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KERR SUPERSPINARS AS AN ALTERNATIVE TO BLACK HOLES
Zdeněk StuchlíkInstitute of Physics, Faculty of Philosophy and Science,
Silesian university in Opava
RAGtime Opava, 14.9.2011
Coauthors: Stanislav Hledík, Jan Schee and Gabriel Török
Chapter 1: Keplerian accretion discs orbiting Kerr superspinars
Chapter 2: Evolution of superspinars due to Keplerian accretion discs
Chapter 3: Near-extreme Kerr superspinars as sources of extremely high energy particles
Chapter 4: Epicyclic oscillations of Keplerian discs around superspinars
Chapter 5: Appearances of Keplerian discs orbiting Kerr superspinars-comparison to the Kerr black hole cases
Chapter 6: Profiled lines of thin Keplerian rings in the vicinity of superspinar.
Kerr geometry
The line element in Boyer-Linquist coordinates
where is
Kerr geometry
Black hole ... Naked singularity … Superspinar ...
0 ≤ a ≤ 1
1>aconstand1 =R>a S
The hypothetical surface is at Rs=0.1M .
String theory behind superspinars• Hořava et.al. – interior solution of the Godel type matched to the external Kerr solution• Time machine removed by the internal solution• Exact model constructed for 4+1 SUSY black hole solution• Defects- no limits – even supermassive superspinars possible in early universe, in agreement with cosmic censorSuperspinar - Naked Singularity geometry with R
S= 0.1 M .
Properties of the boundary assumed similar to those of theHorizon – non-radiating, absorbing.
Chapter 1
Keplerian accretion discs orbiting Kerr superspinars
[Stuchlík 1980]
Geodesic structureof KNS circular orbit (Keplerian)
Specific energy and specific angular momentum of circular geodesics
Angular velocity with respect to static observers at infinity
Parameter
Keplerian discs
Keplerian discs
Energy efficiency of accretion
There is jump in for in BH and NS side. MSrEK 1a
42.3%11 MS rEa KBH
157.7%11 MS rEa KNS
Efficiency of Keplerian discs
a: (0,1) <=> (1.66,6.53)identical spectra
(Takahashi&Harada,CQG,2010)
Chapter 2
Evolution of superspinars due to Keplerian accretion discs
[Stuchlik 1981, Stuchlík & Hledík 2010]
(Calvani & Nobili, 1979; Stuchlík 1981)
Evolution of Kerr superspinars and black holes
• Accretion rate:
dm/dt ~ 10^(-8) M/year (BH)dm/dt ~ 10^(-9) M/year (KS)
Conversion due to counterrotating disc by almost three order faster than by corrotating discs
• Energy radiated during conversion:
Erad = mc(a) – M(a)
Corotating discs: Erad / Mi ~ 2.5Counterrotating discs: Erad / Mi ~ 10^(-2)
• Inversion of BH spin: Erad / Mi ~ 0.5
Chapter 3
Near extreme Kerr superspinars as an source of extremely high energy particles
[Stuchlik 2011]
Circular orbits at r =1No fine tuning necessary
Chapter 4
Epicyclic oscillations of Keplerian discs around superspinars
[Torok & Stuchlík 2005]
Epicyclic frequencies in Kerr geometry
Epicyclic frequenciesBlack holes:
Epicyclic frequencies
Black hole Naked singularity
Loci of marginally stable orbits and extrema points of epicyclic frequencies
Loci of marginally stable orbits and extrema points of epicyclic frequencies
Epicyclic frequencies (BH)
Epicyclic frequencies (BH)
Epicyclic frequencies (NS)
Epicyclic frequencies (NS)
Resonant radii (NS)
Discoseismology, trapped oscillations,…Axisymmetric modes:
BH (after Kato, Fukue & Mineshige; Wagoner et al.)
NS
Nonaxisymmetric modes…
NSBH
Strong resonant radii ( )θr ν=ν
Orbital frequencies in discs orbiting Kerr superspinars (summary)
Behaviour of orbital epicyclic frequencies very different from black holes
Existence of three radii giving the same frequency ratio
(but with different frequencies)
Strong resonance radius at r = a^2 where the radial and vertical epicyclic frequencies coincide
Possible instabilities
Chapter 5
Appearance of Keplerian discs orbiting Kerr superspinars
[Stuchlík & Schee 2010]
Optical effects in the field of KS (KNS)
• Null geodesic – Integration of Carter equations
• Radial and latitudinal motion
• Light escape cones of LNRF and GF Silhouette of BH, KNS and KS Appearance of Keplerian discs Captured and trapped photons
Carter equations of motion for the case m=0
where is
Radial and latitudinal motion
where we have introduced impact parameters
Latitudinal motion
Turning points are determined by the condition
The extrema of the function are determined by
At the maxima of function , there is
Latitudinal motion
Radial motion
The reality conditions
and
lead to the restrictions on the impact parameter
where is
Radial motion
Defining functions
- determine extrema of surface
- determine where is fulfilled
- determine where is fulfilled
Radial motion
Radial motion
Light escape cones (LEC)
Locally Non-Rotating Frame (LNRF) tetrad
,
,
where is
Light escape cones (LEC)
Geodetic Frame (GF) tetrad (r-th and -th component same as for LNRF)
where is
Light escape cones (LEC)
We construct LEC of source frame (LNRF, GF) for fixed (r0,
0)
in the following procedure
The silhouette of superspinarThe superspinar silhouette is determined by photons that reach its surface and finish their travel there, contrary to the case of the rim of a black hole silhouette that corresponds to photon trajectories spiralling near the unstable spherical photon orbit around the black hole many times before they reach the observer.
The spiralling photons concentrated around unstable spherical photon orbits will create an additional arc characterizing the superspinar (or a Kerr naked singularity)
The shape of the superspinar silhouette (arc) is the boundary of the no-turning point region in plane of the observer.
a=1.001
a=2.0
a=6.0
0=85o
a=1.001
a=2.0
a=6.0
0=60o
KBH, KNS and KSa/
00.998
60o
1.001KNS1.001KS
85o
Appearance of Keplerian discs
Direct image – photons do not cross the equatorial plane.
InDirect image – photons cross the equatorial plane once.
Transfer function method for the emitted light is used.
Integration of null geodesics
- deformation of isoradial curves
- frequency shift factor
- lensing effect
Keplerian discs
Frequency shift factors for accretion Keplerian discs
Frequency shift is defined as
which in particular case of source on circular geodesic reads
Appearance of Keplerian discsDirect Images
The representative rotational parameters are
a: 0.9981, 1.0001, 1.001, 1.01, 1.1, 1.5, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0
The inclination of the observer: 0=85o
Inner edge of the disk: rMS
=rMS
(a)
Outer edge of the disk: r=20M
a=0.9981rms=1.24M
a=1.0001rms=0.94M
a=1.001rms=0.87M
a=1.01rms=0.76M
a=1.1rms=0.67M
a=1.5rms=0.88M
a=2.0rms=1.26M
a=3.0rms=2.17M
a=4.0rms=3.17M
a=5.0rms=4.25M
a=6.0rms=5.39M
a=7.0rms=6.65M
Appearance of Keplerian discsInDirect Images
The representative rotational parameters are
a: 0.9981, 1.0001, 1.001, 1.01, 1.1, 1.5, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0
The inclination of the observer: 0=85o
Inner edge of the disk: rMS
=rMS
(a)
Outer edge of the disk: r=20M
a=0.9981rms=1.24M
a=1.0001rms=0.94M
a=1.001rms=0.87M
a=1.01rms=0.76M
a=1.1rms=0.67M
a=1.5rms=0.88M
a=2.0rms=1.26M
a=3.0rms=2.17M
a=4.0rms=3.17M
a=5.0rms=4.25M
a=6.0rms=5.39M
a=7.0rms=6.65M
Appearance of Keplerian discsDirect Images
The representative rotational parameters are
a: 0.9981, 1.0001, 1.001, 1.01, 1.1, 1.5, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0
The inclination of the observer: 0=30o
Inner edge of the disk: rMS
=rMS
(a)
Outer edge of the disk: r=20M
a=0.9981rms=1.24M
a=1.0001rms=0.94M
a=1.001rms=0.87M
a=1.01rms=0.76M
a=1.1rms=0.67M
a=1.5rms=0.88M
a=2.0rms=1.26M
a=3.0rms=2.17M
a=4.0rms=3.17M
a=5.0rms=4.25M
a=6.0rms=5.39M
a=7.0rms=6.65M
Appearance of Keplerian discsInDirect Images
The representative rotational parameters are
a: 0.9981, 1.0001, 1.001, 1.01, 1.1, 1.5, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0
The inclination of the observer: 0=30o
Inner edge of the disk: rMS
=rMS
(a)
Outer edge of the disk: r=20M
a=0.9981rms=1.24M
a=1.0001rms=0.94M
a=1.001rms=0.87M
a=1.01rms=0.76M
a=1.1rms=0.67M
a=1.5rms=0.88M
a=2.0rms=1.26M
a=3.0rms=2.17M
a=4.0rms=3.17M
a=5.0rms=4.25M
Captured and trapped photons
Captured and trapped photons
Captured and trapped photons
Chapter 6
Profiled lines of thin Keplerian rings in the vicinity of Kerr superspinars.
[Stuchlík & Schee 2011]
Profiled lines
The flux of radiation from monochromatic and isotropic source reads
where is
The resulting formula takes form
Profiled lines
Profiled lines
Profiled lines
The end… and the beginning…The work must go on.
Thank you for your attention