Post on 18-Mar-2018
Brane Bianchi I, Bianchi III
Kantowski-Sachs
Brane Cosmology on Bianchi I, Bianchi III
and Kantowski-Sachs Spaces
(Ming-Fun Chun)
(Prof. Win-Fun Kao)
Brane Bianchi I, Bianchi III Kantowski-Sachs
Brane Cosmology on Bianchi I, Bianchi III and Kantowski-Sachs
Spaces
StudentMing-Fun Chun
AdvisorWin-Fun Kao
A Thesis
Submitted to Institute of Physics
College of Science
National Chiao Tung University
in partial Fulfillment of the Requirements
For the Degree of
Master
in
Physics
July 2004
Hsinchu, Taiwan, Republic of China
() 92
2
Brane Bianchi I, Bianchi III Kantowski-Sachs
()
()
()
()
:______________
()
94 7 20
94 7 20
9127508
________________
93 7 20
() 92
Brane Bianchi I, Bianchi III Kantowski-Sachs
()
9127508
________________
93 7 20
_
l>B, B, `>B; yV>B;e, B}V1,
J?D]-AdZ2-7A; .vH,
Bh Osamu Seto .w`BDBSdj; A
db2, nDu67B', F
B;d6.}0A7, ?DvDAn
u, >g',.c7nDjj, 6
7tMb4, u'N% ; rzT5I7rv
DBbnDN, '>; dY4uO j5\Bbv
vJN; 4.,, B_Av46?4
nD}-2 s-V, g)| u>7.=
: DEw.c,, 25,, -n, u.)
n; pD+n7., B7jy:7,
Fb yuBbT7v.), J_hGVz, Fb
u`7Bbr; DdQ7rA%}, BR
_Vz, BR k7r; Dkn{2B7d
b:b4 '', 'A, .>
8YH, ;BbHWu}IA[ 0vm
b>U-H7F4, , !_, db, *2 ;HN
7C
b
d2, Bb* Bianchi I,Bianchi III Kantowski-Sachs _|
; _G (Homogeneity) /4 (anisotropy) _; 7)
=gjj, (n =gjjFf#Bbk
m7 5, Bbd2d7sb cq: uBbec
_?D}0_; (perfect fluid) _, _ucqB
bF0u_JFw&vy|_&&v, 7
&v (brane spaces) 4u|2\nO
}&__=g jj(, BbJ__}w
AG (Homogeneity) /4 (Isotropy) 4, 7/w
2, b (cosmology constant) A3Obb; o
2, uBbF5?&v2&b (brane constant) A3O
bb; , Bb6)Bb5?b6}A3O
bbv`, 6u ov, Iwk&b3v5(
2
Contents
1 4
1.1 FRW_ . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 &v 10
2.1 _ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Brane Bianchi type I,Bianchi type III D Kantowski-
Sachs vS . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 }&Dn 19
3.1 . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1.1 Nj5) . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1.2 NjDbMj5 . . . . . . . . . . . . . . . . . . . 24
3.2 4b . . . . . . . . . . . . . . . . . . . . . . . 31
4 ! 34
5 36
5.1 Vacuum Einstein equation . . . . . . . . . . . . . . . . . . . . 36
5.2 Einstein equation with energy momentum tensor . . . . . . . . 47
3
Chapter 1
(Cosmology) w'o%7, r4}
A jD, ukh,., 6rk, /D
7|s_0J V, kDhxX, U
kA_, bW, 7/r 6\
1920, =gT|72, 7! 1929
, +N rm$Ph, 72wFrTkn
7G, 7 3z GGyA
bJ, rAIp7 nD, Bb6uw
1965,PenaiasDWilson7*5 (Cosmology Microwave
Background), 5A2.7 K, /G}0k5
7, y7Bb8, b
eV CMB hBbD? }0uG/
4 s_!4, UBb Friedmann-Roberson-Worker
_ _u5?ABuG/4G, OOh
, Bb CMB } 0wx
8%H, Dwv| jb ,ks=
n~j Bb&v3-brane , u&v\9A_&
_q, 7&&v (Bb bulk) q 7, 7.
!, &@v b ( Planck scale ) h.
7(V, :t Klein uz h9k Kaluza
&2, Fh7j, J_bHV_, *7 p
vjjJ\-yD~6-5H
.j(V. u\wW.
Ou1980H, k$ Kaluza-Klein hh\T|V,
1.1 FRW_
=g&2T|, zp7vDu.}/,
H7*HvH! vDwy1920, =gT|7.&2
(T|72, un _hj, 7/6ABb
'x Bb=gj (Einstein Equation) *T
(Action Principle) ), 77T2$J[:
S =
d4x
g(R
16G+ Lmatter) (1.1)
R F[u-0" (Riemann curvature tensor),G u
b (Gravitational constant), 7Lmatteru Lagrangian. Bbgd
}(, Bb)=gj:
R 12gR = 8GT (1.2)
Tu?" (Energy Momentum Tensor),gu Metric Ten-
sor, Metric Tensor dzVz, O, 7
Ricci scalar(R) Ricci " RsBb FJ=gj
UiuD?, 7D ?SBbU
i[|V
;WBb- ( 100Mpc) h*5 (Cosmic MicrowaveBackground (CMB)) !, BbBbuG (Homogeneity) /
4 (Isotropy) , G4sBb, BbFh 4DwF
P0; 74sBb, Fhj,w4Q B
b s_!4 *=g22, p7
x| 4, 7b,, x|4&
0kb&7 &7JT& 7T
Friedmann-Roberson-Worker(F-R-W) _uJ,HF|,
6
7 _Bb)_H F-R-W _(j:
ds2 = dt2 + R2(t)( dr2
1 kr2 + r2d2 + r2sin2()d2) (1.3)
,2R(t) (Scale Factor), uOv,
6zp7 k 0b, k = 1, 0,1,k = 0v, Bbu_; k = 1v, zBbFdu
_7[; k = 1v, zBbFdu_ r, , 7,t v , (jds22!4, 2 (Lorentz
Transformation) wM\M. sBb, BbdChv, BbJ
_|_ Vdn, 7!u. y6 Metric Tensor g(j
2- trace ,M:
g = gii = (1, R2(t)
1 kr2 , R2(t)r2, R2(t)r2sin2) (1.4)
k F-R-W _4BbJ) Christofol symbol(ijk)
}:
ijk =1
2gil(
gljxk
+glkxj
gjkxl
) (1.5)
0ij =R
Rgij
i0j =R
Rij
1.4 Christofol symbol 2 Christofol symbol xM
|, sBbu4 Ricci tensor 2R = gR ,
7R uF-0, -02OBb (Bb
)0" Ricci Tensor }:
R00 = 3RR
7
Rij = (RR
+ 2(R
R)2 +
2k
R2)
(1.6)
J" Ricci Scalar:
R = 6(RR
+R2
R2+
k
R2) (1.7)
7H R(t) jJ*j=gj):
R 12gR = 8GT (1.8)
TBbu;$, w} T00 = , T 11 = T 22 = T 33 = p.
?,p 9 jjvF)00}j, OVzB
bd Fridmann Equation:
R2
R2+
k
R2=
8G
3 (1.9)
(, BbJ Fridmann Equation Z:
k
H2R2=
3H2
8G
1 1 (1.10)
u? critical density cM
c
, c 3H2
8G
,2,H RRuF+b (Hubble constant). ;Wh *
5e2Bb 1, 6
, h2Bb, =gj.hz=g7
Uj K7pb (O(VwuG7
Chapter 2
&v
Bb&v;o1919AT|,Oo1.e,1980
Hw, k$ Kaluza-Klein hh\T|V,
Su?"T.,E bulk ,^@,
E = Ciajbnanb (2.3)
Ciajbnanb bulk 2&Weyl tensor , Eq2.12?
"S(1&2?") JT[:
S =1
12TT 1
4T T +
1
24g(3T
T T 2) (2.4)
,TBbFu_x4/.$, FJu;
$, ;?"}:
T 00 = T 11 = T 22 = T 33 = p (2.5)
?";$sBb, Bbe5
>T (f... ) ,T = T u?" trace T =
T = T00 + T
11 + T
22 + T
33 ?9 p ,
[:
p = ( 1) (2.6)
_b/w 1 2 .
;Wj Eq2.1, BbJ, E = 0/k5 0v, BbO brane 2=gj, c5?&v_&v
_y2 Bb d2, .5?E bulk ,^@,
IE = 0VdBbn&v_!c q
, Bb67j brane ,6?0, 722?
[:
T = 0 (2.7)
11
u} (Covariant Differentiation), Fu&v2}, T"? "5$:
T T ; = T , + T + T (2.8)
5(Bb},j7=gjD?0jjdj(,B
b)BbF; , j
12
2.1 _
,, Oh, Bb CMB }0wx
Eq2.10, Bb5(3b}un
Eq2.13 4b, BbJ*4bv
Vn__$
14
2.2 Brane Bianchi type I,Bianchi type
III D Kantowski-Sachs vS
Bbbn_G/4_ Bianchi type I,Bianchi
type III D Kantowski-Sachs(5(Bb BI,BII,KS) &-v!Z
Bbl*2JBbF_(jds2$,7(j(B
7_d"
g = (1, a1(t)2, a2(t)2, a2(t)2f()2) (2.14)
z (Metric Tensor) |, Bb-b) Christofol
symbol(ijk) sBbu", (y)[-
0R , y-0) Ricci"RD Ricci 0bR(, Bb|(y
z
,2k = 0, 1,1v}H[O BI,KS D BIII _,v?, 7 1 2b Bb*_j2BbF)uD?[ Ou?OvZU)Bb
U7j, FJBbb5?2? 0:
T = 0 (2.19)
;Wu}T?",%Jl(BbJ)Ov?D O[:
+ (a1a1
+ 2a2a2
) = 0 (2.20)
FJBb?Ov(V = a1a22):
= 0V , 0 = constant > 0 (2.21)
?), BbJ7jBbF)=jj
-uOvZ, FJBbZJn7 B
b7;)j, Bbj 2.16,2.17,2.18ZA:
d
dt(V H1) = V +
1
2k240V
1(2 ) + 112
k4520V
(1 2)(1 ) (2.22)d
dt(V H2) = V +
1
2k240V
1(2 ) + 112
k4520V
(1 2)(1 ) ka1(2.23)
3H + H21 + 2H22 = V +
1
2k240V
(2 3) + 112
k4520V
( 2)(1 3)(2.24)
J,j2, hb2:
H1 =a1a1
, H2 =a2a2
, V =1
3(H1 + 2H2) =
V
3V
16
-V, Bby,? Eq2.21, %JHb
(BbJ) }j:
V = 3V +3
2k240V
1(2 ) + 14k45
20V
12(1 ) + 2ka1(2.25)
Bb, k=0,k=-1,k=1Bb}J) Bianchi type I, a Bianchi type
III,and a Kantowski-Sachs j (;W}}l(, BbJ)
}j:
V 2 = 3V 2 + 3k240V2 +
1
4k45
20V
22 + 4k
a1dV + c
(2.26)
,2,c }b }jBb5(n3bj, Bb
Jp)|V_j u(4, 7/%rtl-, Bb
u)wj&j, , 5(}&Dn 2BbtwNj ,
BbJ_2j:
t t0 =
L(V )12 dV (2.27)
t v,t0v,V 7b L(V) 2:
L(V ) = 3V 2 + 3k240V2 +
1
4k45
20V
22 + 4k
a1dV + c (2.28)
2$, BbJQ)4b A Dai2j:
A = 3K2L(V )1 (2.29)
ai = a0V13 e
Ki
dv
vL(V )12 (2.30)
17
,2, bK2, KiBbJ;Wj Eq2.22,Eq2.23):
Hi = H +KiV
(2.31)
K1 =2
kaidt
3K2 =
1 kaidt3
(2.32)
K2 = 3i=1K2i (2.33)
j Eq2.21,Eq2.28,Eq2.32p Eq2.22,Eq2.23BbJ) K D}
b c 5O[:
K2 =2c
3(2.34)
18
Chapter 3
}&Dn
A,__R(, Bbub, FJBb
n Ov-, }AS$? Dw2
S[? OBbQ) jQj, ]Bb
nSvq, J)Nj 7_Nj
$ u, BbUKU7, Mathematica D Matlab
V)wbMj, (yDBbNjT_ BbNjJ
Wv Bbbj:
V 2 = 3V 2 + 3k240V2 +
1
4k45
20V
22 4k
a1dV (3.1)
BbJJ,j2, K-)NjV , (
BbZbzjp-j 2K-)a1
d
dt(V H1) = V +
1
2k240V
1(2 ) + 112
k4520V
12(1 )(3.2)
Bbu)a1, (BbJ4k
a1dV
, %4k
a1dVbMj(BbZJ__54 ,
19
a2l27):
b(t) =
v(t)a(t)
(3.3)
BbTbM}&vFUKlN|, Bbcqa(t0) = a0 ,
b(t0) = b0 , b(t0) = b0 , v(t0) = v0 t0 v, 7b)_,
v Bbb#8cqM: = 32,3 = p = 1 , 3k240 = m = 1 ,
14k45
20 = n = 1kKBb#:t0 = 0, a0 = 0.1, b0 = 0.1, b0 = 1. B
bhVz, Bb.UBbUKS, ]
u_b,cq 7Bb*cq|n 2, Bb
4b (b) Vn___44
74b:
A 3
i=1
(Hi H)23H2
(3.4)
n4b6uU7_jV48$
, Bb = 1, = 32, = 2, VT__}&
20
3.1
BbI = 32, /UM0.001VTbM}&M, 7w
FbqlF2,BbndNjbkwF10,100,1000
B10000IVdBbK, Bb63k240d
(, BbJwNS _-Bb
Nj8u/v(v
2. 14k45
20V
223,
3. 3k240V23, Njj:
V 2 ' 3k240V 2 = mV 2 (3.13)
Bb)Nj$:
v(t) = (
m
2(t t)) 2
6)a1(t),a2(t):
b(t) = c3(t t)23 , a(t) =
1
c23(
m
2)
2 (t t) 23 (3.14)
b:t = t0 2mv20 , c3 = b0(
2
mv
20 )
23 = 3
2v, Bb
)
v(t) =3
43 (m)
23 (t t) 432
83
b(t) = c4(t t) 49 , a(t) = 3103 (m)
23 (t t) 49
283 c23
Bbn
f(t) | 3V2 + 1
4k45
20V
22 + 4k
a1dV
3k240V2 | 1 (3.15)
(BbwvS Bb_-, Bb
Njvu v'oq, /ks_Nj
v5
23
3.1.2 NjDbMj5
Bbn__, b}u Kantowski-Sachs moddle,Bianchi Type
I D Bianchi Type III _. QOBbnFb3V 2,14k45
20V
22J
k240V2'bv, bF) Nj, 1DbM}
&Fd|VjWn, ;W k}530QVdBb}&
nYW. QOBbZ J.3vVWn.
BbI = 3/2, /UM0.001VTbM}& M, 7B
bbq = 1/3, k240 = 1000/3 J112
k4520 = 1/3. Bb
nbk240Vv Bbk240 > 1000/3v8,
7Bbk240 = 1000/3Vd yzpBbFn
[:
V (t)2 = 3V 2 + 3k240V (t)2 +
1
4k45
20V (t)
22 + 4k
a1dV (3.16)
QOBbbM}&)(DNjd, ;WBb
}j23vn, Bb|10%JqA3D
bMj$,Bb Error2 Error = (NumericalaproximationNumerical
)100%, J-Bb}._VT}&:
(1)KS: 3.1 KS _bMj(BbJcA
}Avq 7j; 3.2b
3bMjD_j, BbJ vdDbM
_(N; 3.3 brane 3vN$, Bb*$2
J)|VvVQvbM_DBbNjV
, Jc)v Qv brane Tub; J3.4
b3v`NjDbMj$
24
0 1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
9
10x 10
5
Time
Vol
ume
Sca
le F
acto
r (V
)
Numerical Plotting
Numerical
Figure 3.1: KSv(: = 1/3, k240 = 1000/3 J 1
12k45
20 = 1/3, = 3/2
5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 100
1
2
3
4
5
6
7
8
9
10x 10
5
Vo
lum
e S
cale
Fac
tor
(V)
Dominance of Cosmological Constant
5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 100
0.005
0.01
0.015
0.02
0.025
0.03
Time
Err
or
(V)
data1
NumericanceApproxivation
Figure 3.2: KS_3V 23vNjD(: = 1/3,k240 = 1000/3 J
112
k4520 = 1/3, = 3/2
25
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x 105
1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4x 10
3
Vo
lum
e S
cale
Fac
tor
(V)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x 105
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Time
Err
or
(%)
Dominance of 5D Gravitational Constant
Error
NumericanceApproxivation
Figure 3.3: 14k45
20V
223vNjvqD KS (: = 1/3, k240 = 1000/3 J
112
k4520 = 1/3, = 3/2
0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.060.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4Dominance of 4D Gravitational Constant
Vo
lum
e S
cale
Fac
tor
(V)
0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.060
2
4
6
8
10
12
14
16
Time
Err
or
(%)
Error
NumericanceApproxivation
Figure 3.4: 3k240V23vNjvqD KS
(: = 1/3, k240 = 1000/3 J112
k4520 = 1/3, = 3/2
26
(2)BI: 3.5 BI _bMj(; 3.6b
3bMjDNj, *_BbJ3vvu
v((Vv3Nj}}_(; 3.7
brane 3v$, BbJc brane vQv
; 3.8b3v`NjDbMj$, Bb
bbvBb J)yN8Jy
0 1 2 3 4 5 6 7 8 9 10
1
2
3
4
5
6
7
8
9
10x 10
5 Numercal Plotting of Bianchi Type I
Time
Vo
lum
e S
cale
Fac
tor
Numericance
Figure 3.5: BIv(: = 1/3, k240 = 1000/3 J 1
12k45
20 = 1/3, = 3/2
3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
9
10x 10
5 Dominance of Cosmological Constant
time
Vo
lum
e S
cale
Fac
tor
(V)
3 4 5 6 7 8 9 100
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Err
or
(%)
NumericanceApproxivation
Error
Figure 3.6: 3V 23vNjvqD BI (: = 1/3, k240 = 1000/3 J
112
k4520 = 1/3, = 3/2
27
0 0.2 0.4 0.6 0.8 1 1.2
x 105
1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4x 10
3
Vo
lum
e S
cale
Fac
tor
(V)
0 0.2 0.4 0.6 0.8 1 1.2
x 105
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5Dominance of 5D Gravitaional Constant
Time
Err
or
(%)
Error
NumericanceApproxivation
Figure 3.7: 14k45
20V
223vNjvqD BI (: = 1/3, k240 = 1000/3 J
112
k4520 = 1/3, = 3/2
0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.060.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
Time
Vo
lum
e S
cale
Fac
tor
(V)
0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.060
2
4
6
8
10
12
14
16Dominance of 4D Gravitational Constant
Err
or
(%)
Error
NumericanceApproxivation
Figure 3.8: 3k240V23vNjvqD BI
(: = 1/3, k240 = 1000/3 J112
k4520 = 1/3, = 3/2
28
(3)BIII: 3.9_bMj(; 3.10b3bMjD_
j; 3.11 brane 3v$J3.12b3
v`NjDbMj$
0 1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
9
10x 10
5
Time
Vol
ume
Sca
le F
acto
r (V
)
Numerical Plotting
Numerical
Figure 3.9: BIIIv(: = 1/3, k240 = 1000/3 J 1
12k45
20 = 1/3, = 3/2
5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 100
1
2
3
4
5
6
7
8
9
10x 10
5
Vo
lum
e S
cale
Fac
tor
(V)
5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 100
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045Dominance of Cosmological Constant
Time
Err
or
(%)
NumericanceApproxivation
Error
Figure 3.10: 3V 23vNjvqD BIII (: = 1/3, k240 = 1000/3 J
112
k4520 = 1/3, = 3/2
29
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
x 105
1
1.5
2
2.5x 10
3
Vo
lum
e S
cale
Fac
tor
(V)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
x 105
0
2
4
6
8
10
12
Time
Err
or
(%)
Dominance of 5D Gravitational Constant
Error
NumericanceApproxivation
Figure 3.11: 14k45
20V
223vNjvqD BIII (: = 1/3, k240 = 1000/3 J
112
k4520 = 1/3, = 3/2
0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.0650.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
Vo
lum
e S
cale
Fac
tor
(V)
0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.0650
2
4
6
8
10
12
14
16
Time
Err
or
(%)
Dominance of 4D Gravitational Constant
Error
NumericanceApproxivation
Figure 3.12: 3k240V23vNjvqD BIII
(: = 1/3, k240 = 1000/3 J112
k4520 = 1/3, = 3/2
Bb, DMv BbNjJbMj'
N, uIAE, [Bb uW ,VzBbJ
__ KS,BI,BIII 3k240d_|c(, xA.v
Nj, 7__3b.
4k
a1dV TuU__wNjv KS wk BI
wk BIII 7o BIII ok BI ok KS *
}j26J|4k
a1dVT.}T, .}3v`
30
3.2 4b
42:
A 3
i=1
(Hi H)23H2
*l2BbJ4bBbF_-2$:
A = 3K2L(V )1
7w2K2DL(V )
L(V ) = 3V 2 + 3k240V2 +
1
4k45
20V
22 + 4k
a1dV + c
K2 =2c
3
n4b6uU Mathematica 2bM7_j
V48$ , Bb = 1, = 32, = 2, VT__
}& Bbb u: = 1/3, k240 = 1/3,112
k4520 = 1/3, = 3/2
c = 1 %7QbM}&(, BbJ)__4b
(, 3.13 KS (; 3.14 BI (; 3.15 BIII
(F
31
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time
Ani
sotr
opic
Par
amet
er
KantowskiSachs
gamma=1gamma=2gamma=3/2
Figure 3.13: Kantowski-Sachs__4: = 1/3, k240 =1/3, 1
12k45
20 = 1/3, = 3/2, c = 1
0 0.5 1 1.5 2 2.5 3 3.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time
Ani
sotr
opy
Anisotropy of Bianchi Type I on Brane Cosmology
23/21
Figure 3.14: Bianchi type I __4: = 1/3, k240 = 1/3,112
k4520 = 1/3, = 3/2, c = 1
32
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time
Ani
sotr
opic
Par
amet
er
Bianchi Type III
gamma=1gamma=2gamma=3/2
Figure 3.15: Bianchi type III __4: = 1/3, k240 =1/3, 1
12k45
20 = 1/3, = 3/2, c = 1
;WJ,
Chapter 4
!
Bbd7rn, ]iII7, czpZ
1. Bb__ KS,BI,BIII 3k240d_|c(, xA.
v Nj wv, Bb)
ju
V = V0e
3t
*j2Jc)bw27b o,
Bb)j:
v(t) = (
n(t t)) 1
,2n = 14k45
20_v, u brane ,bO3b
7s_v5, BbJv3k240d_|c(,
j:
v(t) = (
m
2(t t)) 2
,2m = 3k240, vubrObiH
34
2. 7__3b.4k
a1dVTuU__w N
jv KS wk BI wk BIII 7o BIII ok BI
ok KS *BbbM_ 26J|4k
a1dVT.}
T, .}3v`OFz, }v A
Chapter 5
5.1 Vacuum Einstein equation
_2#(j:
ds2 = gdxdx (5.1)
= d(t)2dt2 + a(t)2dr2 + b(t)2(d2 + f()2d2)
d":
g =
d(t)2 0 0 00 a(t)2 0 00 0 b(t)2 00 0 0 b(t)2f()2
d"5}:g11 = d(t)2, g22 = a(t)2, g33 = b(t)2, g44 = b(t)2f()2
:
g =
1d(t)2 0 0 0
0 1a(t)2
0 0
0 0 1b(t)2
0
0 0 0 1b(t)2f()2
36
d"5}:
g11 = 1d(t)2 , g22 = 1
a(t)2, g33 = 1
b(t)2, g44 = 1
b(t)2f()2
_2:
f() =
, Bianchi type Isin , Kantowski-Sachssinh , Bianchi type III
l:
1. ) Christofol symble j:
2. ) Ricci tensor R
Curvature tensor 2: R = mm ( )R R
3. 7) Ricci scalar :
R gR
4. |(J) Einstein equation :
R 12gR = kT g
1. I
Christofol symbol 2:
=1
2g(g + g g)
=1
2g(g, + g, g,)
Bbt) Christofol symbol } :
(a)
000 =1
2g0(g0,0 + g0,0 g00,) = 1
2g0(g0,0)
=1
2g0g00,0 =
1
2
1
d2(t)d(d2(t))
dt
=d(t)
d(t)
(b)
00i =1
2g0(g0,i + gi,0 g0i,) = 1
2g0(g0,i + gi,0)
=1
2g00g00,i +
1
2g0kgki,0
= 0
(c)
i00 =1
2gi(g0,0 + g0,0 g00,) = 1
2gi(2g0,0 g00,)
= gi0g00,0 12gi0g00,0
= 0
38
(d)
0ij =1
2g0(gi,j + gj,i gij,)
=1
2g00(g0i,j + g0j,i) 1
2g00gij,0
= 12g00gij,0
0rr =aa
d2
0 =bb
d2
0 =bbf 2()
d2
(e)
i0j =1
2gi(g0,j + gj,0 g0j,) = 1
2gi(g0,j + gj,0)
=1
2gi0g00,j +
1
2gigj,0
=1
2gikgkj,0
r0r =1
2grrgrr,0 =
1
2a2 2aa = a
a
0 =1
2gg,0 =
b
b
0 =1
2gg,0 =
1
2
1
b2f 2() 2bbf 2() = b
b
(f)
ijk =1
2gi(gj,k + gk,j gjk,) = 1
2gil(glj,k + glk,j gjk,l)
rrr =1
2grr(grr,r + grr,r grr,r) = 1
2grrgrr,r = 0
39
rr =1
2grr(grr, + gr,r gr,r) = 0
rr = 0 r =
1
2grr(gr, + gr, g,r) = 0
r = 0 r =
1
2grr(gr, + gr, g,r) = 0
= 0 r = 0 =
1
2gl(gl, + gl, g,l) = 0
rr = 0 r = 0 =
1
2gl(gl, + gl, g,l) = 1
2gg, = ff,
= 0 r = 0 =
1
2gl(gl, + gl, g,l) = 1
2gg, =
f,f
rr = 0 r = 0 = 0
FJBb) }:
000 =dd
0rr =aad2
0 =bbd2
0 =bbf2
d2r0r =
aa
0 =bb
0 =bb
=f,f
= ff,
40
2. II
Curvature tensor 2:
R = , , + mm mm
Bbl Ricci tensor }:
R R
(a)
R00 = R00
= R0000 + Ri0i0
= {000,0 000,0 + m0000m m0000m}+{i00,i i0i,0 + m00iim m0ii0m}
= i00,i i0i,0 + m00iim m0ii0m= i00,i i0i,0 + 000ii0 + j00iij 00ii00 j0ii0j= i0i,0 + 000ii0 j0ii0j= [r0r,0 + 0,0 + 0,0] + 000[rr0 + 0 + 0]
[(r0r)2 + (0)2 + (0)2]
= [( aa)0 + 2(
b
b)0] +
d
d[a
a+ 2
b
b] [( a
a)2 + 2(
b
b)2]
Ha aa
,Hb bb
R00 = [ ddHa 2 d
dHb + Ha + H
2a + 2Hb + 2H
2b ]
41
(b)
Rij = Rij
= R0i0j + Rkikj
= {0ji,0 00i,j + mij 00m mi00jm}+{kji,k kki,j + mij kkm mikkjm}
= [0ji,0 00i,j + 0ij000 + lij00l 0i00j0 li00jl]= +[kji,k kki,j + 0ijkk0 + lijkkl 0ikkj0 likkjl]= [0ji,0 +
0ij
000 li00jl]
= +[kji,k kki,j + 0ijkk0 + lijkkl 0ikkj0 likkjl] R0i0j = 0ji,0 + 0ij000 li00jl Rkikj = kji,k kki,j + 0ijkk0 + lijkkl 0ikkj0 likkjl
i.
R0i0j = 0ji,0 +
0ij
000 li00jl
R0r0r = 0rr,0 + 0rr000 rr00rr= (
aa
d2),0 + (
aa
d2)(
d
d) ( a
a)(
aa
d2)
= (aa
d2),0 + (
aad
d3) ( a
2
d2)
R00 = 0,0 + 0000 00= (
bb
d2),0 + (
bb
d2)(
d
d) ( b
b)(
bb
d2)
= (bb
d2),0 + (
bbd
d3) ( b
2
d2)
R00 = 0,0 + 0000 00= (
bbf 2
d2),0 + (
bbf 2
d2)(
d
d) ( b
b)(
bbf 2
d2)
= f 2[R00]
42
ii.
Rkikj = kij,k kki,j + 0ijkk0 + lijkkl 0ikkj0 likkjl
Rkrkr = Rrrrr + Rrr + Rrr= rrr,r rrr,r + 0rrrr0 + lrrrrl 0rrrr0 lrrrrl
+rr, r,r + 0rr0 + lrrl 0rr0 lrrl+rr, r,r + 0rr0 + lrrl 0rr0 lrrl
= 0rr0 +
0rr
0
= (aa
d2)(
b
b+
b
b)
= (2aab
d2b)
Rkk = Rrr + R + R= Rrr + R
= r,r rr, + 0rr0 + lrrl 0rr0 lrrl+, , + 00 + ll 00 ll
= 0rr0 , + 00 ll
= (bb
d2)(
a
a) (f,
f), + (
bb
d2)(
b
b) (f,
f)2
= (bb
d2)(
a
a) + (
b
d)2 f,,
f
Rkk = Rrr + R + R= Rrr + R
= r,r rr, + 0rr0 + lrrl 0rr0 lrrl+, , + 00 + ll 00 ll
= 0rr0 +
, +
0
0 ll
= (f 2bb
d2)(
a
a) (ff,), + (f 2 bb
d2)(
b
b) (f,
f)(ff,)
= f 2[(bb
d2)(
a
a+
b
b) f,,
f]
43
)Rij }: :
Rrr = R0r0r + R
krkr
= (a
d)2[ d
dHa + Ha + 2HaHb + H
2a ]
R = R00 + R
krr
= (b
d)2[ d
dHb + Hb + HaHb + 2H
2b ]
f,,f
R = R00 + R
krr
= (bf
d)2[ d
dHb + Hb + HaHb + 2H
2b ] ff,,
44
3. III )0b:
R = gR
= g00R00 + grrRrr + g
R + gR
= 2 1d2{ dd(Ha + 2Hb) Ha 2Hb 2HaHb H2a 3H2b }+
2k
b2
hence k = f,, f
4. IV , The vacuum Einstein equation :
R 12gR = g
=gj} (component):
(a) 00 component
R00 12g00R = g00
= 1d2
[H2b + 2HaHb] +k
d2b2(5.2)
(b) rr component
Rrr 12grrR = grr
= 1d2
[ dd(2Hb) + 2Hb + 3H
2b ] +
k
b2
(5.3)
(c) component
R 12gR = g
45
= 1d2
[ dd(Ha + Hb) + Ha + Hb + HaHb + H
2a + H
2b ]
(5.4)
(d) component
R 12gR = g
= 1d2
[ dd(Ha + Hb) + Ha + Hb + HaHb + H
2a + H
2b ]
(5.5)
d = 1, BbJ):
=
H2b + 2HaHb +kb2
, 00 component
2Hb + 3H2b +
kb2
, rr component
Ha + Hb + HaHb + H2a + H
2b , or component
(5.6)
00 component uF Friedmann equation .
46
5.2 Einstein equation with energy momen-
tum tensor
Energy momentum tensor }:
T 00 = , T 11 = T 22 = T 33 = p p ( 1)
T 2:
T = T = p3
Bb,?"(=gjA::
R 12gR = g + k24T (5.7)
k24 = 8G u 4&b.
47
1. The solution of Einstein equation : T = T g , FJ:
(a)
T00 = T00 g00 = ()(d2)
;W A.2 00 }Jh Einstein equation B.1 BbJ):
1
d2[H2b + 2HaHb] +
k
d2b2= + k24
as d = 1 H2b + 2HaHb +k
b2= + k24 (5.8)
(b)
Trr = Trr grr = pa
2 = ( 1)a2
;W A.3 rr }Jh Einstein equation B.1 BbJ):
2Hb + 3H2b +
k
b2= + k24(1 )
(5.9)
(c)
T = T g = pb
2 = ( 1)b2
;W A.2 }Jh Einstein equation B.1 BbJ):
Ha + Hb + HaHb + H2a + H
2b = + k
24(1 )
(5.10)
48
(d)
T = T g = pb
2f 2 = ( 1)b2f 2
;W,}Jh Einstein equation BbJ):
Ha + Hb + HaHb + H2a + H
2b = + k
24(1 )
(5.11)
c(Bb)_j:
H2b + 2HaHb +kb2
= + k24
2Hb + 3H2b +
kb2
= + k24(1 )Ha + Hb + HaHb + H
2a + H
2b = + k
24(1 )
49
(Zj:
(a2a2
)2 + 2(a1a1
)(a2a2
) +k
a22= + k24 (5.12)
2a2a2
+ (a2a2
)2 +k
a22= + k24(1 )
(5.13)a1a1
+a2a2
+ (a1a1
)(a2a2
) = + k24(1 )(5.14)
b2:
Ha =a1a1
, Hb =a2a2
50
2. Energy conservation Low
T = 0
Bb:
T T ; = T , + T + T
:
T = T g T 00 = , T rr = p
a21, T =
p
a22, T =
p
a22f2
FJ:
T = 0T 0 +rT r +T T = (a) + (b) + (c) + (d)
(a)
0T 0 = T 00,0 + T00 + T 00= + (T i0i0 + T
0000) + (T0000 + T
0ii0)
=
(b)
rT r = T rr,r + T rr + T rr= 0 + (T 00r0r + T
0irir) + (Tr00r + T
riir)
= T 00r0r + Triir
51
= T 00r0r + Trr0rr
= a1a1
+p
a21a1a1
= a1a1
(c)
T = T , + T + T = 0 + T 000 + T
0
= a2a2
+p
a22a2a2
= a2a2
(d)
T = T , + T + T = 0 + T 000 + T
+ T + T
0
= a2a2
+p
a22
f,f
+p
a22
ff,f 2
+p
a22
f 2a2a2f
= a2a2
+ pa2a2
= a2a2
T = 0T 0 +rT r +T T
= (a) + (b) + (c) + (d)
= + a1a1
+ a2a2
+ a2a2
= 0
52
+ (a1a1
+ 2a2a2
) = 0 (5.15)
;?Ovt: (V = a1a22)
= 0V , 0 = constant > 0
53
3. BbJZ Eq5.12 , Eq5.13 , Eq5.14 , :
d
dt(V Ha) = V +
1
2k240V
1(2 ) (5.16)d
dt(V Hb) = V +
1
2k240V
1(2 ) ka1(5.17)
3H + H2a + 2H2b = V +
1
2k240V
(2 3) (5.18)
b2:
Ha =a1a1
, Hb =a2a2
, V =1
3(Ha + 2Hb) =
V
3V
;Ws_j Eq5.16 , Eq5.17 , V jJA:
V = 3V +3
2k240V
1(2 ) 2ka1 (5.19)
and
Ha = H +2
3VK
Hb = H 13V
K
K =
ka1dt
j%}}(:
V 2 = 3V 2 + 3k240V2 4k
a1dV (5.20)
54
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55
[15] Ryan and Shepley ,Homogeneous Relativistic Cosmologies ,Princeton
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56