General Solution of Braneworld with the Schwarzschild ansatzK. Akama, T. Hattori, and H. Mukaida

Ref. K. Akama, T. Hattori, and H. Mukaida, arXiv:1109.0840 [gr-qc] submitted to Japanese Physical Society meeting in 2011 spring.

Abstract

The arbitrariness may affect the predictive powers on the Newtonian andthe post-Newtonian evidences.

We derive the general solution of the fundamental equations of the braneworld under the Schwarzschild ansatz. It is expressed in power series of the brane normal coordinate in terms of on-brane functions, which should obey essential on-brane equations including the equation of motion of the brane. They are solved in terms of arbitrary functions on the brane.

Ways out of the difficulty are discussed.

picture that we live in 3+1 brane dynamics with the Einstein Hilbert action on the brane does not reproduce Einstein gravity.

Braneworld, a long history, brief Fronsdal('59), Josesh('62)

Regge,Teitelboim('75)

dynamical model of with braneworld via a higher-dim. solitonwith brane induced gravity K.A.('82)

with trapped massless modes Rubakov,Shaposhnikov('83)

embedding models Maia('84), Pavsic('85)

D-brane: brane where the string endpoints reside, and which is a higer-dim. soliton in the dual picture

dynamical models Visser('85), Gibbons,Wiltschire ('87)

applied to the superstring

Polchinski('95)

Antoniadis('91), Horava,Witten('96)

applied to hierarchy problemsArkani-Hamed,Dimopolos,Dvali('98), Randall,Sundrum('99)

(×^

×)

Introduction

Einstein gravity successfully explaines

②observations on light deflections due to solar gravity, the planetary perihelion precessions, etc. precisely.

(^_^)

It is based on the Schwarzschild solution with the ansatzstaticity, sphericity,

asymptotic flatness, emptiness except for the core

Can the braneworld theory inherit the successes ① and ②?

"Braneworld"

To examine it, we derive the general solution of the fundamental dynamics of the brane under the Schwarzschild anzats.

( ,_ ,)?

①the origin of the Newtonian gravity

: our 3+1 spacetime is embedded in higher dim.

Garriga,Tanaka (00), Visser,Wiltshire('03) Casadio,Mazzacurati('03), Bronnikov,Melnikov,Dehnen('03)

spherical sols. ref.

Motivation

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

constant

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. Nambu-Goto eq.0)

~~~( ; IYTg

IJIJIJ gRgR )2/( 0 IJT

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

We first consider the empty bulk Einstein equation alone.

bulk Einstein eq. Nambu-Goto eq.

222222 )sin( dzddkhdrfdtdxdxg JIIJ

empty

0)~~~

( ; IYTg

zXXXrXtX 43210 ,,,,

alone,

IJIJIJ gRgR )2/( 0 IJT0empty

Nambu-Goto eq.

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

zXXXrXtX 43210 ,,,,

IJIJIJ gRgR )2/( 0222222 )sin( dzddkhdrfdtdxdxg JI

IJ

bulk Einstein eq. alone, empty

curvaturetensor

affineconnection

substituting gIJ, write RIJKL with of f, h, k.

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

RIJKL=GIJK,L-GIJL,K+gAB(GAIKGBJL-GAILGBJK), GIJK=(gIJ,K+gIK,J -gJK,I)/2zXXXrXtX 43210 ,,,,

use again later

Nambu-Goto eq.0)

~~~( ; IYTg

IJIJIJ gRgR )2/( 0

222222 )sin( dzddkhdrfdtdxdxg JIIJ use again later

bulk Einstein eq. alone, empty

covariant derivative

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

current conservation

If we assume implies

if are guaranteed.

Therefore, the independent equations are

Def.

2/IJIJIJ gRRE

with

0IJJ ED

04414221100 RRRRR

=

JD 2/IJIJ gRR ( ) 0

Nambu-Goto eq.0)

~~~( ; IYTg

IJIJIJ gRgR )2/( 0

3/2 IJIJIJ gR R

Bianchi identity

222222 )sin( dzddkhdrfdtdxdxg JIIJ

then

, then

bulk Einstein eq. alone, empty

equivalent equation

independent equations

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

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

Def.

Nambu-Goto eq.0)

~~~( ; IYTg

IJIJIJ gRgR )2/( 0

3/2 IJIJIJ gR R 00 IJIJ RE

222222 )sin( dzddkhdrfdtdxdxg JIIJ

bulk Einstein eq.

IJEalone, empty

=

independent equations

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

Nambu-Goto eq.0)

~~~( ; IYTg

IJIJIJ gRgR )2/( 0

3/2 IJIJIJ gR R 00 IJIJ RE

222222 )sin( dzddkhdrfdtdxdxg JIIJ

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

03

2

f00R

0

][ )(),(n

nn zrFzrFexpansion

n

k

kknn GFFG0

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

000 3/ 2 R f

khkf

hhf

fhf

hf

kkf

hhf

fff rrrrrrrzzzzzzz

24422442 2

22

2]0[ rk using diffeo.

bulk Einstein eq.

IJEalone, empty

=

power seriessolution in z

0

][ )(),(n

nn zrFzrFexpansion

n

k

kknn GFFG0

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

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

(

3

2

2442244 2

22 fkhkf

hhf

fhf

hf

kkf

hhf

ff rrrrrrrzzzzz

zzf22 2 2

f3

4

2222 2

22 fkhkf

hhf

fhf

hf

kkf

hhf

ff rrrrrrrzzzzz

[n][n-2]1

n(n -1)

2]0[ rk using diffeo.

2 2 2 2 2 2 2

zz

[n-2]

Def.

Nambu-Goto eq.0)

~~~( ; IYTg

IJIJIJ gRgR )2/( 0

3/2 IJIJIJ gR R 00 IJIJ RE

222222 )sin( dzddkhdrfdtdxdxg JIIJ

00R 000 3/ 2 R f

khkf

hhf

fhf

hf

kkf

hhf

fff rrrrrrrzzzzzzz

24422442 2

22

03

2

f

2 4

n(n -1)

bulk Einstein eq.

IJEalone, empty

=

power seriessolution in z

0

][ )(),(n

nn zrFzrFexpansion

f3

4

2222 2

22 fkhkf

hhf

fhf

hf

kkf

hhf

ff rrrrrrrzzzzz

[n] 1

n(n -1)

[n-2]

2]0[ rk

2]0[ rk using diffeo.

n

k

kknn GFFG0

][][][)(reduction rule

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

Nambu-Goto eq.0)

~~~( ; IYTg

IJIJIJ gRgR )2/( 0

3/2 IJIJIJ gR R 00 IJIJ RE

222222 )sin( dzddkhdrfdtdxdxg JIIJ

bulk Einstein eq.

IJEalone, empty

=

power seriessolution in z

Nambu-Goto eq.0)

~~~( ; IYTg

power seriessolution in z

0

][ )(),(n

nn zrFzrF

n

k

kknn GFFG0

][][][)(

expansionreduction

rule

f3

4

2222 2

22 fkhkf

hhf

fhf

hf

kkf

hhf

ff rrrrrrrzzzzz

[n] 1

n(n -1)

]2[

2

22][

3

4

2222)1(

1

n

rrrrrrrzzzzzn fkhkf

hhf

fhf

hf

kkf

hhf

ff

nnf

[n-2]

2]0[ rk

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

IJIJIJ gRgR )2/( 0

3/2 IJIJIJ gR R 00 IJIJ RE

222222 )sin( dzddkhdrfdtdxdxg JIIJ

bulk Einstein eq.

IJEalone, empty

=

0

][ )(),(n

nn zrFzrF

n

k

kknn GFFG0

][][][)(

expansionreduction

rule

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[

2

2

2

22][

3

4

22

2

22)1(

1

n

rrrrrrrrrrzzzzzn hkhkh

fhhf

kk

ff

kk

ff

kkh

fhf

hh

nnh

]2[

2

22][

3

4

2222)1(

1

n

rrrrrrrzzzzzn fkhkf

hhf

fhf

hf

kkf

hhf

ff

nnf

2]0[ rk

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

IJIJIJ gRgR )2/( 0

3/2 IJIJIJ gR R 00 IJIJ RE

222222 )sin( dzddkhdrfdtdxdxg JIIJ

bulk Einstein eq.

IJEalone, empty

=

0

][ )(),(n

nn zrFzrF

n

k

kknn GFFG0

][][][)(

expansionreduction

rule

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[

2][

3

42

2222)1(

1

n

rrrrrrzzzzn khkh

fhkf

hk

hkh

fkf

nnk

]2[

2

22][

3

4

2222)1(

1

n

rrrrrrrzzzzzn fkhkf

hhf

fhf

hf

kkf

hhf

ff

nnf

]2[

2

2

2

22][

3

4

22

2

22)1(

1

n

rrrrrrrrrrzzzzzn hkhkh

fhhf

kk

ff

kk

ff

kkh

fhf

hh

nnh

2]0[ rk

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

IJIJIJ gRgR )2/( 0

3/2 IJIJIJ gR R 00 IJIJ RE

222222 )sin( dzddkhdrfdtdxdxg JIIJ

bulk Einstein eq.

IJEalone, empty

=

0

][ )(),(n

nn zrFzrF

n

k

kknn GFFG0

][][][)(

expansionreduction

rule

here.

]2[

2

22][

3

4

2222)1(

1

n

rrrrrrrzzzzzn fkhkf

hhf

fhf

hf

kkf

hhf

ff

nnf

]2[

2

2

2

22][

3

4

22

2

22)1(

1

n

rrrrrrrrrrzzzzzn hkhkh

fhhf

kk

ff

kk

ff

kkh

fhf

hh

nnh

]2[

2][

3

42

2222)1(

1

n

rrrrrrzzzzn khkh

fhkf

hk

hkh

fkf

nnk

Use this are written with &the lower.

]1[]1[]1[ ,, nnn khf

give recursive definitions of ][][][ ,, nnn khf

They

These

2]0[ rk

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

IJIJIJ gRgR )2/( 0

3/2 IJIJIJ gR R 00 IJIJ RE

222222 )sin( dzddkhdrfdtdxdxg JIIJ

bulk Einstein eq.

IJEalone, empty

=

0

][ )(),(n

nn zrFzrF

n

k

kknn GFFG0

][][][)(

expansionreduction

rule

recursive definition

for .2n

)2( n

2]0[ rk

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

IJIJIJ gRgR )2/( 0

3/2 IJIJIJ gR R 00 IJIJ RE

222222 )sin( dzddkhdrfdtdxdxg JIIJ

bulk Einstein eq.

IJEalone, empty

=

0

][ )(),(n

nn zrFzrF

n

k

kknn GFFG0

][][][)(

expansionreduction

rule

khf ,,,, ]0[]0[ hf .,, ]1[]1[]1[ khfwhose coefficients are written with

Thus, we obtained in the forms of power series of z,

]2[

2

22][

3

4

2222)1(

1

n

rrrrrrrzzzzzn fkhkf

hhf

fhf

hf

kkf

hhf

ff

nnf

]2[

2

2

2

22][

3

4

22

2

22)1(

1

n

rrrrrrrrrrzzzzzn hkhkh

fhhf

kk

ff

kk

ff

kkh

fhf

hh

nnh

]2[

2][

3

42

2222)1(

1

n

rrrrrrzzzzn khkh

fhkf

hk

hkh

fkf

nnk

recursive definition )2( n

use again lateruse again later

2]0[ rk

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

IJIJIJ gRgR )2/( 0

3/2 IJIJIJ gR R 00 IJIJ RE

222222 )sin( dzddkhdrfdtdxdxg JIIJ

bulk Einstein eq.

IJEalone, empty

=

0

][ )(),(n

nn zrFzrF

n

k

kknn GFFG0

][][][)(

expansionreduction

rule

khf ,,,, ]0[]0[ hf .,, ]1[]1[]1[ khfwhose coefficients are written with

Thus, we obtained in the forms of power series of z,

branemetric

extrinsiccurvature

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

2]0[ rk IJIJIJ gRgR )2/( 0

222222 )sin( dzddkhdrfdtdxdxg JIIJ

bulk Einstein eq.

IJEalone, empty

=

0

][ )(),(n

nn zrFzrF

n

k

kknn GFFG0

][][][)(

expansionreduction

rule

khf ,,,, ]0[]0[ hf .,, ]1[]1[]1[ khfwhose coefficients are written with

Thus, we obtained in the forms of power series of z,

branemetric

extrinsiccurvature

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

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

khf ,,

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

2]0[ rk IJIJIJ gRgR )2/( 0

222222 )sin( dzddkhdrfdtdxdxg JIIJ

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

0| 014 zR

0| 044 zR]0[

14R

=

03

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

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

[0]

bulk Einstein eq.

IJEalone, empty

=

0

][ )(),(n

nn zrFzrF

n

k

kknn GFFG0

][][][)(

expansionreduction

rule

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

2]0[ rk IJIJIJ gRgR )2/( 0

222222 )sin( dzddkhdrfdtdxdxg JIIJ

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

0

][ )(),(n

nn zrFzrF

n

k

kknn GFFG0

][][][)(

expansionreduction

ruleIJEalone, empty

=bulk Einstein eq.

0

4

2]1[

2]0[

]1[]1[

2]0[

]1[]1[

]0[]0[

]1[]1[

2 4224

1

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

3

2]0[44

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

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| 014 zR

]2[

2

22][

3

4

2222)1(

1

n

rrrrrrrzzzzzn fkhkf

hhf

fhf

hf

kkf

hhf

ff

nnf

]2[

2

2

2

22][

3

4

22

2

22)1(

1

n

rrrrrrrrrrzzzzzn hkhkh

fhhf

kk

ff

kk

ff

kkh

fhf

hh

nnh

]2[

2][

3

42

2222)1(

1

n

rrrrrrzzzzn khkh

fhkf

hk

hkh

fkf

nnk

recursive definition[2]

[2]

[2]

2

2

2

[0]

[0]

[0]

The solution includes three arbitrary functions.

,2/ ]0[]1[ ffu ,2/ ]0[]1[ hhv 2]1[ 2/ rkw Let be u v w arbitrary.

khf ,,We have

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

2]0[ rk

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

0| 044 zR

=

0| 014 zE

014 |zE

IJIJIJ gRgR )2/( 0222222 )sin( dzddkhdrfdtdxdxg JI

IJ

3

]1[

]0[

]1[

]0[]0[

]0[]1[

2]0[

]0[]1[

2

]1[

]0[

]1[

442 rk

rhh

fhfh

fff

rk

ff rrrr

4

2]1[

2]0[

]1[]1[

2]0[

]1[]1[

]0[]0[

]1[]1[

2 4224

1

rk

rhkh

rfkf

hfhf

r

0

bulk Einstein eq.

IJEalone, empty

=

0

][ )(),(n

nn zrFzrF

n

k

kknn GFFG0

][][][)(

expansionreduction

rule

,2/ ]0[]1[ ffu ,2/ ]0[]1[ hhv 2]1[ 2/ rkw Let be u v w arbitrary.

khf ,,We have if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 0| 044 zE

2]0[ rk 014 |zE

IJIJIJ gRgR )2/( 0222222 )sin( dzddkhdrfdtdxdxg JI

IJ

3

]1[

]0[

]1[

]0[]0[

]0[]1[

2]0[

]0[]1[

2

]1[

]0[

]1[

442 rk

rhh

fhfh

fff

rk

ff rrrr

The solution includes three arbitrary functions.

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

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

4

2]1[

2]0[

]1[]1[

2]0[

]1[]1[

]0[]0[

]1[]1[

2 4224

1

rk

rhkh

rfkf

hfhf

r

02

]1[

rkr 2]0[

]0[]1[

4 fff r ]0[]0[

]0[]1[

4 fhfh r

rhh

]0[

]1[

3

]1[

rk

0]0[

]1[

2 ffr( )r-u f [0]( )r- 2r 2w

2+ u

2+ v

+

2v +w2

0| 014 zE

0| 044 zE

bulk Einstein eq.

IJEalone, empty

=

0

][ )(),(n

nn zrFzrF

n

k

kknn GFFG0

][][][)(

expansionreduction

rule

The solution includes three arbitrary functions.

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

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

4

2]1[

2]0[

]1[]1[

2]0[

]1[]1[

]0[]0[

]1[]1[

2 4224

1

rk

rhkh

rfkf

hfhf

r

0| 044 zE

khf ,,We have if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 0| 044 zE

2]0[ rk 014 |zE

IJIJIJ gRgR )2/( 0222222 )sin( dzddkhdrfdtdxdxg JI

IJ

]0[

]1[

2 ffr( )r-u f [0]

2

]1[

rkr

( )r- 2r 2w

= =

rhh

]0[

]1[

2]0[

]0[]1[

4 fff r

2

u

-ur

u f [0]

f [0]- r_____ -2wr

4w r

- ___

+ +]0[]0[

]0[]1[

4 fhfh r

2v

2v3

]1[

rk

w2

0+ +rw2

ru rw2]0[

]0[

2 fufr

rv2

]0[

]0[

2 fvfr 0

0| 014 zE

,2/ ]0[]1[ ffu ,2/ ]0[]1[ hhv 2]1[ 2/ rkw Let be u v w arbitrary.

bulk Einstein eq.

IJEalone, empty

=

0

][ )(),(n

nn zrFzrF

n

k

kknn GFFG0

][][][)(

expansionreduction

rule

The solution includes three arbitrary functions.

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

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

4

2]1[

2]0[

]1[]1[

2]0[

]1[]1[

]0[]0[

]1[]1[

2 4224

1

rk

rhkh

rfkf

hfhf

r

0| 044 zE

khf ,,We have if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 0| 044 zE

2]0[ rk 014 |zE

IJIJIJ gRgR )2/( 0222222 )sin( dzddkhdrfdtdxdxg JI

IJ

Let r

w2ru rw2

]0[

]0[

2 fufr

rv2

]0[

]0[

2 fvfr 0

]0[

]0[

2

)(

ffvu r rwv /)(2 rr wu 2 v w r2vu ]0[

rf]0[2 f

]0[rf

]0[2 f2 r

　 　 　

rwvwu rr /)(22 [ ]

vu ]0[f2]0[

rf

0| 014 zE

vu ,2/ ]0[]1[ ffu ,2/ ]0[]1[ hhv 2]1[ 2/ rkw Let be u v w arbitrary.

for a while

bulk Einstein eq.

IJEalone, empty

=

0

][ )(),(n

nn zrFzrF

n

k

kknn GFFG0

][][][)(

expansionreduction

rule

The solution includes three arbitrary functions.

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

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

4

2]1[

2]0[

]1[]1[

2]0[

]1[]1[

]0[]0[

]1[]1[

2 4224

1

rk

rhkh

rfkf

hfhf

r

0| 044 zE

khf ,,We have if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 0| 044 zE

2]0[ rk 014 |zE

IJIJIJ gRgR )2/( 0222222 )sin( dzddkhdrfdtdxdxg JI

IJ

Let

]0[

]0[

2

)(

ffvu r rwv /)(2 rr wu 2

　 　 　

rwvwu rr /)(22 [ ]

vu ]0[f2]0[

rf

0| 014 zE

vu ,2/ ]0[]1[ ffu ,2/ ]0[]1[ hhv 2]1[ 2/ rkw Let be u v w arbitrary.

for a while

bulk Einstein eq.

IJEalone, empty

=

0

][ )(),(n

nn zrFzrF

n

k

kknn GFFG0

][][][)(

expansionreduction

rule

khf ,,We have if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 0| 044 zE

2]0[ rk 014 |zE

IJIJIJ gRgR )2/( 0222222 )sin( dzddkhdrfdtdxdxg JI

IJ IJE

alone, empty=

,2/ ]0[]1[ ffu ,2/ ]0[]1[ hhv 2]1[ 2/ rkw Let be u v w arbitrary.

　 　 　

rwvwu rr /)(22 [ ]

vu ]0[f2]0[

rf

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

4

2]1[

2]0[

]1[]1[

2]0[

]1[]1[

]0[]0[

]1[]1[

2 4224

1

rk

rhkh

rfkf

hfhf

r

The solution includes three arbitrary functions.

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

u v u w2 v w2 w 2

0| 014 zE

Let vu for a while

0| 044 zE

bulk Einstein eq.

0

][ )(),(n

nn zrFzrF

n

k

kknn GFFG0

][][][)(

expansionreduction

rule

khf ,,We have if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 0| 044 zE

2]0[ rk 014 |zE

IJIJIJ gRgR )2/( 0222222 )sin( dzddkhdrfdtdxdxg JI

IJ

bulk Einstein eq.

IJEalone, empty

=

,2/ ]0[]1[ ffu ,2/ ]0[]1[ hhv 2]1[ 2/ rkw Let be u v w arbitrary.

0| 014 zE

Let vu

　 　 　

rwvwu rr /)(22 [ ]

vu ]0[f2]0[

rf

4

2]1[

2]0[

]1[]1[

2]0[

]1[]1[

]0[]0[

]1[]1[

2 4224

1

rk

rhkh

rfkf

hfhf

r

The solution includes three arbitrary functions.

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

u v u w2 v w2 w 21 / 2r uv uw2 vw2 2w

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[

22

ff

ff r

r

r

rh

]0[

1

rh

]0[

12

]0[

]0[

]0[

]0[

22

ff

ff r

r

r

for a while

0| 044 zE

0

][ )(),(n

nn zrFzrF

n

k

kknn GFFG0

][][][)(

expansionreduction

rule

khf ,,We have if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 0| 044 zE

2]0[ rk 014 |zE

IJIJIJ gRgR )2/( 0222222 )sin( dzddkhdrfdtdxdxg JI

IJ

bulk Einstein eq.

,2/ ]0[]1[ ffu ,2/ ]0[]1[ hhv 2]1[ 2/ rkw Let be u v w arbitrary.

IJEalone, empty

=

　 　 　

rwvwu rr /)(22 [ ]

vu ]0[f2]0[

rf

2]0[

]0[

]0[]0[

]0[ 1

11

4 rrff

hrff rr

The solution includes three arbitrary functions.

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

222 wvwuwuv1 / 2rrh

]0[

12

]0[

]0[

]0[

]0[

22

ff

ff r

r

r

0| 014 zE

Let vu

=U

]0[rf U ]0[f U

U U U U

for a while

0| 044 zE

linear d. e.

0

][ )(),(n

nn zrFzrF

n

k

kknn GFFG0

][][][)(

expansionreduction

rule

khf ,,We have if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 0| 044 zE

2]0[ rk 014 |zE

IJIJIJ gRgR )2/( 0222222 )sin( dzddkhdrfdtdxdxg JI

IJ

bulk Einstein eq.

,2/ ]0[]1[ ffu ,2/ ]0[]1[ hhv 2]1[ 2/ rkw Let be u v w arbitrary.

IJEalone, empty

=

rwvwu rr /)(22 [ ]

vu 2

The solution includes three arbitrary functions.

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

222 wvwuwuv1 / 2r

0| 014 zE

Let vu ]0[

rf U ]0[f Ufor a while

rh

]0[

1

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

24

rh

]0[

1

U / 1/r4

0| 044 zE

linear d. e.

0

][ )(),(n

nn zrFzrF

n

k

kknn GFFG0

][][][)(

expansionreduction

rule

khf ,,We have if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 0| 044 zE

2]0[ rk 014 |zE

IJIJIJ gRgR )2/( 0222222 )sin( dzddkhdrfdtdxdxg JI

IJ

bulk Einstein eq.

,2/ ]0[]1[ ffu ,2/ ]0[]1[ hhv 2]1[ 2/ rkw Let be u v w arbitrary.

IJEalone, empty

=

rwvwu rr /)(22 [ ]

vu 2

222 wvwuwuv1 / 2r

0| 014 zE

Let vu ]0[

rf U ]0[f Ufor a while

U / 1/r4

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

4

rh

]0[

1

]0[

1

h

=P

=Q

rh ]0[

1P

]0[

1

hQ Q

P 0| 044 zE

linear d. e.

linear d. e.!

0

][ )(),(n

nn zrFzrF

n

k

kknn GFFG0

][][][)(

expansionreduction

rule

khf ,,We have if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 0| 044 zE

2]0[ rk 014 |zE

IJIJIJ gRgR )2/( 0222222 )sin( dzddkhdrfdtdxdxg JI

IJ

bulk Einstein eq.

,2/ ]0[]1[ ffu ,2/ ]0[]1[ hhv 2]1[ 2/ rkw Let be u v w arbitrary.

IJEalone, empty

=

rwvwu rr /)(22 [ ]

vu 2

222 wvwuwuv1 / 2r

0| 014 zE

Let vu ]0[

rf U ]0[f Ufor a while

U / 1/r4

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

4

rh ]0[

1P

]0[

1

hQ Q

P 0| 044 zE

linear d. e.

solution ]0[f

r

dre

U]0[h

rdr

e P

1

1

r

Pdrdre rQ

1st order linear differential equations solvable!

linear d. e.!

(GO

G)!

0

][ )(),(n

nn zrFzrF

n

k

kknn GFFG0

][][][)(

expansionreduction

rule

(^O^)

Let vu for a while

rwvwu rr /)(22 [ ]

vu 2

222 wvwuwuv1 / 2r

0| 014 zE

]0[rf U ]0[f U

U / 1/r4

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

4

rh ]0[

1P

]0[

1

hQ Q

P 0| 044 zE

linear d. e.

solution ]0[f

r

dre

U]0[h

rdr

e P

1

1

r

Pdrdre rQ

1st order linear differential equations solvable!

linear d. e.!

The solution include three arbitrary functions.

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

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

4

2]1[

2]0[

]1[]1[

2]0[

]1[]1[

]0[]0[

]1[]1[

2 4224

1

rk

rhkh

rfkf

hfhf

r

0| 044 zE

Let

　 　 　

rwvwu rr /)(22 [ ]

vu ]0[f2]0[

rf

0| 014 zE

vu for a while

]0[

]0[

2

)(

ffvu r rwv /)(2 rr wu 20 =

Remember this page

khf ,,We have if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 0| 044 zE

2]0[ rk 014 |zE

IJIJIJ gRgR )2/( 0222222 )sin( dzddkhdrfdtdxdxg JI

IJ

bulk Einstein eq.

,2/ ]0[]1[ ffu ,2/ ]0[]1[ hhv 2]1[ 2/ rkw Let be u v w arbitrary.

IJE

,2/ ]0[]1[ ffu ,2/ ]0[]1[ hhv 2]1[ 2/ rkw Let be u v w arbitrary.

0

][ )(),(n

nn zrFzrF

n

k

kknn GFFG0

][][][)(

expansionreduction

rule

alone, empty=

rwvwu rr /)(22 0 ,)2/(1

r r drruurw

If u = v

w is no longer arbitrary, and f [0] is arbitrary.

khf ,,We have if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 0| 044 zE

2]0[ rk 044 |zE

IJIJIJ gRgR )2/( 0222222 )sin( dzddkhdrfdtdxdxg JI

IJ

,2/ ]0[]1[ ffu ,2/ ]0[]1[ hhv 2]1[ 2/ rkw Let be u v w arbitrary.

rwvwu rr /)(22 [ ]

vu 2

ULet vu for a while

222 wvwuwuv1 / 2r

U / 1/r4Q

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

4P

solution ]0[f

r

dre

U]0[h

rdr

e P

1

1

r

Pdrdre rQ

]/)(22[2 rwvwu rr vu U

QP /

/

((

(

(

) )

) )

with / ( )22 /1/4/2/ rrUUU r rU /14/

22 22/1 wvwuwuvr rU /14/

, ]0[

r

Udref

1

]0[ 1

r

PdrPdrdrQeeh rr

If vu solution

bulk Einstein eq.

IJEalone, empty

=

0

][ )(),(n

nn zrFzrF

n

k

kknn GFFG0

][][][)(

expansionreduction

rule=rwvwu rr /)(22 0 ,)2/(1

r r drruurw

If u = v

w is no longer arbitrary, and f [0] is arbitrary.

khf ,,We have if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 0| 044 zE

2]0[ rk 044 |zE

IJIJIJ gRgR )2/( 0222222 )sin( dzddkhdrfdtdxdxg JI

IJ

bulk Einstein eq.

,2/ ]0[]1[ ffu ,2/ ]0[]1[ hhv 2]1[ 2/ rkw Let be u v w arbitrary.

]/)(22[2 rwvwu rr vu

,2/ ]0[]1[ ffu ,2/ ]0[]1[ hhv 2]1[ 2/ rkw Let be u v w arbitrary.

If vu

U

solution,

]0[

r

Udref

1

]0[ 1

r

PdrPdrdrQeeh rr

(with / ( )

22 /1/4/2/ rrUUU r

IJEalone, empty

=

Q/

/

(( (

) )

) )

rU /14/ 22 22/1 wvwuwuvr rU /14/

P

0

][ )(),(n

nn zrFzrF

n

k

kknn GFFG0

][][][)(

expansionreduction

rule=rwvwu rr /)(22 0 ,)2/(1

r r drruurw

If u = v

w is no longer arbitrary, and f [0] is arbitrary.

]0[f,u

]0[]0[ / ffU r

,)2/(1

r r drruurw

22 4 wuwu1 / 2r U / 1/r4Q /( (U / 1/r4U r / 2U 2 / U /r 1/ 2r4P /( () )

) )

: arbitrary. w is no longer arbitrary.If u = v1

]0[ 1

r

PdrPdrdrQeeh rr

]/)(22[2 rwvwu rr vu If vu

U

Q

solution,

]0[

r

Udref

1

]0[ 1

r

PdrPdrdrQeeh rr

/

/

((

(

(

) )

) )

with / ( )22 /1/4/2/ rrUUU r rU /14/

22 22/1 wvwuwuvr rU /14/ P

khf ,,We have if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 0| 044 zE

2]0[ rk 044 |zE

IJIJIJ gRgR )2/( 0222222 )sin( dzddkhdrfdtdxdxg JI

IJ

bulk Einstein eq.

,2/ ]0[]1[ ffu ,2/ ]0[]1[ hhv 2]1[ 2/ rkw Let be u v w arbitrary.

IJEalone, empty

=

,2/ ]0[]1[ ffu ,2/ ]0[]1[ hhv 2]1[ 2/ rkw Let be u v w arbitrary.

]0[f,u

]0[]0[ / ffU r

,)2/(1

r r drruurw

22 4 wuwu1 / 2r U / 1/r4Q /( (U / 1/r4U r / 2U 2 / U /r 1/ 2r4P /( () )

) )

: arbitrary. w is no longer arbitrary.If u = v1

]0[ 1

r

PdrPdrdrQeeh rr

khf ,,we obtained

,, ]0[]0[ hf ]1[]1[]1[ ,, khf

the bulk Einstein eq. alone, empty,in power series of z,

in terms of arbitrary

whose coefficients are recursively defined

u v wwritten with , ,

In summary for

.

]2[

2

22][

3

4

2222)1(

1

n

rrrrrrrzzzzzn fkhkf

hhf

fhf

hf

kkf

hhf

ff

nnf

]2[

2

2

2

22][

3

4

22

2

22)1(

1

n

rrrrrrrrrrzzzzzn hkhkh

fhhf

kk

ff

kk

ff

kkh

fhf

hh

nnh

]2[

2][

3

42

2222)1(

1

n

rrrrrrzzzzn khkh

fhkf

hk

hkh

fkf

nnk

recursive definition

functions,

0

][ )(),(n

nn zrFzrF

n

k

kknn GFFG0

][][][)(

expansionreduction

rule

by this

]/)(22[2 rwvwu rr vu If vu

U

Q

solution,

]0[

r

Udref

1

]0[ 1

r

PdrPdrdrQeeh rr

/

/

((

(

(

) )

) )

with / ( )22 /1/4/2/ rrUUU r rU /14/

22 22/1 wvwuwuvr rU /14/ P

]0[f,u

]0[]0[ / ffU r

,)2/(1

r r drruurw

22 4 wuwu1 / 2r U / 1/r4Q /( (U / 1/r4U r / 2U 2 / U /r 1/ 2r4P /( () )

) )

: arbitrary. w is no longer arbitrary.If u = v1

]0[ 1

r

PdrPdrdrQeeh rr

khf ,,We have if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 0| 044 zE

2]0[ rk 044 |zE

IJIJIJ gRgR )2/( 0222222 )sin( dzddkhdrfdtdxdxg JI

IJ

bulk Einstein eq.

,2/ ]0[]1[ ffu ,2/ ]0[]1[ hhv 2]1[ 2/ rkw Let be u v w arbitrary.

IJEalone, empty

=

,2/ ]0[]1[ ffu ,2/ ]0[]1[ hhv 2]1[ 2/ rkw Let be u v w arbitrary.

khf ,,we obtained

,, ]0[]0[ hf ]1[]1[]1[ ,, khf

the bulk Einstein eq. alone, empty,in power series of z,

in terms of arbitrary

whose coefficients are recursively defined

u v wwritten with , ,

In summary for

.functions, (^O^)This gives the solution outside the brane.

0

][ )(),(n

nn zrFzrF

n

k

kknn GFFG0

][][][)(

expansionreduction

rule

like this

khf ,,We have if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 0| 044 zE

2]0[ rk 044 |zE

IJIJIJ gRgR )2/( 0222222 )sin( dzddkhdrfdtdxdxg JI

IJ

bulk Einstein eq.

,2/ ]0[]1[ ffu ,2/ ]0[]1[ hhv 2]1[ 2/ rkw Let be u v w arbitrary.

IJEalone, empty

=

,2/ ]0[]1[ ffu ,2/ ]0[]1[ hhv 2]1[ 2/ rkw Let be u v w arbitrary.

khf ,,we obtained

,, ]0[]0[ hf ]1[]1[]1[ ,, khf

the bulk Einstein eq. alone, empty,in power series of z,

in terms of arbitrary

whose coefficients are recursively defined

u v wwritten with , ,

In summary for

.functions, (^O^)This gives the solution outside the brane.

On the brane Matter is distributed within |z|<d , very small.

0

][ )(),(n

nn zrFzrF

n

k

kknn GFFG0

][][][)(

expansionreduction

rule

IJEalone, empty

=

khf ,,We have if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 0| 044 zE

2]0[ rk 044 |zE

IJIJIJ gRgR )2/( 0222222 )sin( dzddkhdrfdtdxdxg JI

IJ

On the brane

0 IJTNambu-Goto eq.

0)~~~

( ; IYTg

zz

z

zzz khf ,,

,/ ffu z ,/hhv z ,/kkw z ,| zuu

,2/)( uuu wvwv ,,,similarly for uuu

Matter is distributed within |z|<d , very small.

Take the limit d → 0.collective mode dominance in ,IJT .~

IJT

,2/ ]0[]1[ ffu ,2/ ]0[]1[ hhv 2]1[ 2/ rkw Let be u v w arbitrary.

z z z k

bulk Einstein eq. on the brane 3/~ wvu

bulk Einstein eq.

zzzzzz khf ,,

z

u u

u

u u

khf ,,

ratio ratio

Israel Junction condition

≡D

define for short

0

][ )(),(n

nn zrFzrF

n

k

kknn GFFG0

][][][)(

expansionreduction

rule

ratio

0 0 IJT

Nambu-Goto eq.

khf ,,We have if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 0| 044 zE

2]0[ rk 044 |zE

IJIJIJ gRgR )2/(222222 )sin( dzddkhdrfdtdxdxg JI

IJ

Nambu-Goto eq.0)

~~~( ; IYTg

bulk Einstein equation

,2/ ]0[]1[ ffu ,2/ ]0[]1[ hhv 2]1[ 2/ rkw Let be u v w arbitrary.z z z k 02 wvu

0| 044 zE

014 |zE 0

0| 044 zE

014 |zE 0

1 / 2r uv uw2 vw2 2w

2]0[

]0[2

]0[

]0[

]0[

]0[

]0[]0[

]0[

]0[

1

22

11

4

1

rrff

ff

ff

hrff

hrr

r

rr

r

]/)(22[2/)( ]0[]0[ rwvwuffvu rrr

0| 14 zE ]/)(22[2/)( ]0[]0[ rwvwuffvu rrr

2 2 )(22/1 wwvwuvur

2]0[

]0[2

]0[

]0[

]0[

]0[

]0[]0[

]0[

]0[

1

22

11

4

1

rrff

ff

ff

hrff

hrr

r

rr

r

±d

0| 44 zE±d

± ± ± ± ± ±

± ± ± ± ± ± ±connected at the boundary

,/ ffu z ,/hhv z ,/kkw z ,| zuu

,2/)( uuu wvwv ,,,similarly for uuuTake the limit d → 0.collective mode dominance in ,IJT .

~IJT

bulk Einstein eq. on the brane 3/~ wvu

Israel Junction condition

≡D

define for short

3/~ wvu ≡D

holds for the collective modes

khf ,,We have if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 0| 044 zE

2]0[ rk 044 |zE

IJIJIJ gRgR )2/( 0222222 )sin( dzddkhdrfdtdxdxg JI

IJ 0 IJT

Nambu-Goto eq.0)

~~~( ; IYTg

bulk Einstein equation

,2/ ]0[]1[ ffu ,2/ ]0[]1[ hhv 2]1[ 2/ rkw Let be u v w arbitrary.z z z k

0| 044 zE

014 |zE 0

1 / 2r uv uw2 vw2 2w

2]0[

]0[2

]0[

]0[

]0[

]0[

]0[]0[

]0[

]0[

1

22

11

4

1

rrff

ff

ff

hrff

hrr

r

rr

r

]/)(22[2/)( ]0[]0[ rwvwuffvu rrr

0| 14 zE±d

0| 44 zE±d

2 2 )(22/1 wwvwuvur

2]0[

]0[2

]0[

]0[

]0[

]0[

]0[]0[

]0[

]0[

1

22

11

4

1

rrff

ff

ff

hrff

hrr

r

rr

r

± ± ± ± ± ± ±

14 |E |14Ed -d

44 |E |44Ed -d

]/)(22[2/)( ]0[]0[ rwvwuffvu rrr ± ± ± ± ± ±

Nambu-Goto eq. 02 wvu 3/~ wvu ≡D

0)2( wvu

trivially satisfied

satisfied due to

difference of ±

u +v +2w = 0- - -]/)(22[2/)( ]0[]0[ rwvwuffvu rrr ± ± ± ± ± ±D D D D D D

Nambu-Goto eq.

Nambu-Goto eq. 02 wvu 3/~ wvu ≡D

khf ,,We have if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 0| 044 zE

2]0[ rk 044 |zE

IJIJIJ gRgR )2/( 0222222 )sin( dzddkhdrfdtdxdxg JI

IJ 0 IJT

Nambu-Goto eq.0)

~~~( ; IYTg

bulk Einstein equation

,2/ ]0[]1[ ffu ,2/ ]0[]1[ hhv 2]1[ 2/ rkw Let be u v w arbitrary.z z z k

0| 044 zE

014 |zE 0

1 / 2r uv uw2 vw2 2w

2]0[

]0[2

]0[

]0[

]0[

]0[

]0[]0[

]0[

]0[

1

22

11

4

1

rrff

ff

ff

hrff

hrr

r

rr

r

]/)(22[2/)( ]0[]0[ rwvwuffvu rrr

0| 14 zE±d

0| 44 zE±d

2]0[

]0[2

]0[

]0[

]0[

]0[

]0[]0[

]0[

]0[

1

22

11

4

1

rrff

ff

ff

hrff

hrr

r

rr

r

2 2 )(22/1 wwvwuvur

2]0[

]0[2

]0[

]0[

]0[

]0[

]0[]0[

]0[

]0[

1

22

11

4

1

rrff

ff

ff

hrff

hrr

r

rr

r

± ± ± ± ± ± ±

]/)(22[2/)( ]0[]0[ rwvwuffvu rrr ± ± ± ± ± ±

---- --- 6/~2

sum of ±

]0[]0[ /)( ffvu r ]/)(22[2 rwvwu rr - - - - - -

2/1 r 2 )(22 wwvwuvu

14 |E |14Ed -d

44 |E |44Ed -d

2 )(22 wwvwuvu

Nambu-Goto eq. 02 wvu 3/~ wvu ≡D

khf ,,We have if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf obey 0| 044 zE

2]0[ rk 044 |zE

IJIJIJ gRgR )2/( 0222222 )sin( dzddkhdrfdtdxdxg JI

IJ 0 IJT

Nambu-Goto eq.0)

~~~( ; IYTg

bulk Einstein equation

,2/ ]0[]1[ ffu ,2/ ]0[]1[ hhv 2]1[ 2/ rkw Let be u v w arbitrary.z z z k

0| 044 zE

014 |zE 0

1 / 2r uv uw2 vw2 2w

2]0[

]0[2

]0[

]0[

]0[

]0[

]0[]0[

]0[

]0[

1

22

11

4

1

rrff

ff

ff

hrff

hrr

r

rr

r

]/)(22[2/)( ]0[]0[ rwvwuffvu rrr

14 |E |14Ed -d

2/)( vuw

2]0[

]0[2

]0[

]0[

]0[

]0[

]0[]0[

]0[

]0[

1

22

11

4

1

rrff

ff

ff

hrff

hrr

r

rr

r

---- --- 6/~2

]0[]0[ /)( ffvu r ]/)(22[2 rwvwu rr - - - - - -

2/1 r

]/)3([2 rvuvr

4/)323( 22 vvuu

]/)3([2 rvuvr

4/)323( 22 vvuu 6/~2

substitute

substitute

vu , : arbitrary,

44 |E |44Ed -d

14 |E |14Ed -d

]0[]0[ /)( ffvu r- - ]/)3([2 rvuvr

2]0[

]0[2

]0[

]0[

]0[

]0[

]0[]0[

]0[

]0[

1

22

11

4

1

rrff

ff

ff

hrff

hrr

r

rr

r

2/1 r 4/)323( 22 vvuu 6/~2

2/)( vuw vu , : arbitrary,

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

6/~

4/)323(/1 2222 vvuur

equations ]/)3([2/ ]0[]0[ rvuvff rr )( vu 2/)( vuw vu , : arbitrary,

44 |E |44Ed -d

)/(]/)3([2 vurvuvr

If vu U

QP

solution,

]0[

r

Udref

1

]0[ 1

r

PdrPdrdrQeeh rr

/( () )with

]0[f

]0[]0[ / ffU r

4/rCu

6/~

/2 2282 rC1 / 2r U / 1/r4Q /( (U / 1/r4U r / 2U 2 / U /r 1/ 2r4P /( () )

) )

: arbitrary. is no longer arbitrary.

22 /1/4/2/ rrUUU r rU /14/ )/14//(]6/

~4/)323(/1[ 22222 rUvvuur

If1

]0[ 1

r

PdrPdrdrQeeh rr

vu u

C: constant

with

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

6/~

4/)323(/1 2222 vvuur

equations ]/)3([2/ ]0[]0[ rvuvff rr )( vu 2/)( vuw vu , : arbitrary, differs in

from the previous

)/(]/)3([2 vurvuvr

If vu U

QP

solution,

]0[

r

Udref

1

]0[ 1

r

PdrPdrdrQeeh rr

/( () )with

]0[f

]0[]0[ / ffU r

4/rCu

6/~

/2 2282 rC1 / 2r U / 1/r4Q /( (U / 1/r4U r / 2U 2 / U /r 1/ 2r4P /( () )

) )

: arbitrary. is no longer arbitrary.

22 /1/4/2/ rrUUU r rU /14/ )/14//(]6/

~4/)323(/1[ 22222 rUvvuur

If1

]0[ 1

r

PdrPdrdrQeeh rr

vu u

C: constant

with

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

6/~

4/)323(/1 2222 vvuur

equations ]/)3([2/ ]0[]0[ rvuvff rr )( vu 2/)( vuw vu , : arbitrary,

khf ,,we obtained

,|,| ]0[]0[ zz hf zzz khf |,|,| ]1[]1[]1[

in power series of z >0 and of z<0,

in terms of

whose coefficients are recursively defined

written with arbitrary functions, vu &

the bulk Einstein eq. & Nambu Goto eq.In summary for

of r.

]2[

2

22][

3

4

2222)1(

1

n

rrrrrrrzzzzzn fkhkf

hhf

fhf

hf

kkf

hhf

ff

nnf

]2[

2

2

2

22][

3

4

22

2

22)1(

1

n

rrrrrrrrrrzzzzzn hkhkh

fhhf

kk

ff

kk

ff

kkh

fhf

hh

nnh

]2[

2][

3

42

2222)1(

1

n

rrrrrrzzzzn khkh

fhkf

hk

hkh

fkf

nnk

recursive definition Conclusion

like this

by this

)/(]/)3([2 vurvuvr

If vu U

QP

solution,

]0[

r

Udref

1

]0[ 1

r

PdrPdrdrQeeh rr

/( () )with

]0[f

]0[]0[ / ffU r

4/rCu

6/~

/2 2282 rC1 / 2r U / 1/r4Q /( (U / 1/r4U r / 2U 2 / U /r 1/ 2r4P /( () )

) )

: arbitrary. is no longer arbitrary.

22 /1/4/2/ rrUUU r rU /14/ )/14//(]6/

~4/)323(/1[ 22222 rUvvuur

If1

]0[ 1

r

PdrPdrdrQeeh rr

vu u

C: constant

with

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

6/~

4/)323(/1 2222 vvuur

equations ]/)3([2/ ]0[]0[ rvuvff rr )( vu 2/)( vuw vu , : arbitrary,

khf ,,we obtained

,|,| ]0[]0[ zz hf zzz khf |,|,| ]1[]1[]1[

in power series of z >0 and of z<0,

in terms of

whose coefficients are recursively defined

written with arbitrary functions, vu &

the bulk Einstein eq. & Nambu Goto eq.In summary for

of r.

]2[

2

22][

3

4

2222)1(

1

n

rrrrrrrzzzzzn fkhkf

hhf

fhf

hf

kkf

hhf

ff

nnf

]2[

2

2

2

22][

3

4

22

2

22)1(

1

n

rrrrrrrrrrzzzzzn hkhkh

fhhf

kk

ff

kk

ff

kkh

fhf

hh

nnh

]2[

2][

3

42

2222)1(

1

n

rrrrrrzzzzn khkh

fhkf

hk

hkh

fkf

nnk

recursive definition Conclusion

like this

This is the general solution of the system. (^O^)

by this

It has large arbitrariness due to the extrinsic curvature. (×^

×)

wuhf 2,, ]0[]0[ arbitrary

algebraic eq. for wv, solvable

for bulk Einstein eq. alone, empty

3 eqs. for 5 functions

3 arbitrary functions2 eqs. for 5 functions

2 arbitrary functions

non-linear differential eq. for

instead of u,v,w.We can choose

0| 044 zE044 |zE become

for braneworld

]0[]0[ ,hf arbitrary instead ofWe can choose .,vu

0| 044 zE044 |zE vu ,

not solvable, but solution exists.

other choice of the arbitrary functions

(bulk Einstein eq. & Nambu-Goto eq.)

(×^

×)

(^_^)

Discussions

]0[]0[ ,hf be arbitraryLet

1]0[ fThe Newtonian potential becomes arbitrary.

33

22

]0[ )/()/(/1 rararf

33

221

]0[ )/()/(//1/1 rbrbrbrh

In Einstein gravity, 0 ii ba

Assume asymptotic expansion

21 1

Einstein

b

3

2

31 21

Einstein

ab

light deflection by star gravity

planetary perihelion precession

observation

lightstar

0r

Einstein Einstein

Discussions

Here, they are arbitrary.

=arbitrary

=arbitrary

21 1

Einstein

b

3

2

31 21

Einstein

ab

light deflection by star gravity

planetary perihelion precession

observation

lightstar

0r

Einstein Einstein

Discussions

21 1

Einstein

b

3

2

31 21

Einstein

ab

light deflection by star gravity

planetary perihelion precession

observation

lightstar

0r

Einstein Einstein

=arbitrary

=arbitrary

=arbitrary

=arbitrary

Discussions

21 1

Einstein

b

3

2

31 21

Einstein

ab

light deflection by star gravity

planetary perihelion precession

observation

star0r

Einstein Einstein

Einstein gravityThe general solution here

can predict the observed results. includes the case observed,

but, requires fine tuning,and, hence, cannot "predict" the observed results.

1b 22 2ab &0 0 (*)

(^_^)

(×^

×)

Z2 symmetry leaves these arbitrariness unfixed. (×^

×)We need additional physical prescriptions non-dynamical.

Brane induced gravity may by-pass this difficulty. (^O^)

=arbitrary

=arbitrary

light

Summary

We derived the general solution of the fundamental equations of the braneworld under the Schwarzschild ansatz.

Outside, it is expressed in power series of the brane normal coordinate in terms of 5 functions on the brane for each side.

The functions should obey 3 essential on-brane equations including the equation of motion of the brane.

They are solved in terms of 2 arbitrary functions on the brane.

The arbitrariness may affect the predictive powers on the Newtonian and the post-Newtonian evidences.

We need non-dynamical physical prescriptions to recover.Brane induced gravity may by-pass this problem.

(×^

×)

(^O^)

(^O^)

(^O^)

bulk Einstein eq. & Nambu-Goto eq. Summary

Thank you

(^o^)

02 wvu

]0[

]0[

2

)(

ffvu r

rwv /)(2 rr wu 2

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

6/~

22/1 222 wwvwuvur

6/~

4/)()(/1 2222 vuvuvur

6/~

4/)323(/1 2222 vvuur

ruvvvr /)2(

rvuvr /)3(

rwvwu rr /)(22 0/4 ruur

ruur /4

drru

du 4

4Cru

6/~

22/1 2222 wvwuwuvr

6/~

2/1

6/~

22/12222

2222222

ur

uuuur

r r drruurw )2/(1

rdrrrrCr )2/4( 541

r

drCrr 41 3

31 Crr 4Cr

6/~

2/1 22822 rCr

,0221100 RRR 014 |zE 0| 044 zEindependent eqs.Def.

IJIJIJ gRgR )2/(

3/2 IJIJIJ gR R 00 IJIJ RE

222222 )sin( dzddkhdrfdtdxdxg JIIJ

bulk Einstein eq.

IJE = 0 IJTNambu-Goto eq.

0)~~~

( ; IYTg