Squashed KaluzaKlein Black Holes A Window to Extra Dimensions

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Squashed Kaluza-Klein Black Holes - A Window
to Extra Dimensions -

• Hideki Ishihara Jiro Soda

Kyoto University
Osaka City University
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Introduction

• There exist singularities in black holes
• Hence, we need the quantum theory of gravity to
  resolve them
• Superstring theory is a promising candidate of it
• Superstring theory predicts the extra dimensions
• Black holes must be considered in higher
  dimensions
• What kind of black holes exist in higher
  dimensions?

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Kaluza-Klein black holes
To my best knowledge, only two types of BH are
known.
The well known type of Kaluza-Klein BH is
black brane with compact internal space like
black string
Note In the context of the braneworld, we may
have BH localized on the brane.
The other type of BH recently rediscovered by
Ishihara and Matsuno is
Squashed Kaluza-Klein black hole
Purpose of this talk
Squashed black holes could be a window to extra
dimensions in contrast to other type of black
holes.
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Plan of the talk

• Some mathematics
• Squashed black holes
• Hawking radiation of a scalar field
• Generalization?
• Discussion

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How to make
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Torus foliation
With constant
we have foliations
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More intuitive construction

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Hope fibration
Killing vector
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Lens space
Thus, we have shown
U(1) fibre
Lens space
Ninteger
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Squashed Black Holes
Ishihara Matsuno 2006
5d Einstein-Maxwell
Squashed black hole
outer horizon
inner horizon
spatial infinity
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Shape of the horizon
Shape of the horizon is lens space!
p, q integer
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5-d BH embeded in KK
The solution becomes 5-d charged black hole
the horizon is a squashed sphere
For finite
To see the asymptotic geometry, we can perform
the coord. trans.
could be negative
as long as
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Squashed Black Holes Mass, Charge, Surface
gravity
Surface gravity
ADM mass
Charge
Effective 4-d Newton constant
Therefore, we can not measure the size of the
extra dimension.
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What is done, What remains
Done

• 5-d squashed BH without cosmological constant
• 5-d extreme squashed BH with positive
  cosmological constant
• Multi extreme squashed black holes with positive
  cosmological constant

Not yet

• 5-d squshed BH with cosmological constant
• Stability analysis
• 5-d Squashed Kerr-Newman
• Higher dimensional extension

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Scalar field on BH
Hawking radiation is associated to the property
of the horizon
Hence, by looking at the Hawking flux, it would
be possible to know the size of the extra
dimension
Consider the scalar field in the squashed black
hole spacetime
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Spin weighted functions
Using the variable
must be integer
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Radial mode functions
Using new variables,
we have Shrodinger type equation
Except for the zero mode, The potential barrier
is so high.
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Hawking Radiation
Hawking radiation
4-d KK
5-d BH
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Greybody factor
Because of the greybody factor, the zero mode is
dominated
We should solve
with the boundary condition
Equivalently, we can use the following
Ingoing boundary conditions
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Greybody factor
We separate the region into three regions

• Near horizon region
• Intermediate region
• Far region

Near horizon region
Ingoing boundary condition
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Greybody factor
Intermediate region
Far region
Matching conditions
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Greybody factor
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Observability
Luminosity of the Hawking radiation

• M,Q,L determine the size of the extra dimension
• For negative , L is significantly
  enhanced.

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Generalization?
Is it possible to find black hole solution with
the following property
Near the BH horizon
Far from the BH
Is Sasaki-Einstein manifold useful for this
purpose?
At least, the following generalization is
possible
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Conclusion

• Squashed Kaluza-Klein black holes could be a
  window to extra dimensions
• There are many aspects of Kaluza-Klein black
  holes to be explored