ECE Engineering Model
The Basis for Electromagnetic and
1
The Basis for Electromagnetic and Mechanical Applications
Horst Eckardt, AIAS
Version 4.0, 24.12.2013
ECE Field Equations
• Field equations in tensor form
νµν
νµνµ
µε
aa
aa
JF
jF0
1~
=∂
=∂
2
• 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 µ aa JF 0=∂
F~
Properties of Field Equations
• J is not necessarily external current, is defined by spacetime properties completely
• j only occurs if electromagnetism is
3
• 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)
Electromagnetic Field Tensor
• F and are antisymmetric tensors, related tovector components of electromagnetic fields(polarization index omitted)
• Cartesian components are Ex=E1 etc.
F~
4
−−
−−−−
=
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
Potential with polarization directions
• Potential matrix:
ΦΦΦΦ
)3()2()1(
)3(2
)2(2
)1(2
)3(1
)2(1
)1(1
)3()2()1()0(
0
0
0
AAA
AAA
AAA
5
• Polarization vectors:
)3(3
)2(3
)1(30 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
Law Coulomb
Induction of LawFaraday 0
Law Gauss0'
0
0
aea
aeh
aeh
aa
aeh
aeh
a
't
E
jjB
E
B
ρ
µ
ρρµ
=⋅∇
===∂
∂+×∇
===⋅∇
ECE Field Equations – Vector Form
Law Maxwell-Ampère1
Law Coulomb
02
0
ae
aa
ea
tcJ
EB
E
µ
ερ
=∂
∂−×∇
=⋅∇
6
„Material“ Equations
ar
a
ar
a
HB
ED
0
0
µµεε
=
= Dielectric Displacement
Magnetic Induction
ACmA
N
m
sVT
m
V
==
⋅=⋅==
=
][,][
][
][
2
HD
B
E
m
Vs
V
=
=Φ
][
][
Am
1s
1
=
=
][
][ 0
ω
ω
Physical Units
7
m
A
m
C == ][,][2
HD m m
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
Field-Potential Relations IFull Equation Set
bb
aaa
bb
abb
aa
aa
t
AωAB
ωAA
E
×−×∇=
Φ+−∂
∂−Φ−∇= 0ω
8
Potentials and Spin Connections
Aa: Vector potentialΦa: scalar potentialωa
b: Vector spin connectionω0
ab: Scalar spin connection
Please observe the Einstein summation convention!
ECE Field Equations in Terms of Potential I
bb
ab
bab
ba
bb
a
t
AωωA
Aω
0 0)(
)()(
:Induction of LawFaraday
0)(
:Law Gauss
ω =∂
×∂−Φ×∇+×∇−
=×⋅∇
9
ae
bb
aabb
aa
bb
aaa
aeb
bab
baa
a
ttttc
t
t
JωAA
AωAA
ωAA
00
2
2
2
00
)()(1
)()(
:Law Maxwell-Ampère
)()(
:Law Coulomb
µω
ερω
=
∂Φ∂−
∂Φ∂∇+
∂∂+
∂∂+
××∇−∆−⋅∇∇
=Φ⋅∇+⋅∇−∆Φ−∂
∂⋅∇−
∂
Antisymmetry Conditions ofECE Field Equations I
00 =Φ−−∂
∂−Φ∇ bb
abb
aa
a
tωA
A ω
023,32,23 =++
∂∂+
∂∂ b
bab
ba
aa
AAx
A
x
A ωω
Electricantisymmetry constraints:
Magneticantisymmetry
10
0
0
0
12,21,
2
1
1
2
13,31,
3
1
1
3
3,2,
32
=++∂∂+
∂∂
=++∂∂+
∂∂
=++∂
+∂
bb
abb
aaa
bb
abb
aaa
AAx
A
x
A
AAx
A
x
A
AAxx
ωω
ωω
ωωantisymmetryconstraints:
Or simplifiedLindstrom constraint: 0=×+×∇ b
baa AωA
AωAB
ωAA
E
×−×∇=
Φ+−∂∂−Φ−∇= 0ω
t
Field-Potential Relations IIOne Polarization only
11
Potentials and Spin Connections
A: Vector potentialΦ: scalar potentialω: Vector spin connectionω0: Scalar spin connection
ECE Field Equations in Terms of Potential II
t
AωωA
Aω
0 0)(
)()(
:Induction of LawFaraday
0)(
:Law Gauss
ω =∂×∂−Φ×∇+×∇−
=×⋅∇
12e
e
ttttc
t
t
JωAA
AωAA
ωAA
00
2
2
2
00
)()(1
)()(
:Law Maxwell-Ampère
)()(
:Law Coulomb
µω
ερω
=
∂Φ∂−
∂Φ∂∇+
∂∂+
∂∂+
××∇−∆−⋅∇∇
=Φ⋅∇+⋅∇−∆Φ−∂∂⋅∇−
∂
Antisymmetry Conditions ofECE Field Equations II
Electric antisymmetry constraints: Magnetic antisymmetry constraints:
00 =Φ−−∂∂−Φ∇ ωAA ωt
023323
2
2
3
∂∂
=++∂∂+
∂∂
AA
AAx
A
x
A ωω
13
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.
0
0
12212
1
1
2
13313
1
1
3
=++∂∂+
∂∂
=++∂∂+
∂∂
AAx
A
x
A
AAx
A
x
A
ωω
ωω
0=×+×∇ AωAor:
:potentials thefrom calculated becan sconnectionspin Thus
)(2
10 Φ∇+
∂∂−=Φ=
t
AωAω
Relation between Potentials and Spin Connections derived form Antisymmetry Conditions
0
0
:attentiongiven be tohave rsDenominato
)(2
1
)(2
1
220
≠Φ≠
⋅Φ∇+∂∂−=⋅Φ=
Φ∇+∂∂−
Φ=
A
tAA
t
AA
Aω
Aω
ω
14
Alternative I: ECE Field Equations with Alternative Current Definitions (a)
νµνµνµν
νµνµ
µνµ
µνµ
µωε
ω
abaaa
abb
aaa
JTRAF
jTRAF
)0(
0
)0(
:)(
1:)
~~(
~
:like)-(Maxwell currents of definition ECE Standard
=−=∂
=−=∂
1515
νµνµ
µνµ
µνµ
µνµ
νµνµ
µνµ
µνµ
µνµ
νµνµ
µνµ
µνµ
µω
εω
µω
aabb
aaa
aabb
aaa
abb
aaa
JRATFFD
jRATFFD
JTRAF
0)0(
0
)0(
0)0(
:
1:
~~~~
:)maintained derivative (covariant definition eAlternativ
:)(
==+∂=
==+∂=
=−=∂
Alternative I: ECE Field Equations with Alternative Current Definitions (b)
Law Coulomb
Induction of LawFaraday 0
Law Gauss0'
0
0
E
jjB
E
B
=⋅∇
===+×∇
===⋅∇
'dt
d
aea
aeh
aeh
aa
aeh
aeh
a
ερ
µ
ρρµ
1616detector and fieldbetween velocity relative is
with
Law Maxwell-Ampère1
Law Coulomb
02
0
v
v
JE
B
E
∇⋅+∂∂=
=−×∇
=⋅∇
tdt
d
dt
d
ca
e
aa µ
ε
1.pdf)ps/phipps0files/phipsc3/elmag/lfire.com///www.ange:(http
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
1
0
0
)()(
:Law Coulomb
e
e
t
ωA
A
ρω
ερ
=Φ⋅∇+⋅∇−
=∆Φ−∂∂⋅∇−
17
100
2
002
2
2
0
10
)()(1)(
1)(
:Law Maxwell-Ampère
)()(
e
e
e
ttc
ttc
JωA
Aω
JA
AA
ωA
µω
µ
εω
=
∂Φ∂−
∂∂+××∇−
=
∂Φ∂∇+
∂∂+∆−⋅∇∇
=Φ⋅∇+⋅∇−
B
EtωAB
ωA
E
+×∇=
+∂∂−Φ−∇=
Field-Potential Relations IIILinearized Equations
18
Potentials and Spin Connections
A: Vector potentialΦ: scalar potentialωE: Vector spin connection of electric fieldωB: Vector spin connection of magnetic field
ECE Field Equations in Terms of Potential III
BE
B
t
ωω
ω
0
:Induction of LawFaraday
0
:Law Gauss
=∂
∂+×∇
=⋅∇
19e
E
B
eE
tttc
t
t
JωA
ωAA
ωA
02
2
2
0
1
)(
:Law Maxwell-Ampère
:Law Coulomb
µ
ερ
=
∂∂−
∂Φ∂∇+
∂∂+
×∇+∆−⋅∇∇
=⋅∇+∆Φ−∂∂⋅∇−
∂
Electric antisymmetry constraints:
Antisymmetry Conditions ofECE Field Equations III
( )( )21
21
BBB
EEE
ωωω
ωωω
−−=−−=
021 =++∂∂−Φ∇ EEt
ωωA
Define additional vectorsωE1, ωE2, ωB1, ωB2:
20
Electric antisymmetry constraints:
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
ωω
Magnetic antisymmetry constraints:
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
• There is much more structure in ECE than in standard theory (Maxwell-Heaviside)
21
• 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
Examples of Vector Spin Connection
toroidal coil:ω = const
linear coil:ω = 0
Vector spin connection ω represents rotation of plane of A potential
ω
22
A
B
B
A
ECE Field Equations of Dynamics
equation)(Poisson Law sNewton'4
Law magnetic-Gravito0c
G41
Law) Gauss of Equivalent(04
m
mh
mh
Gtc
G
g
jh
g
h
ρπ
πρπ
=⋅∇
==∂∂+×∇
==⋅∇
23
Only Newton‘s Law is known in the standard model.
Law) Maxwell-Ampère of Equivalent(c
G41
equation)(Poisson Law sNewton'4
m
m
tc
G
Jg
h
g
πρπ
=∂∂−×∇
=⋅∇
ECE Field Equations of DynamicsAlternative Form with Ω
equation)(Poisson Law sNewton'4
Law magnetic-Gravito0c
G4
Law) Gauss of Equivalent(0c
G4
mh
mh
Gt
g
jΩ
g
Ω
ρπ
π
ρπ
=⋅∇
==∂∂+×∇
==⋅∇
24
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
22 m
m
tc
G
Jg
Ω
g
πρπ
=∂∂−×∇
=⋅∇
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
25
Constants
G: Newton‘s gravitational constantc: vacuum speed of light, required for correct physical units
Force Equations
Law Torque
Law Force Lorentz
Law Force Torsional
Law ForceNewtonian
L
0
LΘL
M
hvF
TF
gF
×−∂∂=
×===
t
mc
E
m
Θ
26
∂t
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
QωQh
Ω
ωQQ
g
×−×∇==
Φ+−Φ∇−∂∂−=
c
t 0ω
Field-Potential Relations
27
Potentials and Spin Connections
Q=cq: Vector potentialΦ: Scalar potentialω: Vector spin connectionω0: Scalar spin connection
s
s
m
1][
][][2
=
==
Ω
hg 2
3
][skg
mG =
m
1s
1
=
=
][
][ 0
ω
ω
Physical Units
s
ms
m
=
=Φ
][
][2
2
Q
Fields Potentials Spin Connections Constants
28
s m
Mass Density/Current „Gravito-magnetic“ Density/Curren t
sm
kgj
m
kg
m
mh
2
3
][
][
=
=ρ
sm
kgJ
m
kg
m
m
2
3
][
][
=
=ρ
s
Antisymmetry Conditions ofECE Field Equations of Dynamics
:Potenitals ECE and classical
for Relations
t∂∂=Φ∇ Q 0
:sconnectionspin
for Relations
QQ ωωω
−=Φ−= ωQ
292
3
3
2
1
3
3
1
1
2
2
1
x
Q
x
Q
x
Q
x
Q
x
Q
x
Qt
∂∂−=
∂∂
∂∂−=
∂∂
∂∂−=
∂∂
∂
2332
1331
1221
ωωωωωω
−=−=−=
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
30
• 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
Examples of ECE Dynamics
Realisation of gravito-magnetic field hby a rotating mass cylinder(Ampere-Maxwell law)
Detection of h field by mechanical Lorentz force FLv: velocity of mass m
h
31
rotation
h
h
FL
v
Polarization and Magnetization
Electromagnetism
P: PolarizationM: Magnetization
Dynamics
pm: mass polarizationmm: mass magnetization
0
m
pgg m+=
C
PED += 0ε
32
2
0
2
][
][
s
mm
mhhs
mp
m
m
m
=
+=
=
m
AM
MHBm
CP
=
+=
=
][
)(
][
0
2
µ
Note: The definitions of pm and mm, compared to g and h, differ from the electrodynamic analogue concerning constants and units.
Field Equations for Polarizable/Magnetizable Matter
Electromagnetism
D: electric displacementH: (pure) magnetic field
Dynamics
gP: mechanical displacementh0: (pure) gravito-magnetic field
B =⋅∇ 0 h 00 =⋅∇
33
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
=∂∂−×∇
=⋅∇
=∂
∂+×∇
=⋅∇
ECE Field Equations of Dynamicsin Momentum Representation
equation)(Poisson Law sNewton'11
Law magnetic-Gravito02
11
Law) Gauss of Equivalent(02
1
L
jS
L
S
==⋅∇
==∂∂+×∇
==⋅∇
m
hm
mccV
Vtc
cV
ρ
ρ
34
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
pJL
S
L
==∂∂−×∇
==⋅∇
m
m
Vtc
mccVρ
Physical Units
Fields and CurrentsL: orbital angular momentumS: spin angular momentump: linear momentumρm: mass density ρmh: gravito-magn. mass densityJm: mass current jmh: gravito-magn. mass current
35s
mkgs
mkg
⋅=
⋅==
][
][][2
p
SL
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
m mh
V: volume of space [m3] m: mass=integral of mass density
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