1

ECE Engineering Model

The Basis for Electromagnetic and Mechanical Applications

Horst Eckardt, AIAS

Version 4.5, 26.1.2014

2

ECE Field Equations I• Field equations in mathematical form notation

• with– ᶺ: antisymmetric wedge product– Ta: antisymmetric torsion form– Ra

b: antisymmetric curvature form– qa: tetrad form (from coordiate transformation)– ~: Hodge dual transformation– D operator and q are 1-forms, T and R are 2-forms– summation over same upper and lower indices

bbaa

bbaa

qRTD

qRTD

~~

F~

ECE Axioms• Geometric forms Ta, qa are interpreted as

physical quantities• 4-potential A proportional to Cartan tetrade q:

Aa=A(0)qa

• Electromagnetic/gravitational field proportional to torsion:Fa=A(0)Ta

• a: index of tangent space• A(0): constant with physical dimensions

3

4

ECE Field Equations II• Field equations in tensor form

• with– F: electromagnetic field tensor, its Hodge dual, see

later– Ω: spin connection– J: charge current density– j: „homogeneous current density“, „magnetic current“– a,b: polarization indices– μ,ν: indexes of spacetime (t,x,y,z)

ab

baaa

abb

aaa

JFRAF

jFRAF

0)0(

0)0(

:

:~~~

F~F~

5

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)

6

Electromagnetic Field Tensor• F and are antisymmetric tensors, related to

vector components of electromagnetic fields (polarization index omitted)

• Cartesian components are Ex=E1 etc.

00

00

123

132

231

321

cBcBEcBcBEcBcBEEEE

F

00

00

~

123

132

231

321

EEcBEEcBEEcBcBcBcB

F

F~

7

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(

000

AAAAAAAAA

)3(

3

)3(2

)3(1

)3(

)2(3

)1(2

)2(1

)2(

)1(3

)1(2

)1(1

)1( ,,AAA

AAA

AAA

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

JEB

E

jjBE

B

8

ECE Field Equations – Vector Form

„Material“ Equations

ar

a

ar

a

HB

ED

0

0

Dielectric Displacement

Magnetic Induction

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ECE Field Equations – Vector Form without Polarization Index

„Material“ Equations

HBED

0

0

r

r

Dielectric Displacement

Magnetic Induction

Law Maxwell-Ampère1

Law Coulomb

Induction of LawFaraday 0

Law Gauss0'

02

0

0

0

e

e

eheh

eheh

tc

't

JEB

E

jjBE

B

10

mA

mC

mAN

msVT

mV

][,][

][

][

2

2

HD

B

E

mVsV

][

][

Am1s1

][

][ 0

ω

Physical Units

Charge Density/Current „Magnetic“ Density/Current

msAmA

eh

eh

][

][ 2

j

)/(/][

/][22

3

smCmA

mC

e

e

J

2

3

]'[

]'[

mVmVs

eh

eh

j

11

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!

bbaaa

bbab

ba

aaa

tAωAB

ωAAE

0

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ECE Field Equations in Terms of Potential I

ae

bbaab

baa

bbaaa

aeb

bab

baa

a

bba

bbab

ba

bba

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 I

00

b

bab

ba

aa

tωAA

0

0

0

12,21,2

1

1

2

13,31,3

1

1

3

23,32,3

2

2

3

bbab

ba

aa

bbab

ba

aa

bbab

ba

aa

AAxA

xA

AAxA

xA

AAxA

xA

Electricantisymmetry constraints:

Magneticantisymmetryconstraints:

Or simplifiedLindstrom constraint: 0 b

baa AωA

14

AωAB

ωAAE

0t

Field-Potential Relations IIOne Polarization only

Potentials and Spin Connections

A: Vector potentialΦ: scalar potentialω: Vector spin connectionω0: Scalar spin connection

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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

16

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

AAxA

xA

AAxA

xA

AAxA

xA

0 AωAor:

00

:attentiongiven be tohave rsDenominato

)(2

1

)(21

:potentials thefrom calculated becan sconnectionspin Thus

)(21

220

0

A

tAA

t

t

AAAω

Aω

AωA

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Relation between Potentials and Spin Connections derived from Antisymmetry Conditions

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Alternative I: ECE Field Equations with Alternative Current Definitions (a)

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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

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Alternative I: ECE Field Equations with Alternative Current Definitions (b)

19detector andobserver between velocity relative is

with

Law Maxwell-Ampère1

Law Coulomb

Induction of LawFaraday 0

Law Gauss0'

02

0

0

0

v

v

JEB

E

jjBE

B

tdtd

dtd

c

'dtd

ae

aa

aea

aeh

aeh

aa

aeh

aeh

a

1.pdf)ps/phipps0files/phipsc3/elmag/lfire.com///www.ange:(http

20

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

JωAAω

JAAA

ωA

A

21

B

EtωAB

ωAE

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

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Electric antisymmetry constraints:

Antisymmetry Conditions ofECE Field Equations III

21

21

BBB

EEE

ωωωωωω

0

0

21

2

1

1

2

1

3

3

1

3

2

2

3

21

BB

EE

xA

xA

xA

xA

xA

xA

t

ωω

ωωA

Magnetic antisymmetry constraints:

Define additional vectorsωE1, ωE2, ωB1, ωB2:

Curvature Vectors

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2

0

000

1][][)(

: Units :onpolarisatiwithout )(

:field) (magnetic curvatureSpin

1)(

:onpolarisatiwithout

1)(

:field) (electric curvature Orbital

mspin

spin

tcorbital

tcorbital

ba

Bba

EB

bc

ca

ba

ba

ba

B

E

ca

bc

bc

cab

a

ba

ba

ba

E

RRωRR

ωωωRR

ωRR

ωωωRR

Geometrical Definition of Electric Charge/Current Densities

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ba

Bb

ba

Ebb

bab

baa

e

ba

Ebb

baa

e

c

c

RARBωEJ

RAEω

11:current Electric

:density Charge

000

0

BEe

Ee

c

c

RARBωEJ

RAEω

11:current Electric

:density Charge

000

0

With polarization:

Without polarization:

Geometrical Definition of Magnetic Charge/Current Densities

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ba

Eb

ba

Bbb

bab

baa

eh

ba

Bbb

baa

eh

c RAREωBJ

RABω

0'

:current sHomogeneou'

:density charge sHomogeneou

EBeh

Beh

c RAREωBJ

RABω

0':current sHomogeneou

':density charge sHomogeneou

With polarization:

Without polarization:

Additional Field Equations due to Vanishing Homogeneous Currents

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00

bba

ba

Eb

ba

Bbb

bab

ba

ba

Bbb

ba

c

Aω

RARBEω

RABω

With polarization:

Without polarization:

00

AωRARBEω

RABω

EB

B

c

Resonance Equation of Scalar Torsion Field

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abba

a

cRTtT

00

0

With polarization:

Without polarization:

cRTtT

00

0

Physical units:

2

0

1][

1][

mR

mT

Equations of the Free Electromagnetic Field/Photon

29

Field equations:

Spin equations:

010

00

010

0

0

02

0

2

EBω

EωBEω

Bω

EB

E

BE

B

c

tc

t

frequencytimeenergy

Emomentum

vectorwave

numberwave

0

0

E

c

p

ωκpκ

κω

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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

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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(cG41

equation)(Poisson Law sNewton'4

Law magnetic-Gravito0cG41

Law) Gauss of Equivalent(04

m

m

mh

mh

tc

Gtc

G

Jgh

g

jhg

h

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ECE Field Equations of DynamicsAlternative Form with Ω

Alternative gravito-magnetic field:

Only Newton‘s Law is known in the standard model.

chΩ

Law) Maxwell-Ampère of Equivalent(c

G41equation)(Poisson Law sNewton'4

Law magnetic-Gravito0cG4

Law) Gauss of Equivalent(0cG4

22 m

m

mh

mh

tc

Gt

JgΩ

g

jΩg

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

ConstantsG: Newton‘s gravitational constantc: vacuum speed of light, required for correct physical units

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Force Equations

Law Torque

Law Force LorentzLaw Force TorsionalLaw ForceNewtonian

L

0

LΘLM

hvFTF

gF

t

mcEm

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

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QωQhΩ

ωQQg

c

t 0

Field-Potential Relations

Potentials and Spin Connections

Q=cq: Vector potentialΦ: Scalar potentialω: Vector spin connectionω0: Scalar spin connection

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s

sm

1][

][][ 2

Ω

hg 2

3

][skg

mG

m1s1

][

][ 0

ω

Physical Units

Mass Density/Current „Gravito-magnetic“ Density/Current

smkgj

mkg

m

mh

2

3

][

][

smkgJ

mkg

m

m

2

3

][

][

smsm

][

][ 2

2

Q

Fields Potentials Spin Connections Constants

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Antisymmetry Conditions ofECE Field Equations of Dynamics

2

3

3

2

1

3

3

1

1

2

2

1

:Potenitals ECE and classicalfor Relations

xQ

xQ

xQ

xQ

xQ

xQ

t

Q

2332

1331

1221

0

:sconnectionspin for Relations

QQQQQQ

ωQ

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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

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Polarization and MagnetizationElectromagnetism

P: PolarizationM: Magnetization

Dynamics

pm: mass polarizationmm: mass magnetization

2

0

2

0

][

][

smm

mhhsmp

pgg

m

m

m

m

mAM

MHBmCP

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.

42

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

JDH

D

BE

B

0

0

m

m

ctc

Gtc

Jgh

g

hg

h

G414

010

00

0

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ECE Field Equations of Dynamicsin Momentum Representation

None of these Laws is known in the standard model.

Law) Maxwell-Ampère of Equivalent(21

211

equation)(Poisson Law sNewton'21

21

Law magnetic-Gravito0211

Law) Gauss of Equivalent(021

pJLS

L

jSL

S

m

m

m

hm

Vtc

mccV

Vtc

cV

44smkg

smkg

][

][][2

p

SL

Physical Units

Mass Density/Current „Gravito-magnetic“Density/Current

smkgj

mkg

m

mh

2

3

][

][

smkgJ

mkg

m

m

2

3

][

][

Fields

Fields and CurrentsL: 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