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: [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.

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

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

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

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

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

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  

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

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

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

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 ___ [ n  2]

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

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

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

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

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

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

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.

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

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

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

uu vv 2v2v 2w2w 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  

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

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

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

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

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

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

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

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

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

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

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

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

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

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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