. .
(dimstein@list.ru)
, .
, 2007 .
.
- ,
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! !
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# , - -% . ( #" ,
# . * . # # ,
# .
. 0 - ,
, , &
, # # #
(Ωα⋅µν=Ωα⋅[µν]):
(1) Ωα
⋅
µν=∆α
µ
ν−∆α
ν
µ
# ∆α
µ
ν . * . .
∆α
µ
ν # :
(2)
# K
, # #
(K
α
µν= K
[
αµ]
ν), Γµ
α
ν
% ( , . 1-3).
$ # #
" $. # & ( ) ( # ) #:
(3)
dds
2x
2µ µ dxds
α dxds
β= 0
+∆(αβ)
d
2x
µ
(4)
ds
2 +Γαµβ
dxds
α dxds
β= 0
(3) #, (4) .
. $ (3) (4) # #, #,
# :
(5) ∆µ(αβ) =Γαµβ
$ (2) # !:
(6) ∆µ
[
αβ] = K
µ
⋅
αβ
, # #
#. , # (K
α
µν= K
[
αµν]
). . (1) (6)
!
(7)
, #,
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(Ωαµν=Ω[αµν] ). 1
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* .
3.
!" " !"# !-
" $ % ! && #
, & -
- -% , #, ( ),
#, ,
# .
1)
. ( # -
# :
(8) ds
2
= g
µν
dx
µ
dx
ν
g
µ
ν # ∇α
g
µ
ν= 0,
# ∇α
# # x
α
( ,
. 4-5).
2) . .
0 ,
, ",
# & . ,
A
, # # (2)
#:
(9) ∆α
µ
ν=Γµ
α
ν
+ iA
α
⋅
µν
# A
α
µν=−A
µ
αν=−A
α
νµ=−A
ν
µα= A
[
αµν]
. . % #
:
(10)
$ # A
# #:
(11) A
αµν=−εαµνσA
σ
# A
µ
# , εα
βµν 2 3 .
A
µ
# # :
(12) A
µ=−
εµαβγA
αβγ
( # , # # a
µ
:
(13) a
µ
= q
ˆA
µ
# q
ˆ #. . ! (13)
. % q
ˆ #
# ! # , , &
( A
~ A
µ
~ 1/q
ˆ ).
1 " (9) # :
(14) Ωα
⋅
µν= 2∆α
[
µν]
= 2iA
α
⋅
µν
$ # "
. * # ,
#
∆α
µ
ν #
# , # Γµ
α
ν
( , . 6).
3) % .
1 - #
# ( , . 7):
(15) R
α⋅µβν=∂β∆αµν−∂ν∆αµβ+∆ατβ∆τµν−∆ατν∆τµβ
|
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∆α
µ
ν
|
1 - &# " - R
:
(16) R
µν=∂σ∆σµν−∂ν∆σµσ+∆στσ∆τµν−∆στν∆τµσ
|
|
. " (9) - #
( , . 8):
(17) R
µν= R
~µν+ R
ˆµν
~
(18) R
µν=∂σΓµσν−∂νΓµσσ+ΓτσσΓµτν−ΓτσνΓµτσ
(19) R
ˆ
µ
ν= i
∇
~
σA
σ
⋅
µν− A
τ
⋅
σµA
σ
⋅
τν
|
# #
|
~
4# R
µ
ν - ; R
ˆ
µ
ν
|
- ,
|
( ). .
|
∇~
α
|
# (# Γµ
α
ν
).
(11) ,
(20) A
τ⋅σµA
σ⋅τν=−2(A
µA
ν− g
µνA
αA
α)
|
!
|
. (17), (18), (19) (20) -
, #:
~
(21) R
(µν) = R
µν+ 2(A
µA
ν− g
µνA
αA
α)
(22) R
[
µν]
= i
∇~
σA
σ
⋅
µν
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% # (21) (22), -
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# ,
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.
|
, - F
µ
ν, # -
# :
(23) R
µ
ν= R
(
µν)
+ iF
µ
ν
(24) F
µ
ν=∇~
σA
σ
⋅
µν
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1 F
µ
ν , #
F
µν
:
(25) F
µ
ν=
1
εµ
ναβF
αβ
2
* (24) (11), & &#, # - (25) :
(26) F
µν
=∂µ
A
ν
−∂ν
A
µ
, # " .
. (13) (26) " f
µ
ν
# # #
- :
(27) f
µν
=∂µ
a
ν
−∂ν
a
µ
= q
ˆF
µν
. - (21)
# :
(28) R
= g
µνR
(µν) = R
~ − 6 A
αA
α
# R
~ = R
~
µ
⋅µ
.
1 , # , # #
& . *
( ), "
- .
A
µ
# -
F
µ
ν & a
µ
" f
µ
ν, & & .
4.
$ !"( % #$"# #
4 , # -
, ,
:
(29) δ LG
− g d
4
x
= 0
# LG
# . 2 , - , # ,
(29). 2 LG
, ( ! ,
- .
* & - ( , . 9-10)
- :
(30.1) Rc
(30.2) Rc
R
µν
R
αβ
(30.3) Rc
(30.4) Rc
(4) ≡δα⋅β⋅γ⋅λ⋅µνστR
µνR
αβR
στR
γλ
* " & - #
#, , " & #
. & "& & - (30) # # . * Rc
(1)
(30.1) # R
. (28) (13)
:
(31) Rc
(1) = R
= R
~ −6A
αA
α= R
~ −
q
ˆ62 a
αa
α
$ Rc
(2)
(30.2) δα
⋅
β
⋅
µν &
# - ,
(22)
|
(24)
|
!
|
R
[
µν]
= iF
µν.
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!, (25) (27), &# :
|
(32)
Rc
(2)
f
αβ
f
α
β q
ˆ
& (31) (32) #,
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- Rc
(1)
|
Rc
(2)
# &#
|
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~
, #
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" f
αβ
f
α
β,
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. 1
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# & # & & - Rc
(1)
Rc
(2)
, " !
# .
3 LG
. (§ 2). . #
# L
2
(R
) , # :
(33) L
2
=
(R
− R
0
)2
= R
2
− 2R
0
R
+ R
0
2
# R
0
. 2 LG
L
2
&
- :
(34) L
G
= L
2 (R
n
→Rc
(n
) )=Rc
(2) −2R
0Rc
(1) + R
02
$ (34) # # & " #
(33). * R
0
, &# LG
,
# ,
. . " (31) (32) #
#:
(35)
L
G
=− R
0
1q
ˆ2 f
αβf
αβ+ R
~ − q
ˆ62 a
αa
α− R
20
. ,
&#:
(36)
q
ˆ = 8
π
κR
0
(37) Λ= R
0
4
# Λ (Λ ~ 10−56
−2
), κ ( ! . .
" ! (36) # LG
! #:
(38)
LG
=−(f
αβ
f
αβ
+ 6R
0
a
α
a
α
)+ R
~
− 1
R
0
2
, # #
|
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R
0
. ,#
|
, ! (37), R
0
|
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#
|
(38) .
5.
)"#
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(29)
|
(34)
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#
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&
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&
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. " (38)
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(29) #:
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( # #
(39)
δ −(f
αβf
αβ + 6R
0 a
αa
α)+ R
~ − 1 R
0 − g d
4 x
= 0
2
~ = g
# R
(40)
(41)
#
(42)
(43)
G
µ
ν
.
1
# µν
R
~
µν. $ g
µν
, Γµ
α
ν
a
α
( ) ( (10)):
G
µ
∇~σf
µσ+3R
0a
µ= 0
# :
≡ R
~µ
ν − 1 g
µ
νR
~
G
µ
ν
2
T
ˆµν ≡
41π f
a
µa
a
αa
α
( ! , T
ˆ
µ
ν " - . (40) (41), & , # #
# .
#
# (41)
- (43), (40) # ( ! , #
. (41) - ,
& .
,
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. *
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R
0
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&
|
(40)
|
(41)
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& ,
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&
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&
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.
|
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(41)
|
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#
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a
µ
"
|
f
µν
|
. $
|
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, #
|
a
µ
,
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, # f
µ
ν,
|
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.
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-
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T
ˆµν
(43),
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&#
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(40)
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:
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|
#
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(44)
µ
a
µ
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. * #
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(41)
(41)
|
&
#
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&#,
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.
~
∇µ
a
µ
= 0.
|
1 T
ˆ
µ
ν # # #
&, - :
(45) ∇µ
T
ˆµν
= ∇~
µ
T
ˆµν
= 0
$ & (45) (40) #
" # 5 , & .
#
R
0
. . (40) :
(46)
− R
~ + R
0 = − 3κ4πR
0 a
αa
α = −6A
αA
α
, # " (28) &#,
(47) R
0
= R
~
−6A
α
A
α
= R
1 , R
0
. *
(40) ! (47) !.
(40) (41) # ,
, & ( ),
& #. 3 ,
, . $ :
(48) G
µ
(49) ∇~
σ
f
µσ
+3R
0
a
µ
=ξj
µ
# T
µ
ν = T
ˆ
µ
ν +T
~
µν, T
~
µν - , T
µ
ν - , j
µ
, ξ (ξ= 4π/ ).
& & #
, & # :
(50) ∇µ
πµ
= ∇~
µ
πµ
= 0
(51) ∇µ
j
µ
= ∇~
µ
j
µ
= 0
# πµ
= µu
µ
( ), j
µ
= ρu
µ
( #), µ
, ρ # , u
µ
# (dx
µ
d
τ
). $ µ ρ # ,
" . $ & µ, ρ u
µ
, # .
- #
. * # (49) #
& # (51) 2 #
:
(52) ∇µ
a
µ
= ∇~
µ
a
µ
= 0
(
. ( (49), # a
µ
#.
* # # (48)
& # - :
(53) ∇µ
T
µν
= ∇~
µ
T
µν
= 0
. -
:
(54) ∇~µT
~µν = −∇~µT
ˆµν
. " (44) (49) (52) T
~
µν
(54)
! #:
(55)
j
µ
(55) #
& .
1 # , #
# . 1 - # ! #,
~ = µu
µ
u
ν
=πµ
u
ν
,
# & # & , T
µ
ν
# µ #, u
µ
#
# #. # (55) # #
" & (50) #:
(56)
j
µ
+ # # # , # #
# - . $ πµ
=µu
µ
= m
δ(x
− x
0
)u
µ
j
µ
=ρu
µ
= q
δ(x
− x
0
)u
µ
, # m
q
# . $
(56) " , u
β
∇~
β
u
ν
= du
ν
d
τ+ Γα
ν
β
u
α
u
β
, :
du
ν
(57)
+Γα
νβ
u
αu
β=
q f u
β
d
τ mc
( # # . , # , (57)
& # . $
# # 2 , &
& # &.
1 , # ! # ( ) #
|
# #
, # # .
6.
*++%!
|
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. ! & !
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# & # &. $
|
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# # #
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|
(48)
|
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|
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&
|
&
|
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|
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|
&
|
(49),
|
- (43).
|
,# , #
|
R
0
,
|
(49), #
|
. (49)
|
&
|
&
|
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. 1 , #
#, .
|
# (49) #
|
$ -
|
# #
|
( g
00 = −1, g
11 = g
22 = g
33 =1)
(58) ∂2
a
µ
−3R
0
a
µ
= 0
|
(49) #:
|
# ∂2
=∆− −
2
∂t
2
( 0 ). (
|
# #
|
# -
|
, # &
|
# # .
(58) # !, & # & . $
# & # & ! # #:
(59) a
µ
= a
0
µ
sin(kx
−ωt
)
# x
# # # & . *
ω k
!:
(60) ω2
= 2
(k
2
+3R
0
)
# c
# # &
#. . ! (60) & ! # #,
, # ,
# # :
(61)
v
=ω
k
= c
1+ 3
k
R
2
0
> c
(62) v
= d
dk
ω
= c
1− 3R
0
ωc
2
2
< c
1 , # , & (58), # # ! # c
(62). % # (61) (62)
( # ). &
# c
. , c
# & , ! # .
$ - ! (58)
#. . (58)
& # & # :
(64) ϕ =
q
e
−αr
r
# ϕ= a
0
( ), q
#, α= 3R
0
= m
γ
c
/ , r
# # #. - α
(64) « » .
. , &
(58) , ,
! & , m
γ
:
3R
0
(63) m
γ
=
c
* # # (62). .
(63)
. (63) .
* ! (37) , &
:
(64) 3R
0
~10−55
−2
(65) m
γ
~ 10−65
* # # #
. .
# # # # :
(66) m
γ
< 3⋅10−60
1 (65) # . ( ,
# , # " # # , # .
7.
,#%-(
. &
, & # . *
#&#,
# . $ - -% ( ). * -
.
. # #
, ( ),
# & -
. / # # & &
. ( #
, "
# - . * #
# &
.
$ & & & (
) # & & # # &
# #, # #.
, #
, #"
. 3 &
# , ( - ). ) &
# , "
# - . $ & &
. * , # " & &
, # !
2 . $ &, # , , ( ! ( ).
. , " ,
. *
, . * # &#
.
$ , #
, # & &
& ! .
_____________________
"
1. 0 - -% :
∆αµν = Γµαν + K
α⋅µν
K
αµν = −K
µαν
2. ." % :
σ
=∂µ
g
, # g
= det g
µ
ν
Γµσ
2g
3. $ # :
Ωαµν = ∆αµν − ∆ανµ = K
αµν − K
ανµ
K
αµν =
1 (Ωαµν − Ωµαν − Ωναµ)
2
4. :
δu
µ = −∆µαβu
αdx
β, δu
µ = ∆αµβu
αdx
β
5. % # :
∇µu
ν = ∂µu
ν + ∆νσµu
σ, ∇~µu
ν = ∂µu
ν + Γσνµu
σ
∇µu
ν = ∂µu
ν − ∆σνµu
σ, ∇~µu
ν = ∂µu
ν − Γνσµu
σ
6. % # # ∆α
µ
ν = Γµ
α
ν
+ iA
α
⋅
µν:
A
α⋅µα = A
α⋅(µν) = 0, ∆αµα = Γµαα , ∆α(µν) = Γµαν
∇µu
µ = ∂µu
µ+ ∆µσµu
σ = ∂µu
µ+ Γσµµu
σ
∇µ
T
(µν)
= ∂µ
T
(µν)
+∆µσµ
T
(σν)
+ ∆ν(
σµ
)
T
(µσ)
= ∂µ
T
µν + Γσ
µµ
T
(σν)
+ Γσ
νµ
T
(µσ)
7. 1 - :
(∇µ∇ν −∇ν∇µ)u
λ = R
|