arXiv subjects

Georgios Lukes-Gerakopoulos

Publications and source records attributed to Georgios Lukes-Gerakopoulos.

At least 55 records · Page 3Linked to original sources

arXiv2017

Time parameterizations and spin supplementary conditions of the Mathisson-Papapetrou-Dixon equations

The implications of two different time constraints on the Mathisson-Papapetrou-Dixon (MPD) equations are discussed under three spin supplementary conditions (SSC). For this reason the MPD equations are revisited without specifying the affine parameter and several relations are reintroduced in their general form. The latter allows to investigate the consequences of combining the Mathisson-Pirani (MP) SSC, the Tulczyjew-Dixon (TD) SSC and the Ohashi-Kyrian-Semerák (OKS) SSC with two affine parameter types: the proper time on one hand and the parameterizations introduced in [Gen. Rel. Grav. 8, 197 (1977)] on the other. For the MP SSC and the TD SSC it is shown that quantities that are constant of motion for the one affine parameter are not for the other, while for the OKS SSC it is shown that the two affine parameters are the same. To clarify the relation between the two affine parameters in the case of the TD SSC the MPD equations are evolved and discussed.

gr-qc↗

arXiv2017

Recurrence Analysis as a tool to study chaotic dynamics of extreme mass ratio inspiral in signal with noise

Recurrence analysis is a well settled method allowing to discern chaos from order, and determinism from noise. We apply this tool to study time series representing geodesic and inspiraling motion of a test particle in a deformed Kerr spacetime, when deterministic chaos and different levels of stochastic noise are present. In particular, we suggest a recurrence-based criterion to reveal whether the time series comes from a deterministic source and find a noise-level threshold of its applicability.

gr-qc↗

arXiv2017

Spinning test-body orbiting around a Kerr black hole: circular dynamics and gravitational-wave fluxes

In a recent work, [Phys. Rev. D. 94, 104010 (2016)], hereafter Paper I, we have numerically studied different prescriptions for the dynamics of a spinning particle in circular motion around a Schwarzschild black hole. In the present work, we continue this line of investigation to the rotating Kerr black hole. We consider the Mathisson-Papapetrou formalism under three different spin-supplementary-conditions (SSC), the Tulczyjew SSC, the Pirani SSC and the Ohashi-Kyrian-Semerak SSC, and analyze the different circular dynamics in terms of the ISCO shifts and the frequency parameter ${x \equiv (M Ω)^{2/3}}$, where $Ω$ is the orbital frequency and $M$ is the Kerr black hole mass. Then, we solve numerically the inhomogeneous $(2+1)D$ Teukolsky equation to contrast the asymptotic gravitational wave fluxes for the three cases. Our central observation made in Paper I for the Schwarzschild limit is found to hold true for the Kerr background: the three SSCs reduce to the same circular dynamics and the same radiation fluxes for small frequency parameters but differences arise as $x$ grows close to the ISCO. For a positive Kerr parameter $a=0.9$ the energy fluxes mutually agree with each other within a $0.2\%$ uncertainty up to $x<0.14$, while for $a=-0.9$ this level of agreement is preserved up to $x<0.1$. For large frequencies ($x \gtrsim 0.1$), however, the spin coupling of the Kerr black hole and the spinning body results in significant differences of the circular orbit parameters and the fluxes, especially for the $a=-0.9$ case. Instead, in the study of ISCO the negative Kerr parameter $a=-0.9$ results in less discrepancies in comparison with the positive Kerr parameter $a=0.9$.

gr-qc↗

arXiv2017

Dynamics and chaos in the unified scalar field cosmology II. System in a finite box

We revisit the global dynamics of unified dark matter cosmological models and analyze it in a new dynamical system setting. In particular, by defining a suitable set of variables we obtain a bounded variable space, a feature that allows a better control of the critical elements of the system. First, we give a comprehensive cosmological interpretation of the critical points. Then, we turn our focus on particular representative trajectories with physically motivated initial conditions studied in the first paper of the series, and we discuss how the scale factor relates to the equation of state parameter. We review and complement these results in the light of the new variable approach by discussing the issue whether the system is chaotic or not.

gr-qc↗

arXiv2016

Spinning particles moving around black holes: integrability and chaos

The motion of a stellar compact object around a supermassive black hole can be approximated by the motion of a spinning test particle. The equations of motion describing such systems are in general non-integrable, and therefore, chaotic motion should be expected. This article discusses the integrability issue of the spinning particle for the cases of Schwarzschild and Kerr spacetime, and then it focuses on a canonical Hamiltonian formalism where the spin of the particle is included only up to the linear order.

gr-qc↗

arXiv2016

The dynamics of a spinning particle in a linear in spin Hamiltonian approximation

We investigate for order and chaos the dynamical system of a spinning test particle of mass $m$ moving in the spacetime background of a Kerr black hole of mass M. This system is approximated in our investigation by the linear in spin Hamiltonian function provided in [E. Barausse, and A. Buonanno, Phys.Rev. D 81, 084024 (2010)]. We study the corresponding phase space by using 2D projections on a surface of section and the method of color and rotation on a 4D Poincaré section. Various topological structures coming from the non-integrability of the linear in spin Hamiltonian are found and discussed. Moreover, an interesting result is that from the value of the dimensionless spin $S/(m M)=10^{-4}$ of the particle and below, the impact of the non-integrability of the system on the motion of the particle seems to be negligible.

gr-qc↗

arXiv2016

On integrability of certain rank 2 sub-Riemannian structures

We discuss the integrability of rank 2 sub-Riemannian structures on low-dimensional manifolds, and then prove that some structures of that type in dimension 6, 7 and 8 have a lot of symmetry but no integrals polynomial in momenta of low degrees, except for those coming from the Killing fields and the Hamiltonian, thus indicating non-integrability of the corresponding geodesic flows.

math.DG↗

arXiv2016

Comment on "Chaotic orbits for spinning particles in Schwarzschild spacetime"

The astrophysical relevance of chaos for a test particle with spin moving in Schwarzschild spacetime was the objective of \cite{Verhaaren10}. Even if the results of the study seem to be qualitatively in agreement with similar works, the study presented in \cite{Verhaaren10} suffers both from theoretical and technical issues. These issues are discussed in this comment.

gr-qc↗

arXiv2016

Comparing Hamiltonians of a spinning test particle for different tetrad fields

This work is concerned with suitable choices of tetrad fields and coordinate systems for the Hamiltonian formalism of a spinning particle derived in [E. Barausse, E. Racine, and A. Buonanno, Phys. Rev. D 80, 104025 (2009)]. After demonstrating that with the originally proposed tetrad field the components of the total angular momentum are not preserved in the Schwarzschild limit, we analyze other hitherto proposed tetrad choices. Then, we introduce and thoroughly test two new tetrad fields in the horizon penetrating Kerr-Schild coordinates. Moreover, we show that for the Schwarzschild spacetime background the linearized in spin Hamiltonian corresponds to an integrable system, while for the Kerr spacetime we find chaos which suggests a nonintegrable system.

gr-qc↗

arXiv2014

Investigating spinning test particles: spin supplementary conditions and the Hamiltonian formalism

In this paper we report the results of a thorough numerical study of the motion of spinning particles in Kerr spacetime with different prescriptions. We first evaluate the Mathisson-Papapetrou equations with two different spin supplementary conditions, namely, the Tulczyjew and the Newton-Wigner, and make a comparison of these two cases. We then use the Hamiltonian formalism given by Barausse, Racine, and Buonanno in [Phys. Rev. D, 80, 104025 (2009)] to evolve the orbits and compare them with the corresponding orbits provided by the Mathisson-Papapetrou equations. We include a full description of how to treat the issues arising in the numerical implementation.

gr-qc↗

arXiv2014

Non-Linear Effects in Non-Kerr spacetimes

There is a chance that the spacetime around massive compact objects which are expected to be black holes is not described by the Kerr metric, but by a metric which can be considered as a perturbation of the Kerr metric. These non-Kerr spacetimes are also known as bumpy black hole spacetimes. We expect that, if some kind of a bumpy black hole exists, the spacetime around it should possess some features which will make the divergence from a Kerr spacetime detectable. One of the differences is that these non-Kerr spacetimes do not posses all the symmetries needed to make them integrable. We discuss how we can take advantage of this fact by examining EMRIs into the Manko-Novikov spacetime.

gr-qc↗

arXiv2014

Adjusting chaotic indicators to curved spacetimes

In this work, chaotic indicators, which have been established in the framework of classical mechanics, are reformulated in the framework of general relativity in such a way that they are invariant under coordinate transformation. For achieving this, the prescription for reformulating mLCE given by [Y. Sota, S. Suzuki, and K.-I. Maeda, Classical Quantum Gravity 13, 1241 (1996)] is adopted. Thus, the geodesic deviation vector approach is applied, and the proper time is utilized as measure of time. Following the aforementioned prescription, the chaotic indicators FLI, MEGNO, GALI, and APLE are reformulated. In fact, FLI has been reformulated by adapting other prescriptions in the past, but not by adapting the Sota et al. one. By using one of these previous reformulations of FLI, an approximative expression giving MEGNO as function of FLI has been applied on non-integrable curved spacetimes in a recent work. In the present work the reformulation of MEGNO is provided by adjusting the definition of the indicator to the Sota et al. prescription. GALI, and APLE are reformulated in the framework of general relativity for the first time. All the reformulated indicators by the Sota et al. prescription are tested and compared for their efficiency to discern order from chaos.

gr-qc↗

arXiv2013

Comment on "Nonexistence of the final first integral in the Zipoy-Voorhees space-time"

The accuracy of the numerical findings of Lukes-Gerakopoulos PRD 86, 044013 (2012), regarding the existence of additional first integrals in the Zipoy-Voorhees space-time, was recently questioned by Maciejewski et al. PRD 88, 064003 (2013). In this comment, it is shown that the discrepancy between the results of Lukes-Gerakopoulos (2012) and Maciejewski et al. (2013) is not due to issues related to numerical accuracy, as claimed in Maciejewski et al (2013), but due to different choice of coordinates used in Maciejewski et al.(2013).

gr-qc↗

arXiv2013

Mind the Resonances: Final stages of accretion into bumpy black holes

In this article we discuss a possible way of testing the Kerr black hole hypothesis by taking advantage of phenomena correlated with chaotic motion in the final stages of an accretion disk around a bumpy black hole. We anticipate that these phenomena should have an imprint in the electromagnetic spectrum coming from the accretion disk.

gr-qc↗

arXiv2013

Testing the existence of regions of stable orbits at small radii around black hole candidates

Black hole candidates in X-ray binary systems and at the centers of galaxies are expected to be the Kerr black holes of General Relativity, but the actual nature of these objects has yet to be verified. In this paper, we consider the possibility that they are exotic compact objects and we describe their exterior gravitational field with a subclass of the Manko-Novikov metrics, which are exact solutions of the vacuum Einstein's equations and can describe the spacetime geometry around bodies with arbitrary mass-multipole moments. We point out that around a Manko-Novikov object there may exist many disconnected non-plunging regions at small radii, with no counterpart in the Kerr background, and that their existence may be tested. For instance, in the presence of an accretion disk, they may be filled by the accreting gas, forming a ring structure that might remind the rings of Saturn. We suggest that the existence of these regions may have a clear observational signature in the waveform of the gravitational radiation emitted by an EMRI: in the last stage of the inspiral, the waveform would be the combination of "regular chirps", produced when the small object orbits in one of the non-plunging regions, and "bursts", released when the small object jumps from a non-plunging region to another one at smaller radii. Our conclusions are supported by some numerical calculations of trajectories in the geodesic approximation, in which a particle plunges from the ISCO and then seems to get trapped in the potential well at smaller radii.

gr-qc↗

arXiv2012

A Symmetric Integrator for non-integrable Hamiltonian Relativistic Systems

By combining a standard symmetric, symplectic integrator with a new step size controller, we provide an integration scheme that is symmetric, reversible and conserves the values of the constants of motion. This new scheme is appropriate for long term numerical integrations of geodesic orbits in spacetime backgrounds, whose corresponding Hamiltonian system is non-integrable, and, in general, for any non-integrable Hamiltonian system whose kinetic part depends on the position variables. We show by numerical examples that the new integrator is faster and more accurate i) than the standard symplectic integration schemes with or without standard adaptive step size controllers and ii) than an adaptive step Runge-Kutta scheme.

gr-qc↗

arXiv2012

The non-integrability of the Zipoy-Voorhees metric

The low frequency gravitational wave detectors like eLISA/NGO will give us the opportunity to test whether the supermassive compact objects lying at the centers of galaxies are indeed Kerr black holes. A way to do such a test is to compare the gravitational wave signals with templates of perturbed black hole spacetimes, the so-called bumpy black hole spacetimes. The Zipoy-Voorhees (ZV) spacetime (known also as the $γ$ spacetime) can be included in the bumpy black hole family, because it can be considered as a perturbation of the Schwarzschild spacetime background. Several authors have suggested that the ZV metric corresponds to an integrable system. Contrary to this integrability conjecture, in the present article it is shown by numerical examples that in general ZV belongs to the family of non-integrable systems.

gr-qc↗

arXiv2012

Periodic Orbits and Escapes in Dynamical Systems

We study the periodic orbits and the escapes in two different dynamical systems, namely (1) a classical system of two coupled oscillators, and (2) the Manko-Novikov metric (1992) which is a perturbation of the Kerr metric (a general relativistic system). We find their simple periodic orbits, their characteristics and their stability. Then we find their ordered and chaotic domains. As the energy goes beyond the escape energy, most chaotic orbits escape. In the first case we consider escapes to infinity, while in the second case we emphasize escapes to the central "bumpy" black hole. When the energy reaches its escape value a particular family of periodic orbits reaches an infinite period and then the family disappears (the orbit escapes). As this family approaches termination it undergoes an infinity of equal period and double period bifurcations at transitions from stability to instability and vice versa. The bifurcating families continue to exist beyond the escape energy. We study the forms of the phase space for various energies, and the statistics of the chaotic and escaping orbits. The proportion of these orbits increases abruptly as the energy goes beyond the escape energy.

nlin.CD↗