
2
2
m
mg g
c
e
A
B
B
At
ee s =
⎛⎝⎜
⎞⎠⎟
+ +−
±
π 119⎛⎝⎜
⎞⎠⎟
(3.1)
The extension postulated herein is that the principlesexpounded in 'twin paradox' in special relativity are
34applicable to orbiting bodies at the microscopic level. In thiscase, charged particles move in symmetrical vortex patternswithin a superconductor with velocity vε( ) allowed by thedielectric properties of the particular medium. As a result, atime dilation (time±) is generated between this frame ofreference and the stationary frame of reference. Thistransform also results in increase in relativistic mass mt( ) .The trisine symmetry of the medium then allows pseudoparticles (Cooper CPT Charge conjugated pairs) withrelativistic mass mt( ) to move with a corresponding DeBroglie velocity vdx( ) , which expresses itself in a supercurrent J( ) .
At this time, it is not known whether the trisine CPT lattice isthe only Gaussian surface, which would provide such results,but it may be so.
Also, it is important to note, that the trisine surface may ormay not relate to any physical structure in a superconductingmaterial. Although this is so, this theory may provide a basisfor a "jump" in thinking from material withTc ’s below 100degree Kelvin to required anticipated materials withTc ’s onthe order of 1000's of degrees Kelvin.
The present theory predicts the .05% weight loss of the 93degree Kelvin superconducting material in earth'sgravitational field. The presented theory is based on theorydeveloped for predicting gravitational radiation from celestialobjects.
In concert with the weight loss prediction, it is also indicatedthat the superconducting material itself will loose .05% of itweight.
The implications of such a verified effect are profound.Conditions could be postulated as in table 2.9.1, such that amass could be rendered weightless in earth's gravity.Podkletnov and Nieminen alluded to the basic mechanismdeveloped in this report, in that a spinning mass develops itsown gravity as depicted in Figure 2.9.1 although it must benoted that the center of rotation must differ from the center ofmass (nonaxisymmetric spinning) as is the case with thesuperconducting theory presented herein. Also, it is noted thatif somehow the Cooper CPT Charge conjugated pairs could beseparated from the material, the gravitational force shieldingeffect would be much more pronounced or in other words thelower superconducting material density (approaching theCooper CPT Charge conjugated pair density) the greater theeffect.
It is important to note also that the Cooper CPT Chargeconjugated pairs are postulated to travel along a radius ofcurvature (B) with tangential velocity vdT( ) and not in a circle
or in other words the Cooper CPT Charge conjugated pairscenter of masses differs from their center of rotation. Thisvelocity vdT( ) is in three dimensions and the vector sum of x, yand z components. This means that no matter what theorientation of the superconducting material, there will be atangential velocity vector facing the center of the earth asindicated in Figure 2.9.1. In other words, as the earth rotates,there will be a tangential component in the superconductorcircular to the earth's axis as defined by a line from the earth'scenter to the center of rotation of the Cooper CPT Chargeconjugated pairs with a constant gravitational force shieldingeffect.
Also, it is evident from the gravitation shielding principlegraphically presented in figure 6, that it does not matter whatforces hold the rotating mass in orbit whether they bemagnetic, electrical or mechanical. The orbit retainingmechanism postulated in this report is electrical and magnetic.
Also, without quantitative calculations, it is easy to postulatethat a spinning superconducting disc would add to the inherenttangential velocity vdT( ) of the superconductor and thereforeprovide an amplifying effect to the lessening of weight and tothe experimentally observe shielding effect.
As of this time, it is noted that the results of Podkletnov havenot been independently verified. It is thought that the reasonis that a pure YBa Cu O x2 3 7− superconductor having thetheoretically possible super currents is not easily fabricated.Perhaps Podkletnov is the only one who knows how to do thisat this time.
4. CONCLUSION
A momentum and energy conserving (elastic) CPT latticecalled trisine and associated superconducting theory ispostulated whereby electromagnetic and gravitational forcesare mediated by a particle of relativistic velocity transformedmass (110.123 x electron mass) such that the establishedelectron/proton mass is maintained, electron and protoncharge is maintained and the universe radius as computedfrom Einstein's cosmological constant is 2.25E28 cm, theuniverse mass is 3.0E56 gram, the universe density is 6.38E-30 g/cm3 and the universe time is 1.37E10 years.
The cosmological constant is based on a universe surfacetension directly computed from a superconducting energy oversurface area.
The calculated universe mass and density are based on an
35isotropic homogeneous media filling the vacuum of spaceanalogous to the 'aether' referred to in the 19th century (butstill in conformance with Einstein's relativity theory) andcould be considered a candidate for the 'cold dark matter' inpresent universe theories. Universe density is 2/3 ofconventionally calculated critical density. This is inconformance with currently accepted inflationary state of theuniverse.
The 'cold dark matter' emits a CMBR of 2.729 o K as has beenobserved by COBE and later satellites although itstemperature in terms of superconducting critical temperatureis an extremely cold 8.11E-16 o K .
Also, the recently reported results by Podkletnov andNieminen wherein the weight of an object is lessened by .05%are theoretically confirmed where the object is thesuperconductor, although the gravitational shieldingphenomenon of YBa Cu O x2 3 7− is yet to be explained. In futureexperiments, the weight of the superconducting object shouldbe measured. It is understood that the Podkletnov andNieminen experiments have not been replicated at this time. Itcould be that the very high super currents that are requiredhave not be replicated from original experiment. The modelherein interprets the Podkletnov experimental data in thecontext of the virial theorem to which a particularsuperconductor must be engineered.
The trisine model provides a basis for considering thephenomenon of superconductivity in terms of an ideal gaswith 1, 2 or 3 dimensions. The trisine model was developedprimarily in terms of 1 dimension, but experimental data fromMgB2 would indicate a direct translation to 3 dimensions.
Also dimensional guidelines are provided for design of roomtemperature superconductors and beyond. These dimensionalscaling guidelines have been verified by Homes’ Law andgenerally fit within the conventionally describedsuperconductor operating in the “dirty” limit. [54]
The deceleration observed by Pioneer 10 & 11 as they exitedthe solar system into deep space appear to verify the existenceof an space energy density consistent with the amounttheoretically presented in this report. This translationaldeceleration is independent of the spacecraft velocity andindicates that our solar system is comoving with a surroundingor supporting uniform energy or mass density. Also, the samespace density explains the Pioneer spacecraft rotationaldeceleration. This is remarkable in that translational androtational velocities differ by three (3) orders of magnitude.
General conclusions from study of Pioneer deceleraton data areitemized as follows:
1. Dark Matter and Dark Energy are the same thing and arerepresented by a Trisine Elastic Space CPT lattice which iscomposed of virtual particles adjacent Heisenberg Uncertainty
2. Dark Matter and Dark Energy are related to CosmicMicrowave Background Radiation (CMBR) through very largespace dielectric (permittivity) and permeability constants.
3. A very large Trisine dielectric constant and CPT equivalencemakes the Dark Matter/Energy invisible to electromagneticradiation.
4. A velocity independent (v<<c) drag force acts on all objectspassing through the Trisine Space CPT lattice and models thetranslational and rotational deceleration of Pioneer 10 & 11space craft.
5. The Asteroid Belt marks the interface between solar windand Trisine, the radial extent an indication of the dynamics ofthe energetic interplay of these two fields
6. The Trisine Model predicts no Dust in the Kuiper Belt. TheKuiper Belt is made of large Asteroid like objects.
7. The observed Matter Clumping in Galaxies is due tomagnetic or electrical energetics in these particular areas atsuch a level as to destroy the Coherent Trisine.
8. The observed Universe Expansion is due to Trisine SpaceCPT lattice internal pressure.
9. When the Trisine is scaled to molecular dimensions,superconductor parameters such as critical fields, penetrationdepths, coherence lengths, Cooper CPT Charge conjugated pairdensities are modeled.
10. A superconductor is consider electrically neutral withbalanced conjugated charge as per CPT theorem. Observedcurrent flow is due to one CPT time frame being pinnedrelative to observer.
A question logically develops from this line of thinking andthat is “What is the source of the energy or mass density?”.After the energy is imparted to the spacecraft passing throughspace, does the energy regenerate itself locally or essentiallyreduce the energy of universal field? And then there is themost vexing question based on the universal scaledsuperconductivity hypothesis developed in this report and thatis whether this phenomenon could be engineered at anappropriate scale for beneficial use.
One concept for realizing this goal, is to engineer a wavecrystal reactor. Ideally, such a wave crystal reactor would be
36made in the microwave (cm wave length). Perhaps suchmicrowaves would interact at the ω 2 energy level in theTrisine CPT lattice pattern. Conceptually a device asindicated below may be appropriate where the green arearepresents reflecting, generating and polarizing surfaces ofcoherent electromagnetic radiation creating standing waves (inred zone) which interact in the intersecting zone defined bythe trisine characteristic angle of (90 – 22.8) degrees and 120degrees of each other.
Figure 4.1 Conceptual Trisine Generator (Elastic SpaceCPT lattice)
The major conclusion of this report is that an interdisciplinaryteam effort is required to continue this effort involvingphysicists, chemists and engineers. Each has a major role inachieving the many goals that are offered if more completeunderstanding of the superconducting phenomenon is attainedin order to bring to practical reality and bring a more completeunderstanding of our universe.
Figure 4.2 Conceptual Trisine Generator (Elastic SpaceCPT lattice) Within Articulating Motion ControlMechanism
37
5. VARIABLE AND CONSTANT DEFINITIONS
Fundamental Physical Constants are from National Institute of Standards (NIST).
name symbol value units
Attractive Energy V g cm 2 2sec−
Boltzmann constant kb 1.380658120E-16 g cm Kelvin 2 2 1sec− −
Bohr magneton 9.27400949E-21 erg/gauss (gauss cm3)
Bohr radius ao 5.2917724924E-09 cm
De Broglie velocity v v v vdx dy dz dT, , , cm sec−1
Cooper CPT conjugated pair 2 unitless
correlation length ξ cm
critical magnetic fields H H Hc c c1 2, , g cm1
21
2 1 − −sec
critical temperature TcoK
dielectric constant ε unitless
Dirac’s number gd 1.001159652186 unitless
displacement & electric fields D E, g cm1
21
2 1 − −sec
electron mass me 9.10938975E-28 g
earth radius RE 6.371315E+08 cm
earth mass M E 5.979E+27 g
relativistic mass mr g
proton mass mp 1.67262311E-24 g
electron charge e 4.803206799E-10 cm g3
21
2 1sec−
electron gyromagnetic-factor ge 2.0023193043862 unitless
energy Ε g cm2 2sec−
fluxoid Φ 2.0678539884E-7 g cm1
23
2 1sec−
fluxoid dielectric ε( ) mod Φε g cm1
23
2 1sec−
gravitational constant G 6.6725985E-8 cm g3 2 1sec− −
London penetration depth λ cm
momentum vectors p p px y z, , g c m− −1 1sec
natural log base e 2.718281828459050 unitless
Planck's constant h 2π( ) 1.054572675E-27 g cm 2 1sec−
relativistic CPT time± time± second
superconductor gyro-factor gs 1.0096915 unitless
trisine angle θ 22.80 angular degrees
trisine areas section, approach, side cm2
trisine unit cell unitless
trisine constant trisine 2.01091660 unitless
38
trisine density of states D t∈( ) sec2 1 2g cm− −
trisine dimensions A, B, C, P cm
trisine mass mt 1.0031502157E-25 g
transformed mass mt 110.71029 x me g
trisine size ratio B At
( ) 2.37933 unitless
trisine volume per cell cavity cm3
trisine volume per chain chain cm3
trisine wave vectors K K K K KA B C Dn Ds, , , , cm−1
universe absolute viscosity µU g c m− −1 1sec
universe density ρU 1.45E-29 g c m−3
universe kinematic viscosity νU cm2 1sec−
universe radius RU 1.49E+28 cm
universe mass M E 2.01E+56 g
universe pressure pU g c m− −1 2sec
universe time TU sec
velocity of light c 2.997924580E+10 cm sec−1
ε modified velocity of light vε cm sec−1
39
APPENDIX A. TRISINE NUMBER AND B/ARATIO DERIVATION
BCS approach [2]
∆oBCS
TD V
=
∈( )⎛
⎝⎜⎞
⎠⎟
ω
sinh1
(A.1)
Kittel approach [4-Appendix H]
∆o
D Ve T
=−∈( )
2
11
ω (A.2)
Let the following relationship where BCS (equation A.1) andKittel (equation A.2) approach equal each other define aparticular superconducting condition.
k T
D V
e
b c
Euler
T
D VT
=
∈( )⎛
⎝⎜⎞
⎠⎟
−∈( )
e
πω
ω
sinh1
11
⎧⎧
⎨
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
(A.3)
Define:
1
D Vtrisine
T∈( ) = (A.4)
Then:
trisineEuler
= − −⎛⎝⎜
⎞⎠⎟
=
ln
.
21
2 010
e
π991660
(A.5)
Given:
22
m k T
KK
trisine
KK
t b c
Euler
=
( )
( )
ω
ω
πe
e
sinh( )
ttrisine
BK
−
⎧
⎨
⎪⎪⎪
⎩
⎪⎪⎪
⎫
⎬
⎪⎪⎪
⎭
⎪⎪⎪
=
1
2
Ref: [2,4]
(A.6)
And within Conservation of Energy Constraint:
m m
m
m m
mK K K Kp e
p
t e
tB C Ds D
−⎛
⎝⎜⎞
⎠⎟−⎛
⎝⎜⎞⎠⎟
+( ) = +2 2 2nn
2( ) (A.7)
And Within Conservation of Momentum Constraint:
m
m m
m
m mK K K Kp
p e
t
t eB C Ds Dn−
⎛
⎝⎜⎞
⎠⎟ −⎛⎝⎜
⎞⎠⎟
+( ) = +( ) (A.8)
And from KK( ) from table A.1:
KK K KC Ds( ) = +2 2(A.9)
Also from table A.1:
2 2 23 527754
065∆oBCS
b cEuler
Ds
Bk T
K
K= = ≈
πe
. %
. (A.10)
Also:
∆o
b ck T= 2 (A.11)
Also from table A.1:
trisineK K
K KB P
P B
=+2 2
(A.12)
Table A.1 Trisine Wave Vector Multiples KK( ) Compared
To K KB B
KK e
trisine
Euler( )π sinh( )
K KK
K KB B
( ) KK
K KB Btrisine
( )−
1
1e
K
KB
KDn 1.1981 0.1852 1.0946KP 1.3333 0.2061 1.1547KDs 3.1132 0.4812 1.7644KC 3.3526 0.5182 1.8310KA 22.6300 3.4976 4.7571
40
APPENDIX B. DEBYE MODEL NORMAL ANDTRISINE RECIPROCAL LATTICEWAVE VECTORS
This appendix parallels the presentation made in ref [4 pages121 - 122].
N
LK
LK
Dn
Ds
=
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
⎡
⎣
⎢⎢⎢⎢⎢⎢
⎤
⎦
2
4
3
2
33
33
ππ
π
⎥⎥⎥⎥⎥⎥⎥
=
Debye
Spherical
Condition
Debye
Trisinne
Condition
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥
(B.1)
where:
K K K KDs A B P= ( )1
3 (B.2)
dN
d
cavityK
dK
d
cavityK
dK
d
DnDn
DsDs
ω
π ω
π
=2
83
22
32
ωω
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
=
Debye
Spherical
Condition
Debbye
Trisine
Condition
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥
(B.3)
Kv
=ω (B.4)
dK
d vω=
1 (B.5)
dN
d
cavity
v
cavity
v
ω
πω
πω
=
⋅
⎡
⎣
⎢⎢⎢⎢⎢⎢
⎤
⎦
2
3
8
2
2
3
3
2
3
⎥⎥⎥⎥⎥⎥⎥
=⎡
⎣
⎢⎢⎢
⎤
⎦
Debye spherical
Debye trisine
⎥⎥⎥⎥ (B.6)
N
cavity
v
cavity
v
=
⎡
⎣
⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
6
8
2
3
3
3
3
3
πω
πω
⎥⎥
=
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
cavityK
cavityK
Dn
Ds
6
8
23
33
π
π
==
Debye
Spherical
Condition
Debye
Trisine
Condiition
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥
(B.7)
Given that N = 1 Cell
K
K
cavity
cavity
Dn
Ds
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
=
⎛⎝⎜
⎞⎠⎟
6
8
21
3
3
π
π⎛⎛⎝⎜
⎞⎠⎟
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥
=1
3
Debye
Sphericaal
Condition
Debye
Trisine
Condition
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥
(B.8)
APPENDIX C. ONE DIMENSION AND TRISINEDENSITY OF STATES
This appendix parallels the presentation made in ref [4 pages144-155].
Trisine
2
2
3
3
K
L
NC
π⎛⎝⎜
⎞⎠⎟
= (C.1)
One Dimension
∈ =
( )2
22
22 2mN
Bt
π (C.2)
Now do a parallel development of trisine and one dimensiondensity of states.
N
K cavity
m B
C
t
=
⋅
⎛⎝⎜
⎞⎠⎟
∈
⎡
⎣
⎢⎢⎢⎢⎢⎢
2
8
2 2
3
3
2
1
21
2
π
π⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥
=
Trisine
Dimensional
Condition
One
Dimensional
Condition
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
(C.3)
41
dN
d
cavity m
m
t
t
∈=
⋅ ⎛⎝⎜
⎞⎠⎟
∈
⎛⎝⎜
⎞
2
8
2 3
2
2
3 2
3
21
2
2
π
⎠⎠⎟∈
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥
=
−1
21
22 1
2
B
π
Trisine
Dimmensional
Condition
One
Dimensional
Condition
⎡⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥
(C.4)
dN
d
cavity m
m
t
t
∈=
⋅ ⎛⎝⎜
⎞⎠⎟
∈
⎛⎝⎜
⎞
2
8
2 3
2
2
3 2
3
21
2
2
π
⎠⎠⎟∈
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥
=
−1
21
22 1
2
B
π
Trisine
Dimmensional
Condition
One
Dimensional
Condition
⎡⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥
(C.5)
dN
d
cavity m mt t
∈=
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
−
23
2 8
2 23 2
3
2
2
1
π22
2
1
2
2
1
22 2 1
2
2 1
K
m B m
K
C
t t
B
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
⎡
⎣
⎢⎢⎢
π
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥
=
Trisine
Dimensional
Conddition
One
Dimensional
Condition
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥
(C.6)
dN
dD
cavity mK
m
K
tC
t
B
∈= ∈( ) =
⎛⎝⎜
⎞⎠⎟
⎡
⎣
3
8
2
2
3 2
2 2
π⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥
=
Trisine
Dimensional
Conndition
One
Dimensional
Condition
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥
(C.7)
Now equating trisine and one dimensional density of statesresults in equation C.8.
Kcavity K
A
CB
=⋅ ⋅
=
8
3
4
3 3
3
2
π
π
(C.8)
∈ = energy D ∈( ) = density of states
APPENDIX D. GINZBURG-LANDAUEQUATION AND TRISINESTRUCTURE RELATIONSHIP
This appendix parallels the presentation made in ref [4Appendix I].
ψ αβ
2 =
= Cooper
Cavity
(D.1)
αβ
2
2 2
2
2
2
=
=⋅
=
k T
chain
K
m chain
H
b c
B
t
c
88π
(D.2)
λπ ψ
βπ α
ε
ε
ε
=⋅( )
=
=
m v
Cooper e
m v
q
m v
t
t
t
2
2 2
2
2
4
4
22
24
cavity
Cooper e Cooperπ ⋅( )
(D.3)
β α
α
=
=
2
2 2
2
2
2
2
m chain
K
chain
k T
t
B
b c
(D.4)
42
α β
α
=
=
Cooper
cavity
m chain
K
Cooper
cat
B
2
2 2
2
2 vvity
chain
k T
Cooper
cavityb c
=α 2
2
(D.5)
α =
=
k Tcavity
chain
K
m
cavity
chain
b c
B
t
2 2
2
(D.6)
ξα
22
2
2 2
2
2
2
≡
=⋅⋅
m
m
m chain
K cavity
e
e
t
B
=m
m K
chain
cavityt
e B
12
(D.7)
Hsection
m
m
cavity
cha
c
t
e
2
2
2 4=
⋅ ⋅
=
φ π
φπξ
ε
ε iin
(D.8)
APPENDIX E. LORENTZ EINSTEINTRANSFORMATIONCONSEQENCES
Given relativistic CPT times± ∆t•( ) and ∆t( ) then define theLorentz Einstein Transformation Consequence as:
∆∆ ∆
∆t
t
v
c
andt
t
v
c=
−= −
• •
1
12
2
2
2
(E.1)
then
1 1 12
2− =
−= − −
• •∆∆
∆ ∆∆
t
t
t t
t
v
c
' (E.2)
Then
∆∆ ∆
∆ ∆ ∆
tt t
v
c
and t t tc
vwhe
=−
− −
= −( )
•
•
1 1
2
2
2
2
2
rre v c <<
(E.3)
Given relativistic lengths ∆l•( ) and ∆l( ) then define theLorentz Einstein Transformation Consequence as:
∆∆ ∆
∆l
l
v
c
andl
l
v
c•
•=−
= −1
12
2
2
2
(E.4)
then
1 1 12
2− =
−= − −•
•
•
∆∆
∆ ∆∆
l
l
l l
l
v
c
(E.5)
then
∆∆ ∆
∆ ∆ ∆
ll l
v
c
and l l lc
vw
••
• •
=−
− −
= −( )
1 1
2
2
2
2
2 hhere v c <<
(E.6)
Given relativistic masses ∆m•( ) and ∆m( ) then define the
Lorentz Einstein Einstein Transformation Consequence as:
∆∆ ∆
∆m
m
v
c
andm
m
v
c•
•=−
= −1
12
2
2
2
(E.7)
then
1 1 12
2− =
−= − −•
•
•
∆∆
∆ ∆∆
m
m
m m
m
v
c
(E.8)
then
∆∆ ∆
∆ ∆ ∆
mm m
v
c
and m m mc
vwh
••
• •
=−
− −
= −( )
1 1
2
2
2
2
2
eere v c <<
(E.9)
43
6. REFERENCES
[1] William Melis and Richard Saam, "U. S. Patent4,526,691", Separator Apparatus, July 2, 1985.
[2] J. Bardeen, L. N. Cooper and J. R. Schrieffer, "Theory ofSuperconductivity", Physical Review, Vol 28, Number6, December 1, 1957, pages 1175-1204.
[3] E. Schrödinger, "An Undulatory Theory of theMechanics of Atoms and Molecules", Physical Review,Vol 108, Number 5, December , 1926, pages 1049-1069.
[4] Charles Kittel, Introduction to Solid State Physics,Seventh Edition, John Wiley & Sons, Inc., New York,N.Y., 1996.
[5] W. A. Little, "Experimental Constraints on Theories ofHigh Transition Temperature Superconductors" ,Department of Applied Physics, Stanford University,1987.
[6] Leonard Eyges, "The Classical Electromagnetic Field" ,Dover Edition, New York, New York, 1980.
[7] E. Podkletnov and R. Nieminen, "A Possibility ofGravitational Force Shielding by Bulk YBa Cu O x2 2 7−
Superconductor" , Physica C 203, (1992), 414-444.
[8] Sir Arthur Eddington, "The Universe and the Atom" fromThe Expanding Universe, Cambridge University Press.
[9] Giovanni Modanese, "Theoretical Analysis of a ReportedWeak Gravitational Shielding Effect" , MPI-PhT/95-44,May 1995, Max-Planck-Institut für Physik, Werner-Heisenberg-Institut, Föhringer Ring 6, D 80805 München(Germany) to appear in Europhysical Letters.
[10] Giovanni Modanese, "Role of a 'Local' CosmologicalConstant in Euclidian Quantum Gravity", UTF-368/96Jan 1996, Gruppo Collegato di Trento, Dipartimento diFisica dell'Universita I-38050 POVO (TN) - Italy
[11] J. E. Sonier et al, "Magnetic Field Dependence of theLondon Penetration Depth in the Vortex State ofYBa Cu O2 2 6 95. ", Triumf, Canadian Institute for AdvancedResearch and Department of Physics, University ofBritish Columbia, Vancouver, British Columbia, Canada,V6T.
[12] T. R. Camp and P. C. Stein, "Velocity Gradients andInternal Work in Fluid Motion", J. Boston Soc. CivilEngrs., 30, 219 (1943).
[13] Miron Smoluchowski, Drei Vorträge über Diffusion,Brownsche Molekularbewegung und Koagulation vonKolloidteilchen (Three Lectures on Diffusion, BrownianMotion, and Coagulation of Colloidal Particles), Phys. Z.,17, 557 (1916); Versuch einer Mathematischen Theorieder Koaguationskinetik Kolloider Lösugen (Trial of aMathematical Theory of the Coagulation Kinetics ofColloidal Solutions), Z. Physik. Chem., 92, 129, 155(1917).
[14] Fair, Geyer and Okun, Water and WastewaterEngineering, John Wiley & Sons, Inc. New York, 1968,pages 22-9 to 22-14.
[15] Georg Joos, Theoretical Physics, translated from the firstGerman Edition by Ira Freeman, Hafner PublishingCompany, Inc., New York 1934.
[16] R. Weast, Handbook of Chemistry and Physics, 53Edition, The Chemical Rubber Company, Cleveland,Ohio, 1972-73.
[17] D. R. Harshman and A. P. Mills, Jr., "Concerning theNature of High-Tc Superconductivity: Survey ofExperimental Properties and Implications for interlayerCoupling", Physical Review B, 45,18,1May 92.
[18] J. Vennard, Fluid Mechanics, John Wiley & Sons, Inc.,New York, NY, 1961.
[19] W. Moore, Physical Chemistry, Prentice-Hall, Inc.,Englewood Cliffs, NJ, 1962.
[20] Takeshi Fukuyama, "Late-Time Mild Inflation-a possiblesolution of dilemma: cosmic age and the Hubbleparameter", Department of Physics, RitsumeikanUniversity, Kusatsu Shiga, 525-77 JAPAN, August 14,1996.
[21] Hayasaka and Takeuchi, "Anomalous Weight Reductionon a Gyroscope's Right Rotations around the VerticalAxis on the Earth", Physical Review Letters, 63, 25,December 18, 1989, p 2701-4.
[22] Faller, Hollander, and McHugh, "Gyroscope-Weighing
44Experiment with a Null Result", Physical Review Letters,64, 8, February 19, 1990, p 825-6.
[23] Nitschke and Wilmarth, "Null Result for the WeightChange of a Spinning Gyroscope", Physical ReviewLetters, 64, 18, April 30, 1990, p 2115-16.
[24] Cram and Hammond, Organic Chemistry, McGraw-HillBook Company, New York, NY 1964.
[25] Kimberly, Chanmugan, Johnson, & Tohline, MillisecondPulsars Detectable Sources of Continuous GravitationalWaves, Astrophysics J., 450:757, 1995.
[26] Staggs, Jarosik, Meyer and Wilkinson, Enrico FermiInstitue, University of Chicago, Chicago, Illinois 60637,Sept 18, 1996.
[27] Copi, Schramm and Turner, Science, 267, January 13,1995, pp 192-8.
[28] Albert Einstein, The Meaning of Relativity, Fifth Edition,MJF Books, New York, NY 1956.
[29] Albert Einstein, The Special and the General Theory,Three Rivers Press, New York, NY 1961.
[30] Wolfgang Pauli, The Theory of Relativity, DoverPublications, New York, NY 1958.
[31] Robert M. Wald, General Relativity, University ofChicago Press, Chicago, 1984.
[32] Ray D'Inverno, Introducing Einstein's Relativity,Clarendo Press, Oxford, 1995.
[33] Paul A. M. Dirac, General Theory of Relativity,Princeton University Press, Princeton, NJ, 1996.
[34] Barrett O'Neill, The Geometry of Kerr Black Holes, A KPeters, Wellesley, MA, 1995.
[35] Charles W. Misner, Kip S. Thorne & John A. Wheeler,Gravitation, W. H. Freeman and Company, New York,1998.
[36] Rutherford Aris, Vectors, Tensors, and the BasicEquations of Fluid Mechanics, Dover Publications, NewYork, 1962.
[37] Heinrich W. Guggenheimer, Differential Geometry,Dover Publications, New York, 1977.
[38] Hans Schneider, George P. Barker, Matrices and LinearAlgebra, Dover Publications, New York, 1962.
[39] Bernard Schutz, Geometrical Methods of MathematicalPhysics, Cambridge University Press, Cambridge, 1980.
[40] Richard P. Feynman, Six Easy Pieces, Addison-WesleyPublishing Company, New York, 1995.
[41] Jed Z. Buchwald, From Maxwell to Microphysics,University of Chicago Press, Chicago, 1973.
[42] Thomas K. Simpson, Maxwell on the ElectromagneticField, Rutgers University Press, New Brunswick, NJ,1997.
[42] Leonard Eyges, The Classical Electromagnetic Field,Dover Publications, New York, 1972.
[43] L. D. Landau and E. M. Lifshitz, The Classical Theory ofFields, Volume 2, Butterworth-Heinenann, Oxford, 1998.
[44] C. K. Birdsall & A. B. Langdon, Plasma Physics viaComputer Simulation, Institute of Physics Publishing,Philadelphia, 1991.
[45] Peter W. Milonni, The Quantum Vacuum, AnIntroduction to Quantum Electrodynamics, AcademicPress, New York, 1994.
[46] Claude Cohen-Tannoudju, Jacques Dupont-Roc &Gilbert Grynberg, Photons & Atoms, Introduction toQuantum Electrodynamics, John Wiley & Sons, Inc.,New York, 1989.
[47] Eugen Merzbacher, Quantum Mechanics, Third Edition,John Wiley & Sons, Inc., New York, 1998.
[48] James P. Runt and John J. Fitzgerald, DielectricSpectroscopy of Polymeric Materials, Fundamentals andApplications, American Chemical Society, Washington,DC, 1997.
[49] Michael Tinkham, Introduction to Superconductivity,McGraw-Hill Companies, New York, NY, 1996.
[50] Eugene Hecht, Optics Third Edition, Addison-Wesley,
45New York, NY, 1998.
[51] C. C. Homes et al, Universal Scaling Relation in High-Temperature Superconductors. Nature, Vol 430, 29 July2004.http://xxx.lanl.gov/abs/cond-mat/0404216
[52] Jan Zaanen, Why the Temperature is High, Nature, Vol430, 29 July 2004.
[53] Study of the anomalous acceleration of Pioneer 10 and11, John D. Anderson, Philip A. Laing, Eunice L. Lau,Anthony S. Liu, Michael Martin Nieto, and Slava G.Turyshev, July 9, 2004http://arxiv.org/abs/gr-qc/0104064
[54] C. C. Homes et al, Are high-temperature superconductorsin the dirty limit.http://xxx.lanl.gov/abs/cond-mat/0410719
[55] N. Klein, B.B. Jin, J. Schubert, M. Schuster, H.R. Yi,Forschungszentrum Jülich, Institute of Thin Films andInterfaces, D-52425 Jülich, Germany, A. Pimenov, A.Loidl, Universität Augsburg, Experimentalphysik V,EKM, 86135 Augsburg, Germany, S.I.Krasnosvobodtsev, P.N. Lebedev Physics Institute,Russian Academy of Sciences, 117924 Moscow, Russia,http://xxx.lanl.gov/abs/cond-mat/0107259, Energy gapand London penetration depth of MgB2 films determinedby microwave resonator , Submitted to Physical ReviewLetters, June 29, 2001. MgB2 superconductivityoriginally discovered by Nagamatsu et al. 2001Superconductivity at 39 K in magnesium diboride Nature410 63
[56] Directly Measured Limit on the Interplanetary MatterDensity from Pioneer 10 and 11 Michael Martin Nieto,Slava G. Turyshev,b and John D. Andersonhttp://arxiv.org/abs/astro-ph/0501626
[57] Study of the Pioneer Anomaly: A Problem Set, Slava G.Turyshev, Michael Martin Nieto, and John D.Anderson http://arxiv.org/abs/astro-ph/0502123
[58] Heliocentric Trajectories for Selected Spacecraft, Planets,and Comets, NSSDC Goddard National SpaceflightCenter, http://nssdc.gsfc.nasa.gov/space/helios/heli.html
[59] Scaling of the superfluid density in high-temperaturesuperconductors Authors: C. C. Homes, S. V. Dordevic,
T. Valla, M. Stronginhttp://xxx.lanl.gov/abs/cond-mat/0410719
[60] Frank M. White, Viscous Fluid Flow, Second Edition,McGraw Hill. NY, NY, 1974
[61] F. W. Sears, M. W. Zemansky, University Physics, ThirdEdition, Addison-Wesley. Reading, MA, 1964
[62] James L. Anderson, Thompson Scattering in anExpanding Universe,http://xxx.lanl.gov/abs/gr-qc/9709034
[63] Michael Martin Nieto, Slava G. Turyshev, Measuring theInterplanetary Medium with a Solar Sail,http://xxx.lanl.gov/abs/gr-qc/0308108
Acknowledgements: Jiri Kucera, Joseph Krueger,Chris Tolmie, William Rieken,Alan Schwartz
