Лабораторная работа: Единая геометрическая теория классических полей
..
(dimstein@list.ru)
,.
, 2007.
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(Ωα⋅µν=Ωα⋅[µν]):
(1) Ωα⋅ µν=∆αµ ν−∆αν µ
# ∆αµ ν –. * ’..
∆αµ ν #:
(2)
# K –, # #
(K α µν= K [ αµ] ν), Γµαν – % (,. 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)
, #, | (Ωαµν=Ω[αµν] ). 1 | , |
$ # | . | |
%, # | (7) | . |
* ’.
3. !" " !"# !- " $ %! && #
, & —
— -%, #, ( ),
’ #,,
#.
1) . ( # —
#:
(8) ds 2 = g µνdx µdx ν
g µ ν # ∇αg µ ν= 0,
# ∇α – # # x α (,
. 4-5).
2) . . 0,
, ",
# &.,
A , # # (2)
#:
(9) ∆αµ ν=Γµαν + 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
$ # "
. * #,
#
∆αµ ν #
#, # Γµαν (,. 6).
3) % . 1 — # # (,. 7): (15) R α⋅µβν=∂β∆αµν−∂ν∆αµβ+∆ατβ∆τµν−∆ατν∆τµβ | ∆αµ 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 ˆµ ν= i ∇~ σA σ⋅ µν− A τ⋅ σµA σ⋅ τν | # # |
~ 4# R µ ν – — ; R ˆµ ν – | — , |
( ). . | ∇~α |
# (# Γµαν ). (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= 1 εµ ναβF αβ
2
* (24) (11), & &#, # — (25):
(26) F µν =∂µA ν −∂νA µ
, # " ’.
. (13) (26) " ’ f µ ν
# # #
— :
(27) f µν =∂µa ν −∂νa 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 µνR 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. | ’ | !, (25) (27), &#: |
(32) Rc (2) f αβf α β q ˆ
& (31) (32) #, | — Rc (1) | |
Rc (2) # &# | # | # |
. $ | R ~ , # | &# |
(!, # | " f αβf α β, | |
’ | . 1 |
# & # & & — Rc (1)Rc (2) , "!
#.
3 LG
. (§ 2).. ’ #
# L 2 (R ), #:
(33) L 2 = (R − R 0 )2 = R 2 − 2R 0R + R 02
# 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 01q ˆ2 f αβf αβ+ R ~ − q ˆ62 a αa α− R 20
. ’,
&#:
(36) q ˆ = 8π
κR 0
(37) Λ= R 0
4
# Λ – (Λ ~ 10−56 −2 ), κ – (!..
"! (36) # LG! #:
(38) LG =−(f αβf αβ + 6R 0 a αa α )+ R ~ − 1 R 0
2
, # # ’ | ||
R 0 . ,# | ,! (37), R 0 | |
# | (38). 5. )"# | |
(29) | (34) | , |
# | — | , |
’, | & | & |
. | . " (38) | |
(29) #: |
( # #
(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
’
’
# µνR ~ µν. $ g µν, Γµανa 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 ˆµ ν – " ’ — ’. (40) (41), &, # #
’ #.
#
’ # (41)
— ’ (43), (40) # (!, #
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&.
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& | (40) | (41) | &, |
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# | a µ " | f µν | |
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, # f µ ν, |
’ | . | |||
- | T ˆµν (43), | &# | (40) | |
’ - : | ’ | . % | # | ’ |
(44) µa µ
. * # | (41) (41) | & # | &#, | . ~ ∇µa µ = 0. |
1 T ˆµ ν # # ’ #
&, ’ — :
(45) ∇µT ˆµν = ∇~µT 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 αA α = R
1, R 0. *
(40)! (47)!..
(40) (41) #,
, & ( ),
& #. 3,
,. $:
(48) G µ
(49) ∇~σf µσ +3R 0a µ =ξj µ
# T µ ν = T ˆµ ν +T ~ µν, T ~ µν – ’ — , T µ ν – ’ — , j µ –, ξ – (ξ= 4π/ ).
& & #
, & #:
(50) ∇µ πµ = ∇~µ πµ = 0
(51) ∇µj µ = ∇~µj µ = 0
# πµ = µu µ ( ), j µ = ρu µ ( #), µ –
, ρ – #, u µ –
# (dx µd τ ). $ µ ρ #,
". $ & µ, ρ u µ, #.
— #
. * # (49) #
& # (51) 2 #
’:
(52) ∇µa µ = ∇~µa µ = 0
(
. ( ’ (49), # a µ #.
* # # (48)
& # ’ — :
(53) ∇µT µν = ∇~µT µν = 0
. ’ ’ —
:
(54) ∇~µT ~µν = −∇~µT ˆµν
. " (44) (49) (52) T ~µν (54)
! #:
(55) j µ
(55) #
&.
1 #, #
#. 1 ’ — #! #,
~ = µu µu ν =πµu ν,
# & # &, T µ ν
# µ – #, u µ – #
# #. # (55) # ’ #
" & (50) #:
(56) j µ
+ # # #, # #
# ’-. $ ’ πµ =µu µ = m δ(x − x 0 )u µj µ =ρu µ = q δ(x − x 0 )u µ, # m q – #. $
(56) ", u β ∇~βu ν = du νd τ+ Γανβu αu β,:
du ν
(57) +Γα νβu αu β= q f u β
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) ∂2a µ −3R 0 a µ = 0 | (49) #: |
# ∂2 =∆− −2 ∂t 2 ( ’0 ). ( | # # |
# - | , # & |
# #.
(58) # !, & # &. $
# & # & ’! # #:
(59) a µ = a 0µ sin(kx −ωt )
# x – # # # &. *
’ ω k !:
(60) ω2 = 2 (k 2 +3R 0 )
# c – # # &
#..! (60) ’ &! # #,
, # ’ ’,
# #:
(61) v =ω k = c 1+ 3k R 20 > c
(62) v = d dk ω = c 1− 3R 0 ωc 22 < 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 γc /, 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. ,#%-(
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’, ’ ’ & #. *
#&#,
# ’. $ — -% ( ). * ’ —
.
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_____________________
"
1. 0 — -%:
∆αµν = Γµαν + K α⋅µν
K αµν = −K µαν
2. ." %:
σ =∂µg , # 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. % # # ∆αµ ν = Γµαν + 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
∇µT (µν) = ∂µT (µν) +∆µσµT (σν) + ∆ν(σµ)T (µσ) = ∂µT µν + Γσ µµT (σν) + Γσ νµT (µσ)
7. 1 — :
(∇µ∇ν −∇ν∇µ)u λ = R λ⋅σµνu σ + Ωσ⋅µν∇σu λ
R α⋅βµν = ∂µ∆αβν − ∂ν∆αβµ+ ∆ατµ∆τβν − ∆ατν∆τβµ
Ωα⋅ µν = ∆αµ ν − ∆αν µ
8. — -:
R
+∇~ α −∇~νK α⋅βµ+ K α⋅τµK τ⋅βν− K α⋅τνK τ⋅βµ
µK ⋅βν
9. 1 2 3:
εαβγλ = g [αβγλ], εαβγλ =− 1 [αβγλ]
+1, αβγλ — " 0123
[αβγλ ]= −1, αβγλ — " 0123
0, αβγλ #
10. * ’-:
δα⋅β⋅γ⋅ λ⋅µνστ ≡ −εαβγλεµνστ δα⋅β⋅γ⋅µνσ ≡ −εαβγτεµνστ
!"#!&"#
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).