Post on 21-Jan-2016
description
1
ECE Engineering Model
The Basis for Electromagnetic and Mechanical Applications
Horst Eckardt, AIAS
Version 4.1, 11.1.2014
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)
aa
aa
JF
jF
0
0
1~
F~
3
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)
4
Electromagnetic Field Tensor
• F and are antisymmetric tensors, related to vector components of electromagnetic fields (polarization index omitted)
• Cartesian components are Ex=E1 etc.
0
0
0
0
123
132
231
321
cBcBE
cBcBE
cBcBE
EEE
F
0
0
0
0
~
123
132
231
321
EEcB
EEcB
EEcB
cBcBcB
F
F~
5
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(
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
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
6
ECE Field Equations – Vector Form
„Material“ Equations
ar
a
ar
a
HB
ED
0
0
Dielectric Displacement
Magnetic Induction
7
m
A
m
CmA
N
m
sVT
m
V
][,][
][
][
2
2
HD
B
E
m
Vs
V
][
][
Am
1s
1
][
][ 0
ω
Physical Units
Charge Density/Current „Magnetic“ Density/Current
ms
Am
A
eh
eh
][
][2
j
)/(/][
/][22
3
smCmA
mC
e
e
J
2
3
]'[
]'[
m
Vm
Vs
eh
eh
j
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
0
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
10
Antisymmetry Conditions ofECE Field Equations I
00
b
bab
ba
aa
tωA
A
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
Electricantisymmetry constraints:
Magneticantisymmetryconstraints:
Or simplifiedLindstrom constraint: 0 b
baa AωA
11
AωAB
ωAA
E
0t
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
13
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
AAx
A
x
A
AAx
A
x
A
AAx
A
x
A
0 AωAor:
0
0
:attentiongiven be tohave rsDenominato
)(2
1
)(2
1
:potentials thefrom calculated becan sconnectionspin Thus
)(2
1
220
0
A
tAA
t
t
AA
Aω
Aω
AωA
14
Relation between Potentials and Spin Connections derived from Antisymmetry Conditions
15
Alternative I: ECE Field Equations with Alternative Current Definitions (a)
15
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
16
Alternative I: ECE Field Equations with Alternative Current Definitions (b)
16detector andobserver between velocity relative is
with
Law Maxwell-Ampère1
Law Coulomb
Induction of LawFaraday 0
Law Gauss0'
02
0
0
0
v
v
JE
B
E
jjB
E
B
tdt
d
dt
d
c
'dt
d
ae
aa
aea
aeh
aeh
aa
aeh
aeh
a
1.pdf)ps/phipps0files/phipsc3/elmag/lfire.com///www.ange:(http
17
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
JωA
Aω
JA
AA
ωA
A
18
B
EtωAB
ωA
E
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
19
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
20
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
BB
EE
x
A
x
Ax
A
x
Ax
A
x
A
t
ωω
ωωA
Magnetic antisymmetry constraints:
Define additional vectorsωE1, ωE2, ωB1, ωB2:
Geometrical Definition of Charges/Currents
21
ba
Bbb
ba
ba
Ebb
baa
e
ba
Ebb
baa
e
ccc
c
RAEωREJ
RAEω
200
0
:current Electric
:density Charge
BEe
Ee
ccc
c
RAEωREJ
RAEω
200
0
:current Electric
:density Charge
With polarization:
Without polarization:
Curvature Vectors
22
2
1][][
:Units
)(
:field) (magnetic curvatureSpin
)(
:field) (electric curvature Orbital
m
spin
orbital
ba
Bba
E
ba
ba
B
ba
ba
E
RR
RR
RR
Additional Field Equations due to Vanishing Homogeneous Current
23
0
0
bb
a
ba
Eb
ba
Bbb
bab
ba
ba
Bbb
ba
c
Aω
RAREωB
RABω
With polarization:
Without polarization:
00
Aω
RAREωB
RABω
EB
B
c
Resonance Equation of Scalar Torsion Field
24
abb
aa
cRTt
T
0
0
0
With polarization:
Without polarization:
cRTt
T
0
0
0
Physical units:
2
0
1][
1][
mR
mT
25
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
26
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
27
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
28
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
29
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
30
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
31
QωQh
Ω
ωQQ
g
c
t 0
Field-Potential Relations
Potentials and Spin Connections
Q=cq: Vector potentialΦ: Scalar potentialω: Vector spin connectionω0: Scalar spin connection
32
s
s
m
1][
][][2
Ω
hg 2
3
][skg
mG
m
1s
1
][
][ 0
ω
Physical Units
Mass Density/Current „Gravito-magnetic“ Density/Current
sm
kgj
m
kg
m
mh
2
3
][
][
sm
kgJ
m
kg
m
m
2
3
][
][
s
ms
m
][
][2
2
Q
Fields Potentials Spin Connections Constants
33
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
Q
2332
1331
1221
0
:sconnectionspin
for Relations
ωQ
34
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
35
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
36
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
Note: The definitions of pm and mm, compared to g and h, differ from the electrodynamic analogue concerning constants and units.
37
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
JD
H
D
BE
B
0
0
m
m
ctc
Gtc
Jg
h
g
hg
h
G41
4
01
0
00
0
38
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
39s
mkgs
mkg
][
][][2
p
SL
Physical Units
Mass Density/Current „Gravito-magnetic“Density/Current
sm
kgj
m
kg
m
mh
2
3
][
][
sm
kgJ
m
kg
m
m
2
3
][
][
Fields
Fields and Currents
L: 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