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Braneworld の基本方程と 一般解の構造 Akama, T. Hattori, and H. Mukaida Ref.(partial) K. Akama, T. Hattori, and H. Mukaida, arXiv:1109.0840 [gr-qc] 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.

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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 constants g ( Y ) Y I, Y J, g IJ ( Y ) matter action ~ / ~ indicates brane quantity bulk en.mom.tensor coord. 0 g IJ YIYI bulk Ricci tensor

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bulk Einstein eq. Nambu-Goto eq. bulk Einstein eq. Nambu-Goto eq.

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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 dominance of the collective mode Y I among matters bulk Einstein eq. Nambu-Goto eq. empty

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bulk Einstein eq. 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

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bulk Einstein eq. 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 初めに bulk Einstein eq. alone の一般解を求めま す。 use again later

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bulk Einstein eq. Nambu-Goto eq. En.-mom. conservation If we assume we have if are guaranteed. Therefore, the independent equations are Def. covariant derivative with

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bulk Einstein eq. Nambu-Goto eq. Def. Therefore, the independent equations are

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bulk Einstein eq. Nambu-Goto eq. independent eqs. Therefore, the independent equations are

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The only independent non-trivial components expansion reduction rule & derivatives) bulk Einstein eq. Nambu-Goto eq. independent eqs. ( using diffeo.

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expansion reduction rule & derivatives) bulk Einstein eq. Nambu-Goto eq. independent eqs. ( 2 2 2 [n][n] [ n 2] 1 n ( n 1) using diffeo. 2 ___ 2 2 2 2 2 2 2 2 [ n 2]

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expansion bulk Einstein eq. Nambu-Goto eq. independent eqs. [n][n] 1 n ( n 1) rule [ n 2] using diffeo. reduction rule

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expansion bulk Einstein eq. Nambu-Goto eq. independent eqs. [n][n] 1 n ( n 1) rule [ n 2]

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The only independent non-trivial components expansion bulk Einstein eq. Nambu-Goto eq. independent eqs. rule

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The only independent non-trivial components expansion bulk Einstein eq. independent eqs. rule Nambu-Goto eq.

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recursion formulae for if obey in the bulk Thus, we have expansion bulk Einstein eq. independent eqs. rule Nambu-Goto eq.

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bulk Einstein eq. We have Thus, we have recursion formulae for if obey in the bulk if obey Nambu-Goto eq. use again later

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The only independent non-trivial components bulk Einstein eq. Nambu-Goto eq. We have if obey [0] [1] [0] [1] [0] [1] [0] Nambu-Goto eq.

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The only independent non-trivial components Nambu-Goto eq. bulk Einstein eq. [0] [1] [2] 222 [0] substitute if obey We have [2] 2 2 2 [0]

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bulk Einstein eq. We have The solution include three arbitrary functions. Two equations for five functions if obey Nambu-Goto eq.

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bulk Einstein eq. Nambu-Goto eq. We have The solution include three arbitrary functions. Two equations for five functions if obey ( uf [0] ) r uu vv 2v2v 2w2w 2 2 (2 r 2 w ) r u r 2wr2wr u ( ) ( v w )

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uu vv 2v2v 2w2w 2 2 u r 2wr2wr u ( ) ( v w ) bulk Einstein eq. Nambu-Goto eq. We have The solution include three arbitrary functions. Two equations for five functions if obey

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bulk Einstein eq. Nambu-Goto eq. We have The solution include three arbitrary functions. Two equations for five functions if obey uv 2 uw 2 vw w2w2 u v u w 2 v w 2 w 2

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bulk Einstein eq. Nambu-Goto eq. We have The solution include three arbitrary functions. Two equations for five functions if obey u v u w 2 v w 2 w 2

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bulk Einstein eq. Nambu-Goto eq. We have The solution include three arbitrary functions. Two equations for five functions if obey

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bulk Einstein eq. Nambu-Goto eq. We have The solution include three arbitrary functions. Two equations for five functions if obey

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bulk Einstein eq. Nambu-Goto eq. We have The solution include three arbitrary functions. Two equations for five functions if obey use again later

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bulk Einstein eq. Nambu-Goto eq. We have The solution include three arbitrary functions. Two equations for five functions if obey :arbitrary

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bulk Einstein eq. Nambu-Goto eq. We have The solution include three arbitrary functions. Two equations for five functions if obey U U UU U :arbitrary

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U U U U bulk Einstein eq. Nambu-Goto eq. We have The solution include three arbitrary functions. Two equations for five functions if obey :arbitrary

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bulk Einstein eq. Nambu-Goto eq. We haveif obey P Q solution :arbitrary 1st order linear differential equationssolvable!

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bulk Einstein eq. Nambu-Goto eq. We haveif obey : written with :arbitrary solution 1st order linear differential equationssolvable!

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bulk Einstein eq. Nambu-Goto eq. We haveif obey : written with solution :arbitrary solution

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:arbitrary bulk Einstein eq. Nambu-Goto eq. We haveif obey : written with solution

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bulk Einstein eq. Nambu-Goto eq. We haveif obey : arbitrary : written with solution : written with solution :arbitrary

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bulk Einstein eq. Nambu-Goto eq. We haveif obey : arbitrary : linear, solvable : arbitrary algebraic eq. for solvable braneworld bulk Einstein eq. +Nambu-Goto eq. 3 eqs. for 5 functions 3 arbitrary functions 2 eqs. for 5 functions 2 arbitrary functions 2 of u, v, w arbitrary: linear, solvable arbitrarynon-linear differential eq. for u, v, w (1) u, v, w arbitrary collective mode dominance (2) (1) (2) Israel junction condition

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