Some Novel Properties and
Applications of Heun’s
Functions to Physical Problems
Department of Theoretical Physics
University of Sofia
Department of Functions and Functional Analysis
Moscow State University
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
Edgardo S. Cheb-Terrab,
MITACS and Maplesoft, 2004
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
3. Recently discovered collimated one-way running waves
as a basis for novel models of relativistic jets.
1 Albert Einstein Institute, Potsdam
2 NASA Goddard Space Flight Center
3 Jet Propulsion Laboratory, California Institute of Technology
4 Massachussets Institute of Technology
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
Classical general relativist:
Eternal black holes certainly exist mathematically.
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]
What about the EXACT SOLUTIONS
arXiav:0902.3151 [astro-ph.HE]: Critical Thoughts on Cosmology
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
6 Nov 2009
A typical observed real rotation curve
For v = const :
Instabilities of BH:
A hyper nova 08.09.05 (distance 11.7 bills lys)
Formation of WHAT : BH, OR
Series of explosions observed !
Abbott et all, to appear in
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.
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).
Jets from GRB060418 and
~ 200 Earth masses with
2009 FERMI News:
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 !!!
due to unknown
⇒ Acceleration to v = 0.99999966 c (gamma=1220)
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
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
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.
Jets from Neutron Star
Discovered by NASA's
Spitzer Space Telescope
object Herbig-Haro 49/50,
the shockwaves of powerful
protostellar jet hitting
PPF, D. Staicova,
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 gas knot
is 214 lightyears
The Jet in Centaurus A
3C321 Jet :
Black Hole Fires at Neighboring Galaxy
Other observed jets:
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
around z = 0 :
problem is still
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
1. Hidrogen Molecule
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:
Three-confluent Heun’s function:
Heun’s problems in gravity: perturbations of
metric: PPF, CQG,2006, J Phys C, 2007
2. Kerr metric PPF, gr-qc/0902.1277,
3. Reisner-Nortstrom metric .
4. Kerr-Newman metric.
For all spins:
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.
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.
Pure imaginary discrete spectrum:
Two series: n=0,…,6; and
n=7,… exist. The eigenvalues
In them are placed around the
Dependence of eigenvalues on the mixing angle
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:
– no outgoing waves
– 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:
Simple non-algebraic differential invariants
for the Kerr solution:
is the Weyl tensor, - its dual
The density for the Euler
The density for the Chern Pontryagin
do not see the horizons => coordinate singularities
NO ! Let
Then in Boyer - Lindquist coordinates:
The differential invariants:
- 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):
0,1/2, 1, 3/2, 2
of Kerr and
in terms of Weyl
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
A more general integral representation of the solutions: PF gr-qc/0902.1277
TRE explicit form:
16 Exact Local Solutions of TRE: PPF gr-qc/0902.1277, 0908.4234
16 Exact Local Solutions of TAE:PPF gr-qc/0902.1277, 0908.4234
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:
Novel relations for confluent Heun’s functions and
their Derivatives, PF:
Self-adjoint form of confluent Heun’s operator:
Chain of confluent Heun’s operators:
The - condition
A Novel Identity:
Universal form of the Teukolsky-Starobinsky Identities
For the above special values of the parameters all solutions
turn to be -solutions. As a result the universal 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:
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
In the case of
we have more
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 :
For spin =1:
NO continuous spectrum for |s|=2 =>
PPF gr-qc/0908.4234: For spin ½ the kernel is:
General solution to TME – one-way-running waves:
- an arbitrary analitic function of two complex variables
Bounded one-way-runing waves with spin ½ :
for the modified Bessels’s functions:
=> Bounded solution => selfadjoint TAE operator
Overall kernels for solutions to Teukolsky’s Master Equation
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
The QNM spectrum as a function of
the parameter a:
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:
Jet solutions of the angular TME
The phase wave:
The gas knot
is 214 lightyears
Novel spectra for jets from BH and Naked Singularities
As a function
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.
for your attention