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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. Outline. Higher dimensional spacetime: ADD, Randall-Sundrum models AdS/CFT

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Mohamed Anber HEP Bag Lunch April 1st With Lorenzo Sorbo

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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


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


Higher Dimensional Spacetime

  • Kaluza–Klein 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).


Higher Dimensional Spacetime: ADD-model


Randall-Sundrum model

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

  • Model I: 2 branes


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


AdS/CFT

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


AdS/CFT

  • Maldacena conjecture (AdS/CFT) (1997)

  • Conformal=scale invariance, Roughly speaking


AdS/CFT

Randall-Sundrum model provides

a naturla set up to study

AdS/CFT

On the RS brane: gravity+CFT


AdS/CFT

  • Black holes localized on RS-branes: great opportunity to study BH evaporation (classical BH+CFT corrections)

  • This is important to shed some light on quantum gravity

  • But no localized BH on RS 3-brane (4+1 D spacetime) so far!!

  • Emparan, Horowitz and Myers (EHM) (1999) found BH on RS-2 brane (3+1 D)


EHM construction

  • First we need to understand gravity in 2+1 D

  • Gravity is not dynamical in 2+1 D

  • Vacuum Solution: conical singularity, no BH

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


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?


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)


EHM construction

  • What we get?

    For flat brane (at x=0)

  • 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!!


New Solutions M. Anber and L. Sorbo arXiv:0803.2242

  • 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-


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?


New Solutions: Accelerated BHs on RS-2 brane

  • We take

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


In the bulk

On the brane

This metric describes accelerating BHs

This is the first accelerating BH solution on a brane

New Solutions: Accelerated BHs on RS-2 brane


New Solutions: Accelerated BHs on RS-2 brane

  • What is the CFT picture?

Tr T=0

CFT

Cosmological constant on brane


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


New Solutions: Accelerated BHs on RS-2 brane

  • F=mA+ O(mAG)

Classical force

Quantum effect


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


New Solutions: Lump of dark radiation

  • The energy momentum tensor

Cosmological constant

radiation


New Solutions: Lump of dark radiation

  • This metric is very complicated

  • We study the limiting case with a flat barne

  • For m=0 we obtain

  • This is a flat metric!

  • The energy momentum tensor to first order of m


New Solutions: Lump of dark radiation

Radiation density

radius


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)


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


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