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 **The**ory

Denitsa Staicova, Plamen Fiziev,

S**of**ia University

8-th **of** April 2010, IRC-CoSiM , S**of**ia

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

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Maximal redshift z=8.2 = 13.01

billions **of** light years distance

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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

**The**ories **of** GRB origin; Fireball model

Collapsar model:

(long GRBs)

-galaxies with

active star

formation

-s**of**ter 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

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arXiv:astro-ph/0711.1163v2

GRB 070201 – short hard GRB located in

Andromeda Galaxy(M31)

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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 pr**of**ile:

**The** Angular Equation:

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u=cos(θ), Ω=aω, s=-1, m=0,±1,±2,... **and** can be

solved in terms **of** HeunC functions...

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**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

**The**oretical 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-s**of**ia.bg/research

Bibliography

●

Burrows, D. N., et al. 2006, ApJ, 653, 468

●

M. Nysew**and**er, 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-s**of**ia.bg/conf/GAS/files/Plamen Fiziev.pdf

http://tcpa.uni-s**of**ia.bg/conf/GAS/files/GRB Central Engine.pdf

http://tcpa.uni-s**of**ia.bg/research/DStaicova Lesvos.pdf

http://tcpa.uni-s**of**ia.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\