Ãëàâíàÿ Ó÷åáíèêè - Ðàçíûå Ëåêöèè (ðàçíûå) - ÷àñòü 28
. .
(dimstein@list.ru)
, . , 2007 . .
- ,
- .
-
! ! ! ! .
" ! ! ,
# $$# !.
% & ,
! ,
! & .
1.
! " # # . $ # - , – # . % , # # & & – ’ . ( ( # &# & & ) # . ) # , ’ # . & # # , ’ # &. * ’ ’ # # , - , . * & #&# # # & & – & & – # !" # " (# ) . +# # # & #, # ’ # # # . ( # (’ # - , # ). ( & # & # # & & – ’ . ,# , & , ’ # #. ) # , # ’ , # #. * # # ( ! ( ’ # ), # ’ . $ ’ # & # . 2.
# &# ( ! , & ’ . ) – - -% "& - [14]. * ’ &# # # & & & & , # &: # #, # # ( , #). , - -% , 24 . # # &# , # # . * # # & . % ( ! # - ( ). " . # , # , ’ , (−,+,+,+). , # # % # & ! # . # # A) * - ’ . B) / # - ’ # . # C) * # - # #. #, # * A ’ - , & , B # ’ . ) , $ ( ! , ! # , - -% . ( #" , # ’ . * . # # , # . . 0 - , ’ , , & (Ωα⋅µν=Ωα⋅[µν]): # ∆α
µ
ν – . * ’ . . ∆α
µ
ν # : (2) # K
(K
α
µν= K
[
αµ]
ν), Γµ
α
ν
– % ( , . 1-3). $ # # " $. # & ( ) ’ ( ’ # ) #: (3) +∆(αβ) d
2x
µ (4) (3) #, (4) ’ . . $ (3) (4) # #, #, # : (5) ∆µ(αβ) =Γαµβ $ (2) ’ # !: (6) ∆µ
[
αβ] = K
µ
⋅
αβ , # # #. , # (K
α
µν= K
[
αµν]
). . (1) (6) ’ ! (7) , #, (Ωαµν=Ω[αµν] ). 1 , $ # . % , # (7) . * ’ . , & - - -% , #, ( ), ’ #, , # . 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
#: # 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=− ( # ’ , # # ’ a
µ
: (13) a
µ
= q
ˆA
µ
# q
ˆ – ’ #. . ! (13) ’ . % q
ˆ # # ! # , , & ( A
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 ∆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 % # (21) (22), - # , . , - F
µ
ν, # - # : (23) R
µ
ν= R
(
µν)
+ iF
µ
ν (24) F
µ
ν=∇~
σA
σ
⋅
µν 1 F
µ
ν , # F
µν
: (25) F
µ
ν= 2 * (24) (11), & &#, # - (25) : , # " ’ . . (13) (26) " ’ f
µ
ν # # # - : . - (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 , # - , , : # LG
– # . 2 , - , # , (29). 2 LG
, ( ! , - . * & ’- ( , . 9-10) - : (30.1) Rc
(30.2) Rc
(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
~ − $ Rc
(2)
(30.2) δα
⋅
β
⋅
µν & # - , (22) (24) ’ ! R
[
µν]
= iF
µν. ’ !, (25) (27), &# : (32) & (31) (32) #, - Rc
(1) Rc
(2)
# &# # # . $ R
~
, # &# ( ! , # " f
αβ
f
α
β, ’ . 1 # & # & & - 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) . ’ , &#: (36) κR
0
(37) Λ= R
0
4 # Λ – (Λ ~ 10−56
−2
), κ – ( ! . . (38) 2 , # # ’ R
0
. ,# , ! (37), R
0
# (38) . 5.
)"# (29) (34) , # - , ’ , & & . . " (38) (29) #: (39) 2 ~ = g
# R
(40) (41) # (42) (43) G
µ
ν – . ’ 1 ’ ’ # µν
R
~
µν. $ g
µν
, Γµ
α
ν
a
α
( ) ( (10)): ∇~σf
µσ+3R
0a
µ= 0 # :
G
µ
ν 2 T
ˆµν ≡ ( ! , T
ˆ
µ
ν – " ’ - ’ . (40) (41), & , # # ’ # . # ’ # (41) - ’ (43), (40) # ( ! , # . ’ (41) - , & . , , , ’ . * ’ . $ R
0
& (40) (41) & , & & ’ . (41) # a
µ
" f
µν . $ , # a
µ
, , # f
µ
ν, ’ . - T
ˆµν
(43), &# (40) ’ - : ’ . % # ’ (44) . * # (41) (41) & # &#, . ~ ∇µ
a
µ
= 0. 1 T
ˆ
µ
ν # # ’ # &, ’ - : (45) ∇µ
T
ˆµν
= ∇~
µ
T
ˆµν
= 0 $ & (45) (40) # " # 5 , & . # R
0
. . (40) : (46) , # " (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
" . $ & 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) (55) # & . 1 # , # # . 1 ’ - # ! #, ~ = µu
µ
u
ν
=πµ
u
ν
, # & # & , T
µ
ν # µ – #, u
µ
– # # #. # (55) # ’ # " & (50) #: (56) + # # # , # # # ’- . $ ’ 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
du
ν
(57) d
τ mc
( # # . , # , (57) & # . $ # # 2 , & & # &. 1 , # ! # ( ) # # # , # # . 6.
*++%! , . ! & ! # & # &. $ ’ # # # ’ . (48) # (55) (57) # , & & # . , & ’ (49), ’ - (43). ,# , # R
0
’ , (49), # . (49) & & ’ . 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) (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 = r
(64) « » ’ . . , & ’ (58) , , ! ’ & , m
γ
: 3R
0
|