electrodynamics in friedmann-roberson-walker universe
TRANSCRIPT
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8/7/2019 Electrodynamics in Friedmann-Roberson-Walker Universe
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arXiv:ast
ro-ph/0508566v2
23Jun2006
a
a4
a4
http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2http://arxiv.org/abs/astro-ph/0508566v2 -
8/7/2019 Electrodynamics in Friedmann-Roberson-Walker Universe
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ds2 = dt2 a2(t)
dr2
1 Kr2 + r2 d2 + sin2 d2 ,
a(t) K 0, +1 1 =
dta r
= r/ (1 + Kr2/4) = r/f(r)
ds2
= a2
()
d2
1
f2(r)
dr2
+ r2
d2
+ sin2
d2
.
l =
1, 1f
, 0, 0
, n =a2
2
1, 1
f, 0, 0
, m =
ar2f
(0, 0, 1, i sin ) m = =
f
22ar
cot , = fa2
a
af (f /r)
f /r
, = f
2
a
af+ (f /r)
f /r
= a
2a,
r l
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l D = l =
1a2
D, = n = 12D+, = m = f2ar L, = m =
f2ar L+, D = f r L = isin
r =
dr
f=
2K
Kr
2=
r, K = 02 (r/2), K = 1
2 (r/2), K = 1,
D = r r r 0, r r K = 1
a
F 0 = Flm, 1 =
12F (l
n + mm) , 2 = Fmn
(D 2) 1 = ( + 2) 0 2,( 2) 1 = ( + 2) 0 2,
(D + 2) 2 = ( + 2) 1 0 ( + 2) 2 = ( + 2) 1 0.
(1/f)D [an(f /r)m] = an(f /r)m
1f
D naaf + m (f /r)
(f /r)
,
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D
21ar
f
2
= (L+ + cot ) 0ra3
f ,
L
21
ar
f
2= r
2
f2D+
0
ra3
f
,
r2
f2D
2ar
f2
= L+
21
ra
f
2
(L + cot )
2ar
f2
= D+
21
ra
f
2.
1 0 2 + = 0ra
3/f = 22ar/f,
D
r2
f2D
+ L [L + cot ] = 0.
= (r, )Y1(, )
1sin
sin
2i cos
sin2
+1
sin2
2
2 cot2
1 Y1 = Y1,
Y1 Y+1 Y1 ; = l(l +1);l l = 1, 2, 3,... eim
D
r2
f2
D
= .
D
r2
f2D
= .
Z = r2
f2 D
D D2 = DD = 2
r2
22 ,
D2Z = V Z, V = (f /r)2
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8/7/2019 Electrodynamics in Friedmann-Roberson-Walker Universe
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ei D = r i L = msin R
Z
d2
dr2+ 2
R = V R,
V =
l(l + 1)/r2, K = 0l(l + 1)/ sin2 r, K = 1
l(l + 1)/ sinh2 r, K = 1.
l 1/r2
0 < r < r ir i K = 1 K = 1
R = DR.
K = 1 f(r) = 1 r = r
R = r [A0jl (r) + B0yl (r)] ,
jl yl l 1 j 2jl = h
(1)l + h
(2)l
(i)l+1eir + (i)l+1eir / R+ R
C
n
R = A1 (sin r)l+1 Cl+1l1 (cos r) + B1 (sin r)
l Cl+l (cos r) .
r l + 1
R = A1eir 2F1
l, l + 1, 1 i, 1
2(1 coth r)
+B1eir 2F1
l, l + 1, 1 + i, 1
2(1 coth r)
,
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r r ir i 0 2 1
4T = |0|2 nn + |2|2 ll + 2 |1|2
l(n) + m(m)
401n(m) 412l(m) + 202m(m) +
T
4 = 4T =f2
2r2a4|+|2 + ||2 + 2 |1|2
= 4 Trr + T + T 4Tr =
f3
2r2a4|+|2 ||2 ,
=rr =
=
= a/a
gT = i
gTi g = det g =
a4r2
f3 sin 2
= a8g3 g3
Ea4
= a4
i
g3Ti
= a4
drT
r = a
4
dr
f3
r2|+|2 ||2 ,
E r dr =
r2
f3sin dd
2 |+|2
||
2 = |DZ|2
|Y+
|2
|D+Z
|2
|Y|
2 = dRdr2 + 2 |R|2 |Y+|2 |Y|2
+iW[R, R]|Y+|2 + |Y|2 W Y
R R Ea4 a4
a4
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M
P
0
, P
1
,
Q
0
Q
1
(D + ) P0 + ( + ) P1 = iM2
Q1
,
( + ) P1 + ( + ) P0 = iM2
Q0
,
(D + ) Q0 + ( + ) Q1 = iM2
P1,
( + ) Q1 + ( + ) Q0 = iM2
P0.
2raf P
0, ra2
f P1,
2raf Q
0 , ra2
f Q1
1/2Y1/2, 1/2Y1/2, 1/2Y1/2, 1/2Y1/2
L +
1
2cot
Y1/2 = Y1/2,
rD
1/2 = (f
iMra)
1/2.
Y+1/2 =
L + 12 cot
Y1/2 Y1/2 Y1/2
1
sin
sin
i cos
sin2
+
1
sin2
2
2 1
4cot2 1
2+ 2
Y1/2 = 0
Y = (l + 1/2); l l = n + 1/2 n = 0, 1, 2, 3... Z
= +1/2
1/2 = R
T
1
R
r f
r
R =
1
T
iMa
T = ik,
k = pa
r f
r
r f
r
R =
iMa
iMa
T = k2Z.
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d2
dr2+ k2
R = VR,
V = 2f2
r2 d
dr
f
r
.
V = (l + 1/2)
(l + 1/2 1)/r2, K = 0(l + 1/2 cos r)/ sin2 r, K = 1
(l + 1/2 cosh r)/ sinh2
r, K = 1.
1/r2 V+ l = 1/2
l = 1/2 V+ V 1/2 V+ l = 1/2
1/2 0 r l, V+ cosh r > (l + 1/2), V+min = 12(l + 1/2)
l + 1/2
(l 1/2)(l + 3/2)
cosh r = (l + 1/2)+
(l 1/2)(l + 3/2) r l V+ r l K = 0, 1
K = 0,
R = r A1/2jl1/2(kr) + B1/2yl1/2(kr) . j
P(,)
0 r
R = C1/2 sinl r
cot
r2
1/2P
(l+1/21/2,l+1/21/2)kl1 (cos r).
k ik r ir
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d2
d2+ k2
T = VTT,
VT = M2a2 iMa.
a
a2 = a4 + (1 M R ) a2 + M
(1 v2) a2 + v2 + R,
H20 R,M, c v 0 a = a0 = 1 = 0 R = 0 M v = 1 v = 0 0 < v < 1
VT
0 = 1 0 > 1 0 < 1 VT = 0, e
ik
a a2
d2
da2+
d
da+ k2 + M2a2 iMa
T = 0,
a
a = 1
T+ = A+eiM 2/2U(
ik2
4M+
1
2,
1
2,iM2) = A+
14
iM2W 1
4
1+ ik
2
m
, 1
4
,
U W
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8/7/2019 Electrodynamics in Friedmann-Roberson-Walker Universe
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eik k T+ A+
ei
k22iM k
a aI, aI a aI Mv + R a M MMv + R
T = eiM(tt0)T1
d2T1d2
2iMadT1d
+ k2T1 = 0,
x = 0
a () =
n=0
ann! x
n
an =dna()
dn|0 d = dx x
T1 =
n=0
bnxn
n(n 1)bn = (ik)2bn2 (2iM)n2m=0
amm!
(n m 1) bnm1;
T1 b0 =
1/
2 b1 =
ik/
2 M = 0
T1 = eikx/2
AB =1
2
l m
m n
,
12
J = AB PAPA + QAQB
gJ = 0
J =f2
4r2a4|Z+|2 + |Z|2 |Y+|2 + |Y|2
Jr = f3
4r2a4
Z+Z + Z+Z |Y+|2 + |Y|2 .
n = Jt = aJ na3 = a4J
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8/7/2019 Electrodynamics in Friedmann-Roberson-Walker Universe
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d R+Rdr = d R+Rdr = ik |R+|2 |R|2
d |T+|2d
= d |T|2
d= ik
T+T T+T
.
R+R + R+R = C1
R+R
|T+|2+|T|2 = C2
|Z+|2 + |Z|2 = |R+|2 d|T+|2d +|R|2 d|T|
2
d=
|R+|2 |R|
d|T+|2
d= 1
ik
d(R+R)dr
d|T+|2d
gJ =
g3 (na3) =
14ikf
d(R+R)dr
d
|T+|2
d|Y+|2 + |Y|2
r
(Na3)
=
r0
drdd1
4ikf
d
R+R
dr
d |T+|2d
|Y+|2 + |Y|2 = R+R2ik
d |T+|2d
;
N
r 0 N = N0a30 Na3 =
R+
R2ik |
T+ (0)
|
2
|T+ ()
|
2
Z+Z + Z+Z =
R+RT
+T + R+R
T+T
=R+RT
+T +
C1 R+R
T+T
=R+
Rik
d|T+|2d
+ C1T+T
a4 drJr = R+R2ik d|T+|2d + C1 C1 = 0
|T+|2 eik
|T+|2
T+ =
k2/4M
coshk2
4M
+sinh
k2
4m
eiM 2/2U
ik2
4M +12 ,
12 ,iM
2
T =
1
coshk2
4M
+sinh
k2
4m
eiM 2/2U
ik2
4M,12 ,iM
2
R+R = i |A+|2 r2jl1/2 (kr)jl+1/2 (kr)
r
(Na3) r = 1
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8/7/2019 Electrodynamics in Friedmann-Roberson-Walker Universe
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k j r k l
r
T = Trr =if2
4a4r2
1/2D1/2 1/2D+1/2 Y1/22 + Y1/22 ,
Tr = if3
4a4r2
1/2D1/2 + 1/2D+
1/2
Y1/22 + Y1/22 ,T = T .
T
gT
=
i
gTi T
Tr
ik |T+|2 + |T|2 R+R + R+R
a4
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R+
a = 1
a4 a4
a4
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http://arxiv.org/abs/gr-qc/0401065http://arxiv.org/abs/gr-qc/0405140 -
8/7/2019 Electrodynamics in Friedmann-Roberson-Walker Universe
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1 2r
1
2
r
r r K = 0, K = 1 r r K = 1 r 2 r r
0 1 2 /2
0 1 2r
10
20
V
l = 1, 2 3 K = 0, 1 1 1/r2
l
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8/7/2019 Electrodynamics in Friedmann-Roberson-Walker Universe
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1 2 3r
-0.5
1
2
V+
1 2 3r
1
2
V-
/2 r
10
20V+
/2 r
10
20V-
1 2 3r
10
20
V+
1 2 3r
10
20
V-
l = 1/2, 3/2 5/2 K = 0, 1 1 V+ V l = 1/2, V+
r V+ r = 0
l = 1/2, V+ 1/2 r 0, V 1/2 r
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10 20r
R+
10 20r
R+
10 20r
R+
/2 r
R+
/2 r
R+
/2 r
R+
20 40r
R+
10 20r
R+
5 10r
R+
K = 0, 1 1 k = 1/2, 3/2 5/2 l = 1/2, 3/2 9/2 l = 1/2, 3/2 9/2 k = l + 1, l + 2 l + 4 V+ 0.5, 0.27 0.25 k = 0.1, 0.4 0.6 k = 0.1, 0.24 0.3
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0.5 1a
-2
-1
VT
0.5 1a
-2
-1
VT
0.5 1a
-2
-1
VT
a. K = 0, 1 1
VT+ = VT v0 = c v0 = 0.3c
v0 = 0
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0.5 1 1.5 2
-1
-0.5
0.5
ReT
0.5 1 1.5 2
-1
-0.5
0.5
1
ImT
0.5 1 1.5 2
-0.4
-0.2
0.20.4
0.6
ReT+
0.5 1 1.5 2
-0.4
-0.2
0.2
ImT+
M = 1 k = 1 2 M = 2 k = 1 2
|T+|2 + |T|2 = 1
2 4 6 8 10r
[ Na3]
r = 1 l = 1/2 5/2 1 M = 2 k = 1 2 M k l r = 3 l > 1/2
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0.5 1 1.5 2
[ Na3]
r r = 4 r = 1 l, M k