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Some Novel Properties and

Applications of Heun’s

Functions to Physical Problems

The

Goal: To

Open the

Padlocks

of

Nature!

Plamen Fiziev

Department of Theoretical Physics

University of Sofia

Informal seminar

Department of Functions and Functional Analysis

Moscow State University

February 22,

2010

The Heun family of equations has been popping up with

surprising frequency in applications during the last 10 years,

for example in general relativity, quantum, plasma, atomic,

molecular, and nano physics, to mention but a few.

This has been pressing for related mathematical

developments, and developments for special functions.

From some point of view, it would not be wrong

to think that Heun equations will represent - in the

XXI century - what the hypergeometric equations

represented in the XX century.

That is: a vast source of ideas for linear differential

equations.

Edgardo S. Cheb-Terrab,

MITACS and Maplesoft, 2004

We present:

1. Recent developments in the theory and observations of

astrophysical black holes.

2. Recent developments in the understanding of physics of

Gamma Ray Bursts and Relativistic Jets.

.

1. New developments in the theory of Heun’s functions.

2. All classes of the exact solutions to the Teukolsky

Master Equation in terms of the confluent Heun's

functions.

3. Recently discovered collimated one-way running waves

as a basis for novel models of relativistic jets.

ArXiv:

0903.010

1 Albert Einstein Institute, Potsdam

2 NASA Goddard Space Flight Center

3 Jet Propulsion Laboratory, California Institute of Technology

4 Massachussets Institute of Technology

arXiv:0901.4365v2 [gr-qc]:

Matt Visser, Black holes in general relativity

Do black holes “exist”

“This innocent question is more subtle than one might expect, and the answer depends

very much on whether one is thinking as an observational astronomer,

a classical general relativist, or a theoretical physicist.”

Astronomers have certainly seen things that are small, dark, and

heavy.

Classical general relativist:

Eternal black holes certainly exist mathematically.

Theoretical physicist:

We have not seen direct observational evidence of the event horizon.

The mathematical solutions suffer essential physical shortcomings !

Visser M, Barcelo C, Liberati S, Sonego S: gr-qc/0902.0346

Small, dark, and heavy: But is it a black hole

New important results for more realistic

models of compact slowly rotating bodies:

Cecilia B M H Chireli, Luciano Rezzolla,

How to tell gravastar from black hole, CQG 24: 4191-4206, 2007

Cecilia B M H Chireli, Luciano Rezzolla, On the ergoregion

instabilityin rotating gravastars, Phys. Rev. D 78, 084011, 2008

Using Numerical methods and WKB approximation:

Highly nontrivial stability properties are derived

Pani P, Berti E, Cardoso V, Chen Y, Norte R

Gravitational wave signatures of the absence of an event horizon.

Nonradial oscillations of a thin-shell gravastar arXiv:0909.0287 [gr-qc]

(Numerical study)

What about the EXACT SOLUTIONS

arXiav:0902.3151 [astro-ph.HE]: Critical Thoughts on Cosmology

arXiv:0905.1028 [astro-ph.HE]:

Original: ApJ Lett. 674 L1-4, 2008, MASS FUNCTIONS OF THE ACTIVE BLACK HOLES

IN DISTANT QUASARS FROM THE SLOAN DIGITAL SKY SURVEY DATA RELEASE 3

M. Vestergaard, X. Fan, C. A. Tremonti, Patrick S. Osmer, and Gordon T. Richards

62,185 quazars

--------------------------

arXiv:

0911.1355

astro-ph.HE

6 Nov 2009

A typical observed real rotation curve

For v = const :

arXiv:0902.351

[astro-ph.HE]

arXiv:0905.1028

[astro-ph.HE]

Instabilities of BH:

1.

2004

2.

2004

3.

2005

4.

2009

2009

A hyper nova 08.09.05 (distance 11.7 bills lys)

Formation of WHAT : BH, OR

VU6APFLG.mov

Series of explosions observed !

astro-ph/0711.11163

Abbott et all, to appear in

ApJ, 2008

Origin of the GRB070201

from LIGO observations

SEARCH FOR GRAVITATIONAL-WAVE INSPIRAL SIGNALS

ASSOCIATED WITH SHORT GAMMA-RAY BURSTS

DURING LIGO’S FIFTH AND VIRGO’S FIRST SCIENCE RUN

Progenitor scenarios for short gamma-ray bursts (short GRBs) include

coalescenses of two neutron stars or aneutron star and black hole,

which would necessarily be accompanied by the emission of strong

gravitational waves. We present a search for these known

gravitational-wave signatures in temporal and directional coincidence

with 22 GRBs that had sufficient gravitational-wave data available in

multiple instruments during LIGO’s fifth science run, S5, and Virgo’s

first science run, VSR1. We find no statistically significant

gravitational-wave candidates within a [-5; +1) s window around the

time of any GRB. Using the Wilcoxon–Mann–Whitney U test, we find no

evidence for an excess of weak gravitational-wave signals in our

sample of GRBs. We exclude neutron star–black hole progenitors to a

median 90% CL exclusion distance of 6.7 Mpc.

arXiv:

1001.0165

astroph.HE

4 Jan

2010

The Central Engine Problem

-- the physical object producing the GRB

-- before SWIFT epoch usually it was considered

to be a Kerr black hole (KBH)

-- the result of the death of a massive star:

for the long GRB (T > 2 s),

or as part of a binary merger of compact objects

(BH-BH, NS-BH, or NS-NS; where NS=neutron star)

for short GRB (T < 2 s ).

The first hypotheses seems to be hardly compatible

with the observed flares, produced via energy

injection by the central engine.

The second hypotheses was recently refuted by

the existing detectors of gravitational waves.

The latest analysis shows that the short and long GRB

may have a similar central engine (except for its duration).

2006HST

2007

HST, ESA

2007News:

Jets from GRB060418 and

GRB060607A:

~ 200 Earth masses with

velocity 0.999997c

gamma=408

=============

2008 News:

GRB 080916C

velocity 0.999999c

gamma=707

=============

2009 FERMI News:

GRB 090510

velocity 0.999999875

875c

gamma=2000

=============

20xx

Natural GR

scales for relativistic jets from AGN:

Power: WG = c^5 / G = 3.6 x 10^59 erg/sec !!!

Force: FG = c^4 / G =1.2 x 10^59 N !!!

Relativistic

uniform

acceleration

due to unknown

GR process:

=>

⇒ Acceleration to v = 0.99999966 c (gamma=1220)

for Time

TEarth = 2 x 10 ^ (-8) sec

TJupiter = 5 x 10 ^ (-6) sec

TSun = 6 x 10 ^ (-3) sec

The processes may be pure gravitational !

NO MAGNETIC FIELDS are needed !!!

The Relativistic Jets:

The Most Powerful and

Misterious Phenomenon

in the Universe,

which is observed at very different scales:

1. Around Single Brown Dwarf (~ 0.01 AU)

2. Around Single Neutron Star (~10-1000 AU

3. In Binary BH–Star, and Star-Star systems

4. In Gamma Ray Burst (GRB) (~1 kPs)

5. Around Galactic Nuclei (~1 MPs)

6. Around Galactic Collisions (~10 MPs)

7. Around Galactic Clusters (~200 Mps)

=> UNIVERSAL NATURE

Nature 435, 652-654 (2 June 2005)

A resolved outflow of matter from a

brown dwarf

Emma T. Whelan, Thomas P. Ray,

Francesca Bacciotti, Antonella Natta,

Leonardo Testi & Sofia Randich

The remains of supernova SNR 0104-72.3 - a double jet

from thermonuclear explosion of a white dwarf.

NASA's

Chandra

X-ray

Observatoy

June 2009

Jets from Neutron Star

Crab Nebula

Discovered by NASA's

Spitzer Space Telescope

``tornado-like``

object Herbig-Haro 49/50,

created from

the shockwaves of powerful

protostellar jet hitting

the circum-stellarmedium.

PPF, D. Staicova,

astro-ph:HE/0902.2408

astro-ph:HE/0902.2411

BAJ, 2009, v.11

A jet of gas firing out of a very young star can be seen ramming into a wall of material

in this infrared image from NASA's Spitzer Space Telescope.

The young star, called HH 211-mm, is cloaked in dust and can't be seen. But streaming away

from the star are bipolar jets, color-coded blue in this view. The pink blob at the end of the jet

to the lower left shows where the jet is hitting a wall of material. The jet is hitting the wall so

hard that shock waves are being generated, which causes ice to vaporize off dust grains.

The shock waves are also heating material up, producing energetic ultraviolet radiation.

The ultraviolet radiation then breaks the water vapor molecules apart.

The Jet

from M87

M87

Jet

Time

Evolution

The gas knot

is 214 lightyears

from the

galaxy’s core.

Some typical

AGN

The Jet in Centaurus A

3C321 Jet :

Black Hole Fires at Neighboring Galaxy

Other observed jets:

Heun’s Differential

Equation:

A KEY

for

Huge

amount

of

Physical

Problems

found

By

Born in Weisbaden April 3, 1859

Died in Karsruhe January 10, 1929

Zur Theorie der Riemann'schen

Functionen zweiter Ordnung

mit Vier Verzweigungs-punkten

Math. Ann. 31 (1889) 161-179

Limit =>

=>

Confluent

Heun

Equation:

The UNIQUE

Frobenius solution

around z = 0 :

Recurrence

relation:

The connection

problem is still

UNSOLVED !

More detail about the three-term recurrence:

In the limit :

An important factor:

An essential GENERALIZATION:

S. Yu. Slavyanov – A Theorem for all Painleve class of classical equations !

Note: All Painleve equations are Euler-Lagrange equations: Slavyanov 1966

Hamilton structure of the Painleve equations : Malmquist, 1922

Some Exactly Soluble

in terms of Heun’s functions

physical problems:

1. Hidrogen Molecule

2. Wasserstoffmoleculeon

3. Two-centre problem in QM (Helium).

4. Anharmonic Oscillators in QM and QFT

5. Stark Effect

6. Repulsion and Attraction of Quantum Levels,

7. 3D Hydrodinamical Waves in non-isotermal Atmosphere

8. Quantum Diffusion of Kinks

9. Cristalline Materials

10. In celestial Mechanics: Moon’s motion

11. Cologero-Moser-Sutherland System

12. Bethe ansatz systems

…

Exact Quantum Anharmonic oscillator:

The

Eigenvalue

Problem:

Three-confluent Heun’s function:

Since the

geodesic

equations

are

solved

in elliptic

functions

we solve

the wave

equations

in Heun’s

functions

Heun’s problems in gravity: perturbations of

1.Schwarzshild

metric: PPF, CQG,2006, J Phys C, 2007

2. Kerr metric PPF, gr-qc/0902.1277,

gr-qc/ 0980.4234.

3. Reisner-Nortstrom metric .

4. Kerr-Newman metric.

For all spins:

|s|=0,1/2,1,3/2,2,…

5. De Sitter metric.

6. Reisner-Nortstrom-de Sitter metric.

7. Interior perturbations of all above solutions of EE.

- for Schwarzschild: PPF gr-qc/0603003.

8. QNM of nonrotating and rotating stars and other compact

objects: naked singularities, superspinars, gravastars,

boson stars, soliton stars, quark stars, fuzz-balls, dark stars…

9. All D-type metrics - Batic D, Schmid H, 2007 JMP 48

10. Relativistic jets: PPF, Staicova D, BAJ, 2009, v.11, pp. 3-12,

BAJ, 2009, v.11, pp.13-21.

arXiv:1002.0480 [astro-ph.HE]

11. Continuous spectrum in TME for s =1/2, 1

Borissov, PPF, gr-qc/0902.3617

Detweiler S, 1980, ApJ 239, 292-295

PPF, CQG (2006):

Perturbations of the BH interior

Matzner (1980), PPF gr-qc/0603003, PPF JournalPhys C, 66, 2006.

Continuous spectrum:

Polynomial solutions:

Pure imaginary discrete spectrum:

Two series: n=0,…,6; and

n=7,… exist. The eigenvalues

In them are placed around the

lines

and

Dependence of eigenvalues on the mixing angle

NEW PHENOMENON

Attraction and Repulsion of the Levels:

Schwarzschild with a thin massles shell at the horizon

The mixing angle alpha

describes the ratio

of the amplitudes

of the waves,

going in and going out

of the horizon:

alpha=0

– no outgoing waves

alpha= Pi/2

– no ingoing waves

Kerr (1963) Metric (a = J/M)

Event horizon, the ergosphere, Cauchy

horizon and the ring singularity

Black hole Naked singularity

g tt = 0, a =.99 g tt = 0, a =1.44

In Boyer – Lindquist (1967) coordinates:

Let

Simple non-algebraic differential invariants

for the Kerr solution:

is the Weyl tensor, - its dual

The density for the Euler

Characteristic class:

The density for the Chern Pontryagin

Characteristic class:

do not see the horizons => coordinate singularities

NO ! Let

Then in Boyer - Lindquist coordinates:

-two independent

nonalgebraic

invariants

The differential invariants:

CAN LOCALLY

SEE

- The two horizons

- The Ergo-surface

For Schwarzschild , :

,

The horizons are non-coordinate geometrical objects.

Horizons are not singularities of some quantities, but can be for other ones:

NEW Horizon Finders

Teukolsky’s Master Equation PRL,

PRL, 29, , 1115 (1972):

a=J/M

Small

perturbations

of spin-weights

s =-2,-3/2,-1,-1/2

0,1/2, 1, 3/2, 2

of Kerr and

Schwarzschild,

background

in terms of Weyl

invariants

Kinnersley

tetrad

basis:

Cone variables

Teukolsky’s integral representation of the solutions:

with a factorized (in general SINGULAR) kernal:

For the nontrivial factors R and S one obtains

a coupled complex system of ODE’s:

The radial equation (TRE):

and the angular equation (TAE):

with complex separation constants

E and

A more general integral representation of the solutions: PF gr-qc/0902.1277

TRE explicit form:

= 0

16 Exact Local Solutions of TRE: PPF gr-qc/0902.1277, 0908.4234

Parameters:

16 Exact Local Solutions of TAE:PPF gr-qc/0902.1277, 0908.4234

Parameters:

For

and

!

Where we introduce the confluent concomitant Heun function:

Universal Form of the Exact Solutions of TAE, TRE and Regge-Wheeler Eqs.

PF: arXiv:0902.1277, arXiv:0906.5108 [gr-qc],

,

For TAE and:

For TRE and:

x

Regge-Wheeler Equation:

Novel relations for confluent Heun’s functions and

their Derivatives, PF:

Self-adjoint form of confluent Heun’s operator:

The comutator:

Chain of confluent Heun’s operators:

The basic

general

relation:

:

The - condition

=> =>

A Novel Identity:

Note that

=>

= N-polynomial

Universal form of the Teukolsky-Starobinsky Identities

PF:

For the above special values of the parameters all solutions

turn to be -solutions. As a result the universal identities

take place:

Generalized

Teukolsky-

Starobinsky

Identities:

As a result of amazing new symmetry for N+1=2|s| :

if + is a solutions with spin-weight +s,

then is a solution of TE with –s !

For the first time: Teukolsky-Starobinsky like identities for

Solutions of Regge-Wheeler Equation:

and Zerilli Equation:

- obtained

using the relations:

New effective method for calculation of

Starobinsky constant for all spin-weights s

Starobinsky constants for different s coincide up to known factor with

the for Taylor series for confluent Heun’s function .

Hence, we can calculate Starobinsky constants using recurrence

relation :

In the case of

-solutions

we have more

simple

coefficients:

Polynomial solutions: (for |s|=2 – Chandrasekhar 1984)

One-way waves !

In terms of confluent Heun function

Two conditions are to be satisfied:

Two NOVEL classes of polynomial solutions:

First class: the automatically satisfied.

Second class: the yields equidistant spectra:

Sumple examples of polynomial solutions of continuous

spectrum: PPF gr-qc/0902.1277, Borissov, PPF gr-qc/0903.3617

For spin =1/2 :

and contnuous

For spin =1:

and contnuous

=

=

NO continuous spectrum for |s|=2 =>

PPF gr-qc/0908.4234: For spin ½ the kernel is:

Stereographic

projection

wave variable

to

Convergence

conditions

!!!

General solution to TME – one-way-running waves:

- an arbitrary analitic function of two complex variables

!!!

Bounded one-way-runing waves with spin ½ :

Choosing:

and

using the

identity:

we obtain:

for the modified Bessels’s functions:

=> Bounded solution => selfadjoint TAE operator

Overall kernels for solutions to Teukolsky’s Master Equation

256 classes:

Different boundary problems:

1. BH problems

• QNM of KBH

• Jets from KBH

2. Supernovae remnants

3. Other compact objects (CO)

• QNM of CO

• Jets from CO

A New Opportunity:

Studying the spectra of real jets we can decide what is

the Nature of their central engine: BH or

Example QNM of Kerr BH: => regular solutions of TAE

Novel

Spectral

conditions

for

regular

solutiopns

of TAE:

PPF gr-qc/

09084234

The QNM spectrum as a function of

the parameter a:

Making use

of indirect

Methods the

H. Onozawa

1996 result is:

We reproduce the numerical results up to 8 figures

Examples of Relativistic Jets 2

Regular solution of angular TME with three nodes:

The phase wave:

The amplitude

wave:

Jet solutions of the angular TME

The phase wave:

The amplitude

wave:

The Jet

from M87

M87

Jet

Time

Evolution

The gas knot

is 214 lightyears

from the

galaxy’s core.

Novel spectra for jets from BH and Naked Singularities

D.Staicova, PPF

arXiv:1002.0480

The complex

critical

Frequency

As a function

of the

parameter a:

M = ½, m = 0, s = -1

The behavior of the frequencies is in accord

with the bifurcation of the ergo-surface

PPF, gr-qc/0902.1277, 0980.4234.

We

are

steel

looking

for

the

KEY !

Thank You

for your attention