general solution of braneworld dynamics under the schwarzschild anzats k. akama, t. hattori, and h....
TRANSCRIPT
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
IJIJIJIJ TgRgR )2/(1
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
~ indicatesbrane quantity
IJIJIJIJ TgRgR )2/(1
0}~~~
{ ; IYTg
eq. of motion
bulk Einstein eq.
Nambu-Goto eq.
IJIJIJIJ TgRgR )2/(1
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
IJIJIJIJ TgRgR )2/(1
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; IY g~0 Nambu-Gotoeq.bulk Einstein eq. 2/RgR IJIJ
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; IY g~0 Nambu-Gotoeq.bulk Einstein eq. 2/RgR IJIJ
static, spherical, Schwarzschild anz. asymptotically flat, empty
× normal coordinate xbrane polar coordinate t,r,q,f
static, spherical, Schwarzschild anz. asymptotically flat, empty
0; IY g~0 Nambu-Gotoeq.bulk Einstein eq. 2/RgR IJIJ
× 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.
from the bulk Einstein eq.
)sderivative-their &,( rqqFf 000 R
011 R
022 R
044 R
014 R
static, spherical, Schwarzschild anz. asymptotically flat, empty
0; IY g~0 Nambu-Gotoeq.bulk Einstein eq. 2/RgR IJIJ
× normal coordinate xbrane polar coordinate t,r,q,f
vdrdudddkhdrfdtds 2222222 )sin(
vukhf ,,,, : functions of r & xonly
general line element with
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
from the bulk Einstein eq.
)sderivative-their &,( rqqFf 000 R
011 R
022 R
044 R
014 R
static, spherical, Schwarzschild anz. asymptotically flat, empty
0; IY g~0 Nambu-Gotoeq.bulk Einstein eq. 2/RgR IJIJ
× normal coordinate xbrane polar coordinate t,r,q,f
vdrdudddkhdrfdtds 2222222 )sin(
vukhf ,,,, : functions of r & xonly
),,,,( vukhfq
general line element with
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
from the bulk Einstein eq.
)sderivative-their &,( rqqFf 000 R
011 R
022 R
044 R
014 R
static, spherical, Schwarzschild anz. asymptotically flat, empty
0; IY g~0 Nambu-Gotoeq.bulk Einstein eq. 2/RgR IJIJ
× normal coordinate xbrane polar coordinate t,r,q,f
vdrdudddkhdrfdtds 2222222 )sin(
vukhf ,,,, : functions of r & xonly
),,,,( vukhfq
general line element with
with
)sderivative-their &,( rqqHh
)sderivative-their &,( rqqKk
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
from the bulk Einstein eq.
)sderivative-their &,( rqqFf 000 R
011 R
022 R
044 R
014 R
static, spherical, Schwarzschild anz. asymptotically flat, empty
0; IY g~0 Nambu-Gotoeq.bulk Einstein eq. 2/RgR IJIJ
× normal coordinate xbrane polar coordinate t,r,q,f
vdrdudddkhdrfdtds 2222222 )sin(
vukhf ,,,, : functions of r & xonly
),,,,( vukhfq
general line element with
with
)sderivative-their &,( rqqHh
)sderivative-their &,( rqqKk
0)sderivative-their &,( rqqA
0)sderivative-their &,( rqqB
also depend on hgf ,, , which we eliminate.
from the bulk Einstein eq.
)sderivative-their &,( rqqFf 000 R
011 R
022 R
044 R
014 R
static, spherical, Schwarzschild anz. asymptotically flat, empty
0; IY g~0 Nambu-Gotoeq.bulk Einstein eq. 2/RgR IJIJ
),,,,( vukhfq with
)sderivative-their &,( rqqHh
)sderivative-their &,( rqqKk
0)sderivative-their &,( rqqA
0)sderivative-their &,( rqqB
also depend on hgf ,, , which we eliminate.
vdrdudddkhdrfdtds 2222222 )sin(
Ff Hh Kk 0A 0Bbulk Einstein eq.
vdrdudddkhdrfdtds 2222222 )sin(
0
][
n
nnQQ
0
][
n
nnqq
),,,,( BAKHFQ
),,,,( vukhfq expand in x
1~ u 0~ v2~rk
qq ~]0[
QQ~]0[
normal coordinate xpolar cordinate
static, spherical, Schwarzschild anz. asymptotically flat, empty
0; IY g~0 Nambu-Gotoeq.bulk Einstein eq. 2/RgR IJIJ
~ indicatesbrane quantity
vdrdudddkhdrfdtds 2222222 )sin(
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
static, spherical, Schwarzschild anz. asymptotically flat, empty
0; IY g~0 Nambu-Gotoeq.bulk Einstein eq. 2/RgR IJIJ
vdrdudddkhdrfdtds 2222222 )sin(
Ff Hh Kk 0A 0Bbulk Einstein eq.
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
),,,,( vukhfq expand in x
1~ u 0~ v2~rk
qq ~]0[
QQ~]0[
normal coordinate xpolar cordinate
static, spherical, Schwarzschild anz. asymptotically flat, empty
0; IY g~0 Nambu-Gotoeq.bulk Einstein eq. 2/RgR IJIJ
vdrdudddkhdrfdtds 2222222 )sin(
Ff Hh Kk 0A 0Bbulk Einstein eq.
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
: written in terms of & the lower.
]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
),,,,( vukhfq expand in x
1~ u 0~ v2~rk
qq ~]0[
QQ~]0[
normal coordinate xpolar cordinate
static, spherical, Schwarzschild anz. asymptotically flat, empty
0; IY g~0 Nambu-Gotoeq.bulk Einstein eq. 2/RgR IJIJ
vdrdudddkhdrfdtds 2222222 )sin(
Ff Hh Kk 0A 0Bbulk Einstein eq.
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
: written in terms of & the lower.
]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
KK ,Equating the x n terms in
we have
: written in terms of & the lower.
]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
),,,,( vukhfq expand in x
1~ u 0~ v2~rk
qq ~]0[
QQ~]0[
normal coordinate xpolar cordinate
static, spherical, Schwarzschild anz. asymptotically flat, empty
0; IY g~0 Nambu-Gotoeq.bulk Einstein eq. 2/RgR IJIJ
vdrdudddkhdrfdtds 2222222 )sin(
Ff Hh Kk 0A 0Bbulk Einstein eq.
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
KK ,Equating the x n terms in
we have
: written in terms of & the lower.
]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
),,,,( vukhfq expand in x
1~ u 0~ v2~rk
qq ~]0[
QQ~]0[
normal coordinate xpolar cordinate
static, spherical, Schwarzschild anz. asymptotically flat, empty
0; IY g~0 Nambu-Gotoeq.bulk Einstein eq. 2/RgR IJIJ
vdrdudddkhdrfdtds 2222222 )sin(
Ff Hh Kk 0A 0Bbulk Einstein eq.
: 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
KK ,Equating the x n terms in
we have
: written in terms of & the lower.
]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
),,,,( vukhfq expand in x
1~ u 0~ v2~rk
qq ~]0[
QQ~]0[
normal coordinate xpolar cordinate
static, spherical, Schwarzschild anz. asymptotically flat, empty
0; IY g~0 Nambu-Gotoeq.bulk Einstein eq. 2/RgR IJIJ
vdrdudddkhdrfdtds 2222222 )sin(
Ff Hh Kk 0A 0Bbulk Einstein eq.
: written with
]1[]1[][][][ ,,,, nnnnn vukhf & the lower. ][nAQ
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
KK ,Equating the x n terms in
: written in terms of & the lower.
]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
),,,,( vukhfq expand in x
1~ u 0~ v2~rk
qq ~]0[
QQ~]0[
normal coordinate xpolar cordinate
static, spherical, Schwarzschild anz. asymptotically flat, empty
0; IY g~0 Nambu-Gotoeq.bulk Einstein eq. 2/RgR IJIJ
vdrdudddkhdrfdtds 2222222 )sin(
Ff Hh Kk 0A 0Bbulk Einstein eq.
: written with
]1[]1[][][][ ,,,, nnnnn vukhf & the lower. ][nAQ
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
static, spherical, Schwarzschild anz. asymptotically flat, empty
0; IY g~0 Nambu-Gotoeq.bulk Einstein eq. 2/RgR IJIJ
vdrdudddkhdrfdtds 2222222 )sin(
Ff Hh Kk 0A 0Bbulk Einstein eq.
KK ,Equating the x n terms in
we have
: written in terms of & the lower.
]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
static, spherical, Schwarzschild anz. asymptotically flat, empty
0; IY g~0 Nambu-Gotoeq.bulk Einstein eq. 2/RgR IJIJ
vdrdudddkhdrfdtds 2222222 )sin(
Ff Hh Kk 0A 0Bbulk Einstein eq.
,)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
vdrdudddkhdrfdtds 2222222 )sin(
3 equations for 5 functions
Here we solve .,,,~
,~ ]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
vdrdudddkhdrfdtds 2222222 )sin(
Nambu-Goto eq. 0~
/2~
/~
/ ]1[]1[]1[ kkhhff
vdrdudddkhdrfdtds 2222222 )sin(
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
3 equations for 5 functions
Here we solve .,,,~
,~ ]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
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
vdrdudddkhdrfdtds 2222222 )sin(
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
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
vdrdudddkhdrfdtds 2222222 )sin(
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
The general solution is given by that of
It should be modified by the higher order consideration in x.
static, spherical, Schwarzschild anz. asymptotically flat, empty
0; IY g~0 Nambu-Gotoeq.bulk Einstein eq. 2/RgR IJIJ
vdrdudddkhdrfdtds 2222222 )sin(
Ff Hh Kk 0A 0Bbulk Einstein eq.
,)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