1. Elcio Abdalla Perturbations around Black Hole solutions

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

3. 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):

4. Reissner-Nordstrom solution For a Black Hole with mass M and charge q, in 4 dimensions, we have the solution

5. 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

6. Black Holes with nontrivial topology

7. 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 CFT J.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)

8. P-brane solutions In the bosonic sector of 10 dim type II sugra:

9. P-brane solutions In the bosonic sector of 10 dim type II sugra: Gravity in 10-p and p. Dilaton Tensor field

10. P-brane solution

11. Klein Gordon equation in p-brane background

13. Gravitational Perturbations

14. Effective Potential

16. Time evolution of perturbation

18. Effect of mass

19. 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?

20. Gauss Bonnet

22. 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).

23. 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 =  d 10 x  {-g}[-|  - iA  | 2 –m 2 |  | 2 ]

24. P-branes and AdS/CFT The equation of motion for  contains the A-potential: {-g} -1/2 d/dx[{-g} 1/2 g xx d  /dx]- g tt  2  -m 2  =0 Similar equation for , coupled to 

25. P-branes and AdS/CFT At infinity, solution in terms of Bessel functions (J. de Oliveira):  - Q/x 7-p  x (p-7)/2 [C 1 J (7-p)/2 (  {  2 -m 2 }x)+ 2 [C 2 I (7-p)/2 (  {  2 -m 2 }x)]

26. Rotating solutions Rotating cilinders Closed time curves Instabilities Thermodynamical instabilities?

27. 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