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.
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 </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
Basic Equations 5-dimensional Einstein equation Junction condition
4.Propagation of Gravitational Waves Perturbations of the 5-dimensional Einstein equation in the bulk Neumann boundary condition on the brane
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
Einstein Static Brane 5-dim. Anti-de Sitter bulk Brane is located on = b =const. Easy to treat perturbations because the boundary is static
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. </li></ul>avoid gage fixing etc.
Wave Equation Setting we obtain Neumann boundary condition and regularity We solve the eigenvalue problem by shooting method. Klein-Gordon equation where
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 the brane. Expansion of the brane would not modify the waves.
Dispersion relation Frequency on the brane and wavenumber on the brane phase velocity on the brane and group velocity on the brane
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. </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>
Implications Preliminary result : GW Change in wave patterns