The Spectrum of Electromagnetic Jets from Kerr Black Holes and ...

The Spectrum of Electromagnetic Jets from Kerr
Black Holes and Naked Singularities in the Teukolsky
Perturbation Theory
Denitsa Staicova, Plamen Fiziev,
Sofia University
8-th of April 2010, IRC-CoSiM , Sofia

Gamma-Ray Bursts
●
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Collimated jets of ultra-relativistic
particles
Extremely powerful: E tot
~ 10 51 -10 56 erg
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Duration – from seconds to days
●
Maximal redshift z=8.2 = 13.01
billions of light years distance
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●
●
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Two types of GRB – short and long
Two phases – prompt emission and
afterglow
Highly variable in time
Observed by SWIFT,
FERMI, AGILE and
Swift bursts showing various behaviour
patterns: a steep-to-shallow transition (GRB
050315); a large flare (GRB 050502B), and a
gradually declining afterglow (GRB 050826).
ground-based
observatories

Theories of GRB origin; Fireball model
Collapsar model:
(long GRBs)
-galaxies with
active star
formation
-softer emission
-GRB/SN connection
Merger of double
system-NS-NS, NS-BH
or the magnetar
scenario: (short
GRBs)
-older galaxies
-harder emission

The missing gravitational waves
●
arXiv:astro-ph/0711.1163v2
GRB 070201 – short hard GRB located in
Andromeda Galaxy(M31)
●
arXiv:astro-ph/1001.0165v1
“We present a search for gravitational-wave
signatures in temporal and directional
coincidence with 22 GRBs that had sufficient
gravitational-wave data available. (...)
We exclude neutron star–black hole
progenitors to a median 90% CL exclusion
distance of 6.7 Mpc.”

What is the physical nature of the Central Engine
1) We are not able to observe the
motion of objects in the vicinity of
the central engine of GRB.
2) Instead of a quiet phase after
the hypothetical formation of the
KBH, we observe late time engine
activity (flares) which is hardly
compatible with the KBH model.
3) The visible jets are formed at
distance of 20-100 event horizon
radii (K¨enigl (2006))where one
cannot distinguish the exterior field
of a Kerr black hole from the
exterior field of another rotating
massive compact matter object solely
by measuring the parameters of the
metric – M and a.
The only way to discover the nature of the central engine is to study
its spectrum!!!

Our model
Rotating massive object is described by
the Kerr metric
Under certain boundary conditions that object is a black
hole (for aM).

Perturbations
to the metric
is described by
the Teukolsky
Master Equation
(s=spin!):
=e tmi S Rr
separates the Teukolsky
Master Equation into angular and radial part with ω and A
– complex parameters of separation

Confluent Heun Function
Some transformations:
Highly non-trivial profile:

The Angular Equation:
●
u=cos(θ), Ω=aω, s=-1, m=0,±1,±2,... and can be
solved in terms of HeunC functions...
●
The polynomial requirements:
imposed on HeunC allows us
to obtain:
A s=−1, m
=− 2 −2 m±2 2 m
¿

=e tmi S Rr

Radial Equation:
Here, the horizons:
and
are regular singularities, while infinity is irregular
singularity.
The ring singularity is not a singularity of the
radial equation!
● This equation can be also solved in terms of HeunC

The solutions:

Boundary conditions:
●Stability condition: ω = ω R
+ iω I
and ω I
>0
● For the radial equation we impose black hole boundary
conditions:
1. On the BH horizon, we have only incoming waves:
choice of R 1
(r) or R 2
(r)
2. On spatial infinity, the solution is a linear
combination of incoming and outgoing waves.
In order to have only outgoing waves, we need:
For ω=|ω|e iarg(ω) ,r=|r|e iarg(r) :
And then:

Numerical algorithm – Muller's method:
For each x n
, x n-1
, x n-2
-> f n
, f n-1
, f n-2
; x n +1 :
• Works with confluent Heun Function
• A root is found usually with ~10 iterations
• The algorithm exits when :
x n
- x n-1
< precision

The spectra:
s=-1, M=1/2
At a=0, we find a set of complex frequencies:
Jets: D.S.Fiziev,P. (2010) QNM:Kokkotas,Smidth (1999)
(arxiv:astro-ph:HE/1002.0480)
(arXiv:gr-qc/9909058)
Fiziev P P, D.S., 2009 arxiv: astro-ph:HE/0902.2408, BAJ 10, 2009
Fiziev P P, D.S., 2009 arxiv: astro-ph:HE/0902.2411, BAJ 10, 2009

Changing the rotational
parameter a:
n=0,1 – modes:

Our numerical results are best fit by the formula:
Ω +
=a/2Mr +
- angular
velocity of the horizon

m=0

m=-1
The case m=1

Comparison with the QNM case:
Jets:
QNM:

Our results so far:
D.S., Fiziev P. P. 2010
arxiv: astro-ph:HE/1002.0480
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Collimated jets from
the angular equation
A set of complex
frequencies
Theoretical formula
of the first two
modes
Qualitative change of
the behaviour of
ω n=0,1
(a) at a=M.
(transition from
black hole to naked
singularity)
Highly non-trivial
relation ω n
(a)
●
~7645 points

Thank you for your attention!
For more information:
http://tcpa.uni-sofia.bg/research

Bibliography
●
Burrows, D. N., et al. 2006, ApJ, 653, 468
●
M. Nysewander, A.S. Fruchter & A. Pe’er, A Comparison of the Afterglows of Short- and Long-Duration Gamma-Ray
Bursts, astro-ph:HE/0806.3607v2
●
Ferrari V., Gualtieri L., 2008 Class. Quant. Grav. 40 945-970
●
ABBOTT at al., IMPLICATIONS FOR THE ORIGIN OF GRB 070201 FROM LIGO OBSERVATIONS,
arXiv:0711.1163v2[astro-ph]
●
Teukolsly S A, 1972 PRL 16, 1114 ; Teukolsly S A, 1973 ApJ 185, 635; Press W H, Teukolsly S A, 1973 ApJ 185, 649;
Teukolsly S A, Press W H, 1974 ApJ 193, 443
●
Kokkotas K, 1999 Quasi-Normal Modes of Stars and Black Holes, Living Review 22
●
Decarreau A, Dumont-Lepage M C, Maroni P, Robert A, Ronveaux A, 1978, “Formes Canoniques de Equations
confluentes de l’equation de Heun, Annales de la Societe Scientifique de Bruxelles 92, I-II, 53”; Decarreau A, Maroni P and
Robert A, 1978 Ann. Soc. Buxelles 92 151.
●
Ronveaux A (ed.), 1995 Heun’s Differential Equations, Oxford Univ. Press, New York
●
Fiziev P P, 2006 Class. Qunt. Grav., 23, 2447; Fiziev P P, 2006 Exact Solutions of Regge-Wheeler Equation in the
Schwarzschild Black Hole Interior, gr-qc/0603003; Fiziev P P, 2007 Jour, Phys.: Conf. Ser. 66, 012016;
●
Fiziev P P, Staicova D R, 2009 A new model of the Central Engine of GRB and the Cosmic Jets astro-ph:HE/0902.2408 ;
Fiziev P P, Staicova D R, 2009 Toward a New Model of the Central Engine of GRB astro-ph:HE/0902.2411 ;
●
Fiziev P P, 2009 Classes of Exact Solutions to Regge-Wheeler and Teukolsky Equations, gr-qc/0902.1277\

Picture credits:
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http://www.star.le.ac.uk/~julo/research.html
http://www.swift.ac.uk/xrt_curves/
http://heasarc.gsfc.nasa.gov/docs/objects/grbs/grbs.html
http://tcpa.uni-sofia.bg/conf/GAS/files/Plamen Fiziev.pdf
http://tcpa.uni-sofia.bg/conf/GAS/files/GRB Central Engine.pdf
http://tcpa.uni-sofia.bg/research/DStaicova Lesvos.pdf
http://tcpa.uni-sofia.bg/research
http://imagine.gsfc.nasa.gov/docs/science/know_l1/grbs.html
http://cerncourier.com/cws/article/cern/38294
http://en.wikipedia.org/wiki/Gamma-ray_burst
http://nrumiano.free.fr/Estars/int_bh2.html

For more information, check
Fiziev P P, 2009 Classes of Exact Solutions to
Regge-Wheeler and Teukolsky Equations, grqc/0902.1277\