General Solution of Braneworld with the Schwarzschild Ansatz

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:1109.0840; 1208.3303 [gr-qc]; submitted to Japanese Physical Society meeting in 2011 spring. Abstract 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. The arbitrariness may affect the predictive powers on the Newtonian and the post-Newtonian evidences. Ways out of the difficulty are discussed.

is a model of our 3+1 dim. curved spacetime Braneworld higher dim. spacetime where we take it as a membrane-like object • embedded in higher dimensions. This idea has a long history. our spacetime Fronsdal('59), Josesh('62) Regge,Teitelboim('75) K.A.('82) braneworld Rubakov,Shaposhnikov('83) Visser('85), Maia('84), Pavsic('85), Gibbons,Wiltschire('87) Antoniadis('91), Horava,Witten('96) Polchinski('95) applied to hierarchy problems Arkani-Hamed,Dimopolos,Dvali('98), Randall,Sundrum('99)

(^V^) Motivation Einstein gravity successfully explaines ①theorigin of the Newtonian gravity ②post Newtonian evidences: light deflections due to gravity, the planetary perihelion precessions, etc. It is based on the Schwarzschild solution with the ansatz staticity, sphericality, asymptotic flatness, emptiness except for the core "Braneworld" : our 3+1 spacetime is embedded in higher dim. Can the braneworld theory inherit the successes ①and ②? (,_,)? To examine it, we derive the general solution of the fundamental dynamics of the brane under the Schwarzschildanzats. Garriga,Tanaka (00), Visser,Wiltshire('03) Casadio,Mazzacurati('03), Bronnikov,Melnikov,Dehnen('03) spherical sols. ref.

Braneworld Dynamics brane coord. bulk coord. bulk metric dynamical variables brane position ~ gmn(xm)=YI,mYJ,ngIJ(Y) brane metric cannot be a dynamical variable it cannot fully specify the state of the brane constant bulk scalar curvature Action gIJ YI d /d = 0 ~ indicates brane quantity constant matter action bulk en.mom.tensor eq. of motion bulk Ricci tensor bulk Einstein eq. (3+1dim.) brane en.mom.tensor Nambu-Goto eq.

Nambu-Goto eq. bulk Einstein eq. (3+1) bulk Einstein eq. (3+1dim.) Nambu-Goto eq.

Nambu-Goto eq. bulk Einstein eq. off brane (3+1) general solution z under Schwarzschild ansatz static, spherical, empty asymptotically flat on the brane, empty except for the core outside the brane t,r,q,j coordinate system empty brane polar coordinate xm=(t,r,q,j) × normal coordinate z general metric with : functions of r & zonly We first consider the solution outside the brane.

Nambu-Goto eq. bulk Einstein eq. off brane substituting gIJ, write RIJKL with of f, h, k. RIJKL=GIJK,L-GIJL,K+gAB(GAIKGBJL-GAILGBJK), GIJK=(gIJ,K+gIK,J-gJK,I)/2 curvature tensor affine connection The only independent non-trivial components

Nambu-Goto eq. bulk Einstein eq. off brane use again later RIJKL=GIJK,L-GIJL,K+gAB(GAIKGBJL-GAILGBJK), GIJK=(gIJ,K+gIK,J-gJK,I)/2 The only independent non-trivial components use again later

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

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

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

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

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

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

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

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

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

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

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

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

bulk Einstein eq. off brane expansion reduction = rule if obey We have 2 2 8 4f 4f 8f ___ 2f 8f 4f 4f 8 - - - - + + + + + - 2 2 2 4h 4h 4 h h 8 h 2 h 8 4 4 h The only independent non-trivial components 8 8 ___ 2h - - - - + + - __ k k k k k k k ___ 2k - L = 0 + 8 4 4 8 ___ 2 2 4

bulk Einstein eq. off brane expansion reduction = rule if obey We have 2 2 8 2 4 4 4f 4f 8f 8f 4f - - - - + + + + 2 2 4h 2 4 h h 8 h 2 h 8 4 4 h 8 - - - - + + __ k k k k k k - L = 0 - L = 0 4 2 2 4

bulk Einstein eq. off brane expansion reduction = rule if obey We have 2 2 8 2 4 4 4f 8f 4f - + + + 2 2 4 h 8 h 2 h 4 [0] [0] [1] [1] [0] [1] [0] [0] [1] [0] [0] [1] [0] [0] [0] [0] [0] [1] [0] [1] [0] [0] [0] [0] [0] [0] [0] [0] [0] [0] [0] [0] [0] [0] [0] [0] - L = 0 - L = 0 [0] [0] [0] 4 2 2 4

bulk Einstein eq. off brane expansion reduction = rule if obey We have [0] [0] [1] [1] [0] [0] [1] [0] [1] [0] [0] [0] [1] [0] [0] [0] [1] [0] [0] [1] [0] [0] [0] [0] [0] [0] [0] [0] [0] [0] [0] [0] [0] [0] [0] [0] - L = 0 [0] [0] [0] 4 2 2 4

bulk Einstein eq. off brane expansion reduction = rule if obey We have Let v u w 2 v 2 u w w v 2 u w [0] [0] [1] [0] [1] [0] [1] [0] [1] [0] [0] [1] [0] [0] [0] [0] [1] [0] [0] [1] [0] [0] [0] [0] [0] [0] [0] [0] [0] [0] [0] [0] [0] [0] [0] [0] - L = 0 [0] [0] [0] 4

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

bulk Einstein eq. on brane expansion reduction = rule if obey We have Let Let use again later Two differential equations for five functions So far, considered the solution off the braneonly. Next, we turn to the solution inside the brane, and their connection. on brane

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

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

bulk Einstein eq. on brane Nambu-Goto eq. = obey if We have Let v k u w z z z Nambu-Goto eq. ≡D ±d ± ± ± ± ± ± ±d ± ± ± ± ± ± ± difference of ± D D d -d D D D D ± ± ± ± ± ± u +v +2w = 0 trivially satisfied - - - trivially satisfied d -d 3 equations 3 equations 5 equations 2 are trivial

bulk Einstein eq. on brane Nambu-Goto eq. = obey if We have Let v k u w z z z Nambu-Goto eq. ≡D ±d ± ± ± ± ± ± ±d average of ± ± ± ± ± ± ± ± - - - - - - 3 equations - - - - - - -

bulk Einstein eq. on brane Nambu-Goto eq. = obey if We have Let v k u w z z z Nambu-Goto eq. ≡D : arbitrary, substitute use one equation substitute - - - - - - 2 equations 3 equations - - - - - - -

equations differential 2 - - 2 equations

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

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

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

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

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

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

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

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

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

Discussions =arbitrary light =arbitrary light deflection by star gravity (^_^) Einstein gravity can predict the observed results. star The general solution here includes the case observed, & but, requires fine tuning, (*) planetary perihelion precession and, hence, cannot "predict" the observed results. (×^×) We need additional physical prescriptions non-dynamical. (×^×) Z2 symmetry leaves these arbitrariness unfixed. (^O^) Brane induced gravity may by-pass this difficulty. observation

Summary bulk Einstein eq. Nambu-Gotoeq. • The general solution • of the fundamental equations of braneworld • with Schwarzschild ansatz is derived. • 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 • as far as we appropriately choose 2 arbitrary functions. (^V^) • 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^) Thank you for listening. (^O^)

Thank you (^O^)

General Solution of Braneworld with the Schwarzschild Ansatz