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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
gmn(xm)=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
(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 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 solution outside the brane.
bulk Einstein eq. Nambu-Goto eq.
222222 )sin( dzddkhdrfdtdXdXg JIIJ
empty
0)~~~
( ; IYTg
zXXXrXtX 43210 ,,,,
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
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.
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
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. 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
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.
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.
Nambu-Goto eq.0)
~~~( ; IYTg
IJIJIJ gRgR )2/( 0
00 IJIJ RE
222222 )sin( dzddkhdrfdtdXdXg JIIJ
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.
Nambu-Goto eq.0)
~~~( ; IYTg
IJIJIJ gRgR )2/( 0
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(
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
][ )(),(n
nn zrFzrFexpansion
n
k
kknn GFFG0
][][][)(reduction rule& derivatives),,, khfF IJT
,0221100 RRRindependent 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
00 IJIJ RE
222222 )sin( dzddkhdrfdtdXdXg JIIJ
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
][ )(),(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 RRRindependent eqs.Def.
Nambu-Goto eq.0)
~~~( ; IYTg
IJIJIJ gRgR )2/( 0
00 IJIJ RE
222222 )sin( dzddkhdrfdtdXdXg JIIJ
bulk Einstein eq.
IJE =
power seriessolution in z
Nambu-Goto eq.0)
~~~( ; IYTg
power seriessolution in z
0
][ )(),(n
nn zrFzrF
n
k
kknn GFFG0
][][][)(
expansionreduction
rule
3/2 IJIJIJ gR R
off brane
]0[14E ]0[
44E 0
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 RRRindependent eqs.Def.
IJIJIJ gRgR )2/( 0
00 IJIJ RE
222222 )sin( dzddkhdrfdtdXdXg JIIJ
bulk Einstein eq.
IJE =
0
][ )(),(n
nn zrFzrF
n
k
kknn GFFG0
][][][)(
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
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 RRRindependent eqs.Def.
IJIJIJ gRgR )2/( 0
00 IJIJ RE
222222 )sin( dzddkhdrfdtdXdXg JIIJ
bulk Einstein eq.
IJE =
0
][ )(),(n
nn zrFzrF
n
k
kknn GFFG0
][][][)(
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
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 RRRindependent eqs.Def.
IJIJIJ gRgR )2/( 0
00 IJIJ RE
222222 )sin( dzddkhdrfdtdXdXg JIIJ
bulk Einstein eq.
IJE =
0
][ )(),(n
nn zrFzrF
n
k
kknn GFFG0
][][][)(
expansionreduction
rule
3/2 IJIJIJ gR R
off brane
]0[14E ]0[
44E 0
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 RRRindependent eqs.Def.
IJIJIJ gRgR )2/( 0
00 IJIJ RE
222222 )sin( dzddkhdrfdtdXdXg JIIJ
bulk Einstein eq.
IJE =
0
][ )(),(n
nn zrFzrF
n
k
kknn GFFG0
][][][)(
expansionreduction
rule
recursive definition
for .2n
)2( n
3/2 IJIJIJ gR R
off brane
]0[14E ]0[
44E 0
2]0[ rk
,0221100 RRRindependent eqs.Def.
IJIJIJ gRgR )2/( 0
00 IJIJ RE
222222 )sin( dzddkhdrfdtdXdXg JIIJ
bulk Einstein eq.
IJE =
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 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
222222 )sin( dzddkhdrfdtdXdXg JIIJ
bulk Einstein eq.
IJE =
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,
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
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
]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
][ )(),(n
nn zrFzrF
n
k
kknn GFFG0
][][][)(
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
,2/ ]0[]1[ ffu ,2/ ]0[]1[ hhv 2]1[ 2/ rkw Let u v w]/)(22[2 rwvwu rr ]0[]0[ /)( ffvu r
]0[14E 0
2]0[ rk IJIJIJ gRgR )2/( 0
222222 )sin( dzddkhdrfdtdXdXg JIIJ
khf ,,We have
bulk Einstein eq.
IJE =
0
][ )(),(n
nn zrFzrF
n
k
kknn GFFG0
][][][)(
expansionreduction
rule
,, ]0[]0[ hf ]1[]1[]1[ ,, khf obeyif
off brane
0]0[44
]0[14 EE
2]0[ rk IJIJIJ gRgR )2/( 0
222222 )sin( dzddkhdrfdtdXdXg JIIJ
bulk Einstein eq.
IJE =
0
][ )(),(n
nn zrFzrF
n
k
kknn GFFG0
][][][)(
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
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
22 22442 kkk
hkkh
hffh
fff
kk
ff zrrzrzrzrzrz
The only independent non-trivial components
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
-
-
+ - = 0L
__k
]0[
off brane
0]0[44
]0[14 EE
2]0[ rk IJIJIJ gRgR )2/( 0
222222 )sin( dzddkhdrfdtdXdXg JIIJ
bulk Einstein eq.
IJE =
0
][ )(),(n
nn zrFzrF
n
k
kknn GFFG0
][][][)(
expansionreduction
rule
We have ,, ]0[]0[ hf ]1[]1[]1[ ,, khfkhf ,, obeyif
00Rkhkf
hhf
hff
hf
kkf
hhf
fff rrrrrrrzzzzzzz
24 422442 2
22
11Rkhhk
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
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- = 0L
__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
- = 0L
]0[
k1
off brane
0]0[44
]0[14 EE
0]0[44 E
2]0[ rk IJIJIJ gRgR )2/( 0
222222 )sin( dzddkhdrfdtdXdXg JIIJ
bulk Einstein eq.
IJE =
0
][ )(),(n
nn zrFzrF
n
k
kknn GFFG0
][][][)(
expansionreduction
rule
We have ,, ]0[]0[ hf ]1[]1[]1[ ,, khfkhf ,, obeyif
00Rkhkf
hhf
hff
hf
kkf
hhf
fff rrrrrrrzzzzzzz
24 422442 2
22
11Rkhhk
kk
ff
kk
ff
fhhf
kkh
fhf
hhh rrrrrrrrrrzzzzzzz
224242442 2
2
2
22
22R
44R2
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- = 0L
2 4 4 2
2
0222
]0[
44221100
Rk
Rh
Rf
R
- = 0L
]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
hkkrr
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
2]0[ rk IJIJIJ gRgR )2/( 0
222222 )sin( dzddkhdrfdtdXdXg JIIJ
bulk Einstein eq.
IJE =
0
][ )(),(n
nn zrFzrF
n
k
kknn GFFG0
][][][)(
expansionreduction
rule
We have ,, ]0[]0[ hf ]1[]1[]1[ ,, khfkhf ,, obeyif
2
2
2kkz4
- = 0L
2]0[
]0[2
]0[
]0[
]0[
]0[
]0[
1
22
1
rrff
ff
ff
hrr
r
r
4
2]1[
2]0[
]1[]1[
2]0[
]1[]1[
]0[]0[
]1[]1[
2 4224
1
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
hkkrr
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
1
4
1]0[
]0[
]0[0]0[44 E
off brane
0]0[44
]0[14 EE
0]0[44 E
rff
hr
r
1
4
1]0[
]0[
]0[
2]0[
]0[2
]0[
]0[
]0[
]0[
]0[
1
22
1
rrff
ff
ff
hrr
r
r
0]0[14 E
,2/ ]0[]1[ ffu ,2/ ]0[]1[ hhv 2]1[ 2/ rkw Let u v w]/)(22[2 rwvwu rr ]0[]0[ /)( ffvu r
2]0[ rk IJIJIJ gRgR )2/( 0
222222 )sin( dzddkhdrfdtdXdXg JIIJ
bulk Einstein eq.
IJE =
0
][ )(),(n
nn zrFzrF
n
k
kknn GFFG0
][][][)(
expansionreduction
rule
We have ,, ]0[]0[ hf ]1[]1[]1[ ,, khfkhf ,, obeyif
4
2]1[
2]0[
]1[]1[
2]0[
]1[]1[
]0[]0[
]1[]1[
2 4224
1
rk
rhkh
rfkf
hfhf
r
2
2
2kkz4
- = 0Lhfhf zz
kfkf zz
khkh zz
k1
2 4
hfhf rr
2 2
hkhk rr
hff
hff rrr
2
2
4
2
hkfkf rr
2
hkkrr
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
1
4
1]0[
]0[
]0[
2]0[
]0[2
]0[
]0[
]0[
]0[
]0[
1
22
1
rrff
ff
ff
hrr
r
r
,2/ ]0[]1[ ffu ,2/ ]0[]1[ hhv 2]1[ 2/ rkw Let u v w]/)(22[2 rwvwu rr ]0[]0[ /)( ffvu r
2]0[ rk IJIJIJ gRgR )2/( 0
222222 )sin( dzddkhdrfdtdXdXg JIIJ
bulk Einstein eq.
IJE =
0
][ )(),(n
nn zrFzrF
n
k
kknn GFFG0
][][][)(
expansionreduction
rule
We have ,, ]0[]0[ hf ]1[]1[]1[ ,, khfkhf ,, obeyif
4
2]1[
2]0[
]1[]1[
2]0[
]1[]1[
]0[]0[
]1[]1[
2 4224
1
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
2]0[ rk IJIJIJ gRgR )2/( 0
222222 )sin( dzddkhdrfdtdXdXg JIIJ
bulk Einstein eq.
IJE =
0
][ )(),(n
nn zrFzrF
n
k
kknn GFFG0
][][][)(
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
1
4
1]0[
]0[
]0[
2]0[
]0[2
]0[
]0[
]0[
]0[
]0[
1
22
1
rrff
ff
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 later
Two 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
,2/ ]0[]1[ ffu ,2/ ]0[]1[ hhv 2]1[ 2/ rkw Let
2]0[ rk IJIJIJ gRgR )2/( 0
222222 )sin( dzddkhdrfdtdXdXg JIIJ
bulk Einstein eq.
IJE =
0
][ )(),(n
nn zrFzrF
n
k
kknn GFFG0
][][][)(
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
1
4
1]0[
]0[
]0[
2]0[
]0[2
]0[
]0[
]0[
]0[
]0[
1
22
1
rrff
ff
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 later
Two 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
2]0[ rk IJIJIJ gRgR )2/( 0
222222 )sin( dzddkhdrfdtdXdXg JIIJ
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 , 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
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
ratio
obey
on brane
0]0[44
]0[14 EE
(3+1)
0
][ )(),(n
nn zrFzrF
n
k
kknn GFFG0
][][][)(
expansionreduction
rule
IJE =
khf ,,We have if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf
2]0[ rk IJIJIJ gRgR )2/( 0
222222 )sin( dzddkhdrfdtdXdXg JIIJ
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
≡D
define for short
3/~ wvu ≡D
,| zuu
0]0[44 E
0]0[14 E
0]0[44 E
0]0[14 E
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
holds for the collective modes
IJE =
khf ,,We have if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf
2]0[ rk IJIJIJ gRgR )2/( 0
222222 )sin( dzddkhdrfdtdXdXg JIIJ
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
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
trivially satisfied
difference of ±
u +v +2w = 0- - -]/)(22[2/)( ]0[]0[ rwvwuffvu rrr ± ± ± ± ± ±D D D D D D
3 equations5 equations 2 are trivial3 equations
0]0[44 E
0]0[14 E
IJE =
khf ,,We have if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf
2]0[ rk IJIJIJ gRgR )2/( 0
222222 )sin( dzddkhdrfdtdXdXg JIIJ
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
Nambu-Goto eq. 02 wvu 3/~ wvu ≡D
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
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 =
khf ,,We have if ,, ]0[]0[ hf ]1[]1[]1[ ,, khf
2]0[ rk IJIJIJ gRgR )2/( 0
222222 )sin( dzddkhdrfdtdXdXg JIIJ
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
2 )(22 wwvwuvu
Nambu-Goto eq. 02 wvu 3/~ wvu ≡D
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
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,
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[
1
22
11
4
1
rrff
ff
ff
hrff
hrr
r
rr
r
2/1 r 4/)323( 22 vvuu 6/~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/~
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
11
4 rrff
hrff 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 x
x x x x]0[]0[ / ffrLet
( x /4+1/r )
( x /4+1/r )] /[
/P
Q
rh
]0[
1P
]0[
1
hQ
P
Q
solvable!
with arbitrary & v
equations differential2
]0[f ]0[h P Q
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
]/)3([2/ ]0[]0[ rvuvff rr )( vu ]0[]0[ / ffrLet
][][][ ,, nnn khf
and are written with and . v]0[f ]0[h
are written with and vkhf ,, 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[
1
hQ
P
Q
solvable!
with arbitrary & v
222222 )sin( dzddkdrhdtfdXdXg JIIJ )0( z
,0 tY ,1 rY ,2 Y ,3 Y 04 Y
Under the Schwarzschild ansatz,
where
Theorem
,0
][
n
nn zff ,0
][
n
nn zhh
0
][
n
nn 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
222222 )sin( dzddkdrhdtfdXdXg JIIJ )0( z
,0 tY ,1 rY ,2 Y ,3 Y 04 Y
Under the Schwarzschild ansatz,
where
Theorem
,0
][
n
nn zff ,0
][
n
nn zhh
0
][
n
nn 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[]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[
1
hQ
P
Q
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 ][
3
4
2222)1(
1
n
rrrrrrrzzzzzn fhkkf
h
hfhf
fhf
kkf
hhf
ff
nnf
]2[
2
2
2
2 2 ][
3
4
22
2
22)1(
1
n
rrrrrrrrrrzzzzzn hhkkh
hfhf
k
k
f
fkk
ff
kkh
fhf
hh
nnh
]2[
2
][
3
42
2222)1(
1
n
rrrrrrzzzzn k
h
khhfkf
hk
hkh
fkf
nnk
±±
±
± ±
±
± ±
±
±
±
± ± ±
± ± ± ± ±
± ± ±
±±
±
± ±
±
± ± ± ± ± ± ± ± ± ± ±
± ± ± ± ± ± ± ± ±
±± ±
±
± ±
±
±
±
±
±
±
±
±
±
±±
]2[
2
2 2 ][
3
4
2222)1(
1
n
rrrrrrrzzzzzn fhkkf
h
hfhf
fhf
kkf
hhf
ff
nnf
]2[
2
2
2
2 2 ][
3
4
22
2
22)1(
1
n
rrrrrrrrrrzzzzzn hhkkh
hfhf
k
k
f
fkk
ff
kkh
fhf
hh
nnh
]2[
2
][
3
42
2222)1(
1
n
rrrrrrzzzzn k
h
khhfkf
hk
hkh
fkf
nnk
For , are recursively defined by
n
k
kknn GFFG0
][][][)(
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 ][
3
4
2222)1(
1
n
rrrrrrrzzzzzn fhkkf
h
hfhf
fhf
kkf
hhf
ff
nnf
]2[
2
2
2
2 2 ][
3
4
22
2
22)1(
1
n
rrrrrrrrrrzzzzzn hhkkh
hfhf
k
k
f
fkk
ff
kkh
fhf
hh
nnh
]2[
2
][
3
42
2222)1(
1
n
rrrrrrzzzzn k
h
khhfkf
hk
hkh
fkf
nnk
±±
±
± ±
±
± ±
±
±
±
± ± ±
± ± ± ± ±
± ± ±
±±
±
± ±
±
± ± ± ± ± ± ± ± ± ± ±
± ± ± ± ± ± ± ± ±
±± ±
±
± ±
±
±
±
±
±
±
±
±
±
±±
222222 )sin( dzddkdrhdtfdXdXg JIIJ )0( z
,0 tY ,1 rY ,2 Y ,3 Y 04 Y
Under the Schwarzschild ansatz,
where
Theorem
,0
][
n
nn zff ,0
][
n
nn zhh
0
][
n
nn 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[]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
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^)
