ブレイン宇宙における重力波の伝播 石原 秀樹 大阪市立大学 共同研究者 田中 泉
1. Introduction <ul><li>Brane world is a sub-manifold embedded in a higher dimensional bulk spacetime </li></ul><ul><li>Ma...
Closed strings and Open strings
Intrinsic and Extrinsic Geometry  Eratostenes  ( B.C.276‐195 ) Riemann
Randall-Sundram model <ul><li>5-dimensional Anti-de Sitter bulk </li></ul><ul><li>4-dimensional self-gravitating brane  </...
Basic Equations 5-dimensional Einstein equation Junction condition
Short Cut
We study <ul><li>causal structure of the brane universe, </li></ul>and propagation of gravitational waves. H.Ishihara, Phy...
2.Geodesics in the Brane Universe induced metric geodesic on the brane geodesic in the bulk Gauss-Weingarten equation
<ul><li>Stress tensor </li></ul>Vacuum Brane Extrinsic curvature Vacuum branes are flat in null direction!
Friedmann Brane Stress tensor  Branes are concave in null direction Extrinsic curvature Short cuts can exist.  Causality i...
3. Causal Structure of Friedman Brane <ul><li>We draw the Penrose’s conformal diagram with respect to the null geodesics i...
Minkowski Brane
Friedmann-brane Closed  Flat  Open
Short Cuts exist in Friedmann brane !
4.Propagation of Gravitational Waves Perturbations of the 5-dimensional Einstein equation in the bulk Neumann boundary con...
We consider Gravitational wave perturbations of the  Einstein static brane  universe.  Why Einstein Static Brane Universe?...
Einstein Static Brane 5-dim. Anti-de Sitter bulk Brane is located on      =    b =const.   Easy to treat perturbations  ...
Einstein Static Brane is concave Typical dimensions Stress tensor Extrinsic curvature Bulk is negatively curved very much
Perturbations <ul><li>We consider a massless scalar field      living in the bulk which mimics the gravitational wave. ...
Wave Equation Setting we obtain Neumann boundary condition and regularity  We solve the eigenvalue problem by shooting met...
Mode Functions n  :  number of nodes in the extra-dimension k  :  wavenumber along the brane  :  frequency
Eigenfunctions Eigenfunctions for  n=0,1,2  and  k=1000  We consider cases: n=0,1,2 Amplitude of  n=0  mode dominates on t...
Dispersion relation Frequency on the brane and wavenumber on the brane  phase velocity on the brane and group velocity on ...
Phase Velocity The phase velocity for  n=0,1,2  modes are plotted.  b   =5
Group Velocity The group velocity for  n=0,1,2  modes are plotted.  b   =5
<ul><li>Apparent causality vaiolation occurs in the brane universe if null geodesics in the bulk are taken into account.  ...
Implications Preliminary result : GW Change in wave patterns
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Branas

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Branas

  1. 1. ブレイン宇宙における重力波の伝播 石原 秀樹 大阪市立大学 共同研究者 田中 泉
  2. 2. 1. Introduction <ul><li>Brane world is a sub-manifold embedded in a higher dimensional bulk spacetime </li></ul><ul><li>Matter fields live only on the brane </li></ul><ul><li>Gravitational field lives both on the brane and in the bulk </li></ul>Gravitational wave is the most hopeful probe to verify the brane universe Brane-world is a great idea which introduces new aspects on the universe model.
  3. 3. Closed strings and Open strings
  4. 4. Intrinsic and Extrinsic Geometry Eratostenes ( B.C.276‐195 ) Riemann
  5. 5. Randall-Sundram model <ul><li>5-dimensional Anti-de Sitter bulk </li></ul><ul><li>4-dimensional self-gravitating brane </li></ul><ul><li>Z 2 –symmetry </li></ul>Phys.Rev.Lett.83,4690,(1999) <ul><li>Minkowski brane </li></ul><ul><li>4-dimensional gravity on the brane </li></ul>There are lots of works on brane cosmology
  6. 6. Basic Equations 5-dimensional Einstein equation Junction condition
  7. 7. Short Cut
  8. 8. We study <ul><li>causal structure of the brane universe, </li></ul>and propagation of gravitational waves. H.Ishihara, Phys. Rev. Lett. 86, 063513 (2001); H.Ishihara, Phys. Rev. D66 (2002) 023513.
  9. 9. 2.Geodesics in the Brane Universe induced metric geodesic on the brane geodesic in the bulk Gauss-Weingarten equation
  10. 10. <ul><li>Stress tensor </li></ul>Vacuum Brane Extrinsic curvature Vacuum branes are flat in null direction!
  11. 11. Friedmann Brane Stress tensor Branes are concave in null direction Extrinsic curvature Short cuts can exist. Causality in the viewpoint of the brane is apparently violated. If
  12. 12. 3. Causal Structure of Friedman Brane <ul><li>We draw the Penrose’s conformal diagram with respect to the null geodesics in the bulk. </li></ul>
  13. 13. Minkowski Brane
  14. 14. Friedmann-brane Closed Flat Open
  15. 15. Short Cuts exist in Friedmann brane !
  16. 16. 4.Propagation of Gravitational Waves Perturbations of the 5-dimensional Einstein equation in the bulk Neumann boundary condition on the brane
  17. 17. We consider Gravitational wave perturbations of the Einstein static brane universe. Why Einstein Static Brane Universe? Friedmann-Robertson-Walker brane Concave in null direction Expanding brane is a moving boundary
  18. 18. Einstein Static Brane 5-dim. Anti-de Sitter bulk Brane is located on  =  b =const. Easy to treat perturbations because the boundary is static
  19. 19. Einstein Static Brane is concave Typical dimensions Stress tensor Extrinsic curvature Bulk is negatively curved very much
  20. 20. Perturbations <ul><li>We consider a massless scalar field    living in the bulk which mimics the gravitational wave. </li></ul>avoid gage fixing etc.
  21. 21. Wave Equation Setting we obtain Neumann boundary condition and regularity We solve the eigenvalue problem by shooting method. Klein-Gordon equation where
  22. 22. Mode Functions n : number of nodes in the extra-dimension k : wavenumber along the brane  : frequency
  23. 23. Eigenfunctions Eigenfunctions for n=0,1,2 and k=1000 We consider cases: n=0,1,2 Amplitude of n=0 mode dominates on the brane. Expansion of the brane would not modify the waves.
  24. 24. Dispersion relation Frequency on the brane and wavenumber on the brane phase velocity on the brane and group velocity on the brane
  25. 25. Phase Velocity The phase velocity for n=0,1,2 modes are plotted.  b =5
  26. 26. Group Velocity The group velocity for n=0,1,2 modes are plotted.  b =5
  27. 27. <ul><li>Apparent causality vaiolation occurs in the brane universe if null geodesics in the bulk are taken into account. </li></ul>Conclusions <ul><li>The phase velocity on the brane for each mode n=0,1,2 exceeds the light velocity. </li></ul><ul><li>approaches to the light velocity in large as </li></ul><ul><li>The group velocity on the brane for each mode n=0,1,2 is also larger than the light velocity. </li></ul>
  28. 28. Implications Preliminary result : GW Change in wave patterns

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