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Реферат: Единая геометрическая теория классических полей

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(dimstein@list.ru)

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, 2007 .

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(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)

, #,

(Ωαµν=Ω[αµν] ). 1

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(7)

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3. !" " !"# !- " $ % ! && #

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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

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= 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.

!, (25) (27), &# :


(32) Rc (2) f αβ f α β q ˆ

& (31) (32) #,

- Rc (1)

Rc (2) # &#

#

#

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R ~ , #

&#

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. 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) 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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(38) .

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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), & , # #

’ # .

#

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(41)

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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 µj 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 µ u ν ,

# & # & , T µ ν

# µ – #, u µ – #

# #. # (55) # ’ #

" & (50) #:

(56) j µ

+ # # # , # #

# ’- . $ ’ πµu µ = m δ(x x 0 )u µ j µu 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) α νβ u αu β= q f u β

d τ mc

( # # . , # , (57)

& # . $

# # 2 , &

& # &.

1 , # ! # ( ) #

# #

, # # .

6. *++%!

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# (55) (57) #

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#, .

# (49) #

$ -

# #

( g 00 = −1, g 11 = g 22 = g 33 =1) ’

(58) ∂2 a µ −3R 0 a µ = 0

(49) #:

# ∂2 =∆− 2t 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. ,#%-(

. &

’ , ’ ’ & # . *

#&#,

# ’ . $ - -% ( ). * ’ -

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. # #

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# - . * ’ #

# &

’ .

$ & & & (

) # & & # # &

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. * , # " & &

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2 . $ &, # , , ( ! ( ).

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’ .

$ , #

, # & &

& ! .

_____________________

"

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 λ⋅σµνu σ + Ωσ⋅µν∇σu λ

R α⋅βµν = ∂µ∆αβν − ∂ν∆αβµ+ ∆ατµ∆τβν − ∆ατν∆τβµ

α µν = ∆α µ ν − ∆α ν 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).


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