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1
ECE Engineering Model
The Basis for Electromagnetic and Mechanical Applications
Horst Eckardt, AIAS
Version 3.0, 20.7.2009
2
ECE Field Equations
• Field equations in tensor form
• With– F: electromagnetic field tensor, its Hodge dual, see
later– J: charge current density– j: „homogeneous current density“, „magnetic current“– a: polarization index– µ,ν: indexes of spacetime (t,x,y,z)
νµνµ
νµνµ
µε
aa
aa
JF
jF
0
0
~
1
=∂
=∂
F~
3
Properties of Field Equations
• J is not necessarily external current, isdefined by spacetime propertiescompletely
• j only occurs if electromagnetism isinfluenced by gravitation, otherwise =0
• Polarization index „a“ can be omitted iftangent space is defined equal to space of base manifold (assumed from now on)
4
Electromagnetic Field Tensor
• F and F~ are antisymmetric tensors, related to vector components of electromagnetic fields(polarization index omitted)
−−
−−−−
=
0
0
0
0
123
132
231
321
cBcBE
cBcBE
cBcBE
EEE
F µν
−−
−−−−
=
0
0
0
0
~
123
132
231
321
EEcB
EEcB
EEcB
cBcBcB
F µν
5
Potential with polarizationdirections
• Potential matrix:
• Polarization vectors:
ΦΦΦΦ
)3(3
)2(3
)1(3
)3(2
)2(2
)1(2
)3(1
)2(1
)1(1
)3()2()1()0(
0
0
0
AAA
AAA
AAA
=
=
=)3(
3
)3(2
)3(1
)3(
)2(3
)1(2
)2(1
)2(
)1(3
)1(2
)1(1
)1( ,,
A
A
A
A
A
A
A
A
A
AAA
6
ECE Field Equations – Vector Form
„Material“ Equations
ar
a
ar
a
HB
ED
0
0
µµεε
=
= Dielectric Displacement
Magnetic Induction
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
JE
B
E
jjB
E
B
µ
ερ
µ
ρρµ
=∂
∂−×∇
=⋅∇
===∂
∂+×∇
===⋅∇
7
m
A
m
CmA
N
m
sVT
m
V
==
⋅=⋅==
=
][,][
][
][
2
2
HD
B
E
m
Vs
V
=
=Φ
][
][
Am
1s
1
=
=
][
][ 0
ω
ω
Physical Units
Charge Density/Current „Magnetic“ Density/Current
ms
Am
A
eh
eh
=
=
][
][2
j
ρ
)/(/][
/][22
3
smCmA
mC
e
e
==
=
J
ρ
2
3
]'[
]'[
m
Vm
Vs
eh
eh
=
=
j
ρ
8
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!
bb
aaa
bb
abb
aa
aa
t
AωAB
ωAA
E
×−×∇=
Φ+−∂
∂−Φ−∇= 0ω
9
ECE Field Equations in Terms of Potential I
ae
bb
aabb
aa
bb
aaa
aeb
bab
baa
a
bb
ab
bab
ba
bb
a
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
µω
ερω
ω
=
∂Φ∂−
∂Φ∂∇+
∂∂+
∂∂+
××∇−∆−⋅∇∇
=Φ⋅∇+⋅∇−∆Φ−∂
∂⋅∇−
=∂
×∂−Φ×∇+×∇−
=×⋅∇
10
Antisymmetry Conditions ofECE Field Equations I
00 =Φ−−∂
∂−Φ∇ bb
abb
aa
a
tωA
A ω
0
0
0
12,21,
2
1
1
2
13,31,
3
1
1
3
23,32,
3
2
2
3
=++∂∂+
∂∂
=++∂∂+
∂∂
=++∂
∂+∂
∂
bb
abb
aaa
bb
abb
aaa
bb
abb
aaa
AAx
A
x
A
AAx
A
x
A
AAx
A
x
A
ωω
ωω
ωω
Electricantisymmetry constraints:
Magneticantisymmetryconstraints:
Or simplifiedLindstrom constraint: 0=×+×∇ b
baa AωA
11
AωAB
ωAA
E
×−×∇=
Φ+−∂∂−Φ−∇= 0ω
t
Field-Potential Relations IIOne Polarization only
Potentials and Spin Connections
A: Vector potentialΦ: scalar potentialω: Vector spin connectionω0: Scalar spin connection
12
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
µω
ερω
ω
=
∂Φ∂−
∂Φ∂∇+
∂∂+
∂∂+
××∇−∆−⋅∇∇
=Φ⋅∇+⋅∇−∆Φ−∂∂⋅∇−
=∂×∂−Φ×∇+×∇−
=×⋅∇
13
ECE Field Equations in Terms of Potentialwith “cold“ currents II
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
JωA
Aω
JA
AA
ωA
A
µω
µ
ερω
ερ
=
∂Φ∂−
∂∂+××∇−
=
∂Φ∂∇+
∂∂+∆−⋅∇∇
=Φ⋅∇+⋅∇−
=∆Φ−∂∂⋅∇−
14
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 beused to derive special properties.
Electric antisymmetry constraints: Magnetic antisymmetry constraints:
00 =Φ−−∂∂−Φ∇ ωAA ωt
0
0
0
12212
1
1
2
13313
1
1
3
23323
2
2
3
=++∂∂+
∂∂
=++∂∂+
∂∂
=++∂∂+
∂∂
AAx
A
x
A
AAx
A
x
A
AAx
A
x
A
ωω
ωω
ωω
0=×+×∇ AωAor:
15
B
EtωAB
ωA
E
+×∇=
+∂∂−Φ−∇=
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
16
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
µ
ερ
=
∂∂−
∂Φ∂∇+
∂∂+
×∇+∆−⋅∇∇
=⋅∇+∆Φ−∂∂⋅∇−
=∂
∂+×∇
=⋅∇
17
Electric antisymmetry constraints:
Antisymmetry Conditions ofECE Field Equations III
( )( )21
21
BBB
EEE
ωωω
ωωω
−−=−−=
0
0
21
2
1
1
2
1
3
3
1
3
2
2
3
21
=++
∂∂+
∂∂
∂∂+
∂∂
∂∂+
∂∂
=++∂∂−Φ∇
BB
EE
x
A
x
Ax
A
x
Ax
A
x
A
t
ωω
ωωA
Magnetic antisymmetry constraints:
Define additional vectorsωE1, ωE2, ωB1, ωB2:
18
Properties of ECE Equations
• The ECE equations in potential representationdefine a well-defined equation system (8 equations with 8 unknows)
• 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 standardtheory (see red fields)
• Resonance structures (self-enforcingoscillations) are possible in Coulomb and Ampère-Maxwell law
19
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
20
ECE Field Equations of Dynamics
Only Newton‘s Law is known in the standard model.
Law) Maxwell-Ampère of Equivalent(c
G41
equation)(Poisson Law sNewton'4
Law magnetic-Gravito0c
G41
Law) Gauss of Equivalent(04
m
m
mh
mh
tc
Gtc
G
Jg
h
g
jh
g
h
πρπ
πρπ
=∂∂−×∇
=⋅∇
==∂∂+×∇
==⋅∇
21
ECE Field Equations of DynamicsAlternative Form with Ω
Alternative gravito-magnetic field:
Only Newton‘s Law is known in the standard model.
c
hΩ =
Law) Maxwell-Ampère of Equivalent(c
G41
equation)(Poisson Law sNewton'4
Law magnetic-Gravito0c
G4
Law) Gauss of Equivalent(0c
G4
22 m
m
mh
mh
tc
Gt
Jg
Ω
g
jΩ
g
Ω
πρπ
π
ρπ
=∂∂−×∇
=⋅∇
==∂∂+×∇
==⋅∇
22
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
Constants
G: Newton‘s gravitational constantc: vacuum speed of light, required for correct physical units
23
Force Equations
Law Torque
Law Force Lorentz
Law Force Torsional
Law ForceNewtonian
L
0
LΘL
M
hvF
TF
gF
×−∂∂=
×===
t
mc
E
m
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
Θ
24
QωQh
Ω
ωQQ
g
×−×∇==
Φ+−Φ∇−∂∂−=
c
t 0ω
Field-Potential Relations
Potentials and Spin Connections
Q=cq: Vector potentialΦ: Scalar potentialω: Vector spin connectionω0: Scalar spin connection
25
s
s
m
1][
][][2
=
==
Ω
hg 2
3
][skg
mG =
m
1s
1
=
=
][
][ 0
ω
ω
Physical Units
Mass Density/Current „Gravito-magnetic“ Density/Curre nt
sm
kgj
m
kg
m
mh
2
3
][
][
=
=ρ
sm
kgJ
m
kg
m
m
2
3
][
][
=
=ρ
s
ms
m
=
=Φ
][
][2
2
Q
Fields Potentials Spin Connections Constants
26
Antisymmetry Conditions ofECE Field Equations of Dynamics
2
3
3
2
1
3
3
1
1
2
2
1
:Potenitals ECE and classical
for Relations
x
Q
x
Q
x
Q
x
Q
x
Q
x
Qt
∂∂−=
∂∂
∂∂−=
∂∂
∂∂−=
∂∂
∂∂=Φ∇ Q
2332
1331
1221
0
:sconnectionspin
for Relations
ωωωωωω
ω
−=−=−=
Φ−= ωQ
27
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
28
Examples of ECE Dynamics
Realisation of gravito-magnetic field hby a rotating mass cylinder(Ampere-Maxwell law)
rotation
h
Detection of h field bymechanical Lorentz force FL
v: velocity of mass m
h
FL
v
29
Polarization and Magnetization
Electromagnetism
P: PolarizationM: Magnetization
Dynamics
pm: mass polarizationmm: mass magnetization
2
0
2
0
][
][
s
mm
mhhs
mp
pgg
m
m
m
m
=
+=
=
+=
m
AM
MHBm
CP
PED
=
+=
=
+=
][
)(
][
0
2
0
µ
ε
Note: The definitions of pm and mm, compared to g and h, differ from theelectrodynamic analogue concerning constants and units.
30
Field Equations forPolarizable/Magnetizable Matter
Electromagnetism
D: electric displacementH: (pure) magnetic field
Dynamics
gP: mechanical displacementh0: (pure) gravito-magnetic field
e
e
t
t
JD
H
D
BE
B
=∂∂−×∇
=⋅∇
=∂∂+×∇
=⋅∇
ρ
0
0
m
m
ctc
Gtc
Jg
h
g
hg
h
G41
4
01
0
00
0
πρπ
=∂∂−×∇
=⋅∇
=∂
∂+×∇
=⋅∇
31
ECE Field Equations of Dynamicsin Momentum Representation
None of these Laws is known in the standard model.
Law) Maxwell-Ampère of Equivalent(2
1
2
11
equation)(Poisson Law sNewton'2
1
2
1
Law magnetic-Gravito02
11
Law) Gauss of Equivalent(02
1
pJL
S
L
jS
L
S
==∂∂−×∇
==⋅∇
==∂∂+×∇
==⋅∇
m
m
m
hm
Vtc
mccV
Vtc
cV
ρ
ρ
32s
mkgs
mkg
⋅=
⋅==
][
][][2
p
SL
Physical Units
Mass Density/Current „Gravito-magnetic“Density/Current
sm
kgj
m
kg
m
mh
2
3
][
][
=
=ρ
sm
kgJ
m
kg
m
m
2
3
][
][
=
=ρ
Fields
Fields and CurrentsL: orbital angular momentumS: 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
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