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

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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

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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

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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

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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

QQ

QQ

QQ

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

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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