plan 1. introduction: braneworld dynamics general solution of braneworld dynamics under the...

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


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


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


0ˆ

eq. of motion

IJIJIJIJ TGRGR ˆˆ)2/ˆˆ(ˆ 1 bulk Einstein eq.

For later use


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.


,ˆ44

RKKgC

We define this for later convenience

,; YK

0ˆ

2/ˆˆ RGR IJIJ

asym. flat. implies f,h→1 asr→∞


extrinsiccurvature

brane metric

bulk curvature )ˆˆ( 4

44

4

RRJ


),,,(diag ccbaK

),,,(diag wwvuJ

0; IY g0 Nambu-Gotoeq.bulk Einstein eq.

222 2 ddxdxCKgds

line element


),,,(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


),,,(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



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


022

/)1(/)'/()(2/)'')((2

212/12/1

cbcacab

rhrfhffhffh



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




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



(*) 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/)( vuw ,2cab




),3/)2(3/21/(1 22 rcacvrZh

,32/]1)'/[( 22 cacrhru


,2/)( vuw ,2cab

general solutionwith arbitraryfunctions,a,c,v

2/)]1/([ ZQP ),( ZF




),3/)2(3/21/(1 22 rcacvrZh

,32/]1)'/[( 22 cacrhru


,2/)( vuw ,2cab


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 .

(^_^)

(×^

×)




),3/)2(3/21/(1 22 rcacvrZh

,32/]1)'/[( 22 cacrhru


,2/)( vuw ,2cab


,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



,221

0 rZ

rZ

ZZ


,221

rP

rP

P ,221

rQ

rQ

Q 011 QP


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



in

,221

0 rZ

rZ

ZZ


,221

rP

rP

P ,221

rQ

rQ

Q 011 QP


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



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



in

,221

0 rZ

rZ


,33

22

ra

ra

a ,33

22

rc

rc

c 44

33

rv

rv

v

2222


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



in

,221

0 rZ

rZ


,33

22

ra

ra

a ,33

22

rc

rc

c 44

33

rv

rv

v

2222


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


)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


as r→∞0|0K

kK |

23| kJ

02/)1(/)'( 2211 ucbcrhrh



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


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 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.


),3/)2(3/21/(1 22 rcacvrZh

,32/]1)'/[( 22 cacrhru


,2/)( vuw ,2cab


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(×^

×) (^_^)(×^

×)

plan 1. introduction: braneworld dynamics general solution of braneworld dynamics under the...

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