Braneworld の基本方程と一般解の構造 Akama , T. Hattori, and H. Mukaida

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.

Braneworld Dynamics brane coord. bulk coord. bulk metric dynamical variables brane position ~ gmn(Y)=YI,mYJ,n gIJ(Y) brane metric cannot be a dynamical variable it cannot fully specify the state of the brane constants 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. brane en.mom.tensor Nambu-Goto eq.

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

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

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

Nambu-Goto eq. bulk Einstein eq. 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 初めにbulk Einstein eq. alone の一般解を求めます。 use again later

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

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

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

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

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

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

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

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

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

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

Nambu-Goto eq. Nambu-Goto eq. bulk Einstein eq. = We have = 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] obey if

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

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

Nambu-Goto eq. bulk Einstein eq. = obey if We have ur 2wr ( ) + (2r2w)r (uf[0] )r -v -2w -2v -u + u (v -w) + 2 2 Two equations for five functions The solution include three arbitrary functions.

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

Nambu-Goto eq. bulk Einstein eq. = obey if We have 2 v 2 u w w v 2 u w Two equations for five functions The solution include three arbitrary functions. uv 2uw 2vw w2

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

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

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

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

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

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

Nambu-Goto eq. bulk Einstein eq. = obey if We have :arbitrary = P = Q 1st order linear differential equations solvable! solution

Nambu-Goto eq. bulk Einstein eq. = obey if We have :arbitrary : written with 1st order linear differential equations solvable! solution

Nambu-Goto eq. bulk Einstein eq. = obey if We have :arbitrary : written with solution solution

Nambu-Goto eq. bulk Einstein eq. = obey if We have :arbitrary : written with solution : arbitrary

Nambu-Goto eq. bulk Einstein eq. = obey if We have :arbitrary : written with solution : arbitrary : written with solution

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

Thank you

Braneworld の基本方程と一般解の構造 Akama , T. Hattori, and H. Mukaida