54 research outputs found
Persistent junk solutions in time-domain modeling of extreme mass ratio binaries
In the context of metric perturbation theory for non-spinning black holes,
extreme mass ratio binary (EMRB) systems are described by distributionally
forced master wave equations. Numerical solution of a master wave equation as
an initial boundary value problem requires initial data. However, because the
correct initial data for generic-orbit systems is unknown, specification of
trivial initial data is a common choice, despite being inconsistent and
resulting in a solution which is initially discontinuous in time. As is well
known, this choice leads to a "burst" of junk radiation which eventually
propagates off the computational domain. We observe another unintended
consequence of trivial initial data: development of a persistent spurious
solution, here referred to as the Jost junk solution, which contaminates the
physical solution for long times. This work studies the influence of both types
of junk on metric perturbations, waveforms, and self-force measurements, and it
demonstrates that smooth modified source terms mollify the Jost solution and
reduce junk radiation. Our concluding section discusses the applicability of
these observations to other numerical schemes and techniques used to solve
distributionally forced master wave equations.Comment: Uses revtex4, 16 pages, 9 figures, 3 tables. Document reformatted and
modified based on referee's report. Commentary added which addresses the
possible presence of persistent junk solutions in other approaches for
solving master wave equation
Solution of the Dirac equation in the rotating Bertotti-Robinson spacetime
The Dirac equation is solved in the rotating Bertotti-Robinson spacetime. The
set of equations representing the Dirac equation in the Newman-Penrose
formalism is decoupled into an axial and angular part. The axial equation,
which is independent of mass, is solved exactly in terms of hypergeometric
functions. The angular equation is considered both for massless (neutrino) and
massive spin-(1/2) particles. For the neutrinos, it is shown that the angular
equation admits an exact solution in terms of the confluent Heun equation. In
the existence of mass, the angular equation does not allow an analytical
solution, however, it is expressible as a set of first order differential
equations apt for numerical study.Comment: 17 pages, no figure. Appeared in JMP (May, 2008
Conformal Invariance and Near-extreme Rotating AdS Black Holes
We obtain retarded Green's functions for massless scalar fields in the
background of near-extreme, near-horizon rotating charged black holes of
five-dimensional minimal gauged supergravity. The radial part of the
(separable) massless Klein-Gordon equation in such general black hole
backgrounds is Heun's equation, due to the singularity structure associated
with the three black hole horizons. On the other hand, we find the scaling
limit for the near-extreme, near-horizon background where the radial equation
reduces to a Hypergeometric equation whose symmetry signifies
the underlying two-dimensional conformal invariance, with the two sectors
governed by the respective Frolov-Thorne temperatures.Comment: 10 pages, the version published in Phys. Rev.
Analytic structure of radiation boundary kernels for blackhole perturbations
Exact outer boundary conditions for gravitational perturbations of the
Schwarzschild metric feature integral convolution between a time-domain
boundary kernel and each radiative mode of the perturbation. For both axial
(Regge-Wheeler) and polar (Zerilli) perturbations, we study the Laplace
transform of such kernels as an analytic function of (dimensionless) Laplace
frequency. We present numerical evidence indicating that each such
frequency-domain boundary kernel admits a "sum-of-poles" representation. Our
work has been inspired by Alpert, Greengard, and Hagstrom's analysis of
nonreflecting boundary conditions for the ordinary scalar wave equation.Comment: revtex4, 14 pages, 12 figures, 3 table
Singularity Structure and Stability Analysis of the Dirac Equation on the Boundary of the Nutku Helicoid Solution
Dirac equation written on the boundary of the Nutku helicoid space consists
of a system of ordinary differential equations. We tried to analyze this system
and we found that it has a higher singularity than those of the Heun's
equations which give the solutions of the Dirac equation in the bulk. We also
lose an independent integral of motion on the boundary. This facts explain why
we could not find the solution of the system on the boundary in terms of known
functions. We make the stability analysis of the helicoid and catenoid cases
and end up with an appendix which gives a new example where one encounters a
form of the Heun equation.Comment: Version to appear in JM
Physical applications of second-order linear differential equations that admit polynomial solutions
Conditions are given for the second-order linear differential equation P3 y"
+ P2 y'- P1 y = 0 to have polynomial solutions, where Pn is a polynomial of
degree n. Several application of these results to Schroedinger's equation are
discussed. Conditions under which the confluent, biconfluent, and the general
Heun equation yield polynomial solutions are explicitly given. Some new classes
of exactly solvable differential equation are also discussed. The results of
this work are expressed in such way as to allow direct use, without preliminary
analysis.Comment: 13 pages, no figure
Teukolsky-Starobinsky Identities - a Novel Derivation and Generalizations
We present a novel derivation of the Teukolsky-Starobinsky identities, based
on properties of the confluent Heun functions. These functions define
analytically all exact solutions to the Teukolsky master equation, as well as
to the Regge-Wheeler and Zerilli ones. The class of solutions, subject to
Teukolsky-Starobinsky type of identities is studied. Our generalization of the
Teukolsky-Starobinsky identities is valid for the already studied linear
perturbations to the Kerr and Schwarzschild metrics, as well as for large new
classes of of such perturbations which are explicitly described in the present
article. Symmetry of parameters of confluent Heun's functions is shown to stay
behind the behavior of the known solutions under the change of the sign of
their spin weights. A new efficient recurrent method for calculation of
Starobinsky's constant is described.Comment: 8 pages, LaTeX file, no figures, final versio
Novel relations and new properties of confluent Heun's functions and their derivatives of arbitrary order
The present article reveals important properties of the confluent Heun's
functions. We derive a set of novel relations for confluent Heun's functions
and their derivatives of arbitrary order. Specific new subclasses of confluent
Heun's functions are introduced and studied. A new alternative derivation of
confluent Heun's polynomials is presented.Comment: 8 pages, no figures, LaTeX file, final versio
Solutions for the General, Confluent and Biconfluent Heun equations and their connection with Abel equations
In a recent paper, the canonical forms of a new multi-parameter class of Abel
differential equations, so-called AIR, all of whose members can be mapped into
Riccati equations, were shown to be related to the differential equations for
the hypergeometric 2F1, 1F1 and 0F1 functions. In this paper, a connection
between the AIR canonical forms and the Heun General (GHE), Confluent (CHE) and
Biconfluent (BHE) equations is presented. This connection fixes the value of
one of the Heun parameters, expresses another one in terms of those remaining,
and provides closed form solutions in terms of pFq functions for the resulting
GHE, CHE and BHE, respectively depending on four, three and two irreducible
parameters. This connection also turns evident what is the relation between the
Heun parameters such that the solutions admit Liouvillian form, and suggests a
mechanism for relating linear equations with N and N-1 singularities through
the canonical forms of a non-linear equation of one order less.Comment: Original version submitted to Journal of Physics A: 16 pages, related
to math.GM/0002059 and math-ph/0402040. Revised version according to
referee's comments: 23 pages. Sign corrected (June/17) in formula (79).
Second revised version (July/25): 25 pages. See also
http://lie.uwaterloo.ca/odetools.ht
Exact Solutions of Regge-Wheeler Equation and Quasi-Normal Modes of Compact Objects
The well-known Regge-Wheeler equation describes the axial perturbations of
Schwarzschild metric in the linear approximation. From a mathematical point of
view it presents a particular case of the confluent Heun equation and can be
solved exactly, due to recent mathematical developments. We present the basic
properties of its general solution. A novel analytical approach and numerical
techniques for study the boundary problems which correspond to quasi-normal
modes of black holes and other simple models of compact objects are developed.Comment: latex file, 25 pages, 4 figures, new references, new results and new
Appendix added, some comments and corrections in the text made. Accepted for
publication in Classical and Quantum Gravity, 2006, simplification of
notations, changes in the norm in some formulas, corrections in reference
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