arXiv subjects

Jerzy Matyjasek

Publications and source records attributed to Jerzy Matyjasek.

At least 19 recordsLinked to original sources

arXiv2026

Ces\`aro convergence of the high-order WKB method and its applications to black-hole overtones and long-lived modes

We develop a fully automatic Mathematica implementation of the black-hole WKB method at very high orders based on the Bender-Wu algorithm, which in principle is limited only by memory and computational time, and show that when pushed to sufficiently high order and improved by diagonal Pad\'e approximants the method becomes efficient for two regimes which are usually regarded as difficult for the standard low-order WKB treatment: the first several overtones with n>l and the very long-lived quasinormal modes of massive fields. At the same time, we show that this efficiency has a nontrivial limitation: for black-hole metrics belonging to the non-moderate class, especially when higher coefficients of the near-horizon parametrization become large, the WKB sequence may exhibit an apparent convergence to values which are nevertheless far from the accurate quasinormal frequencies. Thus, numerical stabilization of the WKB output alone is not always a sufficient criterion of correctness. However, we observe that although the WKB method with diagonal or near-diagonal Pad\'e approximants does not exhibit monotonic convergence order by order, the corresponding Ces\`aro means become monotonically convergent once a sufficiently high WKB order is reached. This behavior may serve as an internal WKB criterion for the convergence of the method.

gr-qc↗

arXiv2026

An efficient higher-order WKB code for quasinormal modes and greybody factors

The higher-order WKB Mathematica code for computing quasinormal modes, whose accuracy was significantly enhanced through extensions to higher orders and, in particular, through the use of Pad\'e resummation, has been widely employed in numerous studies over the past several years. In this work, we present an updated and optimized version of the code. The main improvement consists in expanding the effective potential in a Taylor series around its maximum, rather than evaluating the full analytic expression of the WKB formula for each specific potential. This modification leads to a substantial reduction in computation time. In cases where the effective potential is complicated and involves non-rational functions, the speed gain can reach several orders of magnitude, while preserving the accuracy of the method.

gr-qc↗

arXiv2025

The shadow and quasinormal modes of the asymptotically flat hairy black holes with a dilaton potential

In this article, the shadow and the quasinormal modes (QNMs) of an exact asymptotically flat hairy electrically charged black hole solution with a dilaton potential are investigated. Using the {constraint} equation among the integration constant $η$ of the gravitational field, the mass $M$, the electric charge $Q$ and the coupling constant $ν$ between the $U(1)$ field and the dilaton field, we find that the shadow radii, the Lyapunov exponent $λ$ and the coordinate angular velocity $Ω_{c}$ only significantly affected by $ν$ if the $Q$ is close to the extremal value, especially when $ν$ approaches to one. Furthermore, the QNMs are numerically computed by using the Hatsuda method and verify with the higher-order WKB approximations with the Padé summation. We find that the QNMs are close to that of the low energy limit of the string theory when $ν$ is large enough. In the eikonal limit, the real and imaginary parts are proved to be given by $Ω_{c}$ and $λ$, respectively.

gr-qc↗

arXiv2025

Quasinormal Modes of Black Holes: Efficient and Highly Accurate Calculations with Recurrence-Based Methods

We discuss new recurrence-based methods for calculating the complex frequencies of the quasinormal modes of black holes. These methods are based on the Frobenius series solutions of the differential equation describing the linearized radial perturbations. Within the general method, we propose two approaches: the first involves calculating the series coefficients, while the second employs generalized continued fractions. Moreover, as a consequence of this analysis, we present a computationally efficient and convenient method that uses double convergence acceleration, consisting of the application of the Wynn algorithm to the approximants obtained from the Hill determinants, with the Leaver-Nollert-Zhidenko-like tail approximations taken into account. The latter is particularly important for stabilizing and enabling the calculations of modes with small real parts as well as higher overtones. The method demonstrates exceptionally high accuracy. We emphasize that Gaussian elimination is unnecessary in all of these calculations. We consider $D$-dimensional ($3<D<10$) Schwarzschild-Tangherlini black holes as concrete examples. Specifically, we calculate the quasinormal modes of the $(2+1)$-dimensional acoustic black hole (which is closely related to the five-dimensional Schwarzschild-Tangherlini black holes), the electromagnetic-vector modes of the six-dimensional black holes and the scalar (gravitational tensor) modes in the seven-dimensional case. We believe that the methods presented here are applicable beyond the examples shown, also outside the domain of the black hole physics.

gr-qc↗

arXiv2024

Accurate quasinormal modes of the analogue black holes

We study the quasinormal modes of the spherically-symmetric $(2+1)$-dimensional analogue black hole, modeled by the ``draining bathtub'' fluid flow, and the $(3+1)$-dimensional canonical acoustic black hole. In the both cases the emphasis is on the accuracy. Formally, the radial equation describing perturbations of the $(2+1)$-dimensional black hole is a special case of the general master equation of the 5-dimensional Tangherlini black hole. Similarly, the $(3+1)$-dimensional equation can be obtained from the master equation of the 7-dimensional Tangherlini black hole. For the $(2+1)$-dimensional analogue black hole we used three major techniques: the higher-order WKB method with the Padé summation, the Hill-determinant method and the continued fraction method, the latter two with the convergence acceleration. In the $(3+1)$-dimensional case, we propose the simpler recurrence relations and explicitly demonstrate that both recurrences, i.e., the eight-term and the six-term recurrences yield identical results. Since the application of the continued-fraction method require five (or three) consecutive Gauss eliminations, we decided not to use this technique in the $(3+1)$-dimensional case. Instead, we used the Hill-determinant method in the two incarnations and the higher-order WKB. We accept the results of our calculations if at least two (algorithmically) independent methods give the same answer to some prescribed accuracy. Our results correct and extend the results existing in the literature and we believe that we approached assumed accuracy of 9 decimal places. In most cases, there is perfect agreement between all the methods; however, in a few cases, the performance of the higher-order WKB method is slightly worse.

gr-qc↗

arXiv2021

Accurate Quasinormal Modes of the Five-Dimensional Schwarzschild-Tangherlini Black Holes

The objective of this paper is to construct the accurate (say, to 11 decimal places) frequencies of the quasinormal modes of the 5-dimensional Schwarzschild-Tangherlini black hole using three major techniques: the Hill determinant method, the continued fractions method and the WKB-Padé method and to discuss the limitations of each. It is shown that for the massless scalar, gravitational tensor, gravitational vector and electromagnetic vector perturbations considered in this paper, the Hill determinant method and the method of continued fractions (both with the convergence acceleration) always give identical results, whereas the WKB-Padé method gives the results that are amazingly accurate in most cases. Notable exception are the gravitational vector perturbations ($j =2$ and $\ell = 2 $), for which the WKB-Padé approach apparently does not work. Here we have interesting situation in which the WKB-based methods (WKB-Padé and WKB-Borel-Le Roy) give the complex frequency that differs from the from the result obtained within the framework of the continued fraction method and the Hill determinant method. For the fundamental mode, deviation of the real part of frequency from the exact value is $0.5\%$ whereas the deviation of the imaginary part is $2.7\%.$ For $\ell \geq 3$ the accuracy of the WKB results is similar again to the accuracy obtained for other perturbations. The case of the gravitational scalar perturbations is briefly discussed.

gr-qc↗

arXiv2020

Quasinormal modes of dirty black holes in the two-loop renormalizable effective gravity

We consider gravitational quasinormal modes of the static and spherically-symmetric dirty black holes in the effective theory of gravity which is renormalizable at the two-loop level. It is demonstrated that using the WKB-Padé summation proposed in \cite{jaOp} one can achieve sufficient accuracy to calculate corrections to the complex frequencies of the quasinormal modes caused by the Goroff-Sagnotti curvature terms. It is shown that the Goroff-Sagnotti correction (with our choice of the sign of the coupling constant) increases damping of the fundamental modes (except for the lowest fundamental mode) and decreases their frequencies. We argue that the methods adopted in this paper can be used in the analysis of the influence of the higher-order curvature terms upon the quasinormal modes and in a number of related problems that require high accuracy.

gr-qc↗

arXiv2020

Stress-energy of the quantized fields in the spacetime of the Damour-Solodukhin wormhole

The static traversable wormhole should made out of some type of exotic matter which satisfies the Morris-Thorne conditions. Although the characteristic size of the region with the exotic matter can be made arbitrary small, the calculations performed so far suggest that the Morris-Thorne conditions are quite restrictive and it is hard to find matter with the desired properties. Traditionally, the quantized fields are considered as the best candidates because they can violate the weak-energy condition. In this paper we employ the Schwinger-DeWitt expansion to construct and examine the approximate stress-energy tensor of the quantized massive scalar (with an arbitrary curvature coupling), spinor and vector field in the spacetime of the Damour-Solodukhin wormhole. We find that for the scalar field there is a region in a parameter space in which the stress-energy tensor has the desired properties. That means that of the twenty-one cases considered so far (the seven types of the wormhole geometries and the three types of the massive fields) only in the four cases (for certain values of the parameters) the stress-energy tensor does satisfy the Morris-Thorne conditions.

gr-qc↗

arXiv2019

Quasinormal modes of black holes. II. Padé summation of the higher-order WKB terms

In previous work [1] we proposed an improvement of the WKB-based semianalytic technique of Iyer and Will for calculation of the quasiormal modes of black holes by constructing the Padé approximants of the formal series for $ω^{2}.$ It has been demonstrated that (within the domain of applicability) the diagonal Padé transforms $\mathcal{P}_{6}^{6}$ and $\mathcal{P}_{7}^{6}$ are always in a very good agreement with the numerical results. In this paper we present a further extension of the method. We show that it is possible to reproduce many known numerical results with a great accuracy (or even exactly) if the Padé transforms are constructed from the perturbative series of a really high order. In our calculations the order depends on the problem but it never exceeds 700. For example, the frequencies of the gravitational mode $l=2,$ $n=0$ calculated with the aid of the Padé approximants and within the framework of the continued fractions method agree to 24 decimal places. The use of such a large number of terms is necessary as the stabilization of the quasinormal frequencies can be slow. Our results reveal some unexpected features of the WKB-based approximations and may shed some fresh light on the problem of overtones.

gr-qc↗

arXiv2018

Quantum fields in Bianchi type I spacetimes. The Kasner metrc

Vacuum polarization of the quantized massive fields in Bianchi type I spacetime is investigated from the point of view of the adiabatic approximation and the Schwinger-DeWitt method. It is shown that both approaches give the same results that can be used in construction of the trace of the stress-energy tensor of the conformally coupled fields. The stress-energy tensor is calculated in the Bianchi type I spacetime and the back reaction of the quantized fields upon the Kasner geometry is studied. A special emphasis is put on the problem of isotropization, studied with the aid of the directional Hubble parameters. Similarities with the quantum corrected interior of the Schwarzschild black hole is briefly discussed.

gr-qc↗

arXiv2018

Vacuum polarization of massive fields in the spacetime of the higher-dimensional black holes

We construct and study the vacuum polarization, $\langle ϕ^{2}\rangle_{D},$ of the quantized massive scalar field with a general curvature coupling parameter in higher-dimensional static and spherically-symmetric black hole spacetimes, with a special emphasis put on the electrically charged Tangherlini solutions and the extremal and ultraextremal configurations. For $4 \leq D \leq 7$ the explicit analytic expressions for the vacuum polarization are given. For the conformally coupled fields the relation between the trace of the stress-energy tensor and the vacuum polarization is examined, which requires knowledge of the higher-order terms in the Schwinger-DeWitt expansion.

gr-qc↗

arXiv2017

Stress-energy tensor of quantized massive fields in static wormhole spacetimes

In order to be traversable, the static Lorentzian wormhole must be made out of some exotic matter that violates the weak energy condition. The quantized fields are the natural candidates as their stress-energy tensor, in many cases, possesses desired properties. In this paper we construct and examine the stress-energy tensor of the quantized massive scalar, spinor and vector fields in six static wormhole spacetimes. We find that in all considered cases the quantum fields violate the Morris-Thorne conditions and do not have the form necessary to support the wormhole throat. This is in concord with the previous results and indicates that the massive quantum fields make the wormholes less operable.

gr-qc↗

arXiv2017

Quasinormal modes of black holes. The improved semianalytic approach

We have extended the semianalytic technique of Iyer and Will for computing the complex quasinormal frequencies of black holes, $ω,$ by constructing the Padé approximants of the (formal) series for $ω^{2}$. It is shown that for the (so far best documented) quasinormal frequencies of the Schwarzschild and Reissner-Nordström black holes the Padé transforms $P_{6}^{6}$ and $P_{7}^{6}$ are, within the domain of applicability, always in excellent agreement with the numerical results. We argue that the method may serve as the black box with the "potential" $Q(x)$ as an input and the accurate quasinormal modes as the output. The generalizations and modifications of the method are briefly discussed as well as the preliminary results for other classes of the black holes.

gr-qc↗

arXiv2015

Inside the Schwarzschild-Tangherlini black holes

The first-order semiclassical Einstein field equations are solved in the interior of the Schwarzschild-Tangherlini black holes. The source term is taken to be the stress-energy tensor of the quantized massive scalar field with arbitrary curvature coupling calculated within the framework of the Schwinger-DeWitt approximation. It is shown that for the minimal coupling the quantum effects tend to isotropize the interior of the black hole (which can be interpreted as an anisotropic collapsing universe) for D=4 and 5, whereas for D=6 and 7 the spacetime becomes more anisotropic. Similar behavior is observed for the conformal coupling with the reservation that for D=5 isotropization of the spacetime occurs during (approximately) the first 1/3 of the lifetime of the interior universe. On the other hand, we find that regardless of the dimension, the quantum perturbations initially strengthen the grow of curvature and its later behavior depends on the dimension and the coupling. It is shown that the Karlhede's scalar can still be used as a useful device for locating the horizon of the quantum-corrected black hole, as expected.

gr-qc↗

arXiv2014

Stress-energy tensor of the quantized massive fields in Schwarzschild-Tangherlini spacetimes. The back reaction

We construct and study the approximate stress-energy tensor of the quantized massive scalar field in higher dimensional Schwarzschild-Tangherlini spacetimes. The stress-energy tensor is calculated within the framework of the Schwinger-DeWitt approach. It is shown that in $N$-dimensional spacetime the main approximation can be obtained from the effective action constructed form the coincidence limit of the Hadamard-DeWitt coefficient $a_{k},$ where $k-1$ is the integer part of $N/2$. The back reaction of the quantized field upon the black hole spacetime is analyzed and the quantum-corrected Komar mass and the Hawking temperature is calculated. It is shown that for the minimal and conformal coupling the increase of the Komar mass of the quantum corrected black hole leads to the decrease of its Hawking temperature. This is not generally true for more exotic values of the coupling parameter. The general formula describing the vacuum polarization, $ \langle ϕ^{2} \rangle,$ is constructed and briefly examined.

gr-qc↗

arXiv2013

Inside the degenerate horizons of regular black holes

The regularized stress-energy tensor of the quantized massive scalar, spinor and vector fields inside the degenerate horizon of the regular charged black hole in the (anti-)de Sitter universe is constructed and examined. It is shown that although the components of the stress-energy tensor are small in the vicinity of the black hole degenerate horizon and near the regular center, they are quite big in the intermediate region. The oscillatory character of the stress-energy tensor can be ascribed to various responses of the higher curvature terms to the changes of the metric inside the (degenerate) event horizon, especially in the region adjacent to the region described by the nearly flat metric potentials. Special emphasis is put on the stress-energy tensor in the geometries being the product of the constant curvature two-dimensional subspaces.

gr-qc↗

arXiv2013

Stress-Energy Tensor of the Quantized Massive Fields in Friedman-Robertson-Walker Spacetimes

The approximate stress-energy tensor of the quantized massive scalar, spinor and vector fields in the spatially flat Friedman-Robertson-Walker universe is constructed. It is shown that for the scalar fields with arbitrary curvature coupling, $ξ,$ the stress-energy tensor calculated within the framework of the Schwinger-DeWitt approach is identical to the analogous tensor constructed in the adiabatic vacuum. Similarly, the Schwinger-DeWitt stress-energy tensor for the fields of spin 1/2 and 1 coincides with the analogous result calculated by the Zeldovich-Starobinsky method. The stress-energy tensor thus obtained are subsequently used in the back reaction problem. It is shown that for pure semiclassical Einstein field equations with the vanishing cosmological constant and the source term consisting exclusively of its quantum part there are no self-consistent exponential solutions driven by the spinor and vector fields. A similar situation takes place for the scalar field if the coupling constant belongs to the interval $ξ\gtrsim 0.1.$ For a positive cosmological constant the expansion slows down for all considered types of massive fields except for minimally coupled scalar field. The perturbative approach to the problem is briefly discussed and possible generalizations of the stress-energy tensor are indicated.

gr-qc↗

arXiv2011

Semiclassical lukewarm black holes

The perturbative solutions to the semiclassical Einstein field equations describing spherically-symmetric and static lukewarm black hole are constructed. The source term is composed of the (classical) stress-energy tensor of the electromagnetic field and the renormalized stress-energy tensor of the quantized massive scalar field in a large mass limit. We used two different parametrizations. In the first parametrization we calculated the zeroth-order solution. Subsequently, making use of the quantum part of the total stress-energy tensor constructed in the classical background we calculated the corrections to the metric potentials and the corrections to the horizons. This procedure can be thought of as switching the quantized field on and analyzing its influence on the classical background via the back-reaction. In the second parametrization we are looking for a self-consistent lukewarm solution from the very beginning. This requires knowledge of a generic tensor which depends functionally on the metric tensor. The transformation formulas relating the line element in both parametrizations are given.

gr-qc↗