New exact solutions on the Randall-Sundrum 2-brane: lumps of dark

radiation and accelerated black holes

Mohamed AnberHEP Bag Lunch

April 1stWith 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 0

T

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

00

RG

20

22222 drdrdtds

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!!

y

Root H(y)

String(conical singularity)

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)

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

New Solutions: Accelerated BHs on RS-2 brane What is the CFT picture?

Cosmological constant on brane

Tr T=0CFT

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)

Quantum effectClassical force

New Solutions: Lump of dark radiation Now we take To find the brane induced metric

With Friedman-like equation

Hubble parameter Cosmological constantcurvature

Thermal radiation

There is a conical singularity

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