arXiv subjects

Tolga Birkandan

Publications and source records attributed to Tolga Birkandan.

14 recordsLinked to original sources

arXiv2026

Non-vacuum metrics for the Newman-Unti-Tamburino background: A coordinate-free approach to diverging and twisting solutions

The geometry of the Newman-Unti-Tamburino (NUT) vacuum solution is characterized as the unique Petrov Type D vacuum metric such that the two double principal null directions form an integrable distribution. We study expanding and twisting non-vacuum Type D metrics in this geometry, with the additional assumption $Φ_{01}=Φ_{12}=0$. We prove that these conditions determine the solutions up to a freedom in $Φ_{11}\pm 3Λ$.

gr-qc↗

arXiv2026

A coordinate-free approach to obtaining exact solutions in general relativity: The Newman-Unti-Tamburino solution revisited

The Newman-Unti-Tamburino (NUT) solution is characterized as the unique Petrov Type $D$ vacuum metric such that the two double principal null directions form an integrable distribution. The uniqueness of the NUT is established by evaluating the integrability conditions of the Newman-Penrose equations up to $SL(2,\mathbb{C})$ transformations, resulting in a coordinate-free characterization of the solution.

gr-qc↗

arXiv2024

Analysis of scalar fields with series convolution

Wave equations for some curved spacetimes may involve functions that prevent a solution in a closed form. In some cases, these functions can be eliminated by transformations and the solutions can be found analytically. In the cases where such transformations are not available, the infinite series expansions of these functions can be convoluted with the power series solution ansatz. We study such an example where the solution is based on a special function.

gr-qc↗

arXiv2023

Solutions of a charged scalar field in five-dimensional helicoid solution with electromagnetic field

We study a charged and massive scalar field in the background of the Nutku-Ghezelbash-Kumar metric which is obtained by the addition of a time coordinate to the Nutku helicoid metric in a non-trivial way. The angular part of the Klein-Gordon equation can be written as a double confluent Heun equation. The radial equation cannot be solved in terms of a known function in its general form. However, in some special cases, the radial equation can also be written explicitly as a double confluent Heun equation. We study the full radial equation numerically and observe that the electromagnetic field parameter defines an effective cut-off on the range of the radial coordinate. Finally, we obtain a quasi-exact solution with an approximation.

gr-qc↗

arXiv2022

Symbolic analysis of second-order ordinary differential equations with polynomial coefficients

The singularity structure of a second-order ordinary differential equation with polynomial coefficients often yields the type of solution. It is shown that the $θ$-operator method can be used as a symbolic computational approach to obtain the indicial equation and the recurrence relation. Consequently, the singularity structure leads to the transformations that yield a solution in terms of a special function, if the equation is suitable. Hypergeometric and Heun-type equations are mostly employed in physical applications. Thus only these equations and their confluent types are considered with SageMath routines which are assembled in the open-source package symODE2.

math-ph↗

arXiv2021

Computations of general Heun functions from their integral series representations

We present a numerical implementation of the recently developed unconditionally convergent representation of general Heun functions as integral series. We produce two codes in Python available for download, one of which is especially aimed at reproducing the output of Mathematica's HeunG function. We show that the present code compares favorably with Mathematica's HeunG and with an Octave/Matlab code of Motygin, in particular when the Heun function is to be evaluated at a large number of points if less accuracy is sufficient. We suggest further improvements concerning the accuracy and discuss the issue of singularities.

math.NA↗

arXiv2018

Symbolic and Numerical Analysis in General Relativity with Open Source Computer Algebra Systems

We study three computer algebra systems, namely SageMath (with SageManifolds package), Maxima (with ctensor package) and Python language (with GraviPy module), which allow tensor manipulation for general relativity calculations along with general algebraic calculations. We present a benchmark of these systems using simple examples. After the general analysis, we focus on the SageMath and SageManifolds system to derive, analyze and visualize the solutions of the massless Klein-Gordon equation and geodesic motion with Hamilton-Jacobi formalism. We compare our numerical result of the Klein-Gordon equation with the asymptotic form of the analytical solution to see that they agree.

gr-qc↗

arXiv2017

Singular inverse square potential in coordinate space with a minimal length

The problem of a particle of mass m in the field of the inverse square potential is studied in quantum mechanics with a generalized uncertainty principle, characterized by the existence of a minimal length. Using the coordinate representation, for a specific form of the generalized uncertainty relation, we solve the deformed Schrödinger equation analytically in terms of confluent Heun functions. We explicitly show the regularizing effect of the minimal length on the singularity of the potential. We discuss the problem of bound states in detail and we derive an expression for the energy spectrum in a natural way from the square integrability condition; the results are in complete agreement with the literature.

quant-ph↗

arXiv2015

Physical Problems Admitting Heun-to-Hypergeometric Reduction

The Heun's equation with its four regular singularities emerges in many applications in science. Despite the growing interest of the scientific community, the literature has many gaps in conceptual mathematical aspects of this equation. Moreover, the translation of the mathematical language for non-mathematicians in making contemporary ideas applied in physical research is not also well developed. In this paper, Maier's Heun-to-hypergeometric reduction cases are studied in detail for four problems in quantum mechanics: (1) The Schrodinger's equation for the Coulomb problem on a 3-sphere, (2) the s-wave bound state equation in the problem of the attractive inverse square potential, (3) the equation for the limit density function for the discrete-time quantum walk, and (4) charged particle under magnetic field and Coulomb force. All these problems give rise to the Heun's equation in their mathematical formulation. A Sage code is given in the appendix for the calculation of some Heun identities.

math-ph↗

arXiv2015

An Analysis of the Wave Equation for the $U(1)^{2}$ Gauged Supergravity Black Hole

We study the massless Klein-Gordon equation in the background of the most general rotating dyonic AdS black hole in gauged $\mathcal{N}=2$, $U(1)^{2}$ gauged supergravity in $D=4$, given by Chow and Compère [Phys. Rev. D\textbf{89} (2014) 065003]. The angular part of the separable wave equation is of the Heun type, while the radial part is a Fuchsian equation with five regular singularities. The radial equation is further analyzed and written in a specific form, that reveals the pole structure of the horizon equation, whose residua are expressed in terms of the surface gravities and angular velocities associated with respective horizons. The near-horizon (near-)extremal limits of the solution are also studied, where the expected hidden conformal symmetry is revealed. Furthermore, we present the retarded Green's functions for these limiting cases. We also comment on the generality of the charge-dependent parts of the metric parameters and address some further examples of limiting cases.

hep-th↗

arXiv2014

Wave Equation for the Wu Black Hole

Wu black hole is the most general solution of maximally supersymmetric gauged supergravity in D=5, containing $U(1)^{3}$ gauge symmetry. We study the separability of the massless Klein-Gordon equation and probe its singularities for a general stationary, axisymmetric metric with orthogonal transitivity, and apply the results to the Wu black hole solution. We start with the zero azimuthal-angle eigenvalues in the scalar field Ansatz and find that the residuum of a pole in the radial equation is associated with the surface gravity calculated at this horizon. We then generalize our calculations to nonzero azimuthal eigenvalues and probing each horizon singularity, we show that the residua of the singularities for each horizon are in general associated with a specific combination of the surface gravity and the angular velocities at the associated horizon. It turns out that for the Wu black hole both the radial and angular equations are general Heun's equations with four regular singularities.

hep-th↗

arXiv2014

Addentum to: Conformal Invariance and Near-extreme Rotating AdS Black Holes

We obtained retarded Green's functions for massless scalar fields in the background of near-extreme, near-horizon rotating charged black hole of five-dimensional minimal gauged supergravity in Phys. Rev. D84, 044018 (2011). For general nonextreme black holes, we also derived the radial part of the massless Klein-Gordon equation with zero azimuthal-angle eigenvalues, and showed that it is a general Heun's equation with a regular singularity at each horizon $u_k$ ($k=1,2,3$) and at infinity. We derived explicitly that the residuum of a pole at each $u_k$ is associated with the surface gravity there. In this addendum, probing regular singularities at each $u_k$ we complete the derivation of the full radial equation with nonzero azimuthal-angle eigenvalues. The residua now include modifications by the angular velocities at respective horizons. This result completes the analysis of the wave equation for the massless Klein-Gordon equation for the general rotating charged black hole of five-dimensional minimal gauged supergravity.

hep-th↗

arXiv2011

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 $SL(2,{\bf R})^2$ symmetry signifies the underlying two-dimensional conformal invariance, with the two sectors governed by the respective Frolov-Thorne temperatures.

hep-th↗

arXiv2002

Three Dimensional Gravity in the Presence of Scalar Fields

We study a scalar field in curved space in three dimensions. We obtain a static perturbative solution and show that this solution satisfies the exact equations in the asymptotic region at infinity. The new solution gives rise to a singularity in the curvature scalar at the origin. Our solution, however, necessitates the excising the region near the origin, thus this naked singularity is avoided.

gr-qc↗