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
2019/4/28

2. 1. Introduction
Brane-world is a great idea which introduces new aspects on the universe model.
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

3. Closed strings and Open strings

4. Intrinsic and Extrinsic Geometry
Riemann
Eratostenes （B.C.276‐195）

5. Randall-Sundram model
Phys.Rev.Lett.83,4690,(1999)
5-dimensional Anti-de Sitter bulk
4-dimensional self-gravitating brane
Z2 –symmetry
Minkowski brane
4-dimensional gravity on the brane
There are lots of works on brane cosmology

6. 5-dimensional Einstein equation
Basic Equations
5-dimensional Einstein equation
Junction condition

7. Short Cut

8. causal structure of the brane universe,
We study
causal structure of the brane universe,
H.Ishihara, Phys. Rev. Lett. 86, (2001);
H.Ishihara, Phys. Rev. D66 (2002)
and
propagation of gravitational waves.

9. 2.Geodesics in the Brane Universe
induced metric
geodesic on the brane
geodesic in the bulk
Gauss-Weingarten equation

10. Vacuum Brane Stress tensor Extrinsic curvature
Vacuum branes are flat in null direction!

11. Friedmann Brane Stress tensor Extrinsic curvature
Branes are concave in null direction
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 c = cb=const.
Easy to treat perturbations
because the boundary is static

19. Einstein Static Brane is concave
Stress tensor
Extrinsic curvature
Typical dimensions
Bulk is negatively curved very much

20. Perturbations
We consider a massless scalar field ,F, living in the bulk which mimics the gravitational wave.
avoid gage fixing etc.

21. Wave Equation Klein-Gordon equation Setting we obtain where
Neumann boundary condition and regularity
We solve the eigenvalue problem by shooting method.

22. Mode Functions n : number of nodes in the extra-dimension
k : wavenumber along the brane
w : frequency

23. Eigenfunctions We consider cases: n=0,1,2
Eigenfunctions for n=0,1,2 and k=1000
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
cb =5
The phase velocity for n=0,1,2 modes are plotted.

26. Group Velocity
cb =5
The group velocity for n=0,1,2 modes are plotted.

27. Conclusions
Apparent causality vaiolation occurs in the brane universe if null geodesics in the bulk are taken into account.
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