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Branas
Branas
Branas
Branas
Branas
Branas
Branas
Branas
Branas
Branas
Branas
Branas
Branas
Branas
Branas
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Branas

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  • 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. 1. Introduction
    • Brane world is a sub-manifold embedded in a higher dimensional bulk spacetime
    • Matter fields live only on the brane
    • Gravitational field lives both on the brane and in the bulk
    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. Closed strings and Open strings
  • 4. Intrinsic and Extrinsic Geometry Eratostenes ( B.C.276‐195 ) Riemann
  • 5. Randall-Sundram model
    • 5-dimensional Anti-de Sitter bulk
    • 4-dimensional self-gravitating brane
    • Z 2 –symmetry
    Phys.Rev.Lett.83,4690,(1999)
    • Minkowski brane
    • 4-dimensional gravity on the brane
    There are lots of works on brane cosmology
  • 6. Basic Equations 5-dimensional Einstein equation Junction condition
  • 7. Short Cut
  • 8. We study
    • causal structure of the brane universe,
    and propagation of gravitational waves. H.Ishihara, Phys. Rev. Lett. 86, 063513 (2001); H.Ishihara, Phys. Rev. D66 (2002) 023513.
  • 9. 2.Geodesics in the Brane Universe induced metric geodesic on the brane geodesic in the bulk Gauss-Weingarten equation
  • 10.
    • Stress tensor
    Vacuum Brane Extrinsic curvature Vacuum branes are flat in null direction!
  • 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. 3. Causal Structure of Friedman Brane
    • We draw the Penrose’s conformal diagram with respect to the null geodesics in the bulk.
  • 13. Minkowski Brane
  • 14. Friedmann-brane Closed Flat Open
  • 15. Short Cuts exist in Friedmann brane !
  • 16. 4.Propagation of Gravitational Waves Perturbations of the 5-dimensional Einstein equation in the bulk Neumann boundary condition on the brane
  • 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. 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. Einstein Static Brane is concave Typical dimensions Stress tensor Extrinsic curvature Bulk is negatively curved very much
  • 20. Perturbations
    • We consider a massless scalar field    living in the bulk which mimics the gravitational wave.
    avoid gage fixing etc.
  • 21. Wave Equation Setting we obtain Neumann boundary condition and regularity We solve the eigenvalue problem by shooting method. Klein-Gordon equation where
  • 22. Mode Functions n : number of nodes in the extra-dimension k : wavenumber along the brane  : frequency
  • 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. Dispersion relation Frequency on the brane and wavenumber on the brane phase velocity on the brane and group velocity on the brane
  • 25. Phase Velocity The phase velocity for n=0,1,2 modes are plotted.  b =5
  • 26. Group Velocity The group velocity for n=0,1,2 modes are plotted.  b =5
  • 27.
    • Apparent causality vaiolation occurs in the brane universe if null geodesics in the bulk are taken into account.
    Conclusions
    • The phase velocity on the brane for each mode n=0,1,2 exceeds the light velocity.
    • approaches to the light velocity in large as
    • The group velocity on the brane for each mode n=0,1,2 is also larger than the light velocity.
  • 28. Implications Preliminary result : GW Change in wave patterns

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