General Solution of Braneworld with the Schwarzschild ansatz

1. 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 [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.

2. Braneworld, a long history, brief Introduction picture that we live in 3+1 brane Fronsdal('59), Josesh('62) dynamics with the Einstein Hilbert action on the brane does not reproduce Einstein gravity. Regge,Teitelboim('75) (×^×) dynamical model of with braneworld via a higher-dim. soliton K.A.('82) with brane induced gravity Rubakov,Shaposhnikov('83) with trapped massless modes embedding models Maia('84), Pavsic('85) dynamical models • Visser('85), Gibbons,Wiltschire('87) Antoniadis('91), Horava,Witten('96) applied to the superstring D-brane: brane where the string endpoints reside, and which is a higer-dim. soliton in the dual picture Polchinski('95) applied to hierarchy problems Arkani-Hamed,Dimopolos,Dvali('98), Randall,Sundrum('99)

3. (^_^) Motivation Einstein gravity successfully explaines ①theorigin of the Newtonian gravity ②observations on light deflections due to solar gravity, the planetary perihelion precessions, etc. precisely. It is based on the Schwarzschild solution with the ansatz staticity, sphericity, 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 Schwarzschild anzats. Garriga,Tanaka (00), Visser,Wiltshire('03) Casadio,Mazzacurati('03), Bronnikov,Melnikov,Dehnen('03) spherical sols. ref.

4. 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 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. brane en.mom.tensor Nambu-Goto eq.

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

6. Nambu-Goto eq. bulk Einstein eq. alone, empty 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 empty bulk Einstein equation alone.

7. Nambu-Goto eq. bulk Einstein eq. alone, empty 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

8. Nambu-Goto eq. bulk Einstein eq. alone, empty 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

9. Nambu-Goto eq. bulk Einstein eq. alone, empty independent equations = equivalent equation Def. Bianchi identity covariant derivative current conservation If we assume , then with implies are guaranteed. then if Therefore, the independent equations are

10. Nambu-Goto eq. bulk Einstein eq. alone, empty independent equations = Def. independent eqs. Therefore, the independent equations are

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

12. Nambu-Goto eq. bulk Einstein eq. alone, empty 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)

13. Nambu-Goto eq. Nambu-Goto eq. bulk Einstein eq. alone, empty 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)

14. bulk Einstein eq. alone, empty expansion reduction = • rule Def. independent eqs. [n-2] 1 [n] n(n -1)

15. bulk Einstein eq. alone, empty expansion reduction = • rule Def. independent eqs. The only independent non-trivial components

16. bulk Einstein eq. alone, empty expansion reduction = • rule Def. independent eqs. The only independent non-trivial components

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

18. bulk Einstein eq. alone, empty expansion reduction = • rule Def. independent eqs. use again later use again later recursive definition Thus, we obtained in the forms of power series of z, whose coefficients are written with

19. bulk Einstein eq. alone, empty expansion reduction = • rule Def. independent eqs. should obey brane metric extrinsic curvature Thus, we obtained in the forms of power series of z, whose coefficients are written with

20. bulk Einstein eq. alone, empty expansion reduction = • rule We have should obey brane metric extrinsic curvature Thus, we obtained in the forms of power series of z, whose coefficients are written with obey if

21. bulk Einstein eq. alone, empty expansion reduction = • rule 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

22. bulk Einstein eq. alone, empty expansion [0] reduction = [2] • rule 2 [0] [2] We have 2 [0] [2] = 2 recursive definition 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

23. bulk Einstein eq. alone, empty expansion reduction = • rule obey if We have Let be • arbitrary. v u w = Two equations for five functions The solution includes three arbitrary functions.

24. bulk Einstein eq. alone, empty expansion reduction = • rule obey if We have Let be • arbitrary. v u w 2 ()r - ()r 2v w 2r2 u f[0] - w v u + + + + 2 2 Two equations for five functions The solution includes three arbitrary functions.

25. bulk Einstein eq. alone, empty expansion reduction = • rule obey if We have Let be • arbitrary. v u w 2 ()r - ()r 2v w 2r2 u f[0] - w v u + + + + 2 2 = = u f[0] 4w r _____ ___ - - r -2wr f[0] -ur Two equations for five functions The solution includes three arbitrary functions.

26. bulk Einstein eq. alone, empty expansion reduction = • rule obey if We have Let be • arbitrary. v u w Let for a while Two equations for five functions The solution includes three arbitrary functions.

27. bulk Einstein eq. alone, empty expansion reduction = • rule obey if We have Let be • arbitrary. v u w Let for a while Two equations for five functions The solution includes three arbitrary functions.

28. bulk Einstein eq. alone, empty expansion reduction = • rule obey if We have Let be • arbitrary. v u w Let for a while 2 v 2 u w w v 2 u w Two equations for five functions The solution includes three arbitrary functions.

29. bulk Einstein eq. alone, empty expansion reduction = • rule obey if We have Let be • arbitrary. v u w Let for a while 2 v 2 u w w v 2 u w Two equations for five functions The solution includes three arbitrary functions.

30. bulk Einstein eq. alone, empty expansion reduction = • rule obey if We have Let be • arbitrary. v u w Let for a while = linear d. e. U U U U U Two equations for five functions The solution includes three arbitrary functions.

31. bulk Einstein eq. alone, empty expansion reduction = • rule obey if We have Let be • arbitrary. v u w Let for a while linear d. e. U U U U Two equations for five functions The solution includes three arbitrary functions.

32. bulk Einstein eq. alone, empty expansion reduction = • rule obey if We have Let be • arbitrary. v u w Let for a while linear d. e. = P = linear d. e.! Q

33. bulk Einstein eq. alone, empty expansion reduction = • rule obey if We have Let be • arbitrary. v u w Let for a while linear d. e. linear d. e.! (GOG)! 1st order linear differential equations solvable! solution

34. bulk Einstein eq. alone, empty expansion If u = v reduction = • rule w is no longer arbitrary, and f [0] is arbitrary. obey if We have Let Let be be • arbitrary. • arbitrary. v v u u w w Let Let for a while for a while linear d. e. 0 = linear d. e.! (^O^) 1st order linear differential equations solvable! Two equations for five functions Remember this page solution The solution include three arbitrary functions.

35. bulk Einstein eq. alone, empty expansion If u = v reduction = = • rule w is no longer arbitrary, and f [0] is arbitrary. obey if We have Let be • arbitrary. v u w solution Let If for a while with solution

36. bulk Einstein eq. alone, empty expansion If u = v reduction = = • rule w is no longer arbitrary, and f [0] is arbitrary. obey if We have Let Let be be • arbitrary. • arbitrary. v v u u w w solution If with If u = v • : arbitrary. • w is no longer arbitrary.

37. bulk Einstein eq. alone, empty expansion reduction = • rule obey if We have Let Let be be • arbitrary. • arbitrary. v v u u w w solution recursive definition If with In summary for the bulk Einstein eq. alone, empty, in power series of z, we obtained If u = v • : arbitrary. • w is no longer arbitrary. by this whose coefficients are recursively defined in terms of arbitrary written with functions, , . v w , u

38. bulk Einstein eq. alone, empty expansion reduction = • rule obey if We have Let Let be be • arbitrary. • arbitrary. v v u u w w solution If with In summary for the bulk Einstein eq. alone, empty, in power series of z, we obtained If u = v • : arbitrary. • w is no longer arbitrary. whose coefficients are recursively defined in terms of arbitrary written with functions, , . v w , u (^O^) This gives the solution outside the brane. like this

39. bulk Einstein eq. alone, empty expansion reduction = • rule obey if We have Let Let be be • arbitrary. • arbitrary. v v u u w w On the brane Matter is distributed within |z|<d , very small. In summary for the bulk Einstein eq. alone, empty, in power series of z, we obtained whose coefficients are recursively defined in terms of arbitrary written with functions, , . v w , u (^O^) This gives the solution outside the brane.

40. bulk Einstein eq. alone, empty expansion Nambu-Goto eq. reduction = • rule obey if We have Let be • arbitrary. v k u w z z z On the brane Matter is distributed within |z|<d , very small. 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

41. bulk Einstein equation Nambu-Goto eq. obey if We have Let be • arbitrary. 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

42. bulk Einstein equation Nambu-Goto eq. obey if We have Let be • arbitrary. 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 - - - d -d satisfied due to Nambu-Goto eq.

43. bulk Einstein equation Nambu-Goto eq. obey if We have Let be • arbitrary. v k u w z z z Nambu-Goto eq. ≡D ±d ± ± ± ± ± ± ±d sum of ± ± ± ± ± ± ± ± - - - - - d -d - d -d - - - - - - -

44. bulk Einstein equation Nambu-Goto eq. obey if We have Let be • arbitrary. v k u w z z z Nambu-Goto eq. ≡D : arbitrary, substitute substitute - - - - - d -d - d -d - - - - - - -

45. : arbitrary, • equations : arbitrary, - - d -d d -d

46. : arbitrary, • differs in • from the previous • equations solution If with If • is no longer arbitrary. • : arbitrary. • C: constant with

47. : arbitrary, Conclusion • equations solution recursive definition If In summary for the bulk Einstein eq. & NambuGoto eq. with in power series of z >0 and of z<0, we obtained whose coefficients are recursively defined by this If • is no longer arbitrary. in terms of • : arbitrary. arbitrary written with functions, of r. • C: constant with like this

48. : arbitrary, Nambu-Goto eq. Conclusion bulk Einstein eq. = • equations Def. independent eqs. solution recursive definition If In summary for the bulk Einstein eq. & NambuGoto eq. with in power series of z >0 and of z<0, we obtained whose coefficients are recursively defined by this If • is no longer arbitrary. in terms of • : arbitrary. arbitrary written with functions, of r. • C: constant (^O^) This is the general solution of the system. with It has large arbitrariness due to the extrinsic curvature. (×^×) like this

49. other choice of the arbitrary functions Discussions for bulk Einstein eq. alone, empty 2 eqs. for 5 functions 3 arbitrary functions arbitrary instead of u,v,w. We can choose algebraic eq. for solvable become (^_^) for braneworld (bulk Einstein eq. & Nambu-Goto eq.) 2 arbitrary functions 3 eqs. for 5 functions We can choose arbitrary instead of non-linear differential eq. for not solvable, but solution exists. (×^×)

50. 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