Braneworld black hole and 4D uniqueness theorem

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Braneworld black hole and 4D uniqueness theorem
12th March 2007_at_Paris

• Tetsuya Shiromizu
• Tokyo Institute of Technology

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Outline

• 1 Introduction
• 2 An exercise
• 3 Some naive trials
• 4 Summary and then

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1. Introduction
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Non-existence conjecture?
Tanaka-san, 2003
Braneworld version of adS/CFT correspondence
(Witten, Gubser, )
Classical braneworld black hole 4-dim. black
hole with Hawking radiation
4-dim. black hole evaporates!
Black hole solutions are always dynamical
5-dim. picture
4-dim. picture
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No holographic-hair conjecture!
If the conjecture is true, we can show the
non-existence of holographic matter outside
black holes in four dimensions.
Holographic matter can exist outside BH?
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additional condition?

• Staticity
• Traceless and linear perturbation (from
  K.Maeda-san)

anisotropic part
cf)Reissner-Nordstrom like solution(Dadhich et
al, 2000)
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As a first step

• Perfect fluid satisfying
• can exist outside BH?

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2 An exercise
Shiromizu, Yamada and Yoshino, JMP 47, 112502,
2006
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Basic question

• Can star composed perfect fluid satisfying energy
  condition exists outside BH in static spacetimes?

should be dynamical
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Theorem
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Assumptions

• energy condition
• condition on EOS

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3 Some naive trials
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3.1 Israels way
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Static spacetimes
metric
Event horizon
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Static spacetime on brane
Einstein equation
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Three equations
Stop here
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If vacuum
Equality holds
(spherical symmetry)
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3.2 Bunting Masood-ul-Alams way
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Black hole spacetimes
metric
3-dimensional Einstein equation
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Asymptotically Flatness
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Proof
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Regularity at V0
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Conformal transformation
Glue at V0
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Paste at V0 surface
Positive energy theorem
(The non-negativity of Ricci is important)
Spherical symmetric (Lindblom)
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Braneworld case
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4. Summary and then
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Summary

• Non-existence failed
• Spherical symmetry failed
• Related extension of 4D Uniqueness theorem
  partially done

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Then

• Imposing the spherical symmetry, we can say
  something more about non-existence?
• New divergence identities
• New conformal transformation

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In this morning
to be discussed in this stay