The System of Equations of Interacting Electromagnetic, Scalar Gravitational and Spinor Fields Anatoliy N. SERDYUKOV Francisk Skorina Gomel State University

•The System of Equations of Interacting Electromagnetic,

Scalar Gravitational and Spinor Fields

•

Anatoliy N. SERDYUKOVFrancisk Skorina Gomel State University

Gomel, Byelorussia

The Classical Particle in the Electromagnetic and Scalar Gravitational Fields

jAc

/cμcΛ1

1 222 v2222 1 U/cμcΛ v

ec

e/cmcL Av222 1 v 2222 1 U/cmcL v

(Comparative analysis) Including of GravitationalInteraction

(By Means of Multiplication)

Including of ElectromagneticInteraction

(By Means of Addition)


jAc

/cμcΛ1

1 222 v2222 1 U/cμcΛ v

ec


t

LL

L

dt

d

LL

dt

d

vv

rv 2222μ 1 U/cmc

dt

dP v

22

22

22

2

/1

/1

c

Umc

c

i

c

Um

P

v

v

v

ec

mc

c

i

c

e

c

m

P

22

2

22

/1

/1

v

vA

v

Avc

edt

dP 1μ

LL

c

iLP

vv

v,





ec


2222μ 1 U/cmcdt

dP v

Avc

edt

dP 1μ






uBc

e

dτ

dpμ ugugum

dτ

dp

22

2

22 /1,

/1 c

mc

c

i

c

mp

vv

v

AAB Ucg ln2 2






22 cu ugugum

dτ

dp

0E

EB

i

iB

gugumA

AA

uAdτ

dpμ

02

2

dτ

dum

dτ

dpu μ

Bc

eA

uBc

e

dτ

dpμ

mup ),()( iηg g

uBc

e

dτ

dpμ ugugum

dτ

dp

),()( iηg g

0E

EB

i

iB

Lor.

221F

v

/c

m

dt

d

v

BvEF

ce

1Lor.

grav

221F

v

/c

m

dt

d

v

η

cc/c

mvgvvgF

11

1222

grav

v






22

2

22 /1,

/1 c

mc

c

i

c

mp

vv

v

The Gauge-Invariance of Gravitational Field

Uc ln2 2

22/ ceU

λ' g

22/ cUeUU

22

22

/1 c

UmcE

v

2222 1 U/cμcΛ v

2/ ceEE'E

2/ ceΛΛ'Λ

Here is an arbitrary real constant

ggg

η

cc/c

mvgvvgF

11

1222

grav

v

) , ()( iηg g

Ucg ln2 2

U

Λ

U

Λ

The Closed Classical Systems: Particles and Fields

2/ ceΛΛ'Λ

2222 1 U/cμcΛ v

24

2222

2

1

UG

c

U/cμcΛ

v

The Massive Particles and the Gravitational Field

A

Λ

A

Λ

jc

B4


The Charged Particles and the Electromagnetic Field


2222

16

11

1

AAjAc

/cμcΛ v

24

2222

2

1

UG

c

U/cμcΛ

v

AAB

(Comparative analysis)

U

Λ

U

Λ

0/12 22

2

Ucμc

Gv

A

Λ

A

Λ

jc

B4




2222

16

11

1

AAjAc

/cμcΛ v

24

2222

2

1

UG

c

U/cμcΛ

v

AAB


U

Λ

U

Λ

0/12 22

2

Ucμc

Gv

A

Λ

A

Λ

jc

B4




2222

16

11

1

AAjAc

/cμcΛ v

24

2222

2

1

UG

c

U/cμcΛ

v

AAB

2222

/142

1cGμg

cg v

Ucg ln2 2


U

Λ

U

Λ

0/12 22

2

Ucμc

Gv

A

Λ

A

Λ

jc

B4

jc

A4




2222

16

11

1

AAjAc

/cμcΛ v

24

2222

2

1

UG

c

U/cμcΛ

v

0 A

Ucg ln2 2

AAB

2222

/142

1cGμg

cg v


U

Λ

U

Λ

0/12 22

2

Ucμc

Gv

A

Λ

A

Λ

jc

B4

jc

A4




2222

16

11

1

AAjAc

/cμcΛ v

24

2222

2

1

UG

c

U/cμcΛ

v

0 A

Ucg ln2 2

AAB

2222

/142

1cGμg

cg v


0/12 22

2

Ucμc

Gv

jc

B4

jc

A4




2222

16

11

1

AAjAc

/cμcΛ v

24

2222

2

1

UG

c

U/cμcΛ

v

2222

/142

1cGμg

cg v


AAB

0 Be

g

0 gg





0 Be 0 gg


jc

B4

uBc

e

dτ

dpμ ugugum

dτ

dp

0/12 22

2

Ucμ

c

Gv

jc

A4

2222

/142

1cGμg

cg v




Three-dimensional Form of Equations

0 Be 0 gg

./142

1 2222

cGμgc

g v

jc

B4

uBc

e

dτ

dpμ ugugum

dτ

dp





0 Be 0 gg

./142

1 2222

cGμgc

g v

jc

B4

0

1

B

BE

tcη

tc

g

g

1

0

4

41

E

jE

Bctc 2222

2/14

2

11cGμη

ct

η

cv

gg





uBc

e

dτ

dpμ ugugum

dτ

dp

BvE

v

ce

/c

m

dt

d 1

1 22v

η

ccc

m

c

m

dt

dvgvvg

v 11

/1/122222 vv





BvE

v

ce

/c

m

dt

d 1

1 22v

η

ccc

m

c

m

dt

dvgvvg

v 11

/1/122222 vv

0

1

B

BE

tcη

tc

g

g

1

0

4

41

E

jE

Bctc 2222

2/14

2

11cGμη

ct

η

cv

gg




Newtonian Limit of Relativistic Equations

The Massive Particlesand the Gravitational Field

η

ccc

m

c

m

dt

dvgvvg

v 11

/1/122222 vv

ηtc

g

g

1

0

22222

/142

11cGμη

ct

η

cv

gg

0

0

t

g

g

01

tc

η

Gμ 4gc

22/2

21, ,

2

ceUcc c

v

gv mmdt

d

cv

Gμ 40/12 22

2

Ucμc

Gv



rc

GMeU c

22/

21

2 0/1

2 222

Ucμc

Gv

22222

/142

11cGμη

ct

η

cv

gg

dVG

UUμc

cM

8

1 2222

2

g

02

22

Uμc

G

Gμc

42

1 22

gg

r

rc

GMr

GM rrg

22

21

rc

GMeU c

22/

21

2

r

rc

GMr

GM rrg

22

21

2/2

8ce

GW

g

2/

22

2

1

ce/c

mcE

v m

mmcE

2

22 v

GW

8

2g2

/

2

2

22

1

21

1

1

2

ce

c/c

c

v

v

12

c

r

GM

rr

GM rrg

2

22/

21

2

ce c

2c

GMr



2)ef(

16

1

AAΛ 2

4)gf(

2U

G

cΛ

The Energy-Momentum Tensor: Particles and Fields

The Gravitational FieldThe Electromagnetic Field

2

222

16

1

1 1

AA

jAc

/cμcΛ v

24

2222

2

1

UG

c

U/cμcΛ

v

AA

ΛΛT

)(

)ef()ef(can U

U

ΛΛT

)(

)gf()gf(can



AA

ΛΛT

)(

)ef()ef(can U

U

ΛΛT

)(

)gf()gf(can

2can

16

1

4

1BBAT

HA4

1

2Bel

16

1

4

1BBBT

canBel TT

22

can

2

1

4ggg

G

UT

24

can

2

1UUU

G

cT

Ugc

U22

1

The Gravitational FieldThe Electromagnetic Field(Comparative analysis)


2Bel

16

1

4

1BBBT

22

can

2

1

4ggg

G

UT

8

1 22 BEW

4

BES

c

4

1BEπ

c

8

1 222 ηUG

W

g gS ηUG

c 2

4

gπ ηUcG

2

4

1

The Gravitational FieldThe Electromagnetic Field(Comparative analysis)

Once more about the Gauge-Invariance of Gravitational Field

λ'

22/ cUeUU

22

22

/1 c

UmcE

v

2222 1 U/cμcΛ v

2/ ceEE'E

2/ ceΛΛ'Λ

ggg

2/ ceT'TT

22

2

1

4ggg

G

UT

Once more about the Gauge-Invariance of Gravitational Field

./142

1 2222

cGμgc

g v

ugugum

dτ

dp

0 gg

0/12 22

2

Ucμ

c

Gv

The System of Gauge-Invariant Equations

The Extended System of Charge Particles, Electromagnetic and Gravitational Fields

jAc

/cμcΛ1

1 222 v

216

1

AA

2222 1 U/cμcΛ v

24

2U

G

c

2/ cΛeΛΛ 22/

λ

cUeU

?


To take into account the presence of gravitation field of system (in the frame of scalar model of gravitation) it is necessary to ensure the next transformation law of complete Lagrangian at gauge transformation Ю '= of gravitation potential.

jAc

/cμcΛ1

1 222 v

216

1

AA 2

4

2U

G

c

2/ cΛeΛΛ

2222 1 U/cμcΛ v


216

1

AA 2

4

2U

G

c

2222 1 U/cμcΛ v

2/ cΛeΛΛ ? 2222 1 U/cμcΛ v 2

4

2U

G

c

jAc

1

22/

λ

cUeU

jAc

/cμcΛ1

1 222 v


216

1

AA 2

4

2U

G

c

2222 1 U/cμcΛ v jAc

/cμcΛ1

1 222 v

2/ cΛeΛΛ

2222 1 U/cμcΛ v 24

2U

G

c

jAc

1

22/

λ

cUeU

2UAA

2UAA


216

1

AA 2

4

2U

G

c


/cμcΛ1

1 222 v

2/ cΛeΛΛ

2222 1 U/cμcΛ v 24

2U

G

c

jAc

1

22/

λ

cUeU


216

1

AA 2

4

2U

G

c


/cμcΛ1

1 222 v

2/ cΛeΛΛ

2222 1 U/cμcΛ v 24

2U

G

c

2/ ceAAA

jAc

1

22/

λ

cUeU


24

2U

G

c


/cμcΛ1

1 222 v

2/ cΛeΛΛ

2222 1 U/cμcΛ v 24

2U

G

c

jAc

1

216

1

AA

216

1

AA

? 22/

λ

cUeU

2/ ceAA 2/222 ceAAAA


24

2U

G

c


/cμcΛ1

1 222 v

2/ cΛeΛΛ

2222 1 U/cμcΛ v 24

2U

G

c

jAc

1

216

1

AA

216

1

AA 2U

2/ ceAA

22/

λ

cUeU


2/ ceAA

22/

λ

cUeU

2224

2222

16

11

21

UAAjAc

UG

cU/cμcΛ v


a

aaa

a

aa ec

eU/ccmL Av2222 1 v

The movement equations of classical particles

2224

2222

16

11

21

UAAjAc

UG

cU/cμcΛ v


aa

LL

dt

d

rv

a

aaa

a

aa ec

eU/ccmL Av2222 1 v


2224

2222

16

11

21

UAAjAc

UG

cU/cμcΛ v

BvEgv

c

eecmU

c

Um

dt

d 222

22

2

/1/1

vv


a

aaa

a

aa ec

eU/ccmL Av2222 1 v


AB

AE

1

tc

2224

2222

16

11

21

UAAjAc

UG

cU/cμcΛ v

BvEgv

c

eecmU

c

Um

dt

d 222

22

2

/1/1

vv

0E

EB

i

iB


a

aaa

a

aa ec

eU/ccmL Av2222 1 v


AB

AE

1

tc

AAB

2224

2222

16

11

21

UAAjAc

UG

cU/cμcΛ v

BvEgv

c

eecmU

c

Um

dt

d 222

22

2

/1/1

vv

HvDvgvvgv

c

eeη

ccc

m

c

m

dt

d 11

/1

/122222 vv

0E

EB

i

iB


a

aaa

a

aa ec

eU/ccmL Av2222 1 v


AB

AE

1

tc

AAB

2224

2222

16

11

21

UAAjAc

UG

cU/cμcΛ v

BvEgv

c

eecmU

c

Um

dt

d 222

22

2

/1/1

vv

HvDvgvvgv

c

eeη

ccc

m

c

m

dt

d 11

/1

/122222 vv

0E

EB

i

iB


a

aaa

a

aa ec

eU/ccmL Av2222 1 v

BHED 22 UU


AB

AE

1

tc

AAB

2224

2222

16

11

21

UAAjAc

UG

cU/cμcΛ v

BvEgv

c

eecmU

c

Um

dt

d 222

22

2

/1/1

vv

0E

EB

i

iB


a

aaa

a

aa ec

eU/ccmL Av2222 1 v

BHED 22 UU


0D

DH

i

iH

2 UBH

AB

AE

1

tc

AAB

2224

2222

16

11

21

UAAjAc

UG

cU/cμcΛ v

BvEgv

c

eecmU

c

Um

dt

d 222

22

2

/1/1

vv

HvDvgvvgv

c

eeη

ccc

m

c

m

dt

d 11

/1

/122222 vv

0E

EB

i

iB

2/ ceBBB


fAAA

2/ ceAA

22/

λ

cUeU

a

aaa

a

aa ec

eU/ccmL Av2222 1 v

BHED 22 UU


HHH

0D

DH

i

iH

2 UBH

AB

AE

1

tc

AAB

2224

2222

16

11

21

UAAjAc

UG

cU/cμcΛ v

BvEgv

c

eecmU

c

Um

dt

d 222

22

2

/1/1

vv

HvDvgvvgv

c

eeη

ccc

m

c

m

dt

d 11

/1

/122222 vv


A

Λ

A

Λ

jc

H4

U

Λ

U

Λ

)(

016

1/1

2 22

222

UH

ccμ

c

Gv

The equations of interacting electromagnetic and gravitational fields

2/ ceBBB

fAAA

2/ ceAA

22/

λ

cUeU

HHH

2222

22

/1442

1cGμH

c

Gg

cg v

0 Be 0 gg

2 UBH

ggg

2224

2222

16

11

21

UAAjAc

UG

cU/cμcΛ v


2224

2222

16

11

21

UAAjAc

UG

cU/cμcΛ v

2

2

/

/

c

c

e

e

BBB

EEE

BH

ED2

2

U

U

0

1

B

BE

tc

4

41

D

jD

Hctc

Three-dimensional form of field equations

22/

λcUeU

22222

222

/1422

11cGμ

c

Gη

ct

η

cv

HDgg

ηηη ggg

t

f

c

f

1

AAA

2

2

/λ

/λ

φφφ c

c

e

e

AAA

02

ψmcUA

c

ei

The System of Equations of Interacting Electromagnetic, Scalar Gravitational and Spinor Fields

ψψmcψAc

eiψcΛ 2

222

4

16

1

2

UAAUG

cΛ

222224

16

1

2ψUψmcψA

c

eiψcUAAU

G

cΛ

02

mcUA

c

eiψ

016

12 222

UH

cψψm

c

G

jc

H4

0 Be

UHB

ψψcej

AABfcieeψψ )/(

fAA 22/ ceUU

2/λ ce

2/ ceAA

ψψ 2/λ ceΛΛ

02

ψmcUA

c

ei

The System of Equations of Interacting Electromagnetic, Scalar Gravitational and Spinor Fields

222224

16

1

2ψUψmcψA

c

eiψcUAAU

G

cΛ

02

mcUA

c

eiψ

016

12 222

UH

cψψm

c

G

jc

H4

0 Be

UHB

Documents

The System of Equations of Interacting Electromagnetic, Scalar Gravitational and Spinor Fields Anatoliy N. SERDYUKOV Francisk Skorina Gomel State University