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Plan 1. Introduction: Braneworld Dynamics General Solution of Braneworld Dynamics under the Schwarzschild Anzats K. Akama, T. Hattori, and H. Mukaida 2.

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Presentation on theme: "Plan 1. Introduction: Braneworld Dynamics General Solution of Braneworld Dynamics under the Schwarzschild Anzats K. Akama, T. Hattori, and H. Mukaida 2."— Presentation transcript:

1 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: [hep-th] 3. General Solution for bulk cosmological constant 4. 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

2 ( brane induced gravity Einstein bulk Einstein Einstein Einstein fine-tuning ) bulk Einstein Schwarzshild (A) bulk (B)waped bulk (C)Einstwin-Hilbert Einstein 3+1 Einstein gravity explaines and why 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 and ? "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 (, _, )? why gravity motions are universal, : our 3+1 spacetime is embedded in higher dim.

3 Braneworld Dynamics dynamical variables brane position bulk metric eq. of motion Action bulk scalar curvature bulk Einstein eq. Nambu-Goto eq. label constant bulk en.mom.tensor brane en.mom.tensor label brane coord. bulk coord. brane metric cannot be a dynamical variable constants g ( Y ) Y I, Y J, G IJ ( Y ) general solution matter action static, spherical, first consider the case under Schwarzschild anzats asymptotically flat on brane empty except for the core

4 first consider the case General solution 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 bulk Einstein eq. Nambu-Goto eq. 2.

5 eq. of motion Nambu-Goto eq. under Schwarzschild anzats assume nothing far outside the brane bulk Einstein eq. Nambu-Goto eq. For later use General solutionfor2.

6 Nambu-Goto eq. bulk Einstein eq. under Schwarzschild anzats assume nothing far outside the brane eq. of motion bulk Einstein eq. For later use General solutionfor2.

7 & normal geodesic coordinate 3+1braneworld in 5dim. bulk take brane polar coordinate t, r,, extrinsic curvature brane metric line element f, h, a, b, c, u, v, w : functions of r bulk curvature spherical symmety implies Nambu-Goto eq. bulk Einstein eq. under Schwarzschild anzats assume nothing far outside the brane We define this for later convenience asym. flat. implies f, h1 as r General solutionfor2.

8 extrinsic curvature brane metric bulk curvature Nambu-Goto eq. bulk Einstein eq. line element spherical symmety implies f, h, a, b, c, u, v, w : functions of r asym. flat. implies f, h1 as r

9 Nambu-Goto eq. bulk Einstein eq. Nambu-Goto eq. bulk Einstein eq. for 8 unknown functions5 eqs. f, h, a, b, c, u, v, & w extrinsic curvature brane metric bulk curvature of r

10 bulk Einstein eq. Nambu-Goto eq. bulk Einstein eq. Nambu-Goto eq. for 8 unknown functions5 eqs. f, h, a, b, c, u, v, & w of r

11 bulk Einstein eq. Nambu-Goto eq. for 8 unknowns5 eqs. f,h,a,b,c,u,v,wf,h,a,b,c,u,v,w for 8 unknown functions5 eqs. f, h, a, b, c, u, v, & w of r

12 differential eq. for h in terms of a, c, & v. chose arbitrary a, c, & v. eliminate f, b, u, & w. with bulk Einstein eq. Nambu-Goto eq. for 8 unknowns 5 eqs. f,h,a,b,c,u,v,wf,h,a,b,c,u,v,w of r with rewrite this into the key equation

13 differential eq. for h in terms of a, c, & v. chose arbitrary a, c, & v. eliminate f, b, u, & w. with rewrite this into the key equation key eq.

14 We solve this eq. around r. change the variable r to 1/ r. A sufficient condition is that We require existence of unique solution with Z Z (0) at 0. F (, Z ) is continuous, & | F / Z | is bounded. (*) solution: with Here we assume the condition(*). Then, The condition (*) implies r ( P + Q )0 Q0 P0 key eq. We assume a, c & v are differentiable. Then so are A, V, P & Q. (**) Then, (**) imply r 2 V, r 2 A0. ra, rc, r 2 v0.Then, recall defs.

15 We solve this eq. around r. change the variable r to 1/ r. A sufficient condition is that We require existence of unique solution with Z Z (0) at 0. F (, Z ) is continuous, & | F / Z | is bounded. (*) solution: with Here we assume the condition(*). Then, The condition (*) implies r ( P + Q )0 Q0 P0 We assume a, c & v are differentiable. Then so are A, V, P & Q. (**) Then, (**) imply r 2 V, r 2 A0. ra, rc, r 2 v0.Then, key eq. Then, a unique solution with Z Z (0) at 0 exists F (, Z ) is continuous & | F / Z | is bounded. (*) We assume (*) implies r (P +Q)r (P +Q) Q r 2V, r 2Ar 2V, r 2Ara, rc, r 2 v0. P,,,, recall defs. To summarize

16 key eq. Then, a unique solution with Z Z (0) at 0 exists F (, Z ) is continuous & | F / Z | is bounded. (*) We assume (*) implies r ( P + Q ), Q, P, r 2 V, r 2 A, ra, rc, r 2 v0. Once given the function Z, we obtain the full general solution: 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 2 w 0 b is from the Nambu-Goto eq. a b 2 c 0 with arbitrary functions, a, c, v

17 general solution: with arbitrary functions, a, b, v key eq. Then, a unique solution with Z Z (0) at 0 exists F (, Z ) is continuous & | F / Z | is bounded. (*) We assume (*) implies r ( P + Q ), Q, P, r 2 V, r 2 A, ra, rc, r 2 v0. general solution with arbitrary functions, a, c, v

18 key eq. Then, a unique solution with Z Z (0) at 0 exists F (, Z ) is continuous & | F / Z | is bounded. (*) We assume (*) implies r ( P + Q ), Q, P, r 2 V, r 2 A, ra, rc, r 2 v0. general solution with arbitrary functions, a, c, v Einstein gravity limit a c v 0. P Q 0. Z arbitrary constant. f h 1 1 / r, u w b 0 Einstein gravity explaines why gravity motions are universal, and why the Newtonian potential 1 f 1/ r. The general solution explaines but not. The Newtonian potential is arbitrary according to a, c & v. (^_^)(^_^) (×^×)(×^×)

19 key eq. Then, a unique solution with Z Z (0) at 0 exists F (, Z ) is continuous & | F / Z | is bounded. (*) We assume (*) implies r ( P + Q ), Q, P, r 2 V, r 2 A, ra, rc, r 2 v0. general solution with arbitrary functions, a, c, v We further impose existence of asymptotic expansion The key eq. implies Z 0 arbitrary, etc. Expand a, c, & v as Then, In genarl, for n 1

20 general solution with arbitrary functions, a, c, v We further impose existence of asymptotic expansion The key eq. implies Z 0 arbitrary, etc. Expand a, c, & v as Then, for the next use general solution with arbitrary functions, a, c, v in

21 We further impose existence of asymptotic expansion The key eq. implies Z 0 arbitrary, etc. Expand a, c, & v as Then, general solution with arbitrary functions, a, c, v in asymptotic expansion Z 0 arbitrary, where, etc., with a i, c i & v i by for the next use

22 general solution with arbitrary functions, a, c, v in asymptotic expansion Z 0 arbitrary, where, etc., with a i, c i & v i by Expansion of f & h (the components of the brane metric) with where substitute arbitrary constant obtain reproduces Einstein gravity differs from Einstein gravity (×^×)(×^×)

23 general solution with arbitrary functions, a, c, v in asymptotic expansion Z 0 arbitrary, where, etc., with a i, c i & v i by

24 light deflection by star gravity planetary perihelion precession observation Einstein gravity light star The general solution here can predict the observed results. includes the case observed, but, requires fine tuning, and, hence, cannot "predict" the observed results. & (*)(*) (^_^)(^_^) (×^×)(×^×)

25 light deflection by star gravity planetary perihelion precession observation Einstein gravity light star The general solution here can predict the observed results. includes the case observed, but, requires fine tuning, and, hence, cannot "predict" the observed results. Physical backgrounds for the condition (*) are desired. Z 2 symmetry: G IJ ( x, ) G IJ ( x, ) & (*)(*) implies a c 0, but leaves v arbitrary,and, hence, still insufficient. (^_^)(^_^) (×^×)(×^×) (×^×)(×^×)

26 The system has the Randall Sundrum type solution withand For | |>, this satisfies empty bulk Einestein eq. For | |, 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 0 in the present coordinate system. ( ** ) (*) (*) From ( * ) & ( ** ), General solution for3.

27 The system has the Randall Sundrum type solution For | |>, this satisfies empty bulk Einestein eq. For | |, 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 0 in the present coordinate system. (*) (*) From ( * ) & ( ** ), withand ( ** ) General solution for3.Randall Sundrum solution (*) (*)

28 Now, we seek for the general solution which tends to ( * ) as r at least near the brane. (*) (*) Randall Sundrum solution Then, as r We assume that the brane-generating interactions are much stronger than the gravity at short distances of O( ), while their gravitations are much weaker than those by the core of the sphere. Then, in | | is independent of r, and so does as r

29 bulk Einstein eq. Nambu-Goto eq. 6 k 2 G IJ 6 k 2 ± ± ± ± ± ± ± ± ±± ± ± ± ± ±± ± ± 0 ± ± ~ ~ ~ ± |0|0 ± as r

30 bulk Einstein eq. Nambu-Goto eq. 6 k 2 G IJ ~ ~ ~ |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, a 0 b 0 c 0 0, but leaves v arbitrary, and, hence, still insufficient. (×^×)(×^×)

31 general solution under the Schwarzschild anzats, For assuming nothing outside of the bulk Einstein eq. +Nambu-Goto eq. key eq. general solution with arbitrary functions, a, c, v P, Q : with a, c, v The general solution includes the case observed, but, requires fine tuning, & (*)(*) For we use Then the same forms of equations give the same solution. Definite physical backgrounds for the condition (*) are desired. 4. Summary Z 2 sym.gapless curvaturebrane induced gravity (×^×)(×^×) (^_^)(^_^) (×^×)(×^×)


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