Реферат: Единая геометрическая теория классических полей - Refy.ru - Сайт рефератов, докладов, сочинений, дипломных и курсовых работ

Единая геометрическая теория классических полей

Рефераты по физике » Единая геометрическая теория классических полей
(dimstein@list)
, .
, 2007 .
.
- ,
- . -
! !
! ! ."
! ! ,
# $ $#
! .%
& ,
! ,
! & .
1.
!
" # # . $
# -
, – # .
% , # # & & –
' . ( ( #
& # & & ) # . )
# ,
' # .
& # #
, '
# & . * '
' # # ,
- ,
. *
& #& # # # & & –
& & – # !" # "
(# ) . +# # #
& # , #
' # # # .
( #
(' # - , #
). ( & # & #
# & & – ' . ,# ,
& , ' # # .
) # , #
' ,
# # .
* # #
( ! (
' # ), #
' . $ '
# & #
.
2.
# & #
( ! ,
& ' . )
– - -%
"& - [14]. * '
& # # # & &
& & , # &:
# # , #
# ( , # ).
, - -%
, 24 .
# # & #
, # #
. * # #
& . % ( ! #
- ( "
).
. # , #
, ' , , # #
(−,+,+,+).
% # & ! # #
# .
A) * - '
.
B) / # - #
' # .
C) * # - # , #
# # .
*
A ' -
, & , B
# ' . ) , $
( ! , !
# , - -% . ( # "
,
# ' . * .
# # ,
# .
. 0 - ,
' , , & ? α ⋅ µν
( ) , # # #
? α = ? α :
⋅ µν ⋅ [ µν ]
(1)
? α = ? α − ? α
⋅ µν µν νµ
# ? – α µν . * '
. .
? α µν # :
(2)
? α µν = Γ µν α + K α ⋅ µν
# K α ⋅ µν – , # #
( )
K αµν = K [ αµ ] ν , Γ – µν α % ( , . 1-3).
$ # #
" $. # & (
) ' ( ' # ) #:
(3) d ds 2 x 2 µ + ? µ ( αβ ) dx ds α dx ds β = 0
(4) d 2 x µ + Γ µ dx α dx β = 0
ds 2 αβ ds ds
(3) # , (4) ' .
. $ (3) (4) # # , # ,
# :
(5)
? µ = Γ µ
( αβ ) αβ
$ (2) ' # ! :
(6)
? µ = K µ
[ αβ ] ⋅ αβ
, # #
( # . ) , #
K = K . . (1) (6) '
αµν [ αµν ]
!
(7)
? α = 2 K α
⋅ µν ⋅ µν
( )
, # , ? αµν = ? [ αµν ] . 1 ,
$ # .
% , # (7) .
* ' .
3. !" " !"# ! -" $ % ! && #
, & -
- -% ,
# , ( ),
' # , ,
# .
1) . ( # -
# :
(8)
ds 2 = g dx µ dx ν
µν
g µν # ∇ α g µν = 0 ,
# ∇ – α # # x α ( ,
. 4-5).
2) . . 0 ,
, " ,
# & . ,
A α ⋅ µν , # # (2)
#:
(9) ? α = Γ α + i A α
µν µν ⋅ µν
# A αµν = − A µαν = − A ανµ = − A νµα = A [ αµν ] . . % #
:
(10) Γ µν α = 2 1 g ασ ( ∂ µ g νσ + ∂ ν g µσ − ∂ σ g µν )
$
# A α ⋅ µν
# # :
(11)
A αµν = − ε αµνσ A σ
# A µ – # , ε αβµν – 2 3 .
A µ # # :
(12) A µ = − 1 ε µαβγ A
6 αβγ
(
# '
, # # '
a µ :
(13)
a µ = qˆ A µ
# qˆ – ' # . . ! (13)
' . % qˆ #
# ! # , , &
( A α ⋅ µν ~ A µ ~ 1 / q ˆ ).
1 " (9) # :
(14) ? α = 2 ? α = 2 i A α
⋅ µν [ µν ] ⋅ µν
$ # "
. * # ,
#
? α µν #
# , # Γ ( µν α , . 6).
3) % . 1 - # ? α µν
# ( , . 7):
(15) R α ⋅ µβν = ∂ β ? α µν − ∂ ν ? α µβ + ? α τβ ? τ µν − ? α τν ? τ µβ
1
- & # " - R σ ⋅ µσν :
(16)
R = ∂ ? σ − ∂ ? σ + ? σ ? τ − ? σ ? τ
µν σ µν ν µσ τσ µν τν µσ
.
" (9) - # # #
( , . 8):
(17)
R µν = R ~ + µν R ˆ µν
(18) R ~ = ∂ Γ σ − ∂ Γ σ + Γ σ Γ τ − Γ σ Γ τ
µν σ µν ν µσ τσ µν τν µσ
(19) R ˆ = i ∇ ~ A σ − A τ A σ
µν σ ⋅ µν ⋅ σµ ⋅ τν
4#
~
R – - ; Rˆ – - ,
µν µν
( ). . ∇~ α
# (# Γ ). µν α !
(11) ,
(20)
( )
A τ ⋅ σµ A σ ⋅ τν = − 2 A µ A ν − g µν A α A α
.
(17), (18), (19) (20) -
, #:
(21)
R ( µν ) = R ~ µν + 2 ( A µ A ν − g µν A α A α )
(22) R [ µν ] = i ∇ ~ σ A σ ⋅ µν
%
# (21) (22), -
# , .
, - F, µν # -
# :
(23)
R = R + i F
µν ( µν ) µν
(24) F = ∇ ~ A σ
µν σ ⋅ µν
1 F µν , #
F µν :
(25) F µν = 2 1 ε µναβ F αβ
*
(24) (11), & & # ,
# - (25) :
(26)
F = ∂ A − ∂ A
µν µ ν ν µ
,
# " ' .
. (13) (26) " ' f µν
# # #
- :
(27)
f µν = ∂ µ a ν − ∂ ν a µ = q ˆ F µν
. - (21)
# :
(28)
R = g µν R ( µν ) = R ~ − 6 A α A α
# R ~ = R ~ ⋅ µ µ – .
1 , # '
, # #
& ' . * ' '
( ), "
' – - .
A µ # -
F µν & ' a µ
" f, & µν & ' .
4. ' $ !" ( % '#$"# #
4 , # -
, ,
:
(29) δ L − g d 4 x = 0
G
# L – # G . 2 ,
- , # ,
(29). 2 L, ( ! ,
G
- .
* & '- ( , . 9-10)
- :
(30.1) Rc ( 1 ) ≡ δ µ α R µ ⋅ α
(30.2) Rc ( 2 ) ≡ δ αβ ⋅ ⋅ µν R µν R αβ
(30.3) Rc ( 3 ) ≡ δ αβγ ⋅ ⋅ ⋅ µνσ R µ ⋅ α R ν ⋅ β R σ ⋅ γ
(30.4) Rc ( 4 ) ≡ δ αβγ λ R µν R R στ R
⋅ ⋅ ⋅ ⋅ µνστ αβ γ λ
*
" & - #
# , , " & #
. & "& & - (30)
# # . * Rc ( 1 )
(30.1) # R. (28) (13)
:
(31) Rc ( 1 ) = R = R ~ − 6 A α A α = R ~ − q ˆ 6 2 a α a α
$
Rc ( 2 ) (30.2) δ αβ ⋅ ⋅ µν &
# - ,
(22) (24) ' !
R [ µν ] = iF µν . ' ! , (25) (27), & # :
(32) Rc ( 2 ) = − δ αβ ⋅ ⋅ µν F µν F αβ = 2 F αβ F αβ = q ˆ 2 2 f αβ f αβ
& (31) (32) # , - Rc ( 1 )
Rc ( 2 ) # & # # #
. $ & #
R~ , #
( ! , # " f αβ f αβ ,
' . 1
# & # & &
- Rc ( 1 ) Rc ( 2 ) , " !
# .
3 L G
. (§ 2). . ' #
# L 2 ( R ) , # :
(33)
L 2 = ( R − R ) 2 = R 2 − 2 R R + R 2
0 0 0
# R – 0 . 2 L G L 2
&
- :
(34)
L = L 2 ( R n → Rc ( n ) ) = Rc ( 2 ) − 2 R Rc ( 1 ) + R 2
G 0 0
$
(34) # # & " #
(33). * R, & # L,
0 G
# ,
. . " (31) (32) #
# :
(35) L = − 1 f αβ f + R ~ − 6 a α a − R 0
G R q ˆ 2 αβ q ˆ 2 α 2
0
.
' ,
& # :
8 π
(36) q ˆ =
κ R
0
R
(37) Λ = 0
4
( )
# Λ – Λ ~ 10 − 56 − 2 , κ – ( ! . .
" ! (36) # L G ! # :
(38) L G = − 8 κ π ( f αβ f αβ + 6 R 0 a α a α ) + R ~ − 2 1 R 0
(
# #
, # # '
R. ,# 0 , ! (37), R 0
# (38) .
5.)"#
(29) (34) ,
# - ,
' , & &
. . " (38)
(29) #:
(39) δ − 8 κ π ( f αβ f αβ + 6 R 0 a α a α ) + R ~ − 1 2 R 0 − g d 4 x = 0
- 9 -
# R ~ = g µν R ~ µν . $ g, µν Γ µν α a α ( )
( (10)):
(40) G + 1 R g = κ T ˆ
µν 4 0 µν µν
(41) ∇ ~ σ f µσ + 3 R 0 a µ = 0
# # :
~ 1 ~
(42) G ≡ R − g R
µν µν 2 µν
(43) T ˆ µν ≡ 4 1 π f ⋅ α µ f αν − 1 4 g µν f αβ f αβ + 3 4 R π 0 a µ a ν − 1 2 g µν a α a α
G – µν ( ! , Tˆ – µν " ' - ' -
. (40) (41), &
, # #
' # .
. #
' # (41)
' - ' (43),
(40) # ( ! , #
. ' (41) - ,
& .
1 , , ,
' ' . * '
. $ R
0
& (40) (41) & ,
& &
' . (41)
# a µ " f µν
' . $ , # a µ ,
# , # f, µν
' .
- Tˆ (43), µν & # (40)
' - ' . % # '
:
(44) ∇ T ˆ µν = ∇ ~ T ˆ µν = − 1 f ν ( ∇ ~ f µσ + 3 R a µ ) + 3 R 0 a ν ∇ ~ a µ
µ µ 4 π ⋅ µ σ 0 4 π µ
. (41) & .
* # (41) # & # , ∇ ~ µ a µ = 0 .
1 Tˆ # µν # ' #
& , ' - :
(45)
∇ µ T ˆ µν = ∇ ~ µ T ˆ µν = 0
$
& (45) (40) #
" # 5 , & .
#
R. . (40) :
0
(46) − R ~ + R 0 = − 3 κ 4 π R 0 a α a α = − 6 A α A α
,
# " (28) & # ,
(47)
R 0 = R ~ − 6 A α A α = R
1
, R . *
0
(40) ! (47) ! .
(40) (41) # ,
, & ( ),
& # . 3 ,
, . $
:
(48) G µν + 1 4 R 0 g µν = κ T µν
(49) ∇ ~ f µσ + 3 R a µ = ξ j µ
σ 0
# T µν = T ˆ + µν T ~ µν , T ~ µν – ' - , T – µν
' - , j – µ , ξ – ( ξ = 4 π / ).
& & #
, & # :
(50) ∇ π µ = ∇ ~ π µ = 0
µ µ
(51) ∇ j µ = ∇ ~ j µ = 0
µ µ
# π µ = µ u µ ( ), j µ = ρ u 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)
∇ T µν = ∇ ~ T µν = 0
µ µ
.
' ' -
:
(54)
∇ ~ µ T ~ µν = − ∇ ~ µ T ˆ µν
.
" (44) (49) (52) T~ (54) µν
! # :
(55) ∇ ~ µ T ~ µν = 1 c f ⋅ ν µ j µ
(55) #
& .
1 # , #
# . 1 ' - # ! # ,
~
# & # & , T µν = µ u µ u ν = π µ u ν ,
# µ – # , u – µ #
# # . # (55) # ' #
" & (50) #:
(56) π µ ∇ ~ µ u ν = 1 c f ⋅ ν µ j µ
+
# # # , # #
# '- . $ ' π µ = µ u µ = m δ ( x − x 0 ) u µ
j µ = ρ u µ = q δ ( x − x 0 ) u µ , # m q – # . $
(56) " , u β ∇~ u ν = du ν d τ + Γ ν u α u β , :
β αβ
(57) du ν + Γ ν u α u β = q f ν u β
d τ αβ mc ⋅ β
(
# # . , # , (57)
& # . $
# # 2 , &
& # & .
1 , # ! # ( ) #
# # ,
, # # .
6. *++ %!
. ! & !
# & # & . $ '
# # #
' . (48)
# (55) (57) # ,
& &
# . ,
& ' (49),
' - (43). ,# , #
R ' , (49), #
0
. (49) &
& '
. 1 , ' # # (49) #
# , .
$ - # #
( g = − 1 , g = g = g = 1 ) ' (49) #:
00 11 22 33
(58) ∂ 2 a µ − 3 R a µ = 0
0
# ∂ 2 = ? − − 2 ∂ 2 t ( ’0 ). ( # #
# - , # &
# # .
(58) # ! , & # & . $
# & # & ' ! # # :
(59) a µ = a µ sin( kx − ω t )
0
# x – # # # & . *
' ω k ! :
(60)
( )
ω 2 = 2 k 2 + 3R
0
# c – # # &
# . . ! (60) ' &
! # # ,
, # ' ' ,
# # :
ω 3 R
(61) v = = c 1 + 0 > c
k k 2
d ω c 2
(62) v = = c 1 − 3 R < c
dk 0 ω 2
1
, ' # , & (58),
' # # ! #
c (62). % # (61) (62)
( # ). &
# c . , c
# & , '
! # .
$ - ! (58)
# . . (58) '
' & # & # :
(64) ? = q e − α r
r
# ? = a 0 (' ), q – ' # , α = 3 R 0 = m γ c / ,
r – # # # . - α
(64) « » '
.
. , &
' (58) , ,
! ' & ,
m :
γ
3 R
(63) m = 0
γ c
*
' # # (62). .
(63)
. (63) '
.
* ! (37) , &
' :
(64) 3 R ~ 10 − 55 − 2
0
(65) m ~ 10 − 65
γ
*
# # #
' . . '
# # # # :
(66) m < 3 ⋅ 10 − 60
γ
1
(65) # ' . ( ,
# ' , # "
# # ' , #
' .
7.,#% -(
. &
' , '
' & # . *
#& # ,
# ' . $
- -%
( ). * ' -
.
. # #
, ' – ( ),
# & -
. / # ' #
& & '
. ( #
, " ' –
# - . * ' #
# &
' .
$ & & & (
) # & & # # &
# # , # # .
, # '
, # "
. 3 ' &
# , ( ' -
). ) &
' # , "
' # ' - ' .
$ & &
. * , # " & &
' , # !
2 . $
& , # , ,
( ! ( ).
. ' ,
" ' ,
. * '
, .
* # & #
' .
$ , #
, # & &
& ! .
_____________________
"
1. 0 - -% :
? α = Γ α + K α
µν µν ⋅ µν
Γ µν α = 1 2 g ασ ( ∂ µ g νσ + ∂ ν g µσ − ∂ σ g µν )
K = − K
αµν µαν
2. . " % :
∂ g
Γ µσ σ = 2 µ g , # g = det g µν
3. $ # :
? = ? − ? = K − K
αµν αµν ανµ αµν ανµ
1 ( )
K = ? − ? − ?
αµν 2 αµν µαν ναµ
4. :
δ u µ = − ? µ αβ u α dx β , δ u µ = ? α µβ u α dx β
5. % # :
∇ µ u ν = ∂ µ u ν + ? ν σµ u σ , ∇~ µ u ν = ∂ µ u ν + Γ σµ ν u σ
∇ µ u ν = ∂ µ u ν − ? σ νµ u σ , ∇~ µ u ν = ∂ µ u ν − Γ νµ σ u σ
6. % # # ? α = Γ α + i A α :
µν µν ⋅ µν
A α = A α = 0 , ? α = Γ α , ? α = Γ α
⋅ µα ⋅ ( µν ) µα µα ( µν ) µν
∇ u µ = ∂ u µ + ? µ u σ = ∂ u µ + Γ µ u σ
µ µ σµ µ σµ
∇ µ T ( µν ) = ∂ µ T ( µν ) + ? µ σµ T ( σν ) + ? ν ( σµ ) T ( µσ ) = ∂ µ T µν + Γ σµ µ T ( σν ) + Γ σµ ν T ( µσ )
7. 1 - :
( ∇ µ ∇ ν − ∇ ν ∇ µ ) u λ = R ⋅ λ σµν u σ + ? σ ⋅ µν ∇ σ u λ
R α ⋅ βµν = ∂ µ ? α βν − ∂ ν ? α βµ + ? α τµ ? τ βν − ? α τν ? τ βµ
? α = ? α − ? α
⋅ µν µν νµ
8. - - :
R α ⋅ βµν = ∂ µ Γ βν α − ∂ ν Γ βµ α + Γ τµ α Γ βν τ − Γ τν α Γ βµ τ +
+ ∇ ~ K α − ∇ ~ K α + K α K τ − K α K τ
µ ⋅ βν ν ⋅ βµ ⋅ τµ ⋅ βν ⋅ τν ⋅ βµ
9. 1 2 3 :
ε αβγλ = g [ αβγλ ] , ε αβγλ = − 1 [ αβγλ ]
g
+ 1 , αβγλ - " 0123
[ αβγλ ] = − 1 , αβγλ - " 0123
0 , αβγλ #
10. * '- :
δ αβγ λ ≡ − ε αβγ λ ε
⋅ ⋅ ⋅ ⋅ µνστ µνστ
δ αβγ ≡ − ε αβγ τ ε
⋅ ⋅ ⋅ µνσ µνστ
δ αβ ≡ − 1 ε αβστ ε = δ α δ β − δ α δ β
⋅ ⋅ µν 2 µνστ µ ν ν µ
δ α ≡ δ α ≡ − 1 ε ανστ ε
⋅ µ µ 6 µνστ
! "#!&"#
1. Einstein A., The Meaning of Relativity, Princeton Univ. Press, Princeton, N.Y, 1950
(* # : (! & ! )., . , 2, ., 1955).
2. ).(! & ! ,. & # , 1. 1-2, #- «) », ., 1966.
3. E. Schrodinger, Space-Time Structure, Cambridge University Press, 1960 (* # :
( . 6 # , * - , , )7 ,
2000).
4. * *.".,* & +.,., 1 , #- «) », ., 1973.
5. C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation, Freeman, San Francisco,
1973 (* # : -., , .. ," ./ , / , #- « », .,
1977).
6. 0.)." $ , .1.% ,)...2 , . : #
, #- «) », ., 1986.
7. E. Cartan, Lecons sur la Geometrie des Espaces de Riemann, Gauthier-Villars, Paris,
1928 and 1946 (* # : % (., -
, #- / , ., 1960).
8. +. Cartan, On Manifolds with an Affine Connection and the Theory of General
Relativity, translated by A. Magnon and A. Ashtekar (Bibliopolis, Naples, 1986).
9. %.%.1 , ) # -
, #- «+# -..», 2002 .
10. 3. .- $ ,0 & # , 7),
1 119. . 3, 1976.
11. Alberto Saa, Einstein-Cartan theory of gravity revisited, gr-qc/9309027 (1993).
12. Hong-jun Xie and Takeshi Shirafuji,Dynamical torsion and torsion potential,
gr-qc/9603006 (1996).
13. V.C. de Andrade and J.G. Pereira, Torsion and the Electromagnetic Field,
gr-qc/9708051 (1999).
14. Yuyiu Lam,Totally Asymmetric Torsion on Riemann-Cartan Manifold,
gr-qc/0211009 (2002).
- 18 -