General Solution of Braneworld with the Schwarzschild Ansatz K. Akama, T. Hattori, and H. Mukaida General Solution of Braneworld with the Schwarzschild Ansatz K. Akama, T. Hattori, and H. Mukaida

General Solution of Braneworld with the Schwarzschild ansatz K. Akama, T. Hattori, and H. Mukaida General Solution of Braneworld with the Schwarzschild ansatz K. Akama, T. Hattori, and H. Mukaida Ref. K. Akama, T. Hattori, and H. Mukaida, arXiv: ; [gr-qc]; submitted to Japanese Physical Society meeting in 2011 spring. Abstract The arbitrariness may affect the predictive powers on the Newtonian and the 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 of our 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 ② post Newtonian evidences: light deflections due to gravity, the planetary perihelion precessions, etc. (^V^)(^V^) It is based on the Schwarzschild solution with the ansatz staticity, 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. (, _, )? ① the 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 Braneworld Dynamics dynamical variables brane position bulk metric eq. of motion Action bulk scalar curvature bulk Einstein eq. Nambu-Goto eq. constant brane en.mom.tensor brane bulk coord. brane metric cannot be a dynamical variable constant g  ( x  )  Y I,  Y J,  g IJ ( Y ) matter action ~  /  ~ indicates brane quantity bulk en.mom.tensor coord.  0 g IJ YIYI bulk Ricci tensor (3+1dim.)

bulk Einstein eq. Nambu-Goto eq. bulk Einstein eq. Nambu-Goto eq. (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 z brane polar coordinate coordinate system x (t,r,,)x (t,r,,) : functions of r & z  only general metric with t,r,,t,r,, z We first consider the solution outside the brane. bulk Einstein eq. Nambu-Goto eq. empty off brane (3+1)

Nambu-Goto eq. R IJKL  IJK,L  IJL,K  g AB  AIK  BJL  AIL  BJK   IJK  g IJ,K  g IK,J  g JK,I  /2 The only independent non-trivial components bulk Einstein eq. curvature tensor affine connection substituting g IJ, write R IJKL with of f, h, k. off brane

The only independent non-trivial components R IJKL  IJK,L  IJL,K  g AB  AIK  BJL  AIL  BJK   IJK  g IJ,K  g IK,J  g JK,I  /2 use again later Nambu-Goto eq. use again later bulk Einstein eq. off brane

covariant derivative If we assume implies if are guaranteed. Therefore, the independent equations are Def. with  Nambu-Goto eq. Bianchi identity then, then bulk Einstein eq. equivalent equation independent equations This & & Owing to is equivalent to off brane

independent eqs. Def. Nambu-Goto eq. bulk Einstein eq.  independent equations Therefore, the independent equations are & off brane

independent eqs. Def. Nambu-Goto eq. The only independent non-trivial components expansion reduction rule & derivatives)( using diffeo. bulk Einstein eq.  power series solution in z off brane

expansion reduction rule & derivatives) independent eqs. ( [n][n] [ n  2] 1 n ( n  1) using diffeo [ n  2] Def. Nambu-Goto eq. 2 4 n ( n  1) bulk Einstein eq.  power series solution in z off brane

expansion [n][n] 1 n ( n  1) [ n  2] using diffeo. reduction rule independent eqs. Def. Nambu-Goto eq. bulk Einstein eq.  power series solution in z Nambu-Goto eq. power series solution in z expansion reduction rule off brane

[n][n] 1 n ( n  1) [ n  2] independent eqs. Def. bulk Einstein eq.  expansion reduction rule off brane

The only independent non-trivial components independent eqs. Def. bulk Einstein eq.  expansion reduction rule off brane

The only independent non-trivial components independent eqs. Def. bulk Einstein eq.  expansion reduction rule off brane

here. Use this are written with &the lower. give recursive definitions of They These independent eqs. Def. bulk Einstein eq.  expansion reduction rule recursive definition for off brane

independent eqs. Def. bulk Einstein eq.  expansion reduction rule whose coefficients are written with Thus, we obtainedin the forms of power series of z, recursive definition use again later used not yet used off brane use again later

bulk Einstein eq.  expansion reduction rule whose coefficients are written with Thus, we obtainedin the forms of power series of z, not yet used We have obey if off brane

The only independent non-trivial components We have  [0] [1] [0] [1] [0] [1] [0] bulk Einstein eq.  expansion reduction rule obey if Let u v w off brane

Let u v w We have bulk Einstein eq.  expansion reduction rule obey if off brane

bulk Einstein eq.  expansion reduction rule We have obey if The only independent non-trivial components 4f4f 8 2 8f8f 4f4f 4f4f 8 2 8f8f 4f4f 4h4h h 4h4h 8 2 h 2 h 4 h 8 h       k k kkk k   ___ 2 f ___ 2 h ___ 2 k ___ 2     __ k off brane

bulk Einstein eq.  expansion reduction rule We have obey if 8f8f 4f4f 4f4f 8 2 8f8f 4f4f 8 h 4h4h 8 2 h 2 h 4 h 8 h     k kkk k   4  __ k   off brane

bulk Einstein eq.  expansion reduction rule We have obey if 4f4f 8 2 8f8f 4f4f 2 h h 8 h   4   422 [1] [0] off brane

bulk Einstein eq.  expansion reduction rule We have obey if 4  422 [1] [0] off brane

Let u v w bulk Einstein eq.  expansion reduction rule We have obey if 4  [1] [0] u v u w 2 v w 2 w 2 off brane

So far, considered the solution Let u v w bulk Einstein eq.  expansion reduction rule We have obey if u v u w 2 v w 2 w 2 Two differential equationsfor five functions Next, we turn to the solution inside the brane, and their connections. off brane on brane off the brane only.

Let bulk Einstein eq.  expansion reduction rule We have obey if Let use again later Two differential equationsfor 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

Let bulk Einstein eq.  expansion reduction rule We have obey if Let use again later Two differential equationsfor 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

 We haveif On the brane, Nambu-Goto eq. similarly for matter is distributed within | z |< ,  : very small. Take the limit  → 0. collective mode dominance in Let u v w z z z k bulk Einstein eq.on the brane bulk Einstein eq. ratio Israel Junction condition ≡≡ define for short ratio obey on brane (3+1) expansion reduction rule

 We haveif Nambu-Goto eq. Let u v w z z z k bulk Einstein eq. obey on brane (3+1) Nambu-Goto eq. similarly for Take the limit  → 0. collective mode dominance in bulk Einstein eq.on the brane Israel Junction condition ≡≡ define for short ≡≡ ±± ±± ± ± ±±±± ± ± ± ± ±± ± connected at the boundary holds for the collective modes

 We haveif Nambu-Goto eq. Let u v w z z z k bulk Einstein eq. obey on brane ±± ±± ± ± ± ± ±± ±   ± ± ±±±± Nambu-Goto eq. ≡≡ trivially satisfied difference of ± u  v  2 w  0  ± ± ±±±±      3 equations 5 equations2 are trivial 3 equations

 We haveif Nambu-Goto eq. Let u v w z z z k bulk Einstein eq. obey on brane Nambu-Goto eq. ≡≡ ±± ±± ± ± ± ± ±± ± ± ± ±±±± average of ± equations

 We haveif Nambu-Goto eq. Let u v w z z z k bulk Einstein eq. obey on brane Nambu-Goto eq. ≡≡ substitute : arbitrary, 3 equations2 equations use one equation

-- 2 equations equations differential 2

where solution linear differential equations     Let (  /4  1/ r ) ] / [ / P Q solvable! with arbitrary & equations differential 2

Let and are written with and. are written with and where solution linear differential equation solvable! with arbitrary &

Under the Schwarzschild ansatz, where Theorem with the coefficients determined by ① and ② below. all the solutions of the braneworld dynamics and (Einstein & Nambu-Goto eqs. in 4+1dim.) are given by

Under the Schwarzschild ansatz, where Theorem with the coefficients determined by ① and ② below. all the solutions of the braneworld dynamics and (Einstein & Nambu-Goto eqs. in 4+1dim.) are given by where solution linear differential equation solvable! with arbitrary & Let and be arbitrary functions of r. ① where Then, we define

For, are recursively defined by where ② We defineand recursive definition ± ± ± ±± ± ±± ± ± ± ±±± ±± ± ±± ±±± ± ± ± ±± ± ±±±±±±±±±±± ±±± ±± ±±±± ± ±± ± ±± ± ± ± ± ± ± ± ± ± ±±

For, are recursively defined by are finally written with where [ n ] obeys the reduction rule where ② and, accordingly, they are written with and. We defineand ± ± ± ±± ± ±± ± ± ± ±±± ±± ± ±± ±±± ± ± ± ±± ± ±±±±±±±±±±± ±±± ±± ±±±± ± ±± ± ±± ± ± ± ± ± ± ± ± ± ±±

Under the Schwarzschild ansatz, where Theorem with the coefficients determined by ① and ② below. all the solutions of the braneworld dynamics and (Einstein & Nambu-Goto eqs. in 4+1dim.) are given by

be arbitrary Let The Newtonian potential becomes arbitrary. In Einstein gravity, Assume asymptotic expansion light deflection by star gravity planetary perihelion precession observation light star Discussions Here, they are arbitrary.  arbitrary

light deflection by star gravity planetary perihelion precession observation light star Discussions light deflection by star gravity planetary perihelion precession observation light star  arbitrary

Discussions light deflection by star gravity planetary perihelion precession observation star Einstein gravity The general solution here can predict the observed results. includes the case observed, but, requires fine tuning, and, hence, cannot "predict" the observed results. & (*)(*) (^_^)(^_^) (×^×)(×^×) Z 2 symmetry leaves these arbitrariness unfixed. (×^×)(×^×) We need additional physical prescriptions non-dynamical. Brane induced gravitymay by-pass this difficulty. (^O^)(^O^)  arbitrary light

Summary The 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^) (^V^)(^V^) bulk Einstein eq. Nambu-Goto eq. as far as we appropriately choose 2 arbitrary functions. with Schwarzschild ansatz is derived. Thank you for listening. (^O^)(^O^)

Thank you (^O^)(^O^)