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General Solution of Braneworld Dynamics under the Schwarzschild Anzats K. Akama, T. Hattori, and H. Mukaida Ref.(partial) K. Akama, T. Hattori, and H. Mukaida, arXiv: [hep-th] Abstract In order to examine how the braneworld theory reproduce the successful predictions of the Einstein gravity theory, we are seeking for the general spherical solution of the system of the bulk Einstein equation and Nambu-Goto equation. Here, we find the general solution at the lowest order. It should be modified by the higher order considerations.

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Einstein gravity successfully explaines at moderate distances, and the post Newtonian tests. (^_^)(^_^) 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 them, we seek for the general spherical solution of the Einstein equation × braneworld dynamics. Introduction (, _, )? the Newtonian gravity : our 3+1 spacetime is embedded in higher dim. Here, we find the general solution at the lowest order.

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Many people considered the picture that we live in 3+1 brane General Spherical solutions on branes are considered by e.g. Garriga,Tanaka (00), Visser,Wiltshire('03), Casadio,Mazzacurati('03), Bronnikov,Melnikov,Dehnen('03), Kanti('04), Creek,Gregory,Kanti,Mistry('06) in higher dimensions from various points of view, e.g. Fronsdal('59), Josesh('62), Regge,Teitelboim('75), K.A.('82), Rubakov,Shaposhnikov('83), Maia('84), Visser('85), Pavsic('85), Gibbons,Wiltschire ('87), Antoniadis('91), Polchinski('95), Horava,Witten('96), Dvali,Shifman('96), Arkani-Hamed,Dimopolos,Dvali('98), Randall,Sundrum('99), Dvali,Gabadadze,Porrati('00), Shiromizu,Maeda,Sasaki(00), Braneworld

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Braneworld Dynamics dynamical variables brane position bulk metric eq. of motion Action bulk scalar curvature bulk Einstein eq. Nambu-Goto eq. label constant 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 ) matter action ~ / g IJ / Y I ~ indicates brane quantity bulk en.mom.tensor

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Braneworld Dynamics dynamical variables brane position bulk metric eq. of motion Action bulk Einstein eq. Nambu-Goto eq. brane coord. bulk coord. brane metric cannot be a dynamical variable g ( Y ) Y I, Y J, g IJ ( Y ) ~ ~ indicates brane quantity Braneworld Dynamics dynamical variables brane position bulk metric eq. of motion Action bulk Einstein eq. Nambu-Goto eq. brane coord. bulk coord. brane metric cannot be a dynamical variable g ( Y ) Y I, Y J, g IJ ( Y ) ~ ~ indicates brane quantity

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eq. of motion bulk Einstein eq. Nambu-Goto eq. general solution static, spherical, here consider the case under Schwarzschild anzats asymptotically flat on the brane, empty except for the core × normal coordinate brane polar coordinate coordinate system t, r,,

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eq. of motion bulk Einstein eq. Nambu-Goto eq. general solution static, spherical, here consider the case under Schwarzschild anzats asymptotically flat on the brane, empty except for the core × normal coordinate brane polar coordinate coordinate system t, r,, Nambu-Goto eq. bulk Einstein eq.

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general solution static, spherical, here consider the case under Schwarzschild anzats asymptotically flat on the brane, empty except for the core × normal coordinate brane polar coordinate coordinate system t, r,, Nambu-Goto eq. bulk Einstein eq. static, spherical, Schwarzschild anz. asymptotically flat, empty × normal coordinate brane polar coordinate t, r,,

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static, spherical, Schwarzschild anz. asymptotically flat, empty Nambu-Goto eq. bulk Einstein eq. × normal coordinate brane polar coordinate t, r,, : functions of r & only general line element with from the bulk Einstein eq.

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static, spherical, Schwarzschild anz. asymptotically flat, empty Nambu-Goto eq. bulk Einstein eq. × normal coordinate brane polar coordinate t, r,, : functions of r & only general line element with Here and hereafter, subscripts and r of functions mean partial differentiations. with

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from the bulk Einstein eq. static, spherical, Schwarzschild anz. asymptotically flat, empty Nambu-Goto eq. bulk Einstein eq. × normal coordinate brane polar coordinate t, r,, : functions of r & only general line element with Here and hereafter, subscripts and r of functions means partial differentiations. with partial differentiations.

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from the bulk Einstein eq. static, spherical, Schwarzschild anz. asymptotically flat, empty Nambu-Goto eq. bulk Einstein eq. × normal coordinate brane polar coordinate t, r,, : functions of r & only general line element with also depend on, which we eliminate.

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from the bulk Einstein eq. static, spherical, Schwarzschild anz. asymptotically flat, empty Nambu-Goto eq. bulk Einstein eq. × normal coordinate brane polar coordinate t, r,, : functions of r & only general line element with also depend on, which we eliminate.

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from the bulk Einstein eq. static, spherical, Schwarzschild anz. asymptotically flat, empty Nambu-Goto eq. bulk Einstein eq. with also depend on, which we eliminate. bulk Einstein eq.

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expand in normal coordinate polar cordinate static, spherical, Schwarzschild anz. asymptotically flat, empty Nambu-Goto eq. bulk Einstein eq. ~ indicates brane quantity bulk Einstein eq.

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expand in normal coordinate polar cordinate static, spherical, Schwarzschild anz. asymptotically flat, empty Nambu-Goto eq. bulk Einstein eq. Equating the n terms in ( n 1)( n 2) [ n 2] [n][n] [n][n] we have : written in terms of & the lower.,,

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expand in normal coordinate polar cordinate static, spherical, Schwarzschild anz. asymptotically flat, empty Nambu-Goto eq. bulk Einstein eq., Equating the n terms in ( n 1)( n 2) [ n 2] [n][n] [n][n] we have : written in terms of & the lower.,,,

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expand in normal coordinate polar cordinate static, spherical, Schwarzschild anz. asymptotically flat, empty Nambu-Goto eq. bulk Einstein eq., Equating the n terms in ( n 1)( n 2) [ n 2] [n][n] [n][n] we have : written in terms of & the lower.,,,,,

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, Equating the n terms in we have : written in terms of & the lower.,,,,, expand in normal coordinate polar cordinate static, spherical, Schwarzschild anz. asymptotically flat, empty Nambu-Goto eq. bulk Einstein eq. [n][n] [n][n] include q [ n 1] use

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, Equating the n terms in we have : written in terms of & the lower.,,,,, expand in normal coordinate polar cordinate static, spherical, Schwarzschild anz. asymptotically flat, empty Nambu-Goto eq. bulk Einstein eq. : written with & the lower. [n][n] [n][n] include q [ n 1] use is just like but the highest terms with are omitted

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, Equating the n terms in we have : written in terms of & the lower.,,,,, expand in normal coordinate polar cordinate static, spherical, Schwarzschild anz. asymptotically flat, empty Nambu-Goto eq. bulk Einstein eq. : written with & the lower., [n][n] [n][n] include q [ n 1] use

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, Equating the n terms in : written in terms of & the lower.,,,,, we have expand in normal coordinate polar cordinate static, spherical, Schwarzschild anz. asymptotically flat, empty Nambu-Goto eq. bulk Einstein eq. : written with & the lower., [n][n] [n][n],

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static, spherical, Schwarzschild anz. asymptotically flat, empty Nambu-Goto eq. bulk Einstein eq., Equating the n terms in we have : written in terms of & the lower.,,,,, : written with & the lower.,,,,

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static, spherical, Schwarzschild anz. asymptotically flat, empty Nambu-Goto eq. bulk Einstein eq.,, 3 equations for 5 functions lowest order determine Nambu-Goto eq. Here we solve Define Then

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3 equations for 5 functions Here we solve Define Then Nambu-Goto eq.

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3 eqs. for 5 functions 3 equations for 5 functions Here we solve Define Then Nambu-Goto eq.

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1) arbitrary functions: two of a, b, c with a b 2 c 0 are 1st-rank linear diff. eqs. for solution solvable 3 eqs. for 5 functions

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2) arbitrary functions: are 2nd order algebraic eqs. for solving without dynamics: choose as arbitrary. solving with dynamics: solvable are 1st-rank linear diff. eqs. for with the elliptic constraint a unique solution exists as far as 3 eqs. for 5 functions The general solution is given by that of It should be modified by the higher order consideration in.

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static, spherical, Schwarzschild anz. asymptotically flat, empty Nambu-Goto eq. bulk Einstein eq.,, Conclusion In order to examine how the braneworld theory reproduce the successful predictions of the Einstein gravity theory, we are seeking for the general spherical solution of the system of the bulk Einstein equation and Nambu-Goto equation. Here we found the general solution of the lowest order in

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Conclusion(cont'd) It should be modified by the higher order consideration in. Thank you for listening and discussions. 3 eqs. for 5 functions 1) with arbitrary functions: two of a, b, c with a b 2 c 0 solution 2) with arbitrary functions: Or, if we assume the ansatz only on the brane (since we know nothing about the outside), this is the general solution! The general sol. is that of

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