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ECE Engineering Model. The Basis for Electromagnetic and Mechanical Applications Horst Eckardt, AIAS. Version 4.5, 26.1.2014. ECE Field Equations I. Field equations in mathematical form notation with ᶺ : antisymmetric wedge product T a : antisymmetric torsion form - PowerPoint PPT Presentation
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1
ECE Engineering Model
The Basis for Electromagnetic and Mechanical Applications
Horst Eckardt, AIAS
Version 4.5, 26.1.2014
2
ECE Field Equations I• Field equations in mathematical form notation
• with– ᶺ: antisymmetric wedge product– Ta: antisymmetric torsion form– Ra
b: antisymmetric curvature form– qa: tetrad form (from coordiate transformation)– ~: Hodge dual transformation– D operator and q are 1-forms, T and R are 2-forms– summation over same upper and lower indices
bbaa
bbaa
qRTD
qRTD
~~
F~
ECE Axioms• Geometric forms Ta, qa are interpreted as
physical quantities• 4-potential A proportional to Cartan tetrade q:
Aa=A(0)qa
• Electromagnetic/gravitational field proportional to torsion:Fa=A(0)Ta
• a: index of tangent space• A(0): constant with physical dimensions
3
4
ECE Field Equations II• Field equations in tensor form
• with– F: electromagnetic field tensor, its Hodge dual, see
later– Ω: spin connection– J: charge current density– j: „homogeneous current density“, „magnetic current“– a,b: polarization indices– μ,ν: indexes of spacetime (t,x,y,z)
ab
baaa
abb
aaa
JFRAF
jFRAF
0)0(
0)0(
:
:~~~
F~F~
5
Properties of Field Equations
• J is not necessarily external current, is defined by spacetime properties completely
• j only occurs if electromagnetism is influenced by gravitation, or magnetic monopoles exist, otherwise =0
• Polarization index „a“ can be omitted if tangent space is defined equal to space of base manifold (assumed from now on)
6
Electromagnetic Field Tensor• F and are antisymmetric tensors, related to
vector components of electromagnetic fields (polarization index omitted)
• Cartesian components are Ex=E1 etc.
00
00
123
132
231
321
cBcBEcBcBEcBcBEEEE
F
00
00
~
123
132
231
321
EEcBEEcBEEcBcBcBcB
F
F~
7
Potential with polarization directions
• Potential matrix:
• Polarization vectors:
)3(3
)2(3
)1(3
)3(2
)2(2
)1(2
)3(1
)2(1
)1(1
)3()2()1()0(
000
AAAAAAAAA
)3(
3
)3(2
)3(1
)3(
)2(3
)1(2
)2(1
)2(
)1(3
)1(2
)1(1
)1( ,,AAA
AAA
AAA
AAA
Law Maxwell-Ampère1
Law Coulomb
Induction of LawFaraday 0
Law Gauss0'
02
0
0
0
ae
aa
aea
aeh
aeh
aa
aeh
aeh
a
tc
't
JEB
E
jjBE
B
8
ECE Field Equations – Vector Form
„Material“ Equations
ar
a
ar
a
HB
ED
0
0
Dielectric Displacement
Magnetic Induction
9
ECE Field Equations – Vector Form without Polarization Index
„Material“ Equations
HBED
0
0
r
r
Dielectric Displacement
Magnetic Induction
Law Maxwell-Ampère1
Law Coulomb
Induction of LawFaraday 0
Law Gauss0'
02
0
0
0
e
e
eheh
eheh
tc
't
JEB
E
jjBE
B
10
mA
mC
mAN
msVT
mV
][,][
][
][
2
2
HD
B
E
mVsV
][
][
Am1s1
][
][ 0
ω
Physical Units
Charge Density/Current „Magnetic“ Density/Current
msAmA
eh
eh
][
][ 2
j
)/(/][
/][22
3
smCmA
mC
e
e
J
2
3
]'[
]'[
mVmVs
eh
eh
j
11
Field-Potential Relations IFull Equation Set
Potentials and Spin Connections
Aa: Vector potentialΦa: scalar potentialωa
b: Vector spin connectionω0
ab: Scalar spin connection
Please observe the Einstein summation convention!
bbaaa
bbab
ba
aaa
tAωAB
ωAAE
0
12
ECE Field Equations in Terms of Potential I
ae
bbaab
baa
bbaaa
aeb
bab
baa
a
bba
bbab
ba
bba
ttttc
t
t
JωAA
AωAA
ωAA
AωωA
Aω
00
2
2
2
00
0
)()(1
)()(
:Law Maxwell-Ampère
)()(
:Law Coulomb
0)()()(
:Induction of LawFaraday 0)(
:Law Gauss
13
Antisymmetry Conditions ofECE Field Equations I
00
b
bab
ba
aa
tωAA
0
0
0
12,21,2
1
1
2
13,31,3
1
1
3
23,32,3
2
2
3
bbab
ba
aa
bbab
ba
aa
bbab
ba
aa
AAxA
xA
AAxA
xA
AAxA
xA
Electricantisymmetry constraints:
Magneticantisymmetryconstraints:
Or simplifiedLindstrom constraint: 0 b
baa AωA
14
AωAB
ωAAE
0t
Field-Potential Relations IIOne Polarization only
Potentials and Spin Connections
A: Vector potentialΦ: scalar potentialω: Vector spin connectionω0: Scalar spin connection
15
ECE Field Equations in Terms of Potential II
e
e
ttttc
t
t
JωAA
AωAA
ωAA
AωωA
Aω
00
2
2
2
00
0
)()(1
)()(:Law Maxwell-Ampère
)()(
:Law Coulomb
0)()()(
:Induction of LawFaraday 0)(
:Law Gauss
16
Antisymmetry Conditions ofECE Field Equations II
All these relations appear in addition to the ECE field equations and areconstraints of them. They replace Lorenz Gauge invariance and can be used to derive special properties.
Electric antisymmetry constraints: Magnetic antisymmetry constraints:
00
ωAA t
0
0
0
12212
1
1
2
13313
1
1
3
23323
2
2
3
AAxA
xA
AAxA
xA
AAxA
xA
0 AωAor:
00
:attentiongiven be tohave rsDenominato
)(2
1
)(21
:potentials thefrom calculated becan sconnectionspin Thus
)(21
220
0
A
tAA
t
t
AAAω
Aω
AωA
17
Relation between Potentials and Spin Connections derived from Antisymmetry Conditions
18
Alternative I: ECE Field Equations with Alternative Current Definitions (a)
18
aabb
aaa
aabb
aaa
abb
aaa
abb
aaa
JRATFFD
jRATFFD
JTRAF
jTRAF
0)0(
0
)0(
0)0(
0
)0(
:
1:~~~~:)maintained derivative (covariant definition eAlternativ
:)(
1:)~~(~:like)-(Maxwell currents of definition ECE Standard
19
Alternative I: ECE Field Equations with Alternative Current Definitions (b)
19detector andobserver between velocity relative is
with
Law Maxwell-Ampère1
Law Coulomb
Induction of LawFaraday 0
Law Gauss0'
02
0
0
0
v
v
JEB
E
jjBE
B
tdtd
dtd
c
'dtd
ae
aa
aea
aeh
aeh
aa
aeh
aeh
a
1.pdf)ps/phipps0files/phipsc3/elmag/lfire.com///www.ange:(http
20
Alternative II: ECE Field Equations with currents defined by curvature only
ρe0, Je0: normal charge density and current
ρe1, Je1: “cold“ charge density and current
100
2
002
2
2
0
10
0
0
)()(1)(
1)(
:Law Maxwell-Ampère
)()(
:Law Coulomb
e
e
e
e
ttc
ttc
t
JωAAω
JAAA
ωA
A
21
B
EtωAB
ωAE
Field-Potential Relations IIILinearized Equations
Potentials and Spin Connections
A: Vector potentialΦ: scalar potentialωE: Vector spin connection of electric fieldωB: Vector spin connection of magnetic field
22
ECE Field Equations in Terms of Potential III
eE
B
eE
BE
B
tttc
t
t
JωA
ωAA
ωA
ωω
ω
02
2
2
0
1
)(:Law Maxwell-Ampère
:Law Coulomb
0
:Induction of LawFaraday 0
:Law Gauss
23
Electric antisymmetry constraints:
Antisymmetry Conditions ofECE Field Equations III
21
21
BBB
EEE
ωωωωωω
0
0
21
2
1
1
2
1
3
3
1
3
2
2
3
21
BB
EE
xA
xA
xA
xA
xA
xA
t
ωω
ωωA
Magnetic antisymmetry constraints:
Define additional vectorsωE1, ωE2, ωB1, ωB2:
Curvature Vectors
24
2
0
000
1][][)(
: Units :onpolarisatiwithout )(
:field) (magnetic curvatureSpin
1)(
:onpolarisatiwithout
1)(
:field) (electric curvature Orbital
mspin
spin
tcorbital
tcorbital
ba
Bba
EB
bc
ca
ba
ba
ba
B
E
ca
bc
bc
cab
a
ba
ba
ba
E
RRωRR
ωωωRR
ωRR
ωωωRR
Geometrical Definition of Electric Charge/Current Densities
25
ba
Bb
ba
Ebb
bab
baa
e
ba
Ebb
baa
e
c
c
RARBωEJ
RAEω
11:current Electric
:density Charge
000
0
BEe
Ee
c
c
RARBωEJ
RAEω
11:current Electric
:density Charge
000
0
With polarization:
Without polarization:
Geometrical Definition of Magnetic Charge/Current Densities
26
ba
Eb
ba
Bbb
bab
baa
eh
ba
Bbb
baa
eh
c RAREωBJ
RABω
0'
:current sHomogeneou'
:density charge sHomogeneou
EBeh
Beh
c RAREωBJ
RABω
0':current sHomogeneou
':density charge sHomogeneou
With polarization:
Without polarization:
Additional Field Equations due to Vanishing Homogeneous Currents
27
00
bba
ba
Eb
ba
Bbb
bab
ba
ba
Bbb
ba
c
Aω
RARBEω
RABω
With polarization:
Without polarization:
00
AωRARBEω
RABω
EB
B
c
Resonance Equation of Scalar Torsion Field
28
abba
a
cRTtT
00
0
With polarization:
Without polarization:
cRTtT
00
0
Physical units:
2
0
1][
1][
mR
mT
Equations of the Free Electromagnetic Field/Photon
29
Field equations:
Spin equations:
010
00
010
0
0
02
0
2
EBω
EωBEω
Bω
EB
E
BE
B
c
tc
t
frequencytimeenergy
Emomentum
vectorwave
numberwave
0
0
E
c
p
ωκpκ
κω
30
Properties of ECE Equations• The ECE equations in potential representation
define a well-defined equation system (8 equations with 8 unknows), can be reduced by antisymmetry conditions and additional constraints
• There is much more structure in ECE than in standard theory (Maxwell-Heaviside)
• There is no gauge freedom in ECE theory• In potential representation, the Gauss and Faraday
law do not make sense in standard theory (see red fields)
• Resonance structures (self-enforcing oscillations) are possible in Coulomb and Ampère-Maxwell law
31
Examples of Vector Spin Connection
toroidal coil:ω = const
linear coil:ω = 0
Vector spin connection ω represents rotation of plane of A potential
A
B
ω
B
A
32
ECE Field Equations of Dynamics
Only Newton‘s Law is known in the standard model.
Law) Maxwell-Ampère of Equivalent(cG41
equation)(Poisson Law sNewton'4
Law magnetic-Gravito0cG41
Law) Gauss of Equivalent(04
m
m
mh
mh
tc
Gtc
G
Jgh
g
jhg
h
33
ECE Field Equations of DynamicsAlternative Form with Ω
Alternative gravito-magnetic field:
Only Newton‘s Law is known in the standard model.
chΩ
Law) Maxwell-Ampère of Equivalent(c
G41equation)(Poisson Law sNewton'4
Law magnetic-Gravito0cG4
Law) Gauss of Equivalent(0cG4
22 m
m
mh
mh
tc
Gt
JgΩ
g
jΩg
Ω
34
Fields, Currents and Constants
g: gravity acceleration Ω, h: gravito-magnetic fieldρm: mass density ρmh: gravito-magn. mass densityJm: mass current jmh: gravito-magn. mass current
Fields and Currents
ConstantsG: Newton‘s gravitational constantc: vacuum speed of light, required for correct physical units
35
Force Equations
Law Torque
Law Force LorentzLaw Force TorsionalLaw ForceNewtonian
L
0
LΘLM
hvFTF
gF
t
mcEm
F [N] ForceM [Nm] TorqueT [1/m] Torsiong, h [m/s2] Accelerationm [kg] Massv [m/s] Mass velocityE0=mc2 [J] Rest energyΘ [1/s] Rotation axis vectorL [Nms] Angular momentum
Physical quantities and units
36
QωQhΩ
ωQQg
c
t 0
Field-Potential Relations
Potentials and Spin Connections
Q=cq: Vector potentialΦ: Scalar potentialω: Vector spin connectionω0: Scalar spin connection
37
s
sm
1][
][][ 2
Ω
hg 2
3
][skg
mG
m1s1
][
][ 0
ω
Physical Units
Mass Density/Current „Gravito-magnetic“ Density/Current
smkgj
mkg
m
mh
2
3
][
][
smkgJ
mkg
m
m
2
3
][
][
smsm
][
][ 2
2
Q
Fields Potentials Spin Connections Constants
38
Antisymmetry Conditions ofECE Field Equations of Dynamics
2
3
3
2
1
3
3
1
1
2
2
1
:Potenitals ECE and classicalfor Relations
xQ
xQ
xQ
xQ
xQ
xQ
t
Q
2332
1331
1221
0
:sconnectionspin for Relations
QQQQQQ
ωQ
39
Properties of ECE Equations of Dynamics
• Fully analogous to electrodynamic case• Only the Newton law is known in classical mechanics• Gravito-magnetic law is known experimentally (ESA
experiment)• There are two acceleration fields g and h, but only g is
known today• h is an angular momentum field and measured in m/s2
(units chosen the same as for g)• Mechanical spin connection resonance is possible as in
electromagnetic case• Gravito-magnetic current occurs only in case of coupling
between translational and rotational motion
40
Examples of ECE Dynamics
Realisation of gravito-magnetic field hby a rotating mass cylinder(Ampere-Maxwell law)
rotation
h
Detection of h field by mechanical Lorentz force FL
v: velocity of mass m
h
FL
v
41
Polarization and MagnetizationElectromagnetism
P: PolarizationM: Magnetization
Dynamics
pm: mass polarizationmm: mass magnetization
2
0
2
0
][
][
smm
mhhsmp
pgg
m
m
m
m
mAM
MHBmCP
PED
][
)(
][
0
2
0
Note: The definitions of pm and mm, compared to g and h, differ from the electrodynamic analogue concerning constants and units.
42
Field Equations for Polarizable/Magnetizable Matter
Electromagnetism
D: electric displacementH: (pure) magnetic field
Dynamics
g: mechanical displacementh0: (pure) gravito-magnetic field
e
e
t
t
JDH
D
BE
B
0
0
m
m
ctc
Gtc
Jgh
g
hg
h
G414
010
00
0
43
ECE Field Equations of Dynamicsin Momentum Representation
None of these Laws is known in the standard model.
Law) Maxwell-Ampère of Equivalent(21
211
equation)(Poisson Law sNewton'21
21
Law magnetic-Gravito0211
Law) Gauss of Equivalent(021
pJLS
L
jSL
S
m
m
m
hm
Vtc
mccV
Vtc
cV
44smkg
smkg
][
][][2
p
SL
Physical Units
Mass Density/Current „Gravito-magnetic“Density/Current
smkgj
mkg
m
mh
2
3
][
][
smkgJ
mkg
m
m
2
3
][
][
Fields
Fields and CurrentsL: orbital angular momentum S: spin angular momentump: linear momentumρm: mass density ρmh: gravito-magn. mass densityJm: mass current jmh: gravito-magn. mass currentV: volume of space [m3] m: mass=integral of mass density