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General Solution of Braneworld Dynamics under the Schwarzschild Anzats K. Akama , T. Hattori, and H. Mukaida. Ref. K. Akama, T. Hattori, and H. Mukaida, arXiv:1008.0066 [hep-th]. Abstract.
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General Solutionof Braneworld Dynamics under the Schwarzschild AnzatsK. Akama, T. Hattori, and H. Mukaida Ref. K. Akama, T. Hattori, and H. Mukaida, arXiv:1008.0066 [hep-th] Abstract General solution of braneworld dynamics under the Schwarzschild anzats is derived. It requires fine tuning to reproduce the successful results of the Einstein gravity. Plan 1. Introduction: Braneworld Dynamics 2. General Solution for 3. General Solution for 4. Summary bulk cosmological constant
1. Introduction: Braneworld Dynamics Einstein gravity explaines ①why gravity motions are universal, (^_^) and ②why the Newtonian potential∝1/distance. It is derived via the Schwarzschild solution under the anzatse empty except for the core asymptotically flat, static, spherical, "Braneworld" : our 3+1 spacetime is embedded in higher dim. Can the braneworld theory reproduce the successes ①and ②? 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. (,_,)? 膜宇宙 3+1時空は高次元時空に埋め込まれた部分時空 膜宇宙理論はEinstein重力の成功を継承できるか。 膜上の重力は膜の力学とbulkEinstein方程式に従う。 Einstein重力の再現はfine-tuning 膜のEinstein方程式ない。 (例外 brane induced gravity 膜上の重力は量子効果) 膜の力学とbulkEinstein方程式の この状況を よく理解・確認 するため Schwarzshild前提下の一般解を求め、 Einstein重力の再現するための条件を導く。 はじめ、(A)平坦bulkの南部後藤膜について求め、 (B)waped bulk、(C)Einstwin-Hilbert膜に拡張する。
Braneworld Dynamics brane coord. bulk coord. label bulk metric dynamical variables label brane position gmn(Y)=YI,mYJ,nGIJ(Y) brane metric cannot be a dynamical variable bulk scalar curvature constants Action matter action constant general solution eq. of motion under Schwarzschild anzats bulk Einstein eq. static, spherical, asymptotically flat on brane Nambu-Goto eq. empty except for the core first consider the case bulk en.mom.tensor brane en.mom.tensor
2. General solution for under Schwarzschild anzats assumenothingfaroutsidethebrane general solution eq. of motion under Schwarzschild anzats bulk Einstein eq. static, spherical, asymptotically flat on brane Nambu-Goto eq. empty except for the core first consider the case
2. General solution for under Schwarzschild anzats assumenothingfaroutsidethebrane eq. of motion bulk Einstein eq. Nambu-Goto eq. For later use Nambu-Goto eq.
2. General solution for under Schwarzschild anzats assumenothingfaroutsidethebrane eq. of motion bulk Einstein eq. For later use bulkEinsteineq. Nambu-Goto eq.
2. General solution for under Schwarzschild anzats assumenothingfaroutsidethebrane 3+1braneworld in 5dim.bulk take brane polar coordinate t,r,q,f & normal geodesic coordinate x line element extrinsic curvature brane metric spherical symmety implies We define this for later convenience bulk curvature f,h,a,b,c,u,v,w : functions of r asym. flat. implies f,h→1 asr→∞ bulkEinsteineq. eq. Nambu-Goto
brane metric extrinsic curvature bulk curvature line element spherical symmety implies f,h,a,b,c,u,v,w : functions of r asym. flat. implies f,h→1 asr→∞ bulkEinsteineq. eq. Nambu-Goto
brane metric extrinsic curvature bulk curvature Nambu-Goto eq. bulk Einstein eq. f,h,a,b,c,u,v, &w 5 eqs. for 8 unknown functions of r bulkEinsteineq. eq. Nambu-Goto
Nambu-Goto eq. bulk Einstein eq. Nambu-Goto eq. bulk Einstein eq. f,h,a,b,c,u,v, &w 5 eqs. for 8 unknown functions of r
Nambu-Goto eq. 5 eqs. for 8 unknowns bulk Einstein eq. f,h,a,b,c,u,v,w of r f,h,a,b,c,u,v, & w 5 eqs. for 8 unknown functions of r
Nambu-Goto eq. 5 eqs. for 8 unknowns bulk Einstein eq. f,h,a,b,c,u,v,w of r chose arbitrary a, c, &v. eliminate f, b, u, &w. with differential eq. for h in terms of a, c, &v. rewrite this into with the key equation
key eq. chose arbitrary a, c, &v. eliminate f, b, u, &w. with differential eq. for h in terms of a, c, &v. rewrite this into with the key equation
key eq. We solve this eq. around r =∞. change the variable r to r =1/r. We require existence of unique solution with Z = Z(0) at r = 0. A sufficientcondition is that F(r,Z )is continuous, & |∂F/∂Z| is bounded. (*) Here we assume the condition(*). Then, with solution: (**) The condition (*) implies Q→0 P→0 r(P+Q)→0 We assume a,c& v are differentiable. Then so are A,V,P & Q. Then, ra, rc, r2v→0. Then, (**) imply r2V,r2A→0. recall defs.
key eq. We solve this eq. around r =∞. We assume F(r,Z )is continuous & |∂F/∂Z| is bounded. (*) change the variable r to r =1/r. Then, a unique solution with Z = Z(0) at r = 0 exists (*) implies , , Q , P , r2V,r2A ra, rc, r2v→0. r(P+Q) We require existence of unique solution with Z = Z(0) at r = 0. A sufficientcondition is that To summarize F(r,Z )is continuous, & |∂F/∂Z| is bounded. (*) Here we assume the condition(*). Then, with solution: (**) The condition (*) implies Q→0 P→0 r(P+Q)→0 We assume a,c& v are differentiable. Then so are A,V,P & Q. Then, ra, rc, r2v→0. Then, (**) imply r2V,r2A→0. recall defs.
key eq. We assume F(r,Z )is continuous & |∂F/∂Z| is bounded. (*) Then, a unique solution with Z = Z(0) at r = 0 exists (*) implies r(P+Q), Q, P, r2V,r2A,ra, rc, r2v→0. Once given the function Z, we obtain the full general solution: with arbitrary functions,a,c,v h is the inversion of the definition: f is from the rr component of the bulk Einstein eq. u is from the tt component of the bulk Einstein eq. w is from the trace of the bulk Einstein eq. u +v +2w = 0 b is from the Nambu-Goto eq. a +b +2c = 0
key eq. We assume F(r,Z )is continuous & |∂F/∂Z| is bounded. (*) Then, a unique solution with Z = Z(0) at r = 0 exists (*) implies r(P+Q), Q, P, r2V,r2A,ra, rc, r2v→0. general solution general solution: with arbitrary functions,a,c,v with arbitrary functions,a,b,v
key eq. We assume F(r,Z )is continuous & |∂F/∂Z| is bounded. (*) Then, a unique solution with Z = Z(0) at r = 0 exists (*) implies r(P+Q), Q, P, r2V,r2A,ra, rc, r2v→0. general solution with arbitrary functions,a,c,v Einstein gravity limit a = c = v = 0. ∴P = Q = 0. ∴Z =m: arbitrary constant. u = w = b = 0 ∴f =h-1=1-m/r, Einstein gravity explaines ①why gravity motions are universal, (^_^) and ②why the Newtonian potential 1-f∝1/r. The general solution explaines ①but not ②. (×^×) The Newtonian potential is arbitrary according to a,c&v .
key eq. We assume F(r,Z )is continuous & |∂F/∂Z| is bounded. (*) Then, a unique solution with Z = Z(0) at r = 0 exists (*) implies r(P+Q), Q, P, r2V,r2A,ra, rc, r2v→0. general solution with arbitrary functions,a,c,v We further impose existence of asymptotic expansion The key eq. implies etc. Z0= arbitrary, Expanda,c, &vas Then, In genarl, for n≧1
in general solution with arbitrary functions,a,c,v for the next use general solution with arbitrary functions,a,c,v We further impose existence of asymptotic expansion The key eq. implies etc. Z0= arbitrary, Expanda,c, &vas Then,
in general solution with arbitrary functions,a,c,v for the next use asymptotic expansion where Z0= arbitrary, , etc., with ai, ci& vi by We further impose existence of asymptotic expansion The key eq. implies etc. Z0= arbitrary, Expanda,c, &vas Then,
in general solution with arbitrary functions,a,c,v substitute asymptotic expansion substitute where Z0= arbitrary, , etc., with ai, ci& vi by obtain Expansion of f& h (the components of the brane metric) where with arbitrary constant reproduces Einstein gravity differs from Einstein gravity (×^×)
in general solution with arbitrary functions,a,c,v asymptotic expansion where Z0= arbitrary, , etc., with ai, ci& vi by
light deflection by star gravity star light planetary perihelion precession observation (^_^) Einstein gravity can predict the observed results. The general solution here includes the case observed, & but, requires fine tuning, (*) and, hence, cannot "predict" the observed results. (×^×)
light deflection by star gravity star light planetary perihelion precession observation (^_^) Einstein gravity can predict the observed results. The general solution here includes the case observed, & 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) implies a =c = 0, but leaves v arbitrary, and, hence, still insufficient. (×^×)
3. General solution for The system has the Randall Sundrumtype solution (*) with and (**) For |x|>d, this satisfies empty bulk Einestein eq. For |x|≦d, matter exists, and F takes appropriate form according to the matter distributions. We do not specify the matter motions except for the collective mode, which is x = 0 in the present coordinate system. From (*)&(**), The Nambu-Goto eq. is satisfied by the collective mode.
3. Randall Sundrumsolution General solution for The system has the Randall Sundrumtype solution (*) (*) with and (**) For |x|>d, this satisfies empty bulk Einestein eq. For |x|≦d, matter exists, and F takes appropriate form according to the matter distributions. We do not specify the matter motions except for the collective mode, which is x = 0 in the present coordinate system. From (*)&(**), The Nambu-Goto eq. is satisfied by the collective mode.
Randall Sundrumsolution (*) Now, we seek for the general solution which tends to (*) as r→∞ at least near the brane. 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 as r→∞
|0 0 0 0 -6k2GIJ -6k2 ~ ± ± 0 0 0 ± ± -6k2 ~ ± 0 ± 0 0 ± ± -6k2 ~ ± ± 0 0 0 ± ± 0 ± 0 0 ± ± Nambu-Goto eq. -6k2 bulk Einsteineq. ± ± 0 ± ± ± 0 ± 0 0 ± 0 0 0 as r→∞
|0 0 0 0 -6k2GIJ ~ 0 0 0 ~ 0 0 0 ~ 0 0 0 0 0 0 Nambu-Goto eq. bulk Einsteineq. 0 0 0 0 0 0 0 the same form as those for solution of the same form The bulk curvature have a gap across the brane. If we require that the bulk curvature is gapless, a0=b0=c0= 0, but leaves v arbitrary, and, hence, still insufficient. (×^×)
4. Summary general solution of the bulk Einstein eq. +Nambu-Goto eq. assumingnothingoutside under the Schwarzschild anzats, key eq. For P, Q: with a,c,v general solution with arbitrary functions,a,c,v includes the case observed, The general solution & but, requires fine tuning, (*) we use For Then the same forms of equations give the same solution. Definite physical backgrounds for the condition (*) are desired. Z2 sym. gapless curvature brane induced gravity (^_^) (×^×) (×^×)
