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192 +2-2 2& 26 & ? &.
22 $2 .&.. 09 + 20 I 2&
j<# Dk 6?10 . 0 /$ ?&60 &2 .0
1
J α0 + α E 2 .
!<"
1 6. I+2&( ?&H2. 62(
+& & 22 /$ J I E &. ? 8
. /$ 62 6/.2 J E # 2 2 2 .62 +&2 2 /&2# /# 6+2 8
(H
&. 22 $2 62( J E 2 2 # ?&6 !<"# & 2( +2/ 2 $0 .
2 $2 .2 6. 2 - 22 m1 # m2 $0 +& ? ?60 <E
<F
6 +&# 62 2H +0( X μ(τ, σ) + ) R1,3 &. $ .. jD# @k
e1 , e2
ω
v1 cos ωσ −
X μ (τ, σ) = a0 τ eμ0 +
sin ωσ (eμ1 cos ωτ + eμ2 sin ωτ ) .
ω
Q1
!D"
g( Q1 Q2 2H 2&
γ
Q1 =
m1
Ẋ 2 (τ, 0),
γ
Q2 =
m2
Ẋ 2 (τ, π),
!@"
/& ω .&.. 2 $ .
(Q1 + Q2 ) ω cos πω = (ω 2 − Q1 Q2 ) sin πω.
!B"
&. $ .. !D" 2 $0 1. + 1.2 +.2 .2
Q1
v1 = ,
ω 2 + Q21
Q2
v2 = .
ω 2 + Q22
!E"
g( ( +&1 c = 1# &&(# vi ∈ [0, 1]
a0 1 !D" 2 m1 Q 1
m2 Q 2
a0 = = .
2
γ 1 − v1
γ 1 − v22
!F"
e0 , e1 , e2 , e3 ?6H 2 ?6 R1,3 2/2
62 ημν = (eμ , eν ) = ^XQY (1; −1; −1; −1)l +2 σ ∈ [0, π] τ ∈ R &1
&. +26$ 2 +0
. !B" ?6H / ?# 622 +&1&(
+. 6.
ω = ωn ,
n = 1, 2, . . .
&/ n = 1 +&. ? -H-. +.2& 8
6 22 $0 n = 2 &1 # / ? 1.
(H # 2 $ 0.. ++&1 1. .2 !E" &/ +6&( n 6 n
&10 6# n − 1 / ? 1. (H 8& B "#!DC @!'-*-E $-$" 9 9#%":9-# "7F"
621 +&1. 9 !D" ++ .6 +&2 0 76/0 0 /&2 19 6 0 K I8
H E &/ & 22 L $0 . !D"
<=
&. +6&( .. &. 22 $2
+&.H. +2-(H &H-0 & ! u" jDkA
Pμ =
pμ (τ, σ) dσ +
L
(Ẋ, X )X μ − X 2 Ẋ μ
pμ (τ, σ) = γ (Ẋ, X )2 − Ẋ 2 X 2
pμi (τ ),
i=1
C
μν
2
2
X μ (τ, σ) pν (τ, σ) − X ν (τ, σ) pμ (τ, σ) dσ +
=
(xμi pνi − xνi pμi ),
!="
!C"
i=1
C
xμi (τ ) = X μ τ, σi pμi (τ ) = mi ẋμi (τ ) ẋ2i (τ ) 2+&( 2
/# C h &H? # .H- . 2 +0 / C &0 !="# !C" +- 6.( & τ = GTR\Z
& 2 +2 1. !D" . !="# +&/2 &H-
1 &. 2+&(A
P μ = Eeμ0 ,
E = πγa0 +
2
i=1
m
i .
1 − vi2
!J"
1 &. & 22 !C" +& /& . &9( 8
&. z 82+A
L
μν
=
L
μν
3 ,
γa20
L=
2ω
v22
v12
+
.
π+
Q1
Q2
!<>"
μ ν
ν μ
μ ν
ν μ
g( μν
3 = e1 e2 − e1 e2 = e é − e é . & & 22 J &. 2 +&.. 12 J = L+S # S h 22 +# .6
22 /2# 2&H-2 9. !J"# !<>" 6H .H 62( L = L(E 2 ) &/8
& 22 !<>" I &. $ .. !D"
+& +&/ +# 62 /&2 n & 7
6/. n# mi γ # .. !D" ?6H +2/ 21
/ +2 21 21 +&(6( &H?H 6 +28
0A ω# a0 # Qi &( +2 1H. /6 6 +2-(H
72& !@" K !F"
7 +&/H-. + I2 62 J = J(E 2 ) ! 1"#
+2 0 +6 <# &. 0 6/ n mi 2H 6&8
# /# +.2&H 2+ + E → ∞
+20 < .1 γ = 0,175 I2 jDk g( 1
+.. & +&1 <# 2 $ 22 L & J
?662 !62.. $0 "# IH 2 ? 62.( I
+& L → 0 &. 0 < &/ a0 # vi # Qi 2..
&H# I. E 2. 2 22&(2 6/H m1 + m2
3&( ? h & 2+/ +. 7$
L(E 2 ) ++&12 !&(&.2" +& E → ∞ &. I
2 &H- +2A
ε1 =
1 − v12 ,
ε2 =
1 − v22 ,
ε = n − ω.
!<<"
<C
L
6
n=1
5
4
n=2
3
2
n=3
1
0
0
5
10
15
2
E
2 " m1 = 1 m2 = 1/2 !"# n
+& E → ∞# 20 $ 2.. 8
!vi → 1"# / ω → n# &&( +2# +&.2 18
.2 !<<" 2.. &H 62 &/ !=" !C" /6 2& +28
ε1 , ε2 , ε.
& + 72& !J"# !<>" +&/2 &H- 1. &.
I & 22A
m1 π(n − ε) 1 − ε21
m1
m2
E=
+
+
,
ε21
ε1
ε2
m1 1 − ε21 m1 1 − ε21
m1 (1 − ε21 ) m2 (1 − ε22 )
L=
(n − ε)π +
+
.
2γε21
ε21
ε1
ε2
!<D"
!<@"
2 9. 21 2 2&2 +22 !<<" 6 8
1. !E" 02A
1 − ε2i
vi
Qi
= =
.
!<B"
2
ω
ε
1 − vi
i
+&(6. 1. !B" !<B"# 02 .6( 21 ε1 , ε2 ε
(Q1 /ω + Q2 /ω)
ε2 1 − ε21 + ε1 1 − ε22
tg πω =
=
.
1 − Q1 Q2 /ω 2
ε1 ε1 − 1 − ε21 1 − ε22
H &
ε2 1 − ε21 + ε1 1 − ε22
πω = πn − πε = πn − arctg
.
1 − ε21 1 − ε22 − ε1 ε1
!<E"
1 !<E" +62 62 ε1 = sin θ1 ,
A
πε = arctg
ε2 = sin θ2 #
<J
+2
sin θ2 cos θ1 + sin θ1 cos θ2
= θ1 + θ2 .
cos θ1 cos θ2 − sin θ1 sin θ2
& ? 62 +&/2 1# .6H- +2 ε, ε1, ε2 A
πε = arcsin ε1 + arcsin ε2 .
!<F"
&.. 1 !<B" !F"# +&/2 9
1 − ε21
1 − ε22
m1
= m2
,
2
ε1
ε22
+6&.H- 6( ε2 /6 ε1 A
−1/2
√ ε2 = ε1 2 4(m1 /m2 )2 (1 − ε21 ) + ε41 + ε21
.
!<="
1. !<=" !<F" +6&. 6( /6 ε1 +2 ε
+&(6. +&/ 9. &. +2 ε ε2 # 02 +8
& ε1 → 0 !E → ∞" 2+/ 1. &. 62 I !<D" & 22 !<@" . + 2&2 +2 ε1 &. 6 nA
E = m1
πn πn 1 + μ32
πn 2 8 + 5μ32 + 3μ52 3 πn 4
+
ε
ε +
ε1 −
ε + ...
−
−
1
ε21
2
3
8 1
60
16 1
m2
L= 1
γ
6(
,
!<C"
πn
14 + 5μ32 + 9μ52
πn
1 + μ32
+
ε1 +
−
−
2ε41
2ε21
3
60
52 + 35μ32 + 42μ52 − 25μ72 3
ε1 + . . . ,
1120
μ2 =
!<J"
m2
.
m1
&H/ 6 . !<J" /2 !<C" +2
ε1 =
1 + μ32
πnm1
1
πnm1 1
+
πnm31 3/2 + . . . ,
1−
E
4 E
6
E
+&/2 2+/ 6&1 & 22 L !<@" + +.2 E A
L = α E 2 + α1 E 1/2 +
α =
α4
α2
α3
α5
+ 3/2 + 2 + . . . ,
+
E
E
E 1/2
E
E → ∞,
3/2
3/2
√ m5/2 + m5/2
2 m +m
2
α2 = πn 1
α1 = − · 1 √ 2 ,
,
3
10γ
γ πn
3/2
3/2 2
7/2
7/2
m2 + m1
m + m2
,
α4 = −(πn)3/2 1
,
α3 =
18γ
112γ
5/2
3/2
3/2
5/2
m4 + m1 m2 + m1 m2 + m42
α5 = πn 1
.
60γ
!D>"
1
,
2γπn
!D<"
D>
6/ !D<" &. & α 1 +& E → ∞
??- 1 2? α = (2πγ)−1 j<# Dk g2( &
α ∼ 1/n 6/. n &&H < I77$ α1 # α2 , . . . 6&8
1 !D>" 6. 2 m1 m2 # / +6&. $( 2&( 6/.
2 . 1. !D>" I+2&(2 8
2 &. I $& ?02 /( + <& )"9 ,A -* $%,,* +G 9-)":
&. / + 20 / !" +22 +0# +&1
?0 j<# Dk &. 26 ? &H/ / + &.H-
&2 22 ? &/2 22 !<>"
J = L + S,
S = s1 + s2 ,
!DD"
s1 , s2 h +$ + ( e3 # 1 ++ I 8
.. !J"# ?&& +8?&(2 622 2H- jDk
E = Ecl + ΔESL ,
ΔESL
2 ω 2
=
1 − 1 − vi si .
a0 i=1
!D@"
g( 1 2 ?6/2 2&2 Ecl &/H IH 22 $0 !J"# !<D" & !<C"
+2-(H 9 !<B" !<=" 02 6&1 ++ !D@" +
+.2 ε1
ΔESL =
=
γε21 s1 (1 − ε1 ) + s2 (1 − ε2 )
=
m1 v1
γε21
s1 + s2 2
ε1 + . . . ,
s1 + s2 − (s1 + μ2 s2 ) ε1 +
m1
2
+&.2 . !<C" 02 ??- 1. !D>" h 6&1
+& & 22 J = L + s1 + s2 + +.2 I E = Ecl + ΔESL A
J = α E 2 + α1 E 1/2 +
α̃4
α̃2
α3
α̃5
+ 3/2 + 2 + . . . ,
+
1/2
E
E
E
E
E → ∞.
!DB"
g( I77$ α # α1 # α3 +H 6/.2 !D<"# /# &
1 α = (2πnγ)−1 6 + I778
$ 6&1 !DB" /2 +0 ++ &/H. 1
!D<" &H-2 ?62A
√ √
√ 3/2
3/2 α̃4 = α4 − 14 (πn)3/2 s1 m1 + s2 m2 ,
πn s1 m1 + s1 m2 ,
√
√ 3/2
3/2 − 3γ(s1 + s2 )2 .
α̃5 = α5 + 16 πn s1 m1 + s1 m2 m1 + m2
α̃2 = α2 +
i 1. +6&.H $( . I+2&(8
2 2 6/. 2 m1 # m2 2&# 6/8
. s1 # s2 +$ + A,-: '9"#%9@#+
D<
j<k ?6 * # )2(. , # v+ ) ?&( 6?1.
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jDk 4 w 2& ? 1 oo 5. 78
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