Braneworld の基本方程と　一般解の構造

Akama, T. Hattori, and H. Mukaida

Ref.(partial) K. Akama, T. Hattori, and H. Mukaida, arXiv:1109.0840 [gr-qc]

Abstract In order to examine how the braneworld theory reproduce the successful predictions of the Einstein gravity theory,

we are seeking for the general spherical solution of the systemof the bulk Einstein equation and Nambu-Goto equation.

Here, we find the general solution.

it cannot fully specify the state of the brane

bulk

1 )))((2()( XdXgRXg NKIJ

K

Braneworld Dynamics

matterS

dynamicalvariables brane position

)( KIJ Xg

)( xY I

bulk metric

brane

4))((~~2 xdxYg K

eq. of motion

Action

,3,2X

0x

1X

0X

x

IJg

)( xY I

bulk scalar curvature

gg ~det~

bulk Einstein eq.

Nambu-Goto eq.

constant

brane en.mom.tensor

g~brane KX xbulk coord.

brane metriccannot be a dynamical variable

constants

gmn(Y)=YI,mYJ

, n gIJ(Y)

matter action

~

S d /d~ indicatesbrane quantity

bulk en.mom.tensor

IJgg det

0)2/( IJIJIJIJ TgRgR

coord.

=

0gIJYI

bulk Ricci tensor

0)~~~

( ; IYTg

0)~~~

( ; IYTg

bulk Einstein eq.

Nambu-Goto eq.

0)2/( IJIJIJIJ TgRgR

bulk Einstein eq.0)2/( IJIJIJIJ TgRgR

Nambu-Goto eq.0)

~~~( ; IYTg

empty

general solution

static, spherical, under Schwarzschild ansatz

asymptotically flat on the brane, empty except for the core outside the brane

× normal coordinate zbrane polar coordinatecoordinate system

x m=(t,r,q,j)

2222222 )sin( dzddkhdrfdtdxdxgds JIIJ

khf ,, : functions of r & z onlygeneral metric with

t,r,q,j

z

dominance of the collective mode Y I among matters

bulk Einstein eq. Nambu-Goto eq.

222222 )sin( dzddkhdrfdtdxdxg JIIJ

0)2/( IJIJIJIJ TgRgR

empty

0)~~~

( ; IYTg

zXXXrXtX 43210 ,,,,

bulk Einstein eq. Nambu-Goto eq.0)2/( IJIJIJIJ TgRgR

00Rkhkf

hhf

fhf

hf

kkf

hhf

fff rrrrrrrzzzzzzz

24422442 2

22

11Rkhkh

kk

ff

kk

ff

fhhf

kkh

fhf

hhh rrrrrrrrrrzzzzzzz

224242442 2

2

2

22

22R 1442442 2

hkh

fhkf

hk

hkh

fkfk rrrrrrzzzzzz

44R2

2

2

2

2

2

24422 kk

hh

ff

kk

hh

ff zzzzzzzzz

14R22 22442 k

kkhkkh

hffh

fff

kk

ff zrrzrzrzrzrz

RIJKL=GIJK,L-GIJL,K+gAB(GAIKGBJL-GAILGBJK), GIJK=(gIJ,K+gIK,J -gJK,I)/2

The only independent non-trivial components

0)~~~

( ; IYTg 222222 )sin( dzddkhdrfdtdxdxg JI

IJ zXXXrXtX 43210 ,,,,

bulk Einstein eq. Nambu-Goto eq.0)2/( IJIJIJIJ TgRgR

RIJKL=GIJK,L-GIJL,K+gAB(GAIKGBJL-GAILGBJK), GIJK=(gIJ,K+gIK,J -gJK,I)/2

0)~~~

( ; IYTg 222222 )sin( dzddkhdrfdtdxdxg JI

IJ

00Rkhkf

hhf

fhf

hf

kkf

hhf

fff rrrrrrrzzzzzzz

24422442 2

22

11Rkhkh

kk

ff

kk

ff

fhhf

kkh

fhf

hhh rrrrrrrrrrzzzzzzz

224242442 2

2

2

22

22R 1442442 2

hkh

fhkf

hk

hkh

fkfk rrrrrrzzzzzz

44R2

2

2

2

2

2

24422 kk

hh

ff

kk

hh

ff zzzzzzzzz

14R22 22442 k

kkhkkh

hffh

fff

kk

ff zrrzrzrzrzrz

The only independent non-trivial components

初めに bulk Einstein eq. alone の一般解を求めます。

zXXXrXtX 43210 ,,,,

use again later

use again later

bulk Einstein eq. Nambu-Goto eq.IJE

00 IJIJ RE

2/2/ 444,1,444,14 RRR U )log( 2hfkU

144,141,1,144,44 /2 RRRR UhV )/log( 2 hfkV

0221100 RRR

0221100 RRR

,0|| 044014 zz RR 04414 RR

0|| 044014 zz RR

0)2/( IJIJIJIJ TgRgR

IJIJ RR ]3/)2([ IJIJ gTT

En.-mom. conservation

If we assume we have

if are guaranteed.

Therefore, the independent equations are

Def.

covariant derivative

2/IJIJIJ gRRE

with

0IJJ ED

04414221100 RRRRR

=

JD 2/IJIJ gRR ( ) 0

0)~~~

( ; IYTg 222222 )sin( dzddkhdrfdtdxdxg JI

IJ

IJT =

00 IJIJ RE

bulk Einstein eq. Nambu-Goto eq.0)2/( IJIJIJIJ TgRgR

Def. 00 IJIJ RE

IJE=0)

~~~( ; IYTg

222222 )sin( dzddkhdrfdtdxdxg JI

IJ

IJIJ RR ]3/)2([ IJIJ gTT IJIJ RR IJT 3/)2( IJIJ gTT IJT

IJT =

0221100 RRR 0|| 044014 zz RRTherefore, the independent equations are

bulk Einstein eq. Nambu-Goto eq.0)2/( IJIJIJIJ TgRgR

,0221100 RRR 014 |zR 0| 044 zRindependent eqs.

IJE=0)

~~~( ; IYTg

222222 )sin( dzddkhdrfdtdxdxg JI

IJ

0221100 RRR 0|| 044014 zz RRTherefore, the independent equations are

00 IJIJ REIJIJ RR IJT 3/)2( IJIJ gTT IJT

00Rkhkf

hhf

fhf

hf

kkf

hhf

fff rrrrrrrzzzzzzz

24422442 2

22

11Rkhkh

kk

ff

kk

ff

fhhf

kkh

fhf

hhh rrrrrrrrrrzzzzzzz

224242442 2

2

2

22

000 T

00R

0

][ )(),(n

nn zrFzrFexpansion

n

k

kknn GFFG0

][][][)(reduction rule& derivatives),,, khfF IJT

bulk Einstein eq. Nambu-Goto eq.0)2/( IJIJIJIJ TgRgR

,0221100 RRR 014 |zR 0| 044 zRindependent eqs.

(

0

IJE=

00R T 00

khkf

hhf

fhf

hf

kkf

hhf

fff rrrrrrrzzzzzzz

24422442 2

22

0)~~~

( ; IYTg

2]0[ rk using diffeo.

222222 )sin( dzddkhdrfdtdxdxg JIIJ

00 IJIJ REIJIJ RR IJT 3/)2( IJIJ gTT IJT

000 T

00R

0

][ )(),(n

nn zrFzrFexpansion

n

k

kknn GFFG0

][][][)(reduction rule& derivatives),,, khfF IJT

bulk Einstein eq. Nambu-Goto eq.0)2/( IJIJIJIJ TgRgR

,0221100 RRR 014 |zR 0| 044 zRindependent eqs.

(

0

IJE=

00R T 00

khkf

hhf

fhf

hf

kkf

hhf

fff rrrrrrrzzzzzzz

24422442 2

22

002

22

2442244

Tkhkf

hhf

fhf

hf

kkf

hhf

ff rrrrrrrzzzzz

zzf 2

- -2

- + +2

- +

f 002

22

22222

Tkhkf

hhf

fhf

hf

kkf

hhf

ff rrrrrrrzzzzz[n]

[n-2]1

n(n -1)

0)~~~

( ; IYTg

2]0[ rk using diffeo.

2

222222 )sin( dzddkhdrfdtdxdxg JIIJ

___2

2 2 2 2 2 2 2

zz

[n-2]

00 IJIJ REIJIJ RR IJT 3/)2( IJIJ gTT IJT

0

][ )(),(n

nn zrFzrFexpansion

bulk Einstein eq. Nambu-Goto eq.0)2/( IJIJIJIJ TgRgR

,0221100 RRR 014 |zR 0| 044 zRindependent eqs.

IJE=

f 002

22

22222

Tkhkf

hhf

fhf

hf

kkf

hhf

ff rrrrrrrzzzzz[n] 1

n(n -1)

n

k

kknn GFFG0

][][][)(rule

[n-2]

0)~~~

( ; IYTg 222222 )sin( dzddkhdrfdtdxdxg JI

IJ 2]0[ rk

2]0[ rk using diffeo.

n

k

kknn GFFG0

][][][)(reduction rule

00 IJIJ REIJIJ RR IJT 3/)2( IJIJ gTT IJT

0

][ )(),(n

nn zrFzrFexpansion

bulk Einstein eq. Nambu-Goto eq.0)2/( IJIJIJIJ TgRgR

,0221100 RRR 014 |zR 0| 044 zRindependent eqs.

IJE=

f 002

22

22222

Tkhkf

hhf

fhf

hf

kkf

hhf

ff rrrrrrrzzzzz[n] 1

n(n -1)

n

k

kknn GFFG0

][][][)(rule]2[

002

22][ 2

2222)1(

1

n

rrrrrrrzzzzzn

khkf

hhf

fhf

hf

kkf

hhf

ff

nnf T

[n-2]

0)~~~

( ; IYTg 222222 )sin( dzddkhdrfdtdxdxg JI

IJ 2]0[ rk 00 IJIJ REIJIJ RR IJT 3/)2( IJIJ gTT IJT

00Rkhkf

hhf

fhf

hf

kkf

hhf

fff rrrrrrrzzzzzzz

24422442 2

22

11Rkhkh

kk

ff

kk

ff

fhhf

kkh

fhf

hhh rrrrrrrrrrzzzzzzz

224242442 2

2

2

22

22R 1442442 2

hkh

fhkf

hk

hkh

fkfk rrrrrrzzzzzz

The only independent non-trivial components

]2[

112

2

2

22][ 2

22

2

22)1(

1

n

rrrrrrrrrrzzzzzn

khkh

fhhf

kk

ff

kk

ff

kkh

fhf

hh

nnh T

0

][ )(),(n

nn zrFzrFexpansion

bulk Einstein eq. Nambu-Goto eq.0)2/( IJIJIJIJ TgRgR

,0221100 RRR 014 |zR 0| 044 zRindependent eqs.

IJE=

n

k

kknn GFFG0

][][][)(rule]2[

002

22][ 2

2222)1(

1

n

rrrrrrrzzzzzn

khkf

hhf

fhf

hf

kkf

hhf

ff

nnf T

0)~~~

( ; IYTg 222222 )sin( dzddkhdrfdtdxdxg JI

IJ 2]0[ rk 00 IJIJ REIJIJ RR IJT 3/)2( IJIJ gTT IJT

00Rkhkf

hhf

fhf

hf

kkf

hhf

fff rrrrrrrzzzzzzz

24422442 2

22

11Rkhkh

kk

ff

kk

ff

fhhf

kkh

fhf

hhh rrrrrrrrrrzzzzzzz

224242442 2

2

2

22

22R 1442442 2

hkh

fhkf

hk

hkh

fkfk rrrrrrzzzzzz

2

2

2

2

2

2

24422 kk

hh

ff

kk

hh

ff zzzzzzzzz

The only independent non-trivial components

]2[

112

2

2

22][ 2

22

2

22)1(

1

n

rrrrrrrrrrzzzzzn

khkh

fhhf

kk

ff

kk

ff

kkh

fhf

hh

nnh T

0

][ )(),(n

nn zrFzrFexpansion

bulk Einstein eq.0)2/( IJIJIJIJ TgRgR

,0221100 RRR 014 |zR 0| 044 zRindependent eqs.

IJE=

n

k

kknn GFFG0

][][][)(rule]2[

002

22][ 2

2222)1(

1

n

rrrrrrrzzzzzn

khkf

hhf

fhf

hf

kkf

hhf

ff

nnf T

]2[

112][ 22

2222)1(

1

n

rrrrrrzzzzn

hkh

fhkf

hk

hkh

fkf

nnk T

222222 )sin( dzddkhdrfdtdxdxg JIIJ 2]0[ rk

00 IJIJ REIJIJ RR IJT 3/)2( IJIJ gTT IJT

Nambu-Goto eq.0)

~~~( ; IYTg

]2[

112][ 22

2222)1(

1

n

rrrrrrzzzzn

hkh

fhkf

hk

hkh

fkf

nnk T

]2[

112

2

2

22][ 2

22

2

22)1(

1

n

rrrrrrrrrrzzzzzn

khkh

fhhf

kk

ff

kk

ff

kkh

fhf

hh

nnh T

recursion formulae for ][][][ ,, nnn khf khf ,,

if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obeyin the bulk

Thus, we have

0

][ )(),(n

nn zrFzrFexpansion

bulk Einstein eq.0)2/( IJIJIJIJ TgRgR

,0221100 RRR 014 |zR 0| 044 zRindependent eqs.

IJE=

n

k

kknn GFFG0

][][][)(rule]2[

002

22][ 2

2222)1(

1

n

rrrrrrrzzzzzn

khkf

hhf

fhf

hf

kkf

hhf

ff

nnf T

014 |zR 0| 044 zR

222222 )sin( dzddkhdrfdtdxdxg JIIJ 2]0[ rk

00 IJIJ REIJIJ RR IJT 3/)2( IJIJ gTT IJT

Nambu-Goto eq.0)

~~~( ; IYTg

0

][ )(),(n

nn zrFzrF

n

k

kknn GFFG0

][][][)(

bulk Einstein eq.0)2/( IJIJIJIJ TgRgR

IJE=

khf ,,We have

]2[

112][ 22

2222)1(

1

n

rrrrrrzzzzn

hkh

fhkf

hk

hkh

fkf

nnk T

]2[

112

2

2

22][ 2

22

2

22)1(

1

n

rrrrrrrrrrzzzzzn

khkh

fhhf

kk

ff

kk

ff

kkh

fhf

hh

nnh T

]2[

002

22][ 2

2222)1(

1

n

rrrrrrrzzzzzn

khkf

hhf

fhf

hf

kkf

hhf

ff

nnf T

khf ,,Thus, we haverecursion formulae for ][][][ ,, nnn khf

if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obeyin the bulk 014 |zR 0| 044 zR

if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 014 |zR 0| 044 zR

222222 )sin( dzddkhdrfdtdxdxg JIIJ 2]0[ rk

Nambu-Goto eq.0)

~~~( ; IYTg

0

][ )(),(n

nn zrFzrF

n

k

kknn GFFG0

][][][)(

use again lateruse again later

00Rkhkf

hhf

fhf

hf

kkf

hhf

fff rrrrrrrzzzzzzz

24422442 2

22

11Rkhkh

kk

ff

kk

ff

fhhf

kkh

fhf

hhh rrrrrrrrrrzzzzzzz

224242442 2

2

2

22

22R 1442442 2

hkh

fhkf

hk

hkh

fkfk rrrrrrzzzzzz

44R2

2

2

2

2

2

24422 kk

hh

ff

kk

hh

ff zzzzzzzzz

14R22 2

2 4 4

2 kkk

khkh

fhfh

fff

kk

ff zrrzrzrzrzrz

The only independent non-trivial components

bulk Einstein eq. Nambu-Goto eq.0)2/( IJIJIJIJ TgRgR

IJE=

khf ,,We have

0| 014 zR

0| 044 zR

0)~~~

( ; IYTg

]0[14

]0[14 TR

=

]0[14T 03

]1[

]0[

]1[

]0[]0[

]0[]1[

2]0[

]0[]1[

2

]1[

]0[

]1[

442 rk

rhh

fhfh

fff

rk

ff rrrr

if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 014 |zR 0| 044 zR

222222 )sin( dzddkhdrfdtdxdxg JIIJ 2]0[ rk

[0][0] [1] [0][1] [0] [1][1] [1] [1] [0]

[0] [0] [0] [0] [0] [0][0] [0]

[0]

Nambu-Goto eq.0)

~~~( ; IYTg

0

][ )(),(n

nn zrFzrF

n

k

kknn GFFG0

][][][)(

00Rkhkf

hhf

fhf

hf

kkf

hhf

fff rrrrrrrzzzzzzz

24422442 2

22

11Rkhkh

kk

ff

kk

ff

fhhf

kkh

fhf

hhh rrrrrrrrrrzzzzzzz

224242442 2

2

2

22

22R 1442442 2

hkh

fhkf

hk

hkh

fkfk rrrrrrzzzzzz

44R2

2

2

2

2

2

2 4 4

22 kk

hh

ff

kk

hh

ff zzzzzzzzz

14R22 22442 k

kkhkkh

hffh

fff

kk

ff zrrzrzrzrzrz

The only independent non-trivial components

Nambu-Goto eq.0)

~~~( ; IYTg

0

][ )(),(n

nn zrFzrF

n

k

kknn GFFG0

][][][)(

3

]1[

]0[

]1[

]0[]0[

]0[]1[

2]0[

]0[]1[

2

]1[

]0[

]1[

442 rk

rhh

fhfh

fff

rk

ff rrrr

bulk Einstein eq. IJE=

)(4224

1 ]0[444

2]1[

2]0[

]1[]1[

2]0[

]1[]1[

]0[]0[

]1[]1[

2 T

rk

rhkh

rfkf

hfhf

r

0| 044 zR

[0][1] [1] [1][2] [2] [2]2 2 2

[0] [0] [0] [0] [0] [0]

[0]

substitute

2]0[ rk 222222 )sin( dzddkhdrfdtdxdxg JIIJ

]0[14T 0

if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 014 |zR 0| 044 zR

0)2/( IJIJIJIJ TgRgR

khf ,,We have

0| 014 zR]2[

112][ 22

2222)1(

1

n

rrrrrrzzzzn

hkh

fhkf

hk

hkh

fkf

nnk T

]2[

112

2

2

22][ 2

22

2

22)1(

1

n

rrrrrrrrrrzzzzzn

khkh

fhhf

kk

ff

kk

ff

kkh

fhf

hh

nnh T

]2[

002

22][ 2

2222)1(

1

n

rrrrrrrzzzzzn

khkf

hhf

fhf

hf

kkf

hhf

ff

nnf T[2]

[2]

[2]

2

2

2

[0]

[0]

[0]

]0[44R

=

)( ]0[44 T

2]0[

]0[

]0[]0[

]0[ 1

11

4 rrff

hrff rr

2]0[

]0[

h

hr2]0[

2]0[

]0[

]0[

4

2 ff

ff rrr

bulk Einstein eq.0)2/( IJIJIJIJ TgRgR

IJE=

khf ,,We have

The solution include three arbitrary functions.

Two equations for five functions ,, ]0[]0[ hf ]1[]1[]1[ ,, khf

0| 014 zR

0| 044 zE

if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 014 |zR 0| 044 zR 0| 044 zE

0]0[14T3

]1[

]0[

]1[

]0[]0[

]0[]1[

2]0[

]0[]1[

2

]1[

]0[

]1[

442 rk

rhh

fhfh

fff

rk

ff rrrr

222222 )sin( dzddkhdrfdtdxdxg JIIJ 2]0[ rk

)(4224

1 ]0[444

2]1[

2]0[

]1[]1[

2]0[

]1[]1[

]0[]0[

]1[]1[

2 T

rk

rhkh

rfkf

hfhf

r

Nambu-Goto eq.0)

~~~( ; IYTg

2]0[

]0[

]0[]0[

]0[ 1

11

4 rrff

hrff rr

2]0[

]0[

h

hr2]0[

2]0[

]0[

]0[

4

2 ff

ff rrr

Nambu-Goto eq.0)

~~~( ; IYTg

0

][ )(),(n

nn zrFzrF

n

k

kknn GFFG0

][][][)(

0| 044 zR

)(4224

1 ]0[444

2]1[

2]0[

]1[]1[

2]0[

]1[]1[

]0[]0[

]1[]1[

2 T

rk

rhkh

rfkf

hfhf

r

3

]1[

]0[

]1[

]0[]0[

]0[]1[

2]0[

]0[]1[

2

]1[

]0[

]1[

442 rk

rhh

fhfh

fff

rk

ff rrrr

bulk Einstein eq. Nambu-Goto eq.0)2/( IJIJIJIJ TgRgR

IJE=

khf ,,We have

The solution include three arbitrary functions.

Two equations for five functions ,, ]0[]0[ hf ]1[]1[]1[ ,, khf

0| 014 zR

0)~~~

( ; IYTg

0| 044 zE

if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 014 |zR 0| 044 zE

0]0[14T

222222 )sin( dzddkhdrfdtdxdxg JIIJ 2]0[ rk

]0[]1[ 2/ ffu ]0[]1[ 2/ hhv 2]1[ 2/ rkw (uf [0] )r -u -v -2v -2w

2 2+(2r 2w)r ur 2wr

+ +u( ) (v -w)

2]0[

]0[

]0[]0[

]0[ 1

11

4 rrff

hrff rr

2]0[

]0[

h

hr2]0[

2]0[

]0[

]0[

4

2 ff

ff rrr

3

]1[

]0[

]1[

]0[]0[

]0[]1[

2]0[

]0[]1[

2

]1[

]0[

]1[

442 rk

rhh

fhfh

fff

rk

ff rrrr 0]0[

14T-u -v -2v -2w2 2

+ ur 2wr+ +u( ) (v -w)

bulk Einstein eq. Nambu-Goto eq.0)2/( IJIJIJIJ TgRgR

IJE=

khf ,,We have

The solution include three arbitrary functions.

Two equations for five functions ,, ]0[]0[ hf ]1[]1[]1[ ,, khf

0| 014 zR

0)~~~

( ; IYTg

0| 044 zE

if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 014 |zR 0| 044 zE

222222 )sin( dzddkhdrfdtdxdxg JIIJ 2]0[ rk

]0[]1[ 2/ ffu ]0[]1[ 2/ hhv 2]1[ 2/ rkw

]0[

]0[

2

)(

ffvu r

rr wu 2 r

wv )(2 ]0[

14T

)(4224

1 ]0[444

2]1[

2]0[

]1[]1[

2]0[

]1[]1[

]0[]0[

]1[]1[

2 T

rk

rhkh

rfkf

hfhf

r

+ +

2]0[

]0[

]0[]0[

]0[ 1

11

4 rrff

hrff rr

2]0[

]0[

h

hr2]0[

2]0[

]0[

]0[

4

2 ff

ff rrr

bulk Einstein eq. Nambu-Goto eq.0)2/( IJIJIJIJ TgRgR

IJE=

khf ,,We have

The solution include three arbitrary functions.

Two equations for five functions ,, ]0[]0[ hf ]1[]1[]1[ ,, khf

0| 014 zR

0)~~~

( ; IYTg

0| 044 zE

if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 014 |zR 0| 044 zE

222222 )sin( dzddkhdrfdtdxdxg JIIJ 2]0[ rk

]0[]1[ 2/ ffu ]0[]1[ 2/ hhv 2]1[ 2/ rkw

]0[

]0[

2

)(

ffvu r

rr wu 2

)(4224

1 ]0[444

2]1[

2]0[

]1[]1[

2]0[

]1[]1[

]0[]0[

]1[]1[

2 T

rk

rhkh

rfkf

hfhf

r

rwv )(2

]0[14T

0)(4224

]0[444

2]1[

2]0[

]1[]1[

2]0[

]1[]1[

]0[]0[

]1[]1[

Tr

krh

khrf

kfhfhf

uv 2uw 2vw w2

u v u w2 v w2 w 2

2]0[

]0[

]0[]0[

]0[ 1

11

4 rrff

hrff rr

2]0[

]0[

h

hr2]0[

2]0[

]0[

]0[

4

2 ff

ff rrr

bulk Einstein eq. Nambu-Goto eq.0)2/( IJIJIJIJ TgRgR

IJE=

khf ,,We have

The solution include three arbitrary functions.

Two equations for five functions ,, ]0[]0[ hf ]1[]1[]1[ ,, khf

0)~~~

( ; IYTg

if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 014 |zR 0| 044 zE

222222 )sin( dzddkhdrfdtdxdxg JIIJ 2]0[ rk

0| 014 zR

]0[]1[ 2/ ffu ]0[]1[ 2/ hhv 2]1[ 2/ rkw

]0[

]0[

2

)(

ffvu r

rr wu 2 r

wv )(2 ]0[

14T

0| 044 zE

)(4224

1 ]0[444

2]1[

2]0[

]1[]1[

2]0[

]1[]1[

]0[]0[

]1[]1[

2 T

rk

rhkh

rfkf

hfhf

ru v u w2 v w2 w 2

2

1

ruv uw2 vw2 2w )( ]0[

44 T

2]0[

]0[

]0[]0[

]0[ 1

11

4 rrff

hrff rr

2]0[

]0[

h

hr2]0[

2]0[

]0[

]0[

4

2 ff

ff rrr

2]0[

]0[

]0[]0[

]0[ 1

11

4 rrff

hrff rr

2]0[

]0[

h

hr2]0[

2]0[

]0[

]0[

4

2 ff

ff rrr

bulk Einstein eq. Nambu-Goto eq.0)2/( IJIJIJIJ TgRgR

IJE=

khf ,,We have

The solution include three arbitrary functions.

Two equations for five functions ,, ]0[]0[ hf ]1[]1[]1[ ,, khf

0)~~~

( ; IYTg

0| 044 zE

if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 014 |zR 0| 044 zE

222222 )sin( dzddkhdrfdtdxdxg JIIJ 2]0[ rk

]0[]1[ 2/ ffu ]0[]1[ 2/ hhv 2]1[ 2/ rkw

0| 014 zR]0[

]0[

2

)(

ffvu r

rr wu 2 r

wv )(2 ]0[

14T

2

]0[

]0[

]0[

]0[

22

ff

ff r

r

r

rh

]0[

1

rh

]0[

12

]0[

]0[

]0[

]0[

22

ff

ff r

r

r

2

1

ruv uw2 vw2 2w )( ]0[

44 T

2]0[

]0[

]0[]0[

]0[ 1

11

4 rrff

hrff rr

bulk Einstein eq. Nambu-Goto eq.0)2/( IJIJIJIJ TgRgR

IJE=

khf ,,We have

The solution include three arbitrary functions.

Two equations for five functions ,, ]0[]0[ hf ]1[]1[]1[ ,, khf

0)~~~

( ; IYTg

0| 044 zE

if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 014 |zR 0| 044 zE

222222 )sin( dzddkhdrfdtdxdxg JIIJ 2]0[ rk

]0[]1[ 2/ ffu ]0[]1[ 2/ hhv 2]1[ 2/ rkw

0| 014 zR]0[

]0[

2

)(

ffvu r

rr wu 2 r

wv )(2 ]0[

14T

rh

]0[

12

]0[

]0[

]0[

]0[

22

ff

ff r

r

r

2

1

ruv uw2 vw2 2w )( ]0[

44 T

2]0[

]0[

]0[]0[

]0[ 1

11

4 rrff

hrff rr

0| 044 zE

0| 014 zR]0[

]0[

2

)(

ffvu r

rr wu 2 r

wv )(2 ]0[

14T

rh

]0[

12

]0[

]0[

]0[

]0[

22

ff

ff r

r

r

2

1

ruv uw2 vw2 2w )( ]0[

44 T

]0[]1[ 2/ ffu ]0[]1[ 2/ hhv 2]1[ 2/ rkw

2]0[

]0[

]0[]0[

]0[ 1

11

4 rrff

hrff rr

bulk Einstein eq. Nambu-Goto eq.0)2/( IJIJIJIJ TgRgR

IJE=

khf ,,We have

The solution include three arbitrary functions.

Two equations for five functions ,, ]0[]0[ hf ]1[]1[]1[ ,, khf

0)~~~

( ; IYTg

0| 044 zE

if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 014 |zR 0| 044 zE

222222 )sin( dzddkhdrfdtdxdxg JIIJ 2]0[ rk

0| 014 zR]0[

]0[

2

)(

ffvu r

rr wu 2 r

wv )(2 ]0[

14T

rh

]0[

12

]0[

]0[

]0[

]0[

22

ff

ff r

r

r

2

1

ruv uw2 vw2 2w )( ]0[

44 T

2]0[

]0[

]0[]0[

]0[ 1

11

4 rrff

hrff rr

0| 044 zE

0| 014 zR]0[

]0[

2

)(

ffvu r

rr wu 2 r

wv )(2 ]0[

14T

rh

]0[

12

]0[

]0[

]0[

]0[

22

ff

ff r

r

r

2

1

ruv uw2 vw2 2w )( ]0[

44 T

]0[]1[ 2/ ffu ]0[]1[ 2/ hhv 2]1[ 2/ rkw

use again lateruse again later

]0[]1[ 2/ ffu ]0[]1[ 2/ hhv 2]1[ 2/ rkw

bulk Einstein eq. Nambu-Goto eq.0)2/( IJIJIJIJ TgRgR

IJE=

khf ,,We have

The solution include three arbitrary functions.

Two equations for five functions ,, ]0[]0[ hf ]1[]1[]1[ ,, khf

0)~~~

( ; IYTg

if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 014 |zR 0| 044 zE

222222 )sin( dzddkhdrfdtdxdxg JIIJ 2]0[ rk

rwv )(2

]0[

]0[

2

)(

ffvu r

rr wu 2 ]0[14T

2

1

ruv uw2 vw2 2w )( ]0[

44 T

]0[14T

]0[14/)(22 Trwvwu rr [ ]

vu ]0[f

2)(2 wv /rvu

2]0[

]0[

]0[]0[

]0[ 1

11

4 rrff

hrff rr

rh

]0[

12

]0[

]0[

]0[

]0[

22

ff

ff r

r

r

wvu ,, :arbitrary

]0[]1[ 2/ ffu ]0[]1[ 2/ hhv 2]1[ 2/ rkw

2]0[

]0[

]0[]0[

]0[ 1

11

4 rrff

hrff rr

rh

]0[

12

]0[

]0[

]0[

]0[

22

ff

ff r

r

r

bulk Einstein eq. Nambu-Goto eq.0)2/( IJIJIJIJ TgRgR

IJE=

khf ,,We have

The solution include three arbitrary functions.

Two equations for five functions ,, ]0[]0[ hf ]1[]1[]1[ ,, khf

0)~~~

( ; IYTg

if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 014 |zR 0| 044 zE

222222 )sin( dzddkhdrfdtdxdxg JIIJ 2]0[ rk

2

1

ruv uw2 vw2 2w )( ]0[

44 T

]0[14/)(22 Trwvwu rr [ ]

vu ]0[

]0[

2

)(

ffvu r

]0[f2

=U

]0[rf U ]0[f U

U U U U

1

vu wvu ,, :arbitrary

]0[]1[ 2/ ffu ]0[]1[ 2/ hhv 2]1[ 2/ rkw

rh

]0[

1

2

1

ruv uw2 vw2 2w )( ]0[

44 T

2]0[

]0[

]0[]0[

]0[ 1

11

4 rrff

hrff rr

2

]0[

]0[

]0[

]0[

22

ff

ff r

r

rU U U U

bulk Einstein eq. Nambu-Goto eq.0)2/( IJIJIJIJ TgRgR

IJE=

khf ,,We have

The solution include three arbitrary functions.

Two equations for five functions ,, ]0[]0[ hf ]1[]1[]1[ ,, khf

0)~~~

( ; IYTg

if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 014 |zR 0| 044 zE

222222 )sin( dzddkhdrfdtdxdxg JIIJ 2]0[ rk

]0[14/)(22 Trwvwu rr [ ]

2

U

)(22 ]0[44

2 Twvwuwuv1 / 2r

U r / 2U 2 /2 U /r 1/ 2rU / 1/r4

24

rh

]0[

1

vu ]0[

rf U ]0[fvu wvu ,, :arbitrary

U / 1/r4

]0[]1[ 2/ ffu ]0[]1[ 2/ hhv 2]1[ 2/ rkw

bulk Einstein eq. Nambu-Goto eq.0)2/( IJIJIJIJ TgRgR

IJE=

khf ,,We have

0)~~~

( ; IYTg

if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 014 |zR 0| 044 zE

222222 )sin( dzddkhdrfdtdxdxg JIIJ 2]0[ rk

]0[14/)(22 Trwvwu rr [ ]

2

U

)(22 ]0[44

2 Twvwuwuv1 / 2r

U r / 2U 2 / U /r 1/ 2rU / 1/r4

4

rh

]0[

1

vu

]0[

1

h

=P

=Q

]0[rf U ]0[f

rh ]0[

1P

]0[

1

hQ

Q

P

solution

vu wvu ,, :arbitrary

U / 1/r4

]0[f

r

dre

U]0[h

rdr

e P

1

1

r

Pdrdre rQ

1st order linear differential equations solvable!

]0[]1[ 2/ ffu ]0[]1[ 2/ hhv 2]1[ 2/ rkw

bulk Einstein eq. Nambu-Goto eq.0)2/( IJIJIJIJ TgRgR

IJE=

khf ,,We have

0)~~~

( ; IYTg

if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 014 |zR 0| 044 zE

222222 )sin( dzddkhdrfdtdxdxg JIIJ 2]0[ rk

]0[rf U ]0[f

rh ]0[

1P

]0[

1

hQ

]0[]0[ Uffr

Qh

Ph r

]0[]0[

11

: written with wvu ,,vu wvu ,, :arbitrary

solution ]0[f

r

dre

U]0[h

rdr

e P

1

1

r

Pdrdre rQ

1st order linear differential equations solvable!

)(22 ]0[44

2 Twvwuwuv1 / 2r

U r / 2U 2 / U /r 1/ 2rU / 1/r4

4

Q

P

U / 1/r4

U P Q, , ]0[14/)(22 Trwvwu rr [ ]

2

Uvu

]0[]1[ 2/ ffu ]0[]1[ 2/ hhv 2]1[ 2/ rkw

bulk Einstein eq. Nambu-Goto eq.0)2/( IJIJIJIJ TgRgR

IJE=

khf ,,We have

0)~~~

( ; IYTg

if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 014 |zR 0| 044 zE

222222 )sin( dzddkhdrfdtdxdxg JIIJ 2]0[ rk

]0[]0[ Uffr

Qh

Ph r

]0[]0[

11

QPU ,, : written with wvu ,,

rUdr

ef ]0[

1

]0[ 1

r

PdrPdrdrQeeh rr

solutionvu wvu ,, :arbitrary

solution ]0[f

r

dre

U]0[h

rdr

e P

1

1

r

Pdrdre rQ

]0[]1[ 2/ ffu ]0[]1[ 2/ hhv 2]1[ 2/ rkw

vu wvu ,, :arbitrary

vu 2/2/ ]0[

14rTruuV r

rVdrrW 1

rUrUU

P r

/4'

)/2'('2'

2

rUTWvWuWuvr

Q/14/'

)(22/1'

]0[44

22

]0[]0[ /' ffU r

: arbitrary

bulk Einstein eq. Nambu-Goto eq.0)2/( IJIJIJIJ TgRgR

IJE=

khf ,,We have

0)~~~

( ; IYTg

if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 014 |zR 0| 044 zE

222222 )sin( dzddkhdrfdtdxdxg JIIJ 2]0[ rk

]0[]0[ Uffr

rUdr

ef ]0[

1

]0[ 1

r

PdrPdrdrQeeh rr

QPU ,,

Qh

Ph r

]0[]0[

11

: written with wvu ,,

solution

Vrw r )(

'1

'1

]0[]0[Q

hP

h r

]0[f

2]0[

]0[

]0[]0[

]0[ 1

11

4 rrff

hrff rr

0| 044 zE

0| 014 zR]0[

]0[

2

)(

ffvu r

rr wu 2 r

wv )(2 ]0[

14T

rh

]0[

12

]0[

]0[

]0[

]0[

22

ff

ff r

r

r

2

1

ruv uw2 vw2 2w )( ]0[

44 T

]0[]1[ 2/ ffu ]0[]1[ 2/ hhv 2]1[ 2/ rkw

bulk Einstein eq. Nambu-Goto eq.0)2/( IJIJIJIJ TgRgR

IJE=

khf ,,We have

0)~~~

( ; IYTg

if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 014 |zR 0| 044 zE

222222 )sin( dzddkhdrfdtdxdxg JIIJ 2]0[ rk

]0[]0[ Uffr

rUdr

ef ]0[

1

]0[ 1

r

PdrPdrdrQeeh rr

QPU ,,

vu

1'']0[ '1

r

drPdrPdreQeh rr

]0[, fu: arbitrary

rVdrrw 1

Qh

Ph r

]0[]0[

11

',', QPV

: written with wvu ,,

solution

Vrw r )(

'1

'1

]0[]0[Q

hP

h r

]0[f : written with

solution

vu wvu ,, :arbitrary

]0[]1[ 2/ ffu ]0[]1[ 2/ hhv 2]1[ 2/ rkw

bulk Einstein eq. Nambu-Goto eq.0)2/( IJIJIJIJ TgRgR

IJE=

khf ,,We have

0)~~~

( ; IYTg

if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 014 |zR 0| 044 zE

222222 )sin( dzddkhdrfdtdxdxg JIIJ 2]0[ rk

vu vu ]0[f : arbitrary

: linear, solvable: linear, solvable

wuhf 2,, ]0[]0[ : arbitrary algebraic eq. for wv, solvable

braneworld

bulk Einstein eq.

bulk Einstein eq. +Nambu-Goto eq.

3 eqs. for 5 functions

3 arbitrary functions2 eqs. for 5 functions

2 arbitrary functions2 of u,v,w arbitrary : linear, solvable

arbitrary]0[]0[ ,hf non-linear differential eq. for u,v,w

(1)u,v,w arbitrary

collective mode dominance

(2)

(1)(2)

Israel junction condition

Thank you