New exact solutions on the Randall-Sundrum 2-brane: lumps of dark radiation and accelerated black holes

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New exact solutions on the Randall-Sundrum
2-brane lumps of dark radiation and accelerated
black holes

• Mohamed Anber
• HEP Bag Lunch
• April 1st
• With Lorenzo Sorbo

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Outline

• Higher dimensional spacetime ADD,
  Randall-Sundrum models
• AdS/CFT
• Building black holes on branes (Emparan, Horwoitz
  and Myers construction)
• New solutions on RS 2-Brane Lump of radiation
  and accelerated black holes
• AdS/CFT interpretation

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Higher Dimensional Spacetime

• KaluzaKlein theory (1921) compactified extra
  dimensions, used extensively in unified field
  theories (string theory, M-theory )
• Large extra dimensions (ADD model-1998)
  Hierarchy problem (elevate it).

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Higher Dimensional Spacetime ADD-model
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Randall-Sundrum model

• Randall-Sundrum (I II) models (1999) Hierarchy
  problem, SM lives on a brane in AdS space.
• Model I 2 branes

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Randall-Sundrum model

• Model II single brane (the world lives on the
  Plank brane), the extra dimension is infinite but
  we have 4-D gravity on brane!!
• String implementation for Add and RS models

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AdS/CFT

• Holography (Bekenstein, Susskind, t Hooft,
  Bousso)

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AdS/CFT

• Maldacena conjecture (AdS/CFT) (1997)
• Conformalscale invariance, Roughly speaking

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AdS/CFT
Randall-Sundrum model provides a naturla set up
to study AdS/CFT
On the RS brane gravityCFT
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AdS/CFT

• Black holes localized on RS-branes great
  opportunity to study BH evaporation (classical
  BHCFT corrections)
• This is important to shed some light on quantum
  gravity
• But no localized BH on RS 3-brane (41 D
  spacetime) so far!!
• Emparan, Horowitz and Myers (EHM) (1999) found BH
  on RS-2 brane (31 D)

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EHM construction

• First we need to understand gravity in 21 D
• Gravity is not dynamical in 21 D
• Vacuum Solution conical singularity, no BH

The same form of this metric in 4D describes a
cosmic string
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EHM construction

• RS brane is accelerating in the AdS
• If you want a BH on this brane, this BH should be
  accelerating as well in AdS
• How to implement this picture?

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EHM construction

• Find some solution that describes an accelerating
  BH in AdS background the AdS-C metric
• Then cut it with an empty brane
• That is
  it!!

String(conical singularity)
Root H(y)
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EHM construction

• What we get?
• For flat brane (at x0)
• The conical singularity is dressed with a horizon
  . This horizon is quantum in nature (quantum
  cosmic censorship)
• EHM also found that other solutions for non-flat
  branes (e.g. with AdS brane there is quantum
  corrected BTZ BH)
• Are there any more allowed solutions?
• Yes!!

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New Solutions M. Anber and L. Sorbo
arXiv0803.2242

• We want the most general brane embedding in the
  AdS-C metric. How to do it?
• Take
• Compute the extrinsic curvature
• Write down Israel junction condition
• We find

K
K-
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New Solutions

• But the AdS-C metric is invariant under the
  transformation
• Then we also have
• These are the only allowed solutions!!
• What is the physics of these new solutions?

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New Solutions Accelerated BHs on RS-2 brane

• We take
• The induced metric on this brane reads (after
  suitable parameter transformations)

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New Solutions Accelerated BHs on RS-2 brane

• In the bulk

• On the brane
• This metric describes accelerating BHs
• This is the first accelerating BH solution on a
  brane

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New Solutions Accelerated BHs on RS-2 brane

• What is the CFT picture?

Tr T0 CFT
Cosmological constant on brane
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New Solutions Accelerated BHs on RS-2 brane

• This black hole is accelerating, there must be a
  string (adge on the brane) that can provide such
  acceleration

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New Solutions Accelerated BHs on RS-2 brane

• FmA O(mAG)

Classical force
Quantum effect
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New Solutions Lump of dark radiation

• Now we take
• To find the brane induced metric
• With Friedman-like equation

There is a conical singularity
Cosmological constant
Thermal radiation
Hubble parameter
curvature
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New Solutions Lump of dark radiation

• The energy momentum tensor

Cosmological constant
radiation
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New Solutions Lump of dark radiation

• This metric is very complicated
• We study the limiting case with
  a flat barne
• For m0 we obtain
• This is a flat metric!
• The energy momentum tensor to first order of m

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New Solutions Lump of dark radiation
Radiation density
radius
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New Solutions Lump of dark radiation

• This is a lump of radiation on top of conical
  geometry
• No horizon, No quantum cosmic censorship! (this
  is time-dependent solution)

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Conclusion

• We have presented new solutions on RS-2 brane
• We hope to learn something about the higher
  dimensional case
• AdS/CFT is very exciting , recently many
  applications have been proposed QCD plasma,
  superconductivity, etc..
• It would be great to see what applications one
  can find in the case of RS-2 brane