kerr superspinars as an alternative to black holes zdeněk stuchlík

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