•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)
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
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,
(Comparative analysis) Including of GravitationalInteraction
(By Means of Multiplication)
Including of ElectromagneticInteraction
(By Means of Addition)
ec
e/cmcL Av222 1 v 2222 1 U/cmcL v
2222μ 1 U/cmcdt
dP v
Avc
edt
dP 1μ
The Classical Particle in the Electromagnetic and Scalar Gravitational Fields
(Comparative analysis) Including of GravitationalInteraction
(By Means of Multiplication)
Including of ElectromagneticInteraction
(By Means of Addition)
uBc
e
dτ
dpμ ugugum
dτ
dp
22
2
22 /1,
/1 c
mc
c
i
c
mp
vv
v
AAB Ucg ln2 2
The Classical Particle in the Electromagnetic and Scalar Gravitational Fields
(Comparative analysis) Including of GravitationalInteraction
(By Means of Multiplication)
Including of ElectromagneticInteraction
(By Means of Addition)
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
The Classical Particle in the Electromagnetic and Scalar Gravitational Fields
(Comparative analysis) Including of GravitationalInteraction
(By Means of Multiplication)
Including of ElectromagneticInteraction
(By Means of Addition)
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 Closed Classical Systems: Particles and Fields
The Charged Particles and the Electromagnetic Field
The Massive Particles and the Gravitational 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
The Closed Classical Systems: Particles and Fields
The Charged Particles and the Electromagnetic Field
The Massive Particles and the Gravitational 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
The Closed Classical Systems: Particles and Fields
The Charged Particles and the Electromagnetic Field
The Massive Particles and the Gravitational Field
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
(Comparative analysis)
U
Λ
U
Λ
0/12 22
2
Ucμc
Gv
A
Λ
A
Λ
jc
B4
jc
A4
The Closed Classical Systems: Particles and Fields
The Charged Particles and the Electromagnetic Field
The Massive Particles and the Gravitational Field
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
(Comparative analysis)
U
Λ
U
Λ
0/12 22
2
Ucμc
Gv
A
Λ
A
Λ
jc
B4
jc
A4
The Closed Classical Systems: Particles and Fields
The Charged Particles and the Electromagnetic Field
The Massive Particles and the Gravitational Field
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
(Comparative analysis)
0/12 22
2
Ucμc
Gv
jc
B4
jc
A4
The Closed Classical Systems: Particles and Fields
The Charged Particles and the Electromagnetic Field
The Massive Particles and the Gravitational Field
2222
16
11
1
AAjAc
/cμcΛ v
24
2222
2
1
UG
c
U/cμcΛ
v
2222
/142
1cGμg
cg v
(Comparative analysis)
AAB
0 Be
g
0 gg
The Closed Classical Systems: Particles and Fields
The Charged Particles and the Electromagnetic Field
The Massive Particles and the Gravitational Field
(Comparative analysis)
0 Be 0 gg
The Closed Classical Systems: Particles and Fields
jc
B4
uBc
e
dτ
dpμ ugugum
dτ
dp
0/12 22
2
Ucμ
c
Gv
jc
A4
2222
/142
1cGμg
cg v
The Charged Particles and the Electromagnetic Field
The Massive Particles and the Gravitational Field
(Comparative analysis)
Three-dimensional Form of Equations
0 Be 0 gg
./142
1 2222
cGμgc
g v
jc
B4
uBc
e
dτ
dpμ ugugum
dτ
dp
The Charged Particles and the Electromagnetic Field
The Massive Particles and the Gravitational Field
(Comparative analysis)
Three-dimensional Form of Equations
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
The Charged Particles and the Electromagnetic Field
The Massive Particles and the Gravitational Field
(Comparative analysis)
Three-dimensional Form of Equations
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
The Charged Particles and the Electromagnetic Field
The Massive Particles and the Gravitational Field
(Comparative analysis)
Three-dimensional Form of Equations
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
The Charged Particles and the Electromagnetic Field
The Massive Particles and the Gravitational Field
(Comparative analysis)
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
Newtonian Limit of Relativistic Equations
The Massive Particlesand the Gravitational Field
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
Newtonian Limit of Relativistic Equations
The Massive Particlesand the Gravitational Field
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
(Comparative analysis)
The Energy-Momentum Tensor: Particles and Fields
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)
The Energy-Momentum Tensor: Particles and Fields
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
?
The Extended System of Charge Particles, Electromagnetic and Gravitational Fields
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
The Extended System of Charge Particles, Electromagnetic and Gravitational Fields
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
The Extended System of Charge Particles, Electromagnetic and Gravitational Fields
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
The Extended System of Charge Particles, Electromagnetic and Gravitational Fields
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
The Extended System of Charge Particles, Electromagnetic and Gravitational Fields
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
2/ ceAAA
jAc
1
22/
λ
cUeU
The Extended System of Charge Particles, Electromagnetic and Gravitational Fields
24
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
216
1
AA
216
1
AA
? 22/
λ
cUeU
2/ ceAA 2/222 ceAAAA
The Extended System of Charge Particles, Electromagnetic and Gravitational Fields
24
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
216
1
AA
216
1
AA 2U
2/ ceAA
22/
λ
cUeU
The Extended System of Charge Particles, Electromagnetic and Gravitational Fields
2/ ceAA
22/
λ
cUeU
2224
2222
16
11
21
UAAjAc
UG
cU/cμcΛ v
The Extended System of Charge Particles, Electromagnetic and Gravitational Fields
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
The Extended System of Charge Particles, Electromagnetic and Gravitational Fields
aa
LL
dt
d
rv
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
BvEgv
c
eecmU
c
Um
dt
d 222
22
2
/1/1
vv
The Extended System of Charge Particles, Electromagnetic and Gravitational Fields
a
aaa
a
aa ec
eU/ccmL Av2222 1 v
The movement equations of classical particles
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
The Extended System of Charge Particles, Electromagnetic and Gravitational Fields
a
aaa
a
aa ec
eU/ccmL Av2222 1 v
The movement equations of classical particles
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
The Extended System of Charge Particles, Electromagnetic and Gravitational Fields
a
aaa
a
aa ec
eU/ccmL Av2222 1 v
The movement equations of classical particles
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
The Extended System of Charge Particles, Electromagnetic and Gravitational Fields
a
aaa
a
aa ec
eU/ccmL Av2222 1 v
BHED 22 UU
The movement equations of classical particles
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
The Extended System of Charge Particles, Electromagnetic and Gravitational Fields
a
aaa
a
aa ec
eU/ccmL Av2222 1 v
BHED 22 UU
The movement equations of classical particles
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
The Extended System of Charge Particles, Electromagnetic and Gravitational Fields
fAAA
2/ ceAA
22/
λ
cUeU
a
aaa
a
aa ec
eU/ccmL Av2222 1 v
BHED 22 UU
The movement equations of classical particles
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
The Extended System of Charge Particles, Electromagnetic and Gravitational Fields
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
The Extended System of Charge Particles, Electromagnetic and Gravitational Fields
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