MEASUREMENT OF BRANY BLACK HOLE PARAMETERS
IN THE FRAMEWORK
OF THE 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 by
Czech grant
MSM 4781305903
Presentation download:
www.physics.cz/research
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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
1. 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
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 brane the solution in the standard
Boyer-Lindquist coordinates takes the form
where
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 brane the 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 .
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 structure depends on the sign of the tidal charge
for
condition:
for extreme horizon
and
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 structure depends on the sign of the tidal charge
for
condition:
for extreme horizon
and
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 structure depends on the sign of the tidal charge
This is not allowed
in the framework
of general relativity !
for
condition:
for extreme horizon
and
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 structure depends 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 limiting photon 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 !
2. Quasiperiodic oscillations (QPOs)
Black hole hi-frequency QPOs in X-ray
Fig. on this page: nasa.gov
2.1. Quasiperiodic oscillations
low-frequency
QPOs
hi-frequency
QPOs
(McClintock & Remillard 2003)
2.1. Quasiperiodic oscillations
(McClintock & Remillard 2003)
2.2. Orbital motion in a strong gravity
Rotating braneworld BH with mass M, dimensionless spin a, 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
xms – radius of the marginally stable orbit
has a local maximum for all values of spin a
- only for rapidly rotating BHs
2.3. Keplerian and epicyclic frequencies
2.3. Keplerian and epicyclic frequencies
- can have a maximum at x = xex !
Notice, that reality condition
must be satisfied
2.3. Keplerian and epicyclic frequencies
- can have a maximum at x = xex !
Can it be located above
• the outher BH horizon xh
• the marginally stable orbit xms?
Notice, that reality condition
must be satisfied
2.3. Keplerian and epicyclic frequencies
- can have a maximum at x = xex !
Can it be located above
• the outher BH horizon xh
• the marginally stable orbit xms?
Notice, that reality condition
must be satisfied
Extreme BHs:
2.3. Keplerian and epicyclic frequencies
2.3. Keplerian and epicyclic frequencies
2.3. Keplerian and epicyclic frequencies
2.4. Digest of orbital resonance models
2.4. Digest of orbital resonance models
2.5. Resonance conditions
– determine implicitly the resonant radius
– must be related to the radius of the innermost stable circular geodesic
2.5. Resonance conditions
2.5. Resonance conditions
2.5. Resonance conditions
2.5. Resonance conditions
2.5. Resonance conditions
2.5. Resonance conditions
2.6. Strong resonant phenomena - "magic" spin
2.6. Strong resonant phenomena - "magic" spin
2.6. Strong resonant phenomena - "magic" spin
2.6. Strong resonant phenomena - "magic" spin
2.6. Strong resonant phenomena - "magic" spin
2.6. Strong resonant phenomena - "magic" spin
2.6. Strong resonant phenomena - "magic" spin
2.6. Strong resonant phenomena - "magic" spin
2.6. Strong resonant phenomena - "magic" spin
2.6. Strong resonant phenomena - "magic" spin
3. Applications to microquasars
GRO J1655-40
3. Applications to microquasars
GRS 1915+105
Török, Abramowicz, Kluzniak,
Stuchlík 2005
3.1. Microquasars data: 3:2 ratio
Török, Abramowicz, Kluzniak,
Stuchlík 2005
3.1. Microquasars data: 3:2 ratio
3.1. Microquasars data: 3:2 ratio
Using known frequencies of the twin peak QPOs and the known mass M of the central BH,
the dimensionless spin a and the tidal charge  can be related assuming a concrete version
of the resonance model.
3.1. Microquasars data: 3:2 ratio
Using known frequencies of the twin peak QPOs and the known mass M of the central BH,
the dimensionless spin a and the tidal charge  can be related assuming a concrete version
of the resonance model.
3.1. Microquasars data: 3:2 ratio
Using known frequencies of the twin peak QPOs and the known mass M of the central BH,
the dimensionless spin a and the tidal charge  can be related assuming a concrete version
of the resonance model.
3.1. Microquasars data: 3:2 ratio
Using known frequencies of the twin peak QPOs and the known mass M of the central BH,
the dimensionless spin a and the tidal charge  can be related assuming a concrete version
of the resonance model.
3.1. Microquasars data: 3:2 ratio
Using known frequencies of the twin peak QPOs and the known mass M of the central BH,
the dimensionless spin a and the tidal charge  can be related assuming a concrete version
of the resonance model.
The most recent angular momentum
estimates from fits of spectral continua:
GRO J1655-40:
GRS 1915+105:
a ~ (0.65 - 0.75)
a > 0.98
a ~ 0.7
- Shafee et al. 2006
- McClintock et al. 2006
- Middleton et al. 2006
3.2. Results for GRO J1655-40
McClintock & Remillard 2004
3.2. Results for GRO J1655-40
Shafee et al. 2006
Possible combinations of mass and spin predicted by individual resonance models for the highfrequency 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.
McClintock & Remillard 2004
3.2. Results for GRO J1655-40
Shafee et al. 2006
Possible combinations of mass and spin predicted by individual resonance models for the highfrequency 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.
McClintock & Remillard 2004
3.2. Results for GRO J1655-40
Shafee et al. 2006
Possible combinations of mass and spin predicted by individual resonance models for the highfrequency 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.
The only model which matches the observational constraints
is the vertical-precession resonance (Bursa 2005)
3.2. Results for GRO J1655-40
3.3. Results for GRS 1915+105
McClintock & Remillard 2004
3.3. Results for GRS 1915+105
McClintock & Remillard 2004
3.3. Results for GRS 1915+105
estimate 1
1 - Middleton et al. 2006
McClintock & Remillard 2004
3.3. Results for GRS 1915+105
estimate 1
estimate 2
1 - Middleton et al. 2006
2 - McClintock et al. 2006
3.3. Results for GRS 1915+105
3.4. Conclusions
3.4. Conclusions
β=0
3.4. Conclusions
-1 < β < 0.51
(βmax for a = 0.7)
3.4. Conclusions
-1 < β < 0.51
3.4. Conclusions

-1 < β < 0.51
3.4. Conclusions

-1 < β < 0.51
3.4. Conclusions
-1 < β < 0.51
3.4. Conclusions

-1 < β < 0.51
3.4. Conclusions
-1 < β < 0.51

there is no specific type of resonance model that could work for both sources
simultaneously
4. References
• Abramowicz, M. A. & Kluzniak, W. 2004, in X-ray Timing 2003: Rossi and Beyond., ed. P. Karet, F. K. Lamb, & J. H. Swank, Vol. 714 (Melville:
NY: American Institute of Physics), 21-28
• Abramowicz, M. A., Kluzniak, W., McClintock, J. E., & Remillard, R. A. 2004, Astrophys. J. Lett., 609, L63
• Abramowicz, M. A., Kluzniak, W., Stuchlík, Z., & Török, G. 2004, in Proceedings of RAGtime 4/5: Workshops on black holes and neutron stars,
Opava, 14-16/13-15 October 2002/2003, ed. S. Hledík & Z. Stuchlík (Opava: Silesian University in Opava), 1-23
• Aliev, A. N., & Gümrükçüoglu, A. E. 2005, Phys. Rev. D 71, 104027
• Aliev, A. N., & Galtsov, D. V. 1981, General Relativity and Gravitation, 13, 899
• Bursa, M. 2005, in Proceedings of RAGtime 6/7: Workshops on black holes and neutron stars, Opava, 16-18/18-20 September 2004/2005, ed. S.
Hledík & Z. Stuchlík (Opava: Silesian University in Opava), 39-45
• McClintock, J. E. & Remillard, R. A. 2004, in Compact Stellar X-Ray Sources, ed. W. H. G. Lewin & M. van der Klis (Cambridge: Cambridge
University Press)
• McClintock, J. E., Shafee, R., Narayan, R., et al. 2006, Astrophys. J., 652, 518
• Middleton, M., Done, C., Gierlinski, M., & Davis, S. W. 2006, Monthly Notices Roy. Astronom. Soc., 373, 1004
• Randall, L., & Sundrum, R. 1999, Phys. Rev. Lett. 83, 4690
• Shafee, R., McClintock, J. E., Narayan, R., et al. 2006, Astrophys. J., 636, L113
• Stuchlík, Z. & Török, G. 2005, in Proceedings of RAGtime 6/7: Workshops on black holes and neutron stars, Opava, 16-18/18-20 September
2004/2005, ed. S. Hledík & Z. Stuchlík (Opava: Silesian University in Opava), 253-263
• Stuchlík, Z., Kotrlová, A., & Török, G. 2007: Black holes admitting strong resonant phenomena, submitted
• Stuchlík, Z., Kotrlová, A., & Török, G. 2007: Multi-resonance model of QPOs: possible high precision determination of black hole spin, in prep.
• Török, G., Abramowicz, M. A., Kluzniak,W. & Stuchlík, Z. 2005, Astronomy and Astrophysics, 436, 1
• Török, G. 2005, Astronom. Nachr., 326, 856
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