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

QQ

QQ

QQ

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