Единая геометрическая теория классических полей (стр. 2 из 3)

(35)

LG =− R₀1qˆ2 f αβfαβ+ R~ − qˆ62 aαaα− R20

. ’ ,

&#:

(36)

qˆ = ^8π

κR₀

(37) Λ= ^R0

4

# Λ – (Λ ~ 10^{−56 −2} ), κ – ( ! . .

" ! (36) # L_G ! #:

(38)

L_G =−(f ^αβf_αβ+ 6R₀ a^αa_α)+ R^~ − ¹ R₀

2

, # # ’
R0 . ,#	, ! (37), R0
#	(38) . 5. )"#
(29)	(34)			,
#	-		,
’ ,	&	&
.	. " (38)
	(29) #:

( # #

(39)

δ −(f αβfαβ + 6R0 aαaα)+ R~ − 1 R0 − g d 4 x = 0

2

~ = g

# R

(40)

(41)

#

(42)

(43)

G_µν –

.

’

1

’

’

# ^µνR^~µν. $ g^µν, Γ_µ^α_νa^α ( ) ( (10)):

Gµ

∇~σf µσ+3R0aµ= 0

# :

≡ R~_µν − 1 g_µν^R~

G_µν

2

Tˆµν ≡

41π f aµa aαaα

( ! , T^ˆ_µν – " ’ - ’ . (40) (41), & , # #

’ # .

#

’ # (41)

- ’ (43), (40) # ( ! , #

. ’ (41) - ,

& .

,	,			,
				’ . *	’
		. $		R0
&	(40)	(41)		& ,
		&		&
’		.		(41)
#				aµ "	fµν
. $				, #	aµ,
				, # fµν,

			’	.
-			Tˆµν (43),	&#		(40)
’ - :			’	. %	#	’

(44)

_µa^µ

. * #				(41) (41)	& #		&#,		. ~ ∇µaµ = 0.

1 ^Tˆ_µν # # ’ #

&, ’ - :

(45) ∇_µTˆ^µν = ∇^~_µTˆ^µν = 0

$ & (45) (40) #

" # 5 , & .

#

R₀ . . (40) :

(46)

− R~ + R0 = − 3κ4πR0 aαaα = −6AαAα

, # " (28) &#,

(47) R₀ = R^~ −6A^αA_α= R

1 , R₀ . *

(40) ! (47) !.

(40) (41) # ,

, & ( ),

& #. 3 ,

, . $ :

(48) G_µ

(49) ∇^~_σf ^µσ+3R₀a^µ=ξj^µ

# T_µν = T^ˆ_µν +T^~µν, T^~µν – ’ - , T_µν – ’ - , j^µ – , ξ – (ξ= 4π/ ).

& & #

, & # :

(50) ∇_µπ^µ = ∇^~_µπ^µ = 0

(51) ∇_µj^µ = ∇^~_µj^µ = 0

# π^µ = µu^µ ( ), j^µ = ρu^µ ( #), µ –

, ρ – # , u^µ –

# (dx^µ

d^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 = −∇~µTˆµν

. " (44) (49) (52) T^~µν (54)

! #:

(55)

j^µ

(55) #

& .

1 # , #

# . 1 ’ - # ! #,

~ = µu_µu_ν =π_µu_ν,

# & # & , T_µν

# µ – #, u_µ – #

# #. # (55) # ’ #

" & (50) #:

(56)

j^µ

+ # # # , # #

# ’- . $ ’ π^µ=µu^µ= mδ(x − x₀ )u^µj^µ=ρu^µ= qδ(x − x₀ )u^µ, # m q – # . $

(56) " , u^β∇^~_βu^ν = du^ν

dτ+ Γ_α^ν_βu^αu^β, :

du^ν

(57)

+Γ_αν_βuαuβ₌ q f uβ

dτ mc

( # # . , # , (57) & # . $ # # 2 , & & # &. 1 , # ! # ( ) #
# # , # # . 6. *++%!	,
. ! & !
# & # &. $	’
# # #
’ .	(48)
# (55) (57) #	,
&	&
#	. ,
& ’	(49),
’ - (43).	,# , #
R0 ’ ,	(49), #
. (49)	&
&	’
. 1 , ’ # #, .	# (49) #
$ -	# #
( g00 = −1, g11 = g22 = g33 =1) ’ (58) ∂2aµ−3R0 aµ= 0	(49) #:
# ∂2 =∆− −2∂t2 ( ’0 ). (	# #
# -	, # &

# # .

(58) # !, & # & . $

# & # & ’ ! # #:

(59) a^µ= a₀^µsin(kx −ωt)

# x – # # # & . *

’ ω k !:

(60) ω² = ² (k ² +3R₀ )

# c – # # &

#. . ! (60) ’ & ! # #,

, # ’ ’ ,

# # :

(61)

v =^ωk = c 1+ ³k^R₂⁰ > c

(62) v = ^ddk^ω= c 1− 3R₀ ω^c2₂ < c

1 , ’ # , & (58), ’ # # ! # c (62). % # (61) (62)

( # ). &

# c. , c

# & , ’ ! # .

$ - ! (58)

#. . (58) ’

’ & # & # :

(64) ϕ =

^q e^−αr

r

# ϕ= a⁰ (’ ), q – ’ #, α= 3R₀ = m_γc/ , r – # # #. - α

(64) « » ’ .

. , &

’ (58) , ,

! ’ & , m_γ:

3R⁰

(63) m_γ=

c

* ’ # # (62). .

(63)

. (63) ’ .

* ! (37) , &

’ :

(64) 3R₀ ~10^{−55 −2}

(65) m_γ ~ 10⁻⁶⁵

* # # #

’ . . ’

# # # # :

(66) m_γ < 3⋅10⁻⁶⁰

1 (65) # ’ . ( ,

# ’ , # " # # ’ , # ’ .

7. ,#%-(

. &

’ , ’ ’ & # . *

#&#,

# ’ . $ - -% ( ). * ’ -

.

. # #

, ’ – ( ),

# & -

. / # ’ # & & ’

. ( #

, " ’ –

# - . * ’ #

# &

’ .

$ & & & (

) # & & # # &

# #, # #.

, # ’

, #"

. 3 ’ &

# , ( ’ - ). ) &