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

2. 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

3. 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).

4. Higher Dimensional Spacetime: ADD-model

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

6. 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

7. AdS/CFT
Holography (Bekenstein, Susskind, ‘t Hooft, Bousso)

8. AdS/CFT Maldacena conjecture (AdS/CFT) (1997)
Conformal=scale invariance, Roughly speaking

9. AdS/CFT Randall-Sundrum model provides a naturla set up to study
On the RS brane: gravity+CFT

10. 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)

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

12. 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?

13. String(conical singularity)
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)

14. 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!!

15. 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-

16. What is the physics of these 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?

17. New Solutions: Accelerated BHs on RS-2 brane
We take
The induced metric on this brane reads (after suitable parameter transformations)

18. 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

19. New Solutions: Accelerated BHs on RS-2 brane
What is the CFT picture?
Tr T=0
CFT
Cosmological constant on brane

20. 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

21. New Solutions: Accelerated BHs on RS-2 brane
F=mA+ O(mAG)
Classical force
Quantum effect

22. 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

23. New Solutions: Lump of dark radiation
The energy momentum tensor
Cosmological constant
radiation

24. 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

25. New Solutions: Lump of dark radiation
Radiation density
radius

26. 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)

27. 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