Elcio Abdalla

Perturbations around Black Hole solutions

Acknowledgements

• Bin Wang (Fudan), C. Molina, A. Pavan, J. de Oliveira, C. E. Pellicer (São Paulo), O. Pavel Fernandez Piedra (Cuba).

The Schwarzschild Black Hole

• Birckhoff Theorem: a static spherically

Symmetric solution must be of the form

• Schwartzschild solution in D=d+1 dimensions (d>2):

Reissner-Nordstrom solution

• For a Black Hole with mass M and charge q, in 4 dimensions, we have the solution

Cosmological Constant

• Einstein Equations with a nonzero cosmological constant are

• Λ>0 corresponds to de Sitter space• Λ<0 corresponds to Anti de Sitter

space

Black Holes with nontrivial topology

Quasi-normal modes in AdS space-time

AdS/CFT correspondence:The BH corresponds to an approximately thermal

state in the field theory, and the decay of the test field corresponds to the decay of the perturbation of the state.

The quasinormal frequencies of AdS BH have direct interpretation in terms of

the dual CFTJ.S.F.Chan and R.B.Mann, PRD55,7546(1997);PRD59,064025(1999)G.T.Horowitz and V.E.Hubeny,

PRD62,024027(2000);CQG17,1107(2000)B.Wang et al,

PLB481,79(2000);PRD63,084001(2001);PRD63,124004(2001); PRD65,084006(2002)

P-brane solutions

• In the bosonic sector of 10 dim type II sugra:

P-brane solutions

• In the bosonic sector of 10 dim type II sugra:

• Gravity in 10-p and p.• Dilaton• Tensor field

P-brane solution

Klein Gordon equation in p-brane background

Klein Gordon equation in p-brane background

Gravitational Perturbations

Effective Potential

Effective Potential

Time evolution of perturbation

Time evolution of perturbation

Effect of mass

AdS/CFT• Further perturbative studies are being

performed in view of the AdS/CFT relation.

• It is possible to study perturbations in the backgrounf of geometry/gauge fields and obtain the CFT counterpart

• Question: what is the interpretation of the tensor field for p>1 in the CFT counterpart?

Gauss Bonnet

Gauss Bonnet

Gauss Bonnet

• The Gauss Bonnet term has also influence in the AdS/CFT superconductor behaviour for dimensions up to 5/4.

• Topology of the Black Hole solution leads to new features on the critical benaviour (to appear).

P-branes and AdS/CFT

• Take the bosonic sector of 10 dim type II sugra and consider a scalar field in such a back ground:

S =d10x {-g}[-|-iA| 2 –m2|| 2]

P-branes and AdS/CFT

• The equation of motion for contains the A-potential:

•{-g}-1/2d/dx[{-g}1/2gxxd/dx]-gtt2 -m2=0

• Similar equation for , coupled to

P-branes and AdS/CFT

• At infinity, solution in terms of Bessel functions (J. de Oliveira):

- Q/x7-p

x(p-7)/2[C1J(7-p)/2({2 -m 2}x)+ 2[C2 I(7-p)/2({2 -m 2}x)]

Rotating solutions

• Rotating cilinders• Closed time curves• Instabilities• Thermodynamical instabilities?

Conclusions and Outlook• Comprehension of Black Holes and its

cosmological consequences• Relation between AdS space and Conformal

Field Theory: condensed matter systems• Is there a relation between dS space and

some Field Theory?• Sounds from gravity at extreme conditions• CFT critical behaviour in terms of quasi

normal perturbations• Bizarre solutions and stability