supported by Czech grant MSM 4781305903

1. MEASUREMENT OF BRANY BLACK HOLE PARAMETERSIN THE FRAMEWORKOFTHE ORBITAL RESONANCE MODEL OF QPOs Zdeněk Stuchlík and Andrea Kotrlová Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, Bezručovo nám. 13, CZ-74601 Opava, CZECH REPUBLIC supported byCzech grantMSM 4781305903 Presentation download: www.physics.cz/research in sectionnews

2. Outline 1. Braneworld, black holes & the 5th dimension 1.1. Rotating braneworld black holes 2. Quasiperiodic oscillations (QPOs) 2.1. Black hole high-frequency QPOs in X-ray 2.2. Orbital motion in a strong gravity 2.3. Keplerian and epicyclic frequencies 2.4. Digest of orbital resonance models 2.5. Resonance conditions 2.6. Strong resonant phenomena - "magic" spin 3. Applications to microquasars 3.1. Microquasars data: 3:2 ratio 3.2. Results for GRO J1655-40 3.3. Results for GRS 1915+105 3.4. Conclusions 4. References

3. Braneworld, black holes & the 5th dimension Braneworld model - Randall & Sundrum (1999): - our observable universe is a slice, a "3-brane" in 5-dimensional bulk spacetime

4. 1.1. Rotating braneworld black holes • Aliev & Gümrükçüoglu (2005): • exact stationary and axisymmetric solutions describing rotating BH localized on a 3-brane in the Randall-Sundrum braneworld • The metric form on the 3-brane • assuming a Kerr-Schild ansatz for the metric on the branethe solution in the standard Boyer-Lindquist coordinates takes the form where

5. 1.1. Rotating braneworld black holes • Aliev & Gümrükçüoglu (2005): • exact stationary and axisymmetric solutions describing rotating BH localized on a 3-brane in the Randall-Sundrum braneworld • The metric form on the 3-brane • assuming a Kerr-Schild ansatz for the metric on the branethe solution in the standard Boyer-Lindquist coordinates takes the form where • looks exactly like the Kerr-Newman solution in general relativity, in which the square of the electric charge Q2 is replaced by a tidal charge parameter .

6. 1.1. Rotating braneworld black holes • The tidal charge • means an imprint of nonlocal gravitational effects from the bulk space, • may take on both positive and negative values! The event horizon: • the horizon structuredepends on the sign of the tidal charge for condition: for extreme horizon and

7. 1.1. Rotating braneworld black holes • The tidal charge • means an imprint of nonlocal gravitational effects from the bulk space, • may take on both positive and negative values! The event horizon: • the horizon structuredepends on the sign of the tidal charge for condition: for extreme horizon and

8. 1.1. Rotating braneworld black holes • The tidal charge • means an imprint of nonlocal gravitational effects from the bulk space, • may take on both positive and negative values! The event horizon: • the horizon structuredepends on the sign of the tidal charge This is not allowed in the framework of general relativity! for condition: for extreme horizon and

9. 1.1. Rotating braneworld black holes • The tidal charge • means an imprint of nonlocal gravitational effects from the bulk space, • may take on both positive and negative values! The event horizon: • the horizon structuredepends on the sign of the tidal charge This is not allowed in the framework of general relativity! for condition: for extreme horizon and • The effects of negative tidal charge • tend to increase the horizon radius rh, the radii of the limitingphoton orbit (rph), the innermost bound (rmb) and the innermost stable circular orbits (rms) for both direct and retrograde motions of the particles, • mechanism for spinning up the black hole so that its rotation parameter exceeds its mass. Such a mechanism is impossible in general relativity!

10. 2. Quasiperiodic oscillations (QPOs) Blackhole hi-frequency QPOsin X-ray Fig. on this page:nasa.gov

11. 2.1. Quasiperiodic oscillations low-frequency QPOs hi-frequency QPOs (McClintock & Remillard 2003)

12. 2.1.Quasiperiodic oscillations (McClintock & Remillard 2003)

13. 2.2. Orbital motion in a strong gravity Rotating braneworld BH with massM, dimensionless spina, and the tidal charge:the formulae for • the Keplerian orbital frequency • and the related epicyclic frequencies (radial , vertical ): ν ~ 1/M Stable circular geodesics exist for has a local maximum for all values of spin a - only for rapidly rotating BHs xms – radius of the marginally stable orbit

14. 2.3. Keplerian and epicyclic frequencies

15. 2.3. Keplerian and epicyclic frequencies - can have a maximum at x = xex! Notice, that reality condition must be satisfied

16. 2.3. Keplerian and epicyclic frequencies • Can it be located above • the outher BH horizon xh • the marginally stable orbit xms? - can have a maximum at x = xex! Notice, that reality condition must be satisfied

17. 2.3. Keplerian and epicyclic frequencies • Can it be located above • the outher BH horizon xh • the marginally stable orbit xms? - can have a maximum at x = xex! Notice, that reality condition must be satisfied Extreme BHs:

18. 2.3. Keplerian and epicyclic frequencies

19. 2.3. Keplerian and epicyclic frequencies

20. 2.3. Keplerian and epicyclic frequencies

21. 2.4. Digest of orbital resonance models

22. 2.4. Digest of orbital resonance models

23. 2.5. Resonance conditions • determine implicitly the resonant radius • must be related to the radius of the innermost stable circular geodesic

24. 2.5. Resonance conditions

25. 2.5. Resonance conditions

26. 2.5. Resonance conditions

27. 2.5. Resonance conditions

28. 2.5. Resonance conditions

29. 2.5. Resonance conditions

30. 2.6. Strong resonant phenomena - "magic" spin

31. 2.6. Strong resonant phenomena - "magic" spin

32. 2.6. Strong resonant phenomena - "magic" spin

33. 2.6. Strong resonant phenomena - "magic" spin

34. 2.6. Strong resonant phenomena - "magic" spin

35. 2.6. Strong resonant phenomena - "magic" spin

36. 2.6. Strong resonant phenomena - "magic" spin

37. 2.6. Strong resonant phenomena - "magic" spin

38. 2.6. Strong resonant phenomena - "magic" spin

39. 2.6. Strong resonant phenomena - "magic" spin

40. 3. Applications to microquasars GRO J1655-40

41. 3. Applications to microquasars GRS 1915+105

42. 3.1. Microquasars data: 3:2 ratio Török, Abramowicz, Kluzniak, Stuchlík 2005

43. 3.1. Microquasars data: 3:2 ratio Török, Abramowicz, Kluzniak, Stuchlík 2005

44. 3.1. Microquasars data: 3:2 ratio Using known frequencies of the twin peakQPOs and the known mass Mof the central BH, the dimensionless spin aand the tidal charge  can be related assuming a concrete version of the resonance model.

45. 3.1. Microquasars data: 3:2 ratio Using known frequencies of the twin peakQPOs and the known mass Mof the central BH, the dimensionless spin aand the tidal charge  can be related assuming a concrete version of the resonance model.

46. 3.1. Microquasars data: 3:2 ratio Using known frequencies of the twin peakQPOs and the known mass Mof the central BH, the dimensionless spin aand the tidal charge  can be related assuming a concrete version of the resonance model.

47. 3.1. Microquasars data: 3:2 ratio Using known frequencies of the twin peakQPOs and the known mass Mof the central BH, the dimensionless spin aand the tidal charge  can be related assuming a concrete version of the resonance model.

48. 3.1. Microquasars data: 3:2 ratio Using known frequencies of the twin peakQPOs and the known mass Mof the central BH, the dimensionless spin aand the tidal charge  can be related assuming a concrete version of the resonance model. The most recent angular momentumestimates from fits of spectral continua: - Shafee et al. 2006 - McClintock et al. 2006 - Middleton et al. 2006 GRO J1655-40:a ~ (0.65 - 0.75) GRS 1915+105:a > 0.98a~ 0.7

49. 3.2. Results for GRO J1655-40

50. 3.2. Results for GRO J1655-40 McClintock & Remillard 2004 Shafee et al. 2006 Possible combinations of mass and spin predicted by individual resonance models for the high-frequency QPOs. Shaded regions indicate the likely ranges for the mass (inferred from optical measurements of radial curves) and the dimensionless spin (inferred from the X-ray spectral data fitting) of GRO J1655-40.