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General Solution of Braneworld with the Schwarzschild ansatz K. 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. - PowerPoint PPT Presentation
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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