Единая геометрическая теория классических полей

∇~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) - ,

& .

(44) µ
a
µ

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

# # .

(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) # ’ . ( ,

# ’ , # " # # ’ , # ’ .

. &

’ , ’ ’ & # . *

#&#,

# ’ . $ - -% ( ). * ’ -

.

. # #

, ’ – ( ),

# & -

. / # ’ # & & ’

. ( #

, " ’ –

# - . * ’ #

# &

’ .

$ & & & (

) # & & # # &

# #, # #.

, # ’

, #"

. 3 ’ &

# , ( ’ - ). ) &

’ # , "

’ # ’ - ’ . $ & &

. * , # " & &

’ , # !

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

. ’ , " ’ ,

. * ’

, . * # &#

’ .

$ , #

, # & &

& ! .

_____________________

"

1. 0 - -% :

∆αµν = Γµαν + K
α⋅µν

K
αµν = −K
µαν

2. ." % :

σ
=∂µ
g
, # g
= det g
µ
ν

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

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).