1. Introduction

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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

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Closed strings and Open strings
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Intrinsic and Extrinsic Geometry
Riemann
Eratostenes (B.C.276-195)
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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
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Basic Equations
5-dimensional Einstein equation
Junction condition
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Short Cut
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We study

• causal structure of the brane universe,

H.Ishihara, Phys. Rev. Lett. 86, 063513
(2001) H.Ishihara, Phys. Rev. D66 (2002) 023513.
and propagation of gravitational waves.
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2.Geodesics in the Brane Universe
induced metric geodesic on the brane geodesic in
the bulk
Gauss-Weingarten equation
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Vacuum Brane

• Stress tensor

Extrinsic curvature
Vacuum branes are flat in null direction!
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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
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3. Causal Structure of Friedman Brane

• We draw the Penroses conformal diagram with
  respect to the null geodesics in the bulk.

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Minkowski Brane
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Friedmann-brane
Closed Flat
Open
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Short Cuts exist in Friedmann brane !
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4.Propagation of Gravitational Waves
Perturbations of the 5-dimensional Einstein
equation in the bulk
Neumann boundary condition on the brane
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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

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Einstein Static Brane
5-dim. Anti-de Sitter bulk
Brane is located on c cbconst.
Easy to treat perturbations because the
boundary is static
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Einstein Static Brane is concave
Stress tensor
Extrinsic curvature
Typical dimensions
Bulk is negatively curved very much
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Perturbations

• We consider a massless scalar field ,F, living in
  the bulk which mimics the gravitational wave.

avoid gage fixing etc.
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Wave Equation
Klein-Gordon equation
Setting
we obtain
where
Neumann boundary condition and regularity
We solve the eigenvalue problem by shooting
method.
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Mode Functions
n number of nodes in the extra-dimension k
wavenumber along the brane w frequency
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Eigenfunctions
We consider cases
n0,1,2
Eigenfunctions for n0,1,2 and k1000

• Amplitude of n0 mode dominates on the brane.
• Expansion of the brane would not modify the
  waves.

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Dispersion relation
Frequency on the brane and wavenumber on the
brane
phase velocity on the brane and group velocity on
the brane
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Phase Velocity
cb 5
The phase velocity for n0,1,2 modes are plotted.
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Group Velocity
cb 5
The group velocity for n0,1,2 modes are plotted.
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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
  n0,1,2 exceeds the light velocity.
• approaches to the light velocity in large
  as
• The group velocity on the brane for each mode
  n0,1,2 is also larger than the light velocity.

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Implications
Preliminary result
GW
Change in wave patterns