Electrodynamics in Friedmann-Roberson-Walker Universe

8/7/2019 Electrodynamics in Friedmann-Roberson-Walker Universe

1/20

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


2/20

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


3/20

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)

,


4/20

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


5/20

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)

,


6/20

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


7/20

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.


8/20

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


9/20

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


10/20

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


11/20

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


12/20

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


13/20

R+

a = 1

a4 a4

a4


14/20
http://arxiv.org/abs/gr-qc/0401065http://arxiv.org/abs/gr-qc/0405140


15/20

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


16/20

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


17/20

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


18/20

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


19/20

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


20/20

0.5 1 1.5 2

[ Na3]

r r = 4 r = 1 l, M k

Documents

Electrodynamics in Friedmann-Roberson-Walker Universe