plan 1. introduction: braneworld dynamics general solution of braneworld dynamics under the...
TRANSCRIPT
Plan 1. Introduction: Braneworld Dynamics
General Solution of Braneworld Dynamics under the Schwarzschild Anzats
K. Akama, T. Hattori, and H. Mukaida
2. General Solution for
Ref. K. Akama, T. Hattori, and H. Mukaida, arXiv:1008.0066 [hep-th]
0ˆ 3. General Solution for 0ˆ
bulk cosmological constant4. Summary
General solution of braneworld dynamics under the Schwarzschild anzats is derived. It requires fine tuning to reproduce the successful results of the Einstein gravity.
Abstract
3+1 時空は高次元時空に埋め込まれた部分時空
Einstein gravity explaines and ②why the Newtonian potential∝1/distance. (^_^)
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 the situations, we derive here the general solution of the braneworld dynamics under the Schwarzschid anzats.
1. Introduction: Braneworld Dynamics
( ,_ ,)?
①why gravity motions are universal,
: our 3+1 spacetime is embedded in higher dim.
Braneworld Dynamics
bulk
1 )))((ˆˆˆ()( XdXGRXGS NKIJ
K
matterS
dynamicalvariables brane position
)( KIJ XG
)( xY I
bulk metric
brane
4))(( xdxYg K
IJIJIJIJ TGRGR ˆˆ)2/ˆˆ(ˆ 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
g
label
brane coord.KX xbulk coord.
brane metriccannot be a dynamical variable
constants
gmn(Y)=YI,mYJ
,nGIJ(Y)
general solution
matter action
static, spherical,
first consider the case 0ˆ
under Schwarzschild anzats
asymptotically flat on brane empty except for the core
first consider the case 0ˆ
General solution 0ˆ under Schwarzschild anzats
eq. of motion general solution
static, spherical, under Schwarzschild anzats
asymptotically flat on brane empty except for the core
assume nothing far outside the brane
for
IJIJIJIJ TGRGR ˆˆ)2/ˆˆ(ˆ 1
0}{ ; IYTg
bulk Einstein eq.
Nambu-Goto eq.
2.
eq. of motion
Nambu-Goto eq. 0; IY g
under Schwarzschild anzatsassume nothing far outside the brane
0ˆ
IJIJIJIJ TGRGR ˆˆ)2/ˆˆ(ˆ 1
0}{ ; IYTg
bulk Einstein eq.
Nambu-Goto eq. For later use
General solution for2.
Nambu-Goto eq. 0; IY gbulk Einstein eq. 2/ˆˆ RGR IJIJ 0
under Schwarzschild anzatsassume nothing far outside the brane
0ˆ
eq. of motion
IJIJIJIJ TGRGR ˆˆ)2/ˆˆ(ˆ 1 bulk Einstein eq.
For later use
General solution for2.
222 2 ddxdxCKgds
44
44
ˆˆ
RRJ
& normal geodesic coordinate x
3+1braneworld in 5dim. bulktake brane polar coordinate t,r,q,f
extrinsic curvature
branemetric
line element
)sin,,,(diag 222 rrhfg
),,,(diag ccbaK
),,,(diag wwvuJ
)( 3O
f,h,a,b,c,u,v,w : functions of r
bulk curvature
spherical symmety implies
0; IY g0 Nambu-Gotoeq.bulk Einstein eq.
under Schwarzschild anzatsassume nothing far outside the brane
,ˆ44
RKKgC
We define this for later convenience
,; YK
0ˆ
2/ˆˆ RGR IJIJ
asym. flat. implies f,h→1 asr→∞
General solution for2.
extrinsiccurvature
brane metric
bulk curvature )ˆˆ( 4
44
4
RRJ
)sin,,,(diag 222 rrhfg
),,,(diag ccbaK
),,,(diag wwvuJ
0; IY g0 Nambu-Gotoeq.bulk Einstein eq.
222 2 ddxdxCKgds
line element
)sin,,,(diag 222 rrhfg
),,,(diag ccbaK
),,,(diag wwvuJ
)( 3O
spherical symmety implies
,; YK
;YK
2/ˆˆ RGR IJIJ
44
44
ˆˆ
RRJ ,ˆ
44
RKKgC
f,h,a,b,c,u,v,w : functions of rasym. flat. implies f,h→1 asr→∞
0; IY g2/ˆˆ RGR IJIJ 0 Nambu-Gotoeq.
02/ˆˆ RGR IJIJ
0; IYg
bulk Einstein eq.
Nambu-Goto eq.
bulk Einstein eq.
02/)1(/)'( 2211 ucbcrhrh
02/)1(/' 221 vcacrhrfhf
0 2/)'/()(2/)'')(( 2/12/1 wbcacabrfhffhffh
022
/)1(/)'/()(2/)'')((2
212/12/1
cbcacab
rhrfhffhffh
02 cba
for 8 unknown functions5 eqs. f,h,a,b,c,u,v, & w
extrinsiccurvature
brane metric
bulk curvature
)sin,,,(diag 222 rrhfg
),,,(diag ccbaK
),,,(diag wwvuJ
;YK
of r
)ˆˆ( 44
44
RRJ
02/ RGR IJIJ
0; IYg
bulk Einstein eq.
Nambu-Goto eq.
02/)1(/)'( 2211 ucbcrhrh
02/)1(/' 221 vcacrhrfhf
0 2/)'/()(2/)'')(( 2/12/1 wbcacabrfhffhffh
022
/)1(/)'/()(2/)'')((2
212/12/1
cbcacab
rhrfhffhffh
02 cba
bulk Einstein eq.Nambu-Goto eq.
02/)1(/)'( 2211 ucbcrhrh02/)1(/' 221 vcacrhrfhf
02 cba
for 8 unknown functions5 eqs. f,h,a,b,c,u,v, & w of r
0 2/)'/()(2/)'')(( 2/12/1 wbcacabrfhffhffh
022
/)1(/)'/()(2/)'')((2
212/12/1
cbcacab
rhrfhffhffh
bulk Einstein eq.Nambu-Goto eq.
02/)1(/)'( 2211 ucbcrhrh02/)1(/' 221 vcacrhrfhf
02 cba
0 2/)'/()(2/)'')(( 2/12/1 wbcacabrfhffhffh212/12/1 /)1(/)'/()(2/)'')(( rhrfhffhffh
022 2 cbcacab
for 8 unknowns5 eqs.f,h,a,b,c,u,v,w
for 8 unknown functions5 eqs. f,h,a,b,c,u,v, & w of r
of r
)/1/(' rZQPZ 2/2/)1(3 31 VrhrZ
22 cacvV
differential eq. for h in terms of a, c, & v.
chose arbitrary a, c, & v. eliminate f, b, u, & w.
with 2)2/'34( rrVVAP 2/)'4)(1( 22 rrVVAVrQ
22 642 cacaA
bulk Einstein eq.Nambu-Goto eq.
02/)1(/)'( 2211 ucbcrhrh02/)1(/' 221 vcacrhrfhf
0 2/)'/()(2/)'')(( 2/12/1 wbcacabrfhffhffh212/12/1 /)1(/)'/()(2/)'')(( rhrfhffhffh
02 cba for 8 unknowns5 eqs.
022 2 cbcacab
f,h,a,b,c,u,v,wof r
3)1(
)'4(21
2
22
VrhArVVr
Vrhr
with'
rewrite this into
the key equation
)/1/(' rZQPZ
22 cacvV
differential eq. for h in terms of a, c, & v.
chose arbitrary a, c, & v. eliminate f, b, u, & w.
with 2)2/'34( rrVVAP 2/)'4)(1( 22 rrVVAVrQ
22 642 cacaA 3)1(
)'4(21
2
22
VrhArVVr
Vrhr
with'
rewrite this into
)/1/(' rZQPZ
the key equation
key eq.
2/2/)1(3 31 VrhrZ
We solve this eq. around r =∞.
change the variable r to r =1/r. )1(22 Z
QPddZ
A sufficient condition is thatWe require existence of unique solution with Z = Z(0) at r = 0.
),( ZF
F(r,Z ) is continuous, & |∂F/∂Z | is bounded. (*)
solution: )(lim nnZZ
dr
ZrQr
PZZr
nn
)1()0()(with
Here we assume the condition(*). Then,
The condition (*) implies r (P +Q)→0 Q→0 P→0
)/1/(' rZQPZ key eq.
We assume a,c & v are differentiable. Then so are A,V,P & Q.
(**)
Then, (**) imply r 2V, r 2A →0. ra, rc, r 2v →0.Then,
22 cacvV 2)2/'34( rrVVAP
2/)'4)(1( 22 rrVVAVrQ 22 642 cacaA recalldefs.
We solve this eq. around r =∞.
change the variable r to r =1/r. )1(22 Z
QPddZ
A sufficient condition is thatWe require existence of unique solution with Z = Z(0) at r = 0.
),( ZF
F(r,Z ) is continuous, & |∂F/∂Z | is bounded. (*)
solution: )(lim nnZZ
dr
ZrQr
PZZr
nn
)1()0()(with
Here we assume the condition(*). Then,
The condition (*) implies r (P +Q)→0 Q→0 P→0We assume a,c & v are differentiable. Then so are A,V,P & Q.
(**)
Then, (**) imply r 2V, r 2A →0. ra, rc, r 2v →0.Then,
22 cacvV 2)2/'34( rrVVAP
2/)'4)(1( 22 rrVVAVrQ 22 642 cacaA
)/1/(' rZQPZ key eq. 2/)]1/([ ZQP ),( ZF P Q )1( Z 2
Then,a unique solution with Z = Z(0) at r = 0 existsF(r,Z ) is continuous & |∂F/∂Z | is bounded. (*)We assume
(*) implies r (P +Q) Q r 2V, r 2A ra, rc, r 2v →0.P, , , ,
recalldefs.
To summarize
)/1/(' rZQPZ key eq.
Then,a unique solution with Z = Z(0) at r = 0 existsF(r,Z ) is continuous & |∂F/∂Z | is bounded. (*)We assume
(*) implies r (P +Q), Q, P, r 2V, r 2A, ra, rc, r 2v →0.
Once given the function Z,
),3/)2(3/21/(1 22 rcacvrZh
,32/]1)'/[( 22 cacrhru
we obtain the full general solution:
2/2/)1(3 31 VrhrZ h is the inversion of the definition: f is from the rr component of the bulk Einstein eq.
,])2(/)1[(exp 2 drrcacvhrhfr
,2/)( vuw ,2cab
u is from the tt component of the bulk Einstein eq. w is from the trace of the bulk Einstein eq. u +v +2w = 0b is from the Nambu-Goto eq. a +b +2c = 0
with arbitraryfunctions,a,c,v
2/)]1/([ ZQP ),( ZF
general solution: with arbitraryfunctions,a,b,v
),3/)2(3/21/(1 22 rcacvrZh
,32/]1)'/[( 22 cacrhru
,])2(/)1[(exp 2 drrcacvhrhfr
,2/)( vuw ,2cab
)/1/(' rZQPZ key eq.
Then,a unique solution with Z = Z(0) at r = 0 existsF(r,Z ) is continuous & |∂F/∂Z | is bounded. (*)We assume
(*) implies r (P +Q), Q, P, r 2V, r 2A, ra, rc, r 2v →0.
),3/)2(3/21/(1 22 rcacvrZh
,32/]1)'/[( 22 cacrhru
,])2(/)1[(exp 2 drrcacvhrhfr
,2/)( vuw ,2cab
general solutionwith arbitraryfunctions,a,c,v
2/)]1/([ ZQP ),( ZF
)/1/(' rZQPZ key eq.
Then,a unique solution with Z = Z(0) at r = 0 existsF(r,Z ) is continuous & |∂F/∂Z | is bounded. (*)We assume
(*) implies r (P +Q), Q, P, r 2V, r 2A, ra, rc, r 2v →0.
),3/)2(3/21/(1 22 rcacvrZh
,32/]1)'/[( 22 cacrhru
,])2(/)1[(exp 2 drrcacvhrhfr
,2/)( vuw ,2cab
general solutionwith arbitraryfunctions,a,c,v
2/)]1/([ ZQP ),( ZF
Einstein gravity limita = c = v = 0.∴P = Q = 0.∴Z =m : arbitrary constant.∴f =h-1=1- m /r,u = w = b = 0Einstein gravity explaines ①why gravity motions are universal,
and ②why the Newtonian potential 1-f ∝1/r. The general solution explaines ① but not .② The Newtonian potential is arbitrary according to a,c & v .
(^_^)
(×^
×)
)/1/(' rZQPZ key eq.
Then,a unique solution with Z = Z(0) at r = 0 existsF(r,Z ) is continuous & |∂F/∂Z | is bounded. (*)We assume
(*) implies r (P +Q), Q, P, r 2V, r 2A, ra, rc, r 2v →0.
),3/)2(3/21/(1 22 rcacvrZh
,32/]1)'/[( 22 cacrhru
,])2(/)1[(exp 2 drrcacvhrhfr
,2/)( vuw ,2cab
general solutionwith arbitraryfunctions,a,c,v
,221
0 rZ
rZ
ZZ
We further impose existence of asymptotic expansion
,221
rP
rP
P ,221
rQ
rQ
Q 011 QP
The key eq. implies ,22101 QPPZZ Z0 = arbitrary, etc. ,
33
22
ra
ra
a ,33
22
rc
rc
c 44
33
rv
rv
v
1Z
Expanda,c, &v as
Then,2
2222
230 7232/ ccaavZ
2/)]1/([ ZQP ),( ZF
,32/]1)'/[( 22 cacrhru ,2/)( vuw ,2cab
),3/)2(3/21/(1 22 rcacvrZh
,])2(/)1[(exp 2 drrcacvhrhfr
general solutionwith arbitraryfunctions,a,c,v
,221
0 rZ
rZ
ZZ
We further impose existence of asymptotic expansion
,221
rP
rP
P ,221
rQ
rQ
Q 011 QP
The key eq. implies ,22101 QPPZZ Z0 = arbitrary, etc. ,
33
22
ra
ra
a ,33
22
rc
rc
c 44
33
rv
rv
v
1Z
Expanda,c, &v as
Then,2
2222
230 7232/ ccaavZ
for the next use
),3/)2(3/21/(1 22 rcacvrZh
,])2(/)1[(exp 2 drrcacvhrhfr
general solutionwith arbitraryfunctions,a,c,v
in
,221
0 rZ
rZ
ZZ
We further impose existence of asymptotic expansion
,221
rP
rP
P ,221
rQ
rQ
Q 011 QP
The key eq. implies ,22101 QPPZZ Z0 = arbitrary, etc. ,
33
22
ra
ra
a ,33
22
rc
rc
c 44
33
rv
rv
v
1Z
Expanda,c, &v as
Then,2
2222
230 7232/ ccaavZ
),3/)2(3/21/(1 22 rcacvrZh
,])2(/)1[(exp 2 drrcacvhrhfr
general solutionwith arbitraryfunctions,a,c,v
in
,221
0 rZ
rZ
ZZasymptotic expansion
,33
22
ra
ra
a ,33
22
rc
rc
c 44
33
rv
rv
v
2222
22301 7232/ ccaavZZ Z0 = arbitrary, where , etc.,
with ai, ci & vi by
for the next use
2211 1
rh
rh
hh22
111
rf
rf
f
,31 vh 2
2222
2432 5222/ ccaavvh
,1 f3/)(2 03 Zv
22
22432 24/3 cavvf
),3/)2(3/21/(1 22 rcacvrZh
,])2(/)1[(exp 2 drrcacvhrhfr
general solutionwith arbitraryfunctions,a,c,v
in
,221
0 rZ
rZ
ZZasymptotic expansion
,33
22
ra
ra
a ,33
22
rc
rc
c 44
33
rv
rv
v
2222
22301 7232/ ccaavZZ Z0 = arbitrary, where , etc.,
with ai, ci & vi by
Expansion of f & h (the components of the brane metric)
with
where
substitute
substitute
arbitrary constant
obtain
reproduces Einstein gravity
differs from Einstein gravity (×
^×)
),3/)2(3/21/(1 22 rcacvrZh
,])2(/)1[(exp 2 drrcacvhrhfr
general solutionwith arbitraryfunctions,a,c,v
in
,221
0 rZ
rZ
ZZasymptotic expansion
,33
22
ra
ra
a ,33
22
rc
rc
c 44
33
rv
rv
v
2222
22301 7232/ ccaavZZ Z0 = arbitrary, where , etc.,
with ai, ci & vi by
,33
22
ra
ra
a ,33
22
rc
rc
c 44
33
rv
rv
v
,33
22
ra
ra
a ,33
22
rc
rc
c 44
33
rv
rv
v
21 3
Einstein
v
2
22
2243
Einstein 3
)2(2
6
51
cavv
light deflection by star gravity
planetary perihelion precession
observationEinstein gravity
lightstar
0r
The general solution herecan predict the observed results.
includes the case observed,
Einstein Einstein
but, requires fine tuning,and, hence, cannot "predict" the observed results.
3v2
22
24 2cav &0 0 (*)
(^_^)
(×^
×)
21 3
Einstein
v
2
22
2243
Einstein 3
)2(2
6
51
cavv
light deflection by star gravity
planetary perihelion precession
observationEinstein gravity
lightstar
0r
The general solution herecan predict the observed results.
includes the case observed,
Einstein Einstein
but, requires fine tuning,and, hence, cannot "predict" the observed results.
Physical backgrounds for the condition (*) are desired.Z2 symmetry: GIJ(xm,x)=GIJ(xm,-x)
3v2
22
24 2cav &0 0 (*)
implies a =c = 0, but leaves v arbitrary, and, hence, still
insufficient.
(^_^)
(×^
×)
(×^
×)
0ˆ
)1,sin,,,(diag 222 FrFrFFG IJ
The system has the Randall Sundrum type solution
||2 keF 0|' 0F ||forwith andFor |x|>d, this satisfies empty bulk Einestein eq.
6/ˆˆ2 k
For |x|≦d, matter exists, and F takes appropriate form according to the matter distributions.
The Nambu-Goto eq. is satisfied by the collective mode.
We do not specify the matter motions except for the collective mode, which is x = 0 in the present coordinate system.
(**)
(*)
From (*) & (**),
kKK ||
23|| kJJ
0|| 00
KK
General solution for3.
)1,sin,,,(diag 222 FrFrFFG IJ
The system has the Randall Sundrum type solution
For |x|>d, this satisfies empty bulk Einestein eq. For |x|≦d, matter exists, and F takes appropriate form according to the matter distributions.
The Nambu-Goto eq. is satisfied by the collective mode.
We do not specify the matter motions except for the collective mode, which is x = 0 in the present coordinate system.
(*)
From (*) & (**),
kKK ||
23|| kJJ
0|| 00
KK
||2 keF 0|' 0F ||forwith and 6/ˆˆ2 k (**)
0ˆ General solution for3. )1,sin,,,(diag 222 FrFrFFG IJ
Randall Sundrum solution
(*)
|K |
J0|
K 0
k
23k
Now, we seek for the general solution which tends to (*)
as r→∞ at least near the brane.
)1,sin,,,(diag 222 FrFrFFG IJ (*)
Randall Sundrum solution
Then, as r→∞
We assume that the brane-generating interactions are much stronger than the gravity at short distances of O(d),while their gravitations are much weaker than those by the core of the sphere. Then, in |x|≦ d is independent of r, and so does
T
0|0K
kK |
23| kJ
0||
KK
kKK 0|| kaa 0 kbb 0 kcc 0
|K |
J0|
K 0
k
23k
kKK 0|| kaa 0 kbb 0 kcc 0
as r→∞0|0K
kK |
23| kJ
02/)1(/)'( 2211 ucbcrhrh
02/)1(/' 221 vcacrhrfhf
0 2/)'/()(2/)'')(( 2/12/1 wbcacabrfhffhffh212/12/1 /)1(/)'/()(2/)'')(( rhrfhffhffh
02 cba0 K
02/ˆˆ RGR IJIJ
0; IYg
bulk Einstein eq.
Nambu-Goto eq.
022 2 cbcacab
-6k2GIJ -6k2
-6k2
-6k2
-6k2
± ±
± ±
±
±
0 0 0
± ± ± ± ± ±
± ± ± ± ± ±
0 ±
±
~
~
~±
|0
±
0 0
0 0 0
0 0 0 0 0 0
0 0 0 0 0 0 0
kKK 0|| kaa 0 kbb 0 kcc 0
as r→∞0|0K
kK |
23| kJ
02 |23|
~
kKkJJ
,23~0
2 kakuu ,23~0
2 kbkvv 02 23~ kckww
0 0 0 00
02/)1(/)'( 2211 ucbcrhrh
02/)1(/' 221 vcacrhrfhf
0 2/)'/()(2/)'')(( 2/12/1 wbcacabrfhffhffh212/12/1 /)1(/)'/()(2/)'')(( rhrfhffhffh
02 cba0 K
02/ˆˆ RGR IJIJ
0; IYg
bulk Einstein eq.
Nambu-Goto eq.
022 2 cbcacab
-6k2GIJ 0 0 0
0~
~
~
|0
0 0
0 0 0
0
0 0 0 0 0 0 0
the same form as those for solution of the same form0ˆ The bulk curvature have a gap across the brane.
4
4R
If we require that the bulk curvature is gapless, a0=b0=c0= 0, but leaves v arbitrary, and, hence, still
insufficient. (×
^×)
02 |23|
~
kKkJJ
,23~0
2 kakuu ,23~0
2 kbkvv 02 23~ kckww
general solutionunder the Schwarzschild anzats,
For
assuming nothing outside
0ˆ
of the bulk Einstein eq. +Nambu-Goto eq.
)/1/(' rZQPZ key eq.
),3/)2(3/21/(1 22 rcacvrZh
,32/]1)'/[( 22 cacrhru
,])2(/)1[(exp 2 drrcacvhrhfr
,2/)( vuw ,2cab
general solutionwith arbitraryfunctions,a,c,v
P, Q: with a,c,v
The general solution includes the case observed, but, requires fine tuning, 3v
22
224 2cav &0 0 (*)
For 0ˆ 02 |23|
~
kKkJJ we use
Then the same forms of equations give the same solution.
Definite physical backgrounds for the condition (*) are desired.
4. Summary
Z2 sym. gapless curvature brane induced gravity(×^
×) (^_^)(×^
×)