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

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

Collimated jets of ultra-relativistic


Extremely powerful: E tot

~ 10 51 -10 56 erg

Duration – from seconds to days

Maximal redshift z=8.2 = 13.01

billions of light years distance

Two types of GRB – short and long

Two phases – prompt emission and


Highly variable in time

Observed by SWIFT,


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).



Theories of GRB origin; Fireball model

Collapsar model:

(long GRBs)

-galaxies with

active star


-softer emission

-GRB/SN connection

Merger of double

system-NS-NS, NS-BH

or the magnetar

scenario: (short


-older galaxies

-harder emission

The missing gravitational waves


GRB 070201 – short hard GRB located in

Andromeda Galaxy(M31)


“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).


to the metric

is described by

the Teukolsky

Master Equation


=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:


are regular singularities, while infinity is irregular


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


● For the radial equation we impose black hole boundary


1. On the BH horizon, we have only incoming waves:

choice of R 1

(r) or R 2


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)



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



The case m=1

Comparison with the QNM case:



Our results so far:

D.S., Fiziev P. P. 2010

arxiv: astro-ph:HE/1002.0480

Collimated jets from

the angular equation

A set of complex


Theoretical formula

of the first two


Qualitative change of

the behaviour of

ω n=0,1

(a) at a=M.

(transition from

black hole to naked


Highly non-trivial

relation ω n


~7645 points

Thank you for your attention!

For more information:


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



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:

 Fiziev.pdf Central Engine.pdf Lesvos.pdf

For more information, check

Fiziev P P, 2009 Classes of Exact Solutions to

Regge-Wheeler and Teukolsky Equations, grqc/0902.1277\

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