general solution of braneworld with the schwarzschild ansatz

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

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; 1208.3303 [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.

higher dim. spacetime

our spacetime

braneworld

Braneworld is a model ofour 3+1 dim. curved spacetime

This idea has a long history.

Fronsdal('59), Josesh('62)Regge,Teitelboim('75) K.A.('82)Rubakov,Shaposhnikov('83)Visser('85), Maia('84), Pavsic('85), Gibbons,Wiltschire ('87)

Polchinski('95) Antoniadis('91), Horava,Witten('96)

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

where we take it as a membrane-like object embedded in higher dimensions.

Einstein gravity successfully explaines

②post Newtonian evidences: light deflections due to gravity,

the planetary perihelion precessions, etc.

(^V^)

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

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

)( KIJ Xg

)( xY I

bulk scalar curvature

gg ~det~

bulk Einstein eq.

Nambu-Goto eq.

constant

brane en.mom.tensor

)(~ xgbrane KX xbulk coord.

brane metriccannot be a dynamical variable

constant

g(x)YI,YJ

, gIJ(Y)

matter action

~

Sd /d~ indicatesbrane quantity

bulk en.mom.tensor

IJgg det

0)2/( IJIJIJIJ TgRgR

coord.0

gIJYI

bulk Ricci tensor

0)~~~( ; IYTg

(3+1dim.)

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

(3+1dim.)

(3+1)

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 (t,r,q,j)

2222222 )sin( dzddkhdrfdtdXdXgds JIIJ jqq

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

t,r,q,j

z

We first consider the solution outside the brane.

bulk Einstein eq. Nambu-Goto eq.

222222 )sin( dzddkhdrfdtdXdXg JIIJ jqq

empty

0)~~~( ; IYTg

zXXXrXtX 43210 ,,,, jq

IJIJIJ gRgR )2/( 0 IJT0off brane (3+1)

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

44R 2

2

2

2

2

2

24422 kk

hh

ff

kk

hh

ff zzzzzzzzz

14R 22 22442 kkk

hkkh

hffh

fff

kk

ff zrrzrzrzrzrz

RIJKLGIJK,LGIJL,KgAB(GAIKGBJLGAILGBJK), GIJK(gIJ,KgIK,J gJK,I)/2

The only independent non-trivial components

0)~~~( ; IYTg

zXXXrXtX 43210 ,,,, jq

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

IJ jqq

bulk Einstein eq.

curvaturetensor

affineconnection

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

off brane

00Rkhkf

hhf

fhf

hf

kkf

hhf

fff rrrrrrrzzzzzzz

24422442 2

22

11Rkhkh

kk

ff

kk

ff

fhhf

kkh

fhf


224242442 2

2

2

22

22R 1442442 2 hkh

fhkf

hk

hkh

fkfk rrrrrrzzzzzz

44R 2

2

2

2

2

2

24422 kk

hh

ff

kk

hh

ff zzzzzzzzz

14R 22 22442 kkk

hkkh

hffh

fff

kk

ff zrrzrzrzrzrz


RIJKLGIJK,LGIJL,KgAB(GAIKGBJLGAILGBJK), GIJK(gIJ,KgIK,J gJK,I)/2zXXXrXtX 43210 ,,,, jq

use again later

Nambu-Goto eq.0)~~~( ; IYTg

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

IJ jqquse again later

bulk Einstein eq. off brane

covariant derivativecovariant 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

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


IJIJIJ gRgR )2/( 0

3/2 IJIJIJ gR R

Bianchi identity


then

, then

bulk Einstein eq.

equivalent equation

independent equations

0| | 044014 zzThis &

&

Owing to

is equivalent to 0221100 RRR

off brane

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


IJIJIJ gRgR )2/( 0

00 IJIJ RE222222 )sin( dzddkhdrfdtdXdXg JI

IJ jqq

bulk Einstein eq.

IJE

independent equations

Therefore, the independent equations are

0| | 044014 zz&0221100 RRR

3/2 IJIJIJ gR R

off brane

3/2 IJIJIJ gR R014 |zE 0| 044 zE,0221100 RRRindependent eqs.

Def.


IJIJIJ gRgR )2/( 0


IJ jqq

00Rkhkf

hhf

fhf

hf

kkf

hhf

fff rrrrrrrzzzzzzz

24422442 2

22

11Rkhkh

kk

ff

kk

ff

fhhf

kkh

fhf


224242442 2

2

2

22

03

2

f00R

0][ )(),(

nnn zrFzrFexpansion

n

kkknn GFFG

0][][][)(reduction rule

& derivatives),,, khfF IJT(

03/ 2 R f

khkf

hhf

fhf

hf

kkf

hhf

fff rrrrrrrzzzzzzz

24422442 2

22

2]0[ rk using diffeo.

bulk Einstein eq.

IJE

power seriessolution in z

00 g00

off brane

]0[14E 0]0[

44E

]0[14E014 | zE ]0[

44E014 | zE

]0[44E

0][ )(),(

nnn zrFzrFexpansion

n

kkknn GFFG


& derivatives),,, khfF IJT,0221100 RRRindependent eqs.

(

32

2442244 2

22 fkhkf

hhf

fhf

hf

kkf

hhf

ff rrrrrrrzzzzz

zzf22 2 2

f3

42222 2

22 fkhkf

hhf

fhf

hf

kkf

hhf

ff rrrrrrrzzzzz

[n][n2]1

n(n 1)


2 2 2 2 2 2 2

zz[n2]

Def.


IJIJIJ gRgR )2/( 0


IJ jqq

khkf

hhf

fhf

hf

kkf

hhf

fff rrrrrrrzzzzzzz

24422442 2

22

03

2

f

2 4

n(n 1)

bulk Einstein eq.

IJE

power seriessolution in z

3/2 IJIJIJ gR R

00R 03/ 2 R f00

off brane

]0[14E ]0[

44E 0

0][ )(),(

nnn zrFzrFexpansion

f3

42222 2

22 fkhkf

hhf

fhf

hf

kkf

hhf

ff rrrrrrrzzzzz

[n] 1n(n 1)

[n2]

2]0[ rk


n

kkknn GFFG


,0221100 RRRindependent eqs.Def.


IJIJIJ gRgR )2/( 0


IJ jqq

bulk Einstein eq.

IJE

power seriessolution in zNambu-Goto eq.

0)~~~( ; IYTg power seriessolution in z

0][ )(),(

nnn zrFzrF

n

kkknn GFFG

0][][][)(

expansionreduction

rule

3/2 IJIJIJ gR R

off brane

]0[14E ]0[

44E 0

f3

42222 2

22 fkhkf

hhf

fhf

hf

kkf

hhf

ff rrrrrrrzzzzz

[n] 1n(n 1)

]2[

2

22][

34

2222)1(1

n

rrrrrrrzzzzzn fkhkf

hhf

fhf

hf

kkf

hhf

ff

nnf

[n2]

2]0[ rk


IJIJIJ gRgR )2/( 0


IJ jqq

bulk Einstein eq.

IJE

0][ )(),(

nnn zrFzrF

n

kkknn GFFG

0][][][)(

expansionreduction

rule

3/2 IJIJIJ gR R

off brane

]0[14E ]0[

44E 0

00Rkhkf

hhf

fhf

hf

kkf

hhf

fff rrrrrrrzzzzzzz

24422442 2

22

11Rkhkh

kk

ff

kk

ff

fhhf

kkh

fhf


224242442 2

2

2

22

22R 1442442 2 hkh

fhkf

hk

hkh

fkfk rrrrrrzzzzzz


]2[

2

2

2

22][

34

222

22)1(1

n

rrrrrrrrrrzzzzzn hkhkh

fhhf

kk

ff

kk

ff

kkh

fhf

hh

nnh

]2[

2

22][

34

2222)1(1

n

rrrrrrrzzzzzn fkhkf

hhf

fhf

hf

kkf

hhf

ff

nnf

2]0[ rk


IJIJIJ gRgR )2/( 0


IJ jqq

bulk Einstein eq.

IJE

0][ )(),(

nnn zrFzrF

n

kkknn GFFG

0][][][)(

expansionreduction

rule

3/2 IJIJIJ gR R

off brane

]0[14E ]0[

44E 0

00Rkhkf

hhf

fhf

hf

kkf

hhf

fff rrrrrrrzzzzzzz

24422442 2

22

11Rkhkh

kk

ff

kk

ff

fhhf

kkh

fhf


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


]2[

2][

342

2222)1(1

nrrrrrrzzzzn k

hkh

fhkf

hk

hkh

fkf

nnk

]2[

2

22][

34

2222)1(1

n

rrrrrrrzzzzzn fkhkf

hhf

fhf

hf

kkf

hhf

ff

nnf

]2[

2

2

2

22][

34

222

22)1(1

n


fhhf

kk

ff

kk

ff

kkh

fhf

hh

nnh

2]0[ rk


IJIJIJ gRgR )2/( 0


IJ jqq

bulk Einstein eq.

IJE

0][ )(),(

nnn zrFzrF

n

kkknn GFFG

0][][][)(

expansionreduction

rule

3/2 IJIJIJ gR R

off brane

]0[14E ]0[

44E 0

here.

]2[

2

22][

34

2222)1(1

n

rrrrrrrzzzzzn fkhkf

hhf

fhf

hf

kkf

hhf

ff

nnf

]2[

2

2

2

22][

34

222

22)1(1

n


fhhf

kk

ff

kk

ff

kkh

fhf

hh

nnh

]2[

2][

342

2222)1(1

nrrrrrrzzzzn k

hkh

fhkf

hk

hkh

fkf

nnk

Use this are written with &the lower.

]1[]1[]1[ ,, nnn khfgive recursive definitions of ][][][ ,, nnn khf

They

These

2]0[ rk


IJIJIJ gRgR )2/( 0


IJ jqq

bulk Einstein eq.

IJE

0][ )(),(

nnn zrFzrF

n

kkknn GFFG

0][][][)(

expansionreduction

rule

recursive definition

for .2n

)2( n

3/2 IJIJIJ gR R

off brane

]0[14E ]0[

44E 0

2]0[ rk


IJIJIJ gRgR )2/( 0


IJ jqq

bulk Einstein eq.

IJE

0][ )(),(

nnn zrFzrF

n

kkknn GFFG

0][][][)(

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][

34

2222)1(1

n

rrrrrrrzzzzzn fkhkf

hhf

fhf

hf

kkf

hhf

ff

nnf

]2[

2

2

2

22][

34

222

22)1(1

n


fhhf

kk

ff

kk

ff

kkh

fhf

hh

nnh

]2[

2][

342

2222)1(1

nrrrrrrzzzzn k

hkh

fhkf

hk

hkh

fkf

nnk

recursive definition )2( n

use again later

3/2 IJIJIJ gR R

used not yet used

off brane

]0[14E ]0[

44E 0use again later

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


bulk Einstein eq.

IJE

0][ )(),(

nnn zrFzrF

n

kkknn GFFG

0][][][)(

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,

not yet used

We have ,, ]0[]0[ hf ]1[]1[]1[ ,, khfkhf ,, obey 0]0[44

]0[14 EEif

off brane

]0[14E ]0[

44E 0

00Rkhkf

hhf

fhf

hf

kkf

hhf

fff rrrrrrrzzzzzzz

24422442 2

22

11Rkhkh

kk

ff

kk

ff

fhhf

kkh

fhf


224242442 2

2

2

22

22R 1442442 2 hkh

fhkf

hk

hkh

fkfk rrrrrrzzzzzz

44R 2

2

2

2

2

2

24422 kk

hh

ff

kk

hh

ff zzzzzzzzz

14R 22 22442 kkk

hkkh

hffh

fff

kk

ff zrrzrzrzrzrz




khf ,,We have

]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.

IJE

0][ )(),(

nnn zrFzrF

n

kkknn GFFG

0][][][)(

expansionreduction

rule

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

]0[14E

,2/ ]0[]1[ ffu ,2/ ]0[]1[ hhv 2]1[ 2/ rkw Let u v w]/)(22[2 rwvwu rr ]0[]0[ /)( ffvu r

off brane

0]0[44

]0[14 EE

0

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[14E 0



khf ,,We have

bulk Einstein eq.

IJE

0][ )(),(

nnn zrFzrF

n

kkknn GFFG

0][][][)(

expansionreduction

rule

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

off brane

0]0[44

]0[14 EE



bulk Einstein eq.

IJE

0][ )(),(

nnn zrFzrF

n

kkknn GFFG

0][][][)(

expansionreduction

rule

We have ,, ]0[]0[ hf ]1[]1[]1[ ,, khfkhf ,, obeyif

0]0[44 E 0

222

]0[

44221100

Rk

Rh

Rf

R

00Rkhkf

hhf

hff

hf

kkf

hhf

fff rrrrrrrzzzzzzz

24 422442 2

22

11Rkhkh

kk

ff

kk

ff

fhhf

kkh

fhf


224242442 2

2

2

22

22R 1442442 2 hkh

fhkf

hk

hkh

fkfk rrrrrrzzzzzz

44R 2

2

2

2

2

2

24422 kk

hh

ff

kk

hh

ff zzzzzzzzz

22 22442 kkk

hkkh

hffh

fff

kk

ff zrrzrzrzrzrz


4f 8 2 8f 4f 4f 8 2 8f 4f

4h 8 2 8 h 4h 8 2 h 2 h4 h8 h4 4 2

k k k k k k

4 4 2 8 8 4

___2f

___2h

___2k

___2

0

__k

]0[

off brane

0]0[44

]0[14 EE



bulk Einstein eq.

IJE

0][ )(),(

nnn zrFzrF

n

kkknn GFFG

0][][][)(

expansionreduction

rule


00Rkhkf

hhf

hff

hf

kkf

hhf

fff rrrrrrrzzzzzzz

24 422442 2

22

11Rkhhk

kk

ff

kk

ff

fhhf

kkh

fhf


224242442 2

2

2

22

22R 1442442 2 hkh

fhkf

hk

hkh

fkfk rrrrrrzzzzzz

44R 2

2

2

2

2

2

24422 kk

hh

ff

kk

hh

ff zzzzzzzzz

22 22442 kkk

hkkh

hffh

fff

kk

ff zrrzrzrzrzrz

8f 4f 4f 8 2 8f 4f

8 h 4h 8 2 h 2 h4 h8 h4 4 2

k k k k k

4 0

__k

hfhf zz

kfkf zz

khkh zz

4 2 2

2 4 4 2

2

fhfrr

hffr

2

2 2fh

hf rr

fkhkf rr

khkrr

2

khhk rr

0222

]0[

44221100

Rk

Rh

Rf

R

0

]0[

k1

off brane

0]0[44

]0[14 EE

0]0[44 E



bulk Einstein eq.

IJE

0][ )(),(

nnn zrFzrF

n

kkknn GFFG

0][][][)(

expansionreduction

rule


00Rkhkf

hhf

hff

hf

kkf

hhf

fff rrrrrrrzzzzzzz

24 422442 2

22

11Rkhhk

kk

ff

kk

ff

fhhf

kkh

fhf


224242442 2

2

2

22

22R

44R 2

2

2

2

2

2

24422 kk

hh

ff

kk

hh

ff zzzzzzzzz

22 22442 kkk

hkkh

hffh

fff

kk

ff zrrzrzrzrzrz

4f 8 2 8f 4f

2 h h8 h4 4 2

4 0

2 4 4 2

2

0222

]0[

44221100

Rk

Rh

Rf

R

0

]0[

hfhf zz

kfkf zz

khkh zz

4 2 2k1

2 4 hfhf rr

2 2 hkhk rr

hff

hff rrr

2

2

4

2

hkfkf rr

2

hk

krr

hkkr

2

2

4

[1] [1] [1][1] [1] [1] [1]

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

[0] [0]

[0]

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

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

[0] [0]

[0] [0] [0]

[0]

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

[0]

off brane

0]0[44

]0[14 EE

0]0[44 E



bulk Einstein eq.

IJE

0][ )(),(

nnn zrFzrF

n

kkknn GFFG

0][][][)(

expansionreduction

rule


2

2

2kkz4

0

2]0[

]0[2

]0[

]0[

]0[

]0[

]0[

122

1rrf

fff

ff

hrr

r

r

4

2]1[

2]0[

]1[]1[

2]0[

]1[]1[

]0[]0[

]1[]1[

2 42241

rk

rhkh

rfkf

hfhf

r

0222

]0[

44221100

Rk

Rh

Rf

R

hfhf zz

kfkf zz

khkh zz

4 2 2k1

2 4 hfhf rr

2 2 hkhk rr

hff

hff rrr

2

2

4

2

hkfkf rr

2

hk

krr

hkkr

2

2

4

[1] [1] [1][1] [1] [1] [1]

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

[0] [0]

[0]

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

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

[0] [0]

[0] [0] [0]

[0]

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

[0]

rff

hr

r

14

1]0[

]0[

]0[0]0[44 E

off brane

0]0[44

]0[14 EE

0]0[44 E

rff

hr

r

14

1]0[

]0[

]0[

2]0[

]0[2

]0[

]0[

]0[

]0[

]0[

122

1rrf

fff

ff

hrr

r

r

0]0[14 E




bulk Einstein eq.

IJE

0][ )(),(

nnn zrFzrF

n

kkknn GFFG

0][][][)(

expansionreduction

rule


4

2]1[

2]0[

]1[]1[

2]0[

]1[]1[

]0[]0[

]1[]1[

2 42241

rk

rhkh

rfkf

hfhf

r

2

2

2kkz4

0hfhf zz

kfkf zz

khkh zz

k1

2 4 hfhf rr

2 2 hkhk rr

hff

hff rrr

2

2

4

2

hkfkf rr

2

hk

krr

hkkr

2

2

4

[1] [1] [1][1] [1] [1] [1]

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

[0] [0]

[0]

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

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

[0] [0]

[0] [0] [0]

[0]

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

[0]

u v u w2 v w2 w 2

off brane

0]0[44

]0[14 EE

0]0[44 E

So far, considered the solution

rff

hr

r

14

1]0[

]0[

]0[

2]0[

]0[2

]0[

]0[

]0[

]0[

]0[

122

1rrf

fff

ff

hrr

r

r




bulk Einstein eq.

IJE

0][ )(),(

nnn zrFzrF

n

kkknn GFFG

0][][][)(

expansionreduction

rule


4

2]1[

2]0[

]1[]1[

2]0[

]1[]1[

]0[]0[

]1[]1[

2 42241

rk

rhkh

rfkf

hfhf

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

Two differential equations ,, ]0[]0[ hf wvu ,,for five functions

0]0[44

]0[14 EE

0]0[44 E

0]0[14 E

Next, we turn to the solution inside the brane, and their connections.

off brane

on brane

off the brane only.

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



bulk Einstein eq.

IJE

0][ )(),(

nnn zrFzrF

n

kkknn GFFG

0][][][)(

expansionreduction

rule

We have ,, ]0[]0[ hf ]1[]1[]1[ ,, khfkhf ,, obeyif,2/ ]0[]1[ ffu ,2/ ]0[]1[ hhv 2]1[ 2/ rkw Let

0]0[44

]0[14 EE

rff

hr

r

14

1]0[

]0[

]0[

2]0[

]0[2

]0[

]0[

]0[

]0[

]0[

122

1rrf

fff

ff

hrr

r

r

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

1 / 2r uv uw2 vw2 2w

0]0[44 E

0]0[14 E

use again laterTwo differential equations ,, ]0[]0[ hf wvu ,,for five functions

Next, we turn to the solution inside the brane, and their connection.

So far, considered the solution off the brane only.

on brane

on brane

IJE

khf ,,We have if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf



On the brane,

0 IJTNambu-Goto eq.

0)~~~( ; IYTg

z zz

zzz khf ,,

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

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

matter is distributed within |z|<d , 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 u v w

dd

z z z k

bulk Einstein eq. on the brane 3/~ wvu

bulk Einstein eq.

zzzzzz khf ,,

z

u u

uu u

khf ,,

d d

ratio ratio

Israel Junction condition

≡

define for short

ratio

obey

on brane

0]0[44

]0[14 EE

(3+1)

0][ )(),(

nnn zrFzrF

n

kkknn GFFG

0][][][)(

expansionreduction

rule

IJE




0 IJTNambu-Goto eq.

0)~~~( ; IYTg

,2/ ]0[]1[ ffu ,2/ ]0[]1[ hhv 2]1[ 2/ rkw Let u v wz z z k

bulk Einstein eq.

obey

on brane

0]0[44

]0[14 EE

(3+1)

Nambu-Goto eq. 02 wvu

,/ ffu z ,/hhv z ,/kkw z,2/)( uuu wvwv ,,,similarly for uuu

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

bulk Einstein eq. on the brane 3/~ wvu

Israel Junction condition

≡

define for short

3/~ wvu ≡

,| d zuu

0]0[44 E

0]0[14 E

0]0[44 E

0]0[14 E


2]0[

]0[2

]0[

]0[

]0[

]0[

]0[]0[

]0[

]0[

122

114

1rrf

fff

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[

122

114

1rrf

fff

ff

hrff

hrr

r

rr

r

±d

0| 44 zE±d

± ± ± ± ± ±

± ± ± ± ± ± ±connected at the boundary

holds for the collective modes

IJE




0 IJTNambu-Goto eq.

0)~~~( ; IYTg


bulk Einstein eq.

obey

on brane

0]0[44

]0[14 EE


2]0[

]0[2

]0[

]0[

]0[

]0[

]0[]0[

]0[

]0[

122

114

1rrf

fff

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[

122

114

1rrf

fff

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 ≡

0)2( wvutrivially satisfied

trivially satisfied

difference of ±

u v 2w 0 ]/)(22[2/)( ]0[]0[ rwvwuffvu rrr ± ± ± ± ± ±

3 equations5 equations 2 are trivial3 equations

0]0[44 E

0]0[14 E

IJE




0 IJTNambu-Goto eq.

0)~~~( ; IYTg


bulk Einstein eq.

obey

on brane

0]0[44

]0[14 EE



2]0[

]0[2

]0[

]0[

]0[

]0[

]0[]0[

]0[

]0[

122

114

1rrf

fff

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[

122

114

1rrf

fff

ff

hrff

hrr

r

rr

r

2 2 )(22/1 wwvwuvur

2]0[

]0[2

]0[

]0[

]0[

]0[

]0[]0[

]0[

]0[

122

114

1rrf

fff

ff

hrff

hrr

r

rr

r± ± ± ± ± ± ±

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

---- --- 6/~2

average of ±

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

2/1 r 2 )(22 wwvwuvu

0]0[14 E

0]0[44 E

3 equations

0]0[44 E

0]0[14 E

IJE




0 IJTNambu-Goto eq.

0)~~~( ; IYTg


bulk Einstein eq.

obey

on brane

0]0[44

]0[14 EE

2 )(22 wwvwuvu



2]0[

]0[2

]0[

]0[

]0[

]0[

]0[]0[

]0[

]0[

122

114

1rrf

fff

ff

hrff

hrr

r

rr

r

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

2/)( vuw

2]0[

]0[2

]0[

]0[

]0[

]0[

]0[]0[

]0[

]0[

122

114

1rrf

fff

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

substitutesubstitute

vu , : arbitrary,

3 equations2 equations

use one equation

0]0[44 E

0]0[14 E

0]0[14 E

0]0[44 E

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

2]0[

]0[2

]0[

]0[

]0[

]0[

]0[]0[

]0[

]0[

122

114

1rrf

fff

ff

hrff

hrr

r

rr

r2/1 r 4/)323( 22 vvuu 6/~2

2]0[

]0[

]0[]0[

]0[ 1 114 rrf

fhrf

f rr

rh

]0[

12

]0[

]0[

]0[

]0[

22

ff

ff r

r

r

6/~4/)323(/1 2222 vvuur

]/)3([2/ ]0[]0[ rvuvff rr )( vu

2 equations

equations differential2

0]0[14 E

0]0[44 E

, ]0[]0[ ffr

)/14//()/1/4/2/( 22 rrrr )/14//(]6/~4/)323(/1[ 22222 rvvuur

),/2/(])/6(2[ rvrvu r

where

, ]0[

r

dref

,1

]0[

r

PdrPdrdrQeCeh rr

solution

linear differential equations

2]0[

]0[

]0[]0[

]0[ 1 114 rrf

fhrf

f rr

rh

]0[

12

]0[

]0[

]0[

]0[

22

ff

ff r

r

r

6/~4/)323(/1 2222 vvuur

]/)3([2/ ]0[]0[ rvuvff rr )( vu

]0[]0[ / ffrLet

( /41/r )

( /41/r )] /[/

P

Q

rh

]0[

1P ]0[

1h

Q

PQ

solvable!

with arbitrary & v

equations differential2

]0[f ]0[h P Q

2]0[

]0[

]0[]0[

]0[ 1 114 rrf

fhrf

f rr

rh

]0[

12

]0[

]0[

]0[

]0[

22

ff

ff r

r

r

6/~4/)323(/1 2222 vvuur

]/)3([2/ ]0[]0[ rvuvff rr )( vu ]0[]0[ / ffrLet

][][][ ,, nnn khf

and are written with and . v]0[f ]0[hare written with and v

khf ,, are written with and v

, ]0[]0[ ffr

)/14//()/1/4/2/( 22 rrrr )/14//(]6/~4/)323(/1[ 22222 rvvuur

),/2/(])/6(2[ rvrvu r

where

, ]0[

r

dref

,1

]0[

r

PdrPdrdrQeCeh rr

solution

linear differential equationrh

]0[

1P ]0[

1h

Q

PQ

solvable!

with arbitrary & v

222222 )sin( dzddkdrhdtfdXdXg JIIJ jqq )0( z

,0 tY ,1 rY ,2 qY ,3 jY 04 Y

Under the Schwarzschild ansatz,

where

Theorem

,0

][

nnn zff ,

0][

nnn zhh

0][

nnn zkk

with the coefficients determined by and below.① ②

all the solutions of the braneworld dynamics

and

(Einstein & Nambu-Goto eqs. in 4+1dim.)are given by


,0 tY ,1 rY ,2 qY ,3 jY 04 Y


where

Theorem

,0

][

nnn zff ,

0][

nnn zhh

0][

nnn zkk



and


, ]0[]0[ ffr

)/14//()/1/4/2/( 22 rrrr )/14//(]6/~4/)323(/1[ 22222 rvvuur

),/2/(])/6(2[ rvrvu r

where

, ]0[

r

dref

,1

]0[

r

PdrPdrdrQeCeh rr

solution

linear differential equationrh

]0[

1P ]0[

1h

Q

PQ

solvable!

with arbitrary & v

Let and be arbitrary functions of r. v①

]0[f ,

r

dre

]0[h ,11

r

PdrPdrdrQee rr ]0[k ,2r

)/14//()/1/4/2/( 22 rrrP r

)/14//(]6/~4/)323(/1[ 22222 rvvuurQ

),/2/(])/6(2[ rvrvu r

where

Then, we define

For , are recursively defined by 2n

,]0[]0[ ff ,]0[]0[ hh ]0[]0[ kk

,)2( ]0[kw ,)2( ]0[hv

,3/~

]1[f ,)2( ]0[fu ]1[h ]1[k

where .2/)( vuw

②

][][][ ,, nnn khf

We define and

recursive definition )2( n

]2[

2

2 2 ][

34

2222)1(1

n

rrrrrrrzzzzzn fhkkf

hhf

hff

hf

kkf

hhf

ff

nnf

]2[

2

2

2

2 2 ][

34

222

22)1(1

n

rrrrrrrrrrzzzzzn hhkkh

hfhf

kk

ff

kk

ff

kkh

fhf

hh

nnh

]2[

2

][

342

2222)1(1

nrrrrrrzzzzn k

hkh

hfkf

hk

hkh

fkf

nnk

±±

±

± ±

±

± ±

±

±

±

± ± ±

± ± ± ± ±

± ± ±

±±

±

± ±

±

± ± ± ± ± ± ± ± ± ± ±

± ± ± ± ± ± ± ± ±

±± ±

±

± ±

±

±

±

±

±

±

±

±

±

±±

]2[

2

2 2 ][

34

2222)1(1

n

rrrrrrrzzzzzn fhkkf

hhf

hff

hf

kkf

hhf

ff

nnf

]2[

2

2

2

2 2 ][

34

222

22)1(1

n


hfhf

kk

ff

kk

ff

kkh

fhf

hh

nnh

]2[

2

][

342

2222)1(1

nrrrrrrzzzzn k

hkh

hfkf

hk

hkh

fkf

nnk

For , are recursively defined by

n

kkknn GFFG

0][][][)(

2n

]1[]1[]1[]0[]0[]0[ ,,,,, khfkhf][][][ ,, nnn khf are finally written with

where [n] obeys the reduction rule

,]0[]0[ ff ,]0[]0[ hh ]0[]0[ kk

,)2( ]0[kw ,)2( ]0[hv

,3/~

]1[f ,)2( ]0[fu ]1[h ]1[k

where .2/)( vuw

②

and, accordingly, they are written with and .

][][][ ,, nnn khf

v

We define and

]2[

2

2 2 ][

34

2222)1(1

n

rrrrrrrzzzzzn fhkkf

hhf

hff

hf

kkf

hhf

ff

nnf

]2[

2

2

2

2 2 ][

34

222

22)1(1

n


hfhf

kk

ff

kk

ff

kkh

fhf

hh

nnh

]2[

2

][

342

2222)1(1

nrrrrrrzzzzn k

hkh

hfkf

hk

hkh

fkf

nnk

±±

±

± ±

±

± ±

±

±

±

± ± ±

± ± ± ± ±

± ± ±

±±

±

± ±

±

± ± ± ± ± ± ± ± ± ± ±

± ± ± ± ± ± ± ± ±

±± ±

±

± ±

±

±

±

±

±

±

±

±

±

±±


,0 tY ,1 rY ,2 qY ,3 jY 04 Y


where

Theorem

,0

][

nnn zff ,

0][

nnn zhh

0][

nnn zkk



and


]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

Einsteinbjj

32

31 21

Einsteinab

light deflection by star gravity

planetary perihelion precession

observation

lightstar

j0r

Einsteinjj Einsteinjj

Discussions

Here, they are arbitrary.

arbitrary

arbitrary

21 1

Einsteinbjj

32

31 21

Einsteinab



observation

lightstar

j0r


Discussions

21 1

Einsteinbjj

32

31 21

Einsteinab



observation

lightstar

j0r


arbitrary

arbitrary

arbitrary

arbitrary

Discussions

21 1

Einsteinbjj

32

31 21

Einsteinab



observation

starj0r


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

SummaryThe general solution of the fundamental equations of braneworld

Off the brane, it is expressed in power series of the normal coordinate on each side.

The coefficients: recursively defined with on-brane functions,which obey solvable differential equations

The arbitrariness may affect the predictive powers on the Newtonian and the post-Newtonian evidences. We need other physical prescriptions to recover them.Brane induced gravity may by-pass this problem.

(×^

×)

(^V^)

(^V^)

bulk Einstein eq. Nambu-Goto eq.

as far as we appropriately choose 2 arbitrary functions.

0221100 RRR

0]0[44

]0[14 EE

, v

with Schwarzschild ansatz is derived.

Thank you for listening. (^O^)

Thank you

(^O^)

general solution of braneworld with the schwarzschild ansatz

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