general solution of braneworld dynamics under the schwarzschild anzats k. akama, t. hattori, and h....

General Solution of Braneworld Dynamics　under the Schwarzschild Anzats

K. Akama, T. Hattori, and H. Mukaida

Ref.(partial) K. Akama, T. Hattori, and H. Mukaida, arXiv:1008.0066 [hep-th]

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 at the lowest order.

It should be modified by the higher order considerations.

Einstein gravity successfully explaines at moderate distances, and ②the post Newtonian tests. (^_^)It is derived via the Schwarzschild solution under the anzatse static, spherical, asymptotically flat, empty except for the core

Can the braneworld theory reproduce the successes ① and ②? "Braneworld"

It is not trivial because we have no Einstein eq. on the brane.The brane metric cannot be dynamical variable of the brane,becaus it cannot fully specify the state of the brane.

The dynamical variable should be the brane-position variable, and brane metric is induced variable from them.In order to clarify them, we seek for the general spherical solution of the Einstein equation × braneworld dynamics.

Introduction

( ,_ ,)?

①the Newtonian gravity

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

Here, we find the general solution at the lowest order.

Many people considered the picture that we live in 3+1 brane

General Spherical solutions on branes are considered by e.g.

Garriga,Tanaka (00), Visser,Wiltshire('03), Casadio,Mazzacurati('03), Bronnikov,Melnikov,Dehnen('03), Kanti('04), Creek,Gregory,Kanti,Mistry('06)

in higher dimensions from various points of view, e.g.

Fronsdal('59), Josesh('62), Regge,Teitelboim('75), K.A.('82), Rubakov,Shaposhnikov('83), Maia('84), Visser('85), Pavsic('85), Gibbons,Wiltschire ('87), Antoniadis('91), Polchinski('95), Horava,Witten('96), Dvali,Shifman('96),Arkani-Hamed,Dimopolos,Dvali('98), Randall,Sundrum('99), Dvali,Gabadadze,Porrati('00), Shiromizu,Maeda,Sasaki(00),

Braneworld

bulk

1 )))((()( XdXgRXg NKIJ

K

Braneworld Dynamics

matterS

dynamicalvariables brane position

)( KIJ Xg

)( xY I

bulk metric

brane

4))((~~xdxYg K

IJIJIJIJ TgRgR )2/(1

0}~~~

{ ; IYTg

eq. of motion

Action

,3,2X

0x

1X

0X

x

IJg

)( xY I

bulk scalar curvature

gg ~det~

IJgg det

bulk Einstein eq.

Nambu-Goto eq.

label

constant

brane en.mom.tensor

g~

label

brane coord.KX xbulk coord.

brane metriccannot be a dynamical variable

constants

gmn(Y)=YI,mYJ

,ngIJ(Y)

matter action

~

S d /dgIJ d /dYI

~ indicatesbrane quantity

bulk en.mom.tensor

bulk

1 )))((()( XdXgRXg NKIJ

K

Braneworld Dynamics

matterS

dynamicalvariables brane position

)( KIJ Xg

)( xY I

bulk metric

brane

4))((~~xdxYg K


0}~~~

{ ; IYTg

eq. of motion

Action

,3,2X

0x

1X

0X

x

IJg

)( xY I

bulk Einstein eq.

Nambu-Goto eq.

g~brane coord.KX xbulk coord.

brane metriccannot be a dynamical variable

gmn(Y)=YI,mYJ

,ngIJ(Y)~

S



0}~~~

{ ; IYTg

eq. of motion

bulk Einstein eq.

Nambu-Goto eq.


0}~~~

{ ; IYTg

eq. of motion

bulk Einstein eq.

Nambu-Goto eq.

general solution

static, spherical,

here consider the case 0

under Schwarzschild anzats

asymptotically flat on the brane, empty except for the core

× normal coordinate x

brane polar coordinate

coordinate system

t,r,q,f


0}~~~

{ ; IYTg

eq. of motion

bulk Einstein eq.

Nambu-Goto eq.

general solution

static, spherical,






coordinate system

t,r,q,f

0; IY g~0 Nambu-Gotoeq.bulk Einstein eq. 2/RgR IJIJ

general solution

static, spherical,






coordinate system

t,r,q,f


static, spherical, Schwarzschild anz. asymptotically flat, empty

× normal coordinate xbrane polar coordinate t,r,q,f




vdrdudddkhdrfdtds 2222222 )sin(

vukhf ,,,, : functions of r & xonly

general line element with

000 R

011 R

022 R

044 R

014 R

from the bulk Einstein eq.


)sderivative-their &,( rqqFf 000 R

011 R

022 R

044 R

014 R







rrrrr

rrrrrr

rrrrr

rr

fhvhu

uvvhvhuuvhu

khuk

hvhuhu

fhuf

huf

hvhu

huvf

kh

kvf

h

vf

fvhu

uvvhu

vuhvhu

uvhkhvk

fhvf

k

kf

hvhu

hfvhu

f

fv

vhu

ufvfu

vhu

vfhf

)()(2

)2(

)(2

2)(2

2

)(2)(2

)(2

)2(

2)()(2

22

2

2

2

2

2

222

2

22

22

Ff

Here and hereafter, subscripts x and r of functions mean partial differentiations.

),,,,( vukhfq with



011 R

022 R

044 R

014 R






),,,,( vukhfq


with

Here and hereafter, subscripts x and r of functions means partial differentiations.

),,,,( vukhfq with

partial differentiations.

)sderivative-their &,( rqqHh

rrrrrrrrrr

rrrrrrrrr

rrrr

rrrrr

uuvhu

vvhuuhkk

ff

hvvuh

kk

kk

ff

ff

hvhu

vhvhu

uhkk

hff

kkh

hvvhf

fhhvvh

vhu

uh

k

kh

f

hf

vvvhu

vvuhvhu

vhu

hvvhhh

)(2

22

2

2

242

)(2

)(22

2

2

2

)(22

22

)(2

2

2

2

2

2

22

2

2

2

22

Hh

)sderivative-their &,( rqqKk

hvhu

vhuhvuvk

vhuhukvhu

vhuhkhu

fhkuf

huk

khv

kvhuh

huvhvuuvhfh

vfh

vhuhuvk

ffh

vk

vhuh

khvuh

f

kfv

vhu

ukvku

vhu

vkhk

rr

rrrrrrrr

rrrrrrr

rr

)(2

)(2

2

)(2

)2(

)(22

2

)(2

2

2)(22

)(2

)2(

2)()(2

2

2

2

2

2

2

22

2

2

22

Kk

partial differentiations.

),,,,( vukhfq ),,,,( vukhfq



011 R

022 R

044 R

014 R






),,,,( vukhfq


with



0)sderivative-their &,( rqqA

0)sderivative-their &,( rqqB

also depend on hgf ,, , which we eliminate.

),,,,( vukhfq

024)(

)2(

242

24)(2

2

2

2)(4

2

4

22

2

2

2

2

22

2

2

22

2

kk

ff

vhuvvhuuhv

kvk

kvk

fvf

fvf

hk

kf

fk

hkk

vhuhvvuhhv

khk

f

hff

vhufhvvuhhv

fhf

rrrrrrrrrrr

rrrrrrr

rrrrr

A

022244

)2(

4224

222224

4224

2

22

2

22

2

2

2

hvv

fhvvf

fuf

huh

fhhuf

hvhu

ff

ffh

hvhu

h

vh

ff

vf

vfvh

vhf

fv

fv

fhvh

h

f

hfh

f

hf

rrrrrrrrrrr

rrrrrrr

rrrrrrr

B



011 R

022 R

044 R

014 R






),,,,( vukhfq


with








011 R

022 R

044 R

014 R



),,,,( vukhfq with







Ff Hh Kk 0A 0Bbulk Einstein eq.


0

][

n

nnQQ

0

][

n

nnqq

),,,,( BAKHFQ

),,,,( vukhfq expand in x

1~ u 0~ v2~rk

qq ~]0[

QQ~]0[

normal coordinate xpolar cordinate






0

][

n

nnQQ

0

][

n

nnqq

),,,,( BAKHFQ


1~ u 0~ v2~rk

qq ~]0[

QQ~]0[






Equating the x n terms in Ff

rrrrr

rrrrrr

rrrrr

rr

fhvhu

uvvhvhuuvhu

khuk

hvhuhu

fhuf

huf

hvhu

huvf

kh

kvf

h

vf

k

kf

f

f

fvhu

uvvhu

vuhvhu

uvhkhvk

fhvf

hvhu

hfvhuv

vhu

ufvfu

vhu

vfhf

)()(2

)2(

)(2

2)(2

2

2

)(2)(2

)(2

)2(

)()(2

22

2

2

2

2

2

2

222

2

2

22

Ff (n+1)(n+2)[n+2] [n]

[n]

]2[nf )2)(1( nnwe have

: 　 written in terms of & the lower.

]1[ nq

,

,

),,,,( vukhfq

][nF

][nF

0

][

n

nnQQ

0

][

n

nnqq

),,,,( BAKHFQ


1~ u 0~ v2~rk

qq ~]0[

QQ~]0[






Hh ,Equating the x n terms in

rrrrrrrrrr

rrrrrrrrr

rrrrr

rrrr

uuvhu

vvhuuhkk

ff

hvvuh

kk

kk

ff

ff

hvhu

vhvhu

uhkk

hff

kkh

hvvhf

fhhvvh

vhu

uh

k

kh

f

hfv

vvhu

vvuhvhu

vhu

hvvhhh

)(2

22

2

2

242

)(2

)(22

2

2

2

)(222

2

)(2

2

2

2

2

2

22

2

2

2

22

Hh (n+1)(n+2) [n+2] [n]

[n]

we have


]1[ nq

Ff ,

,][][ , nn HF ][nF

),,,,( vukhfq

]2[nf )2)(1( nn

][nF ]2[nh )2)(1( nn ,][nH

0

][

n

nnQQ

0

][

n

nnqq

),,,,( BAKHFQ


1~ u 0~ v2~rk

qq ~]0[

QQ~]0[






KK ,Equating the x n terms in

hvhu

vhuhvuvk

vhuhukvhu

vhuhkhu

fhkuf

huk

khv

kvhuh

huvhvuuvhfh

vf

hvhuh

uvkf

fhvk

vhuh

khvuh

f

kf

vvhu

ukvku

vhu

vkhk

rrrrrr

rrrrr

rrrr

rr

rr

)(2

)(2

2

)(2

)2(

)(2

2

2

)(2

2

2

)(22)(2

)2(

2

)()(2

2

22

2

2

2

2

22

2

22

Kk (n+1)(n+2)[n+2] [n]

[n]

we have


]1[ nq

Ff ,

][][][ ,, nnn KHF

Hh ,

),,,,( vukhfq

][][ , nn HF

, ]2[nf )2)(1( nn

][nF ]2[nh )2)(1( nn ,][nH ]2[nk )2)(1( nn ,

][nK


we have


]1[ nq

Ff ,

][][][ ,, nnn KHF

Hh ,

,)2)(1( nn ]2[nf

][nF ]2[nh )2)(1( nn ,][nH ]2[nk )2)(1( nn ,

][nK

0

][

n

nnQQ

0

][

n

nnqq

),,,,( BAKHFQ


1~ u 0~ v2~rk

qq ~]0[

QQ~]0[






0A

024)(

)2(

242

24)(2

2

2

2)(4

2

4

22

2

2

2

2

22

2

2

22

2

kk

ff

vhuvvhuuhv

kvk

kvk

fvf

fvf

hk

kf

fk

hkk

vhuhvvuhhv

khk

f

hff

vhufhvvuhhv

fhf

rrrrrrrrrrr

rrrrrrr

rrrrr

A[n]

[n]

include q [n+1]

use


we have


]1[ nq

Ff ,

][][][ ,, nnn KHF

Hh ,

,)2)(1( nn ]2[nf

][nF ]2[nh )2)(1( nn ,][nH ]2[nk )2)(1( nn ,

][nK

0

][

n

nnQQ

0

][

n

nnqq

),,,,( BAKHFQ


1~ u 0~ v2~rk

qq ~]0[

QQ~]0[






: written with

]1[]1[][][][ ,,,, nnnnn vukhf & the lower. ][nAQ

0A

024)(

)2(

242

24)(2

2

2

2)(4

2

4

22

2

2

2

2

22

2

2

22

2

kk

ff

vhuvvhuuhv

kvk

kvk

fvf

fvf

hk

kf

fk

hkk

vhuhvvuhhv

khk

f

hff

vhufhvvuhhv

fhf

rrrrrrrrrrr

rrrrrrr

rrrrr

A[n]

[n]

include q [n+1]

use

]0,[]1,1[

2/1

]1,1[

2/12/1

]1,1[

2/1

][

]1[]1[]1[

2

]1[]1[

2

]1[

A][][

2~~

~

~2

~

~~2

~~1

~~2

~

~~4

~

~~

~

~2

~4

~ ,

nn

rr

r

n

r

nn

A

rrrrrrnA

nA

Ahn

H

k

k

f

f

k

K

knf

F

fnQ

hk

hk

hf

hf

k

k

k

kk

f

f

f

ffPQuP

][nA

0][][][ nA

nA

n QuPA


we have


]1[ nq

Ff ,

][][][ ,, nnn KHF

Hh ,

,)2)(1( nn ]2[nf

][nF ]2[nh )2)(1( nn ,][nH ]2[nk )2)(1( nn ,

][nK

0

][

n

nnQQ

0

][

n

nnqq

),,,,( BAKHFQ


1~ u 0~ v2~rk

qq ~]0[

QQ~]0[






: written with


0A 0B,

0 22244

)2(

4224

222224

4224

2

22

2

22

2

2

2

hvv

fhvvf

fuf

huh

fhhuf

hvhu

ff

ffh

hvhu

h

vh

ff

vf

vfvh

vhf

fv

fv

fhvh

h

f

hfh

f

hf

rrrrrrrrrrr

rrrrrrr

rrrrrrr

B[n]

[n]

include q [n+1]

use

][][][ nA

nA

n QuPA 0



]1[ nq

Ff ,

][][][ ,, nnn KHF

Hh ,

,)2)(1( nn ]2[nf

][nF ]2[nh )2)(1( nn ,][nH ]2[nk )2)(1( nn ,

][nKwe have

0

][

n

nnQQ

0

][

n

nnqq

),,,,( BAKHFQ


1~ u 0~ v2~rk

qq ~]0[

QQ~]0[






: written with


0A 0B,

0 22244

)2(

4224

222224

4224

2

22

2

22

2

2

2

hvv

fhvvf

fuf

huh

fhhuf

hvhu

ff

ffh

hvhu

h

vh

ff

vf

vfvh

vhf

fv

fv

fhvh

h

f

hfh

f

hf

rrrrrrrrrrr

rrrrrrr

rrrrrrr

[n]

[n]

B

2

~

4

~

~2

~

2~

4 , )4(

4

2]1[

2

]1[]1[

2

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

B][][ R

rkh

rf

kfhrkh

f

hfPQuP n

Bn

B

0][][][ nB

nB

n QuPB

222

2

)4(

1~

~

~

~

~

~~4

~~

~4

~

~2

~

~1~

rrh

rh

h

rf

f

hf

hf

f

f

f

f

hR rrrrrrr

]0,[]1,1[4

]1[

2

]1[

2

]1[

]1,1[2

]1[]1[]1,1[

2

]1[]1[][

1

4

~2

~2

~

2

1

2~

4

1~

2

~

~4

nn

nnnB

BKnr

kh

rf

fhr

h

Hnr

k

f

fF

nrf

kh

f

hQ

][][ , nB

nA QQ

,

][nB

][][][ nA

nA

n QuPA 0






we have


]1[ nq

Ff ,

][][][ ,, nnn KHF

Hh ,,,

)2)(1( nn ]2[nf][nF ]2[nh )2)(1( nn ,

][nH ]2[nk )2)(1( nn

][nK

: written with

]1[]1[][][][ ,,,, nnnnn vukhf & the lower.

0A 0B,][][ , n

Bn

A QQ

,)2)(1( nn ]2[nf

][nF ]2[nh )2)(1( nn ,][nH ]2[nk )2)(1( nn

][nK

,0][][][ nA

nA

n QuPA 0][][][ nB

nA

n QuPB

0][][][ nB

nB

n QuPB ,][][][ n

An

An QuPA 0

,0AP 0BP





,)2)(1( nn ]2[nf

][nF ]2[nh )2)(1( nn ,][nH ]2[nk )2)(1( nn

][nK

3 equations for 5 functions

lowest order ,2/~]2[ Ff ,2/

~]2[ Hh ,2/~]2[ Kk

determine .,, ]2[]2[]2[ khf

Nambu-Goto eq. 0~

/2~

/~

/ ]1[]1[]1[ kkhhff

Here we solve

,0][][][ nA

nA

n QuPA 0][][][ nB

nB

n QuPB

)0( n

.,,,~

,~ ]1[]1[]1[ khfhf

Define ,~

2/]1[ ffa ,~

2/]1[ hhb kkc~

2/]1[0/)(2

~2/

~)(2 rcbffbacaP rrrA

02/~

22 )4(2 RcacbcabPB

02 cba

Then



Here we solve .,,,~

,~ ]1[]1[]1[ khfhf

Define ,~

2/]1[ ffa ,~

2/]1[ hhb kkc~

2/]1[0/)(2

~2/


02/~

22 )4(2 RcacbcabPB

02 cba

Then


Nambu-Goto eq. 0~

/2~

/~

/ ]1[]1[]1[ kkhhff


ffa~

2/]1[

hhb~

2/]1[kkc~

2/]1[

0/)(2~

2/~

)(2 rcbffbacaP rrrA

02/~

22 )4(2 RcacbcabPB

02 cba 3 eqs. for 5 functions


Here we solve .,,,~

,~ ]1[]1[]1[ khfhf

Define ,~

2/]1[ ffa ,~

2/]1[ hhb kkc~

2/]1[0/)(2

~2/


02/~

22 )4(2 RcacbcabPB

02 cba

Then

Nambu-Goto eq. 0~

/2~

/~

/ ]1[]1[]1[ kkhhff

1) arbitrary functions: two of a, b, c with a+b+2c =0

,)(

)222(2

]0[dr

barrcracb rr

ef

0,0 BA PP are 1st-rank linear diff. eqs. for hf~

/1&~

solutiondrQeeh PdrPdr ]0[

)2)(2)((/)]2)((2

124284)43(4

)(4)32(4412862[

2

2222222

222

rrrrrr

rrrrrrrrr

rr

rcracabarcabar

carcbrbarcrarrccba

rbcaracbaacbccbaP

)2)(2(/])22(1)[(2 22rr rcracarrcbcacabbaQ

solvable


ffa~

2/]1[

hhb~

2/]1[kkc~

2/]1[

0/)(2~

2/~

)(2 rcbffbacaP rrrA

02/~

22 )4(2 RcacbcabPB


2) arbitrary functions:

0&0 BA PP are 2nd order algebraic eqs. for

hf~

&~

solving without dynamics: choose as arbitrary.cab 2'

.,, cba

solving with dynamics:solvable

brf

fa

rff

b rrr

3

2

1

2 ]0[

]0[

]0[

]0[

)4(22 ~

2323 Rbaba

are 1st-rank linear diff. eqs. for ba , with the elliptic constraint

a unique solution exists 0~

)4( Ras far as


ffa~

2/]1[

hhb~

2/]1[kkc~

2/]1[

0/)(2~

2/~

)(2 rcbffbacaP rrrA

02/~

22 )4(2 RcacbcabPB


The general solution is given by that of

It should be modified by the higher order consideration in x.





,)2)(1( nn ]2[nf

][nF ]2[nh )2)(1( nn ,][nH ]2[nk )2)(1( nn

][nK

,0][][][ nA

nA

n QuPA 0][][][ nB

nA

n QuPB

Conclusion

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 found the general solution of the lowest order in x

Conclusion(cont'd)

It should be modified by the higher order consideration in x.

Thank you for listening and discussions.

ffa~

2/]1[

hhb~

2/]1[kkc~

2/]1[

0/)(2~

2/~

)(2 rcbffbacaP rrrA

02/~

22 )4(2 RcacbcabPB

02 cba 3 eqs. for 5 functions 1) with arbitrary functions: two of a, b, c with a+b+2c =0

,)(

)222(2

]0[dr

barrcracb rr

ef

solution drQeeh PdrPdr ]0[

2) with arbitrary functions: hf~

&~

,3

2

1

2 ]0[

]0[

]0[

]0[

brf

fa

rff

b rrr

)4(

22 ~2323 Rbaba

Or, if we assume the ansatz only on the brane (since we know nothing about the outside), this is the general solution!

The general sol. is that of

general solution of braneworld dynamics under the schwarzschild anzats k. akama, t. hattori, and h....

Documents

brane metric

brane quantity bulk

motion bulk einstein

bulk einstein equation

brane quantity slide

bulk coord

braneposition variable

motion action bulk einstein