arXiv subjects

Patryk Mach

Publications and source records attributed to Patryk Mach.

At least 55 records · Page 3Linked to original sources

arXiv2016

Rotating systems, universal features in dragging and anti-dragging effects, and bounds onto angular momentum

We consider stationary, axially symmetric toroids rotating around spinless black holes, assuming the general-relativistic Keplerian rotation law, in the first post-Newtonian approximation. Numerical investigation shows that the angular momentum accumulates almost exclusively within toroids. It appears that various types of dragging (anti-dragging) effects are positively correlated with the ratio $M_\mathrm{D}/m$ ($M_\mathrm{D}$ is the mass of a toroid and $m$ is the mass of the black hole) - moreover, their maxima are proportional to $M_\mathrm{D}/m$. The horizontal sizes of investigated toroids range from c. 50 to c. 450 of Schwarzschild radii $R_\mathrm{S}$ of the central black hole; their mass $M_\mathrm{D} \in (10^{-4}m, 40m)$ and the radial size of the system is c. 500 $R_\mathrm{S}$. We found that the relative strength of various dragging (anti-dragging) effects does not change with the mass ratio, but it depends on the size of toroids. Several isoperimetric inequalities involving angular momentum are shown to hold true.

gr-qc↗

arXiv2015

Michel accretion of a polytropic fluid with adiabatic index gamma > 5/3: Global flows versus homoclinic orbits

We analyze the properties of a polytropic fluid which is radially accreted into a Schwarzschild black hole. The case where the adiabatic index gamma lies in the range 1 < gamma <= 5/3 has been treated in previous work. In this article we analyze the complementary range 5/3 < gamma <= 2. To this purpose, the problem is cast into an appropriate Hamiltonian dynamical system whose phase flow is analyzed. While for 1 < gamma <= 5/3 the solutions are always characterized by the presence of a unique critical saddle point, we show that when 5/3 < gamma <= 2, an additional critical point might appear which is a center point. For the parametrization used in this paper we prove that whenever this additional critical point appears, there is a homoclinic orbit.

gr-qc↗

arXiv2015

General-relativistic rotation laws in rotating fluid bodies, new weak-field effects and the post-newtonian expansion

Recent general-relativistic extensions of Newtonian rotation laws for self-gravitating stationary fluids allow one to rederive, in the first post-Newtonian approximation, the well known geometric dragging of frames, and two new weak-field effects within rotating tori. These are the recently discovered anti-dragging and a new effect that measures the deviation from the Keplerian motion and/or the contribution of the fluids selfgravity. They can be applied to the study of the existence of the (post-)Newtonian limits of solutions and in investigations of inequalities relating parameters of rotating black holes.

gr-qc↗

arXiv2015

General-relativistic rotation laws in rotating fluid bodies: constant linear velocity

New rotation laws have been recently found for general-relativistic self-gravitating stationary fluids. It was not clear whether they apply to systems rotating with a constant linear velocity. In this paper we fill this gap. The answer is positive. That means, in particular, that these systems should exhibit the recently discovered general-relativistic weak-field effects within rotating tori: the dynamic anti-dragging and the deviation from the Keplerian motion induced by the fluid selfgravity.

gr-qc↗

arXiv2015

General-relativistic rotation laws in rotating fluid bodies

We formulate new general-relativistic extensions of Newtonian rotation laws for self-gravitating stationary fluids. They have been used to re-derive, in the first post-Newtonian approximation, the well known geometric dragging of frames. We derive two other general-relativistic weak-field effects within rotating tori: the recently discovered dynamic anti-dragging and a new effect that measures the deviation from the Keplerian motion and/or the contribution of the fluids selfgravity. One can use the rotation laws to study the uniqueness and the convergence of the post-Newtonian approximations, and the existence of the post-Newtonian limits.

gr-qc↗

arXiv2015

Homoclinic accretion solutions in the Schwarzschild-anti-de Sitter spacetime

The aim of this paper is to clarify the distinction between homoclinic and standard (global) Bondi-type accretion solutions in the Schwarzschild-anti-de Sitter spacetime. The homoclinic solutions have recently been discovered numerically for polytropic equations of state. Here I show that they exist also for certain isothermal (linear) equations of state, and an analytic solution of this type is obtained. It is argued that the existence of such solutions is generic, although for sufficiently relativistic matter models (photon gas, ultra-hard equation of state) there exist global solutions that can be continued to infinity, similarly to standard Michel's solutions in the Schwarzschild spacetime. In contrast to that global solutions should not exist for matter models with a non-vanishing rest-mass component, and this is demonstrated for polytropes. For homoclinic isothermal solutions I derive an upper bound on the mass of the black hole for which stationary transonic accretion is allowed.

gr-qc↗

arXiv2015

Virial tests for post-Newtonian stationary black-hole-disk systems

We investigated hydrodynamical post-Newtonian models of selfgravitating stationary black-hole-disk systems. The post-Newtonian scheme presented here and also in our recent paper is a continuation of previous, purely Newtonian studies of selfgravitating hydrodynamical disks rotating according to the Keplerian rotation law. The post-Newtonian relativistic corrections are significant even at the 1PN level. The 1PN correction to the angular velocity can be of the order of 10% of its Newtonian value. It can be expressed as a combination of geometric and hydrodynamical terms. Moreover, in contrast to the Newtonian Poincare-Wavre theorem, it depends both on the distance from the rotation axis and the distance from the equatorial plane.

gr-qc↗

arXiv2014

General-relativistic versus Newtonian: geometric dragging and dynamic anti-dragging in stationary disks in the first post-Newtonian approximation

We evaluate general-relativistic effects in motion of stationary accretion disks around a Schwarzschild black hole, assuming the first post-Newtonian (1PN) approximation. There arises an integrability condition, that leads to the emergence of two types of general-relativistic corrections to a Newtonian rotation curve. The well known geometric dragging of frames accelerates rotation but the hitherto unknown dynamic term, that reflects the disk structure, deccelerates rotation. The net result can diminish the Newtonian angular velocity of rotation in a central disk zone but the geometric dragging of frames dominates in the disk boundary zone. Both effects are nonlinear in nature and they disappear in the limit of test fluids. Dust disks can be only geometrically dragged while uniformly rotating gaseous disk are untouched at the 1PN order. General-relativistic contributions can strongly affect rotation periods in Keplerian motion for compact systems.

gr-qc↗

arXiv2014

Global dynamics of a Yang-Mills field on an asymptotically hyperbolic space

We consider a spherically symmetric (purely magnetic) SU(2) Yang-Mills field propagating on an ultrastatic spacetime with two asymptotically hyperbolic regions connected by a throat of radius $α$. Static solutions in this model are shown to exhibit an interesting bifurcation pattern in the parameter $α$. We relate this pattern to the Morse index of the static solution with maximal energy. Using a hyperboloidal approach to the initial value problem, we describe the relaxation to the ground state solution for generic initial data and unstable static solutions for initial data of codimension one, two, and three.

math.AP↗

arXiv2014

Spherically symmetric Riemannian manifolds of constant scalar curvature and their conformally flat representations

All spherically symmetric Riemannian metrics of constant scalar curvature in any dimension can be written down in a simple form using areal coordinates. All spherical metrics are conformally flat, so we search for the conformally flat representations of these geometries. We find all solutions for the conformal factor in 3, 4 and 6 dimensions. We write them in closed form, either in terms of elliptic or elementary functions. We are particularly interested in 3-dimensional spaces because of the link to General Relativity. In particular, all 3-dimensional constant negative scalar curvature spherical manifolds can be embedded as constant mean curvature surfaces in appropriate Schwarzschild solutions. Our approach, although not the simplest one, is linked to the Lichnerowicz-York method of finding initial data for Einstein equations.

gr-qc↗

arXiv2013

Self-gravitating Newtonian disks revisited

Recent analytic results concerning stationary, self-gravitating fluids in Newtonian theory are discussed. We give a theorem that forbids infinitely extended fluids, depending on the assumed equation of state and the rotation law. This part extends previous results that have been obtained for static configurations. The second part discusses a Sobolev bound on the mass of the fluid and a rigorous Jeans-type inequality that is valid in the stationary case.

gr-qc↗

arXiv2013

Stability of relativistic Bondi accretion in Schwarzschild-(anti-)de Sitter spacetimes

In a recent paper we investigated stationary, relativistic Bondi-type accretion in Schwarzschild-(anti-)de Sitter spacetimes. Here we study their stability, using the method developed by Moncrief. The analysis applies to perturbations satisfying the potential flow condition. We prove that global isothermal flows in Schwarzschild-anti-de Sitter spacetimes are stable, assuming the test-fluid approximation. Isothermal flows in Schwarzschild-de Sitter geometries and polytropic flows in Schwarzschild-de Sitter and Schwarzschild-anti-de Sitter spacetimes can be stable, under suitable boundary conditions.

gr-qc↗

arXiv2013

Spherical steady accretion flows -- dependence on the cosmological constant, exact isothermal solutions and applications to cosmology

We investigate spherical, isothermal and polytropic steady accretion models in the presence of the cosmological constant. Exact solutions are found for three classes of isothermal fluids, assuming the test gas approximation. The cosmological constant damps the mass accretion rate and - above certain limit - completely stops the steady accretion onto black holes. A "homoclinic-type" accretion flow of polytropic gas has been discovered in AdS spacetimes in the test-gas limit. These results can have cosmological connotation, through the Einstein--Straus vacuole model of embedding local structures into Friedman-Lemaitre-Robertson-Walker spacetimes. In particular one infers that steady accretion would not exist in the late phases of the Penrose's scenario of the evolution of the Universe, known as the Weyl curvature hypothesis.

gr-qc↗

arXiv2012

Estimating masses of Keplerian disk systems: the case of AGN in NGC 4258

The Keplerian motion of accretion disks in Active Galactic Nuclei (AGN) is usually believed to be generated by a heavy central mass. We investigate accreting disk systems with polytropic gas in Keplerian rotation and obtain a phenomenological formula that relates the Keplerian angular frequency to the ratio of disk and central masses. Central mass approaches the Keplerian value, if the inner boundary of a disk is close to the minimal stable orbit of a black hole. These results are applied to NGC 4258, the unique AGN with a finely measured Keplerian rotation curve of the central disk, with the conclusion that its rotation curve is, in fact, determined by the central black hole. The mass of the accretion disk exceeds 100 solar masses.

astro-ph.HE↗

arXiv2012

All solutions of the n = 5 Lane-Emden equation

All real solutions of the Lane-Emden equation for n = 5 are obtained in terms of Jacobian and Weierstrass elliptic functions. A new family of solutions is found. It is expressed by remarkably simple formulae involving Jacobian elliptic functions only. The general properties and discrete scaling symmetries of these new solutions are discussed. We also comment on their possible applications.

math-ph↗

arXiv2012

(In)finite extent of stationary perfect fluids in Newtonian theory

For stationary, barotropic fluids in Newtonian gravity we give simple criteria on the equation of state and the "law of motion" which guarantee finite or infinite extent of the fluid region (providing a priori estimates for the corresponding stationary Newton-Euler system). Under more restrictive conditions, we can also exclude the presence of "hollow" configurations. Our main result, which does not assume axial symmetry, uses the virial theorem as the key ingredient and generalises a known result in the static case. In the axially symmetric case stronger results are obtained and examples are discussed.

gr-qc↗

arXiv2012

Accretion and Structure of Radiating Disks

We studied a steadily accreting, geometrically thick disk model that selfconsistently takes into account selfgravitation of the polytropic gas, its interaction with the radiation and the mass accretion rate. The accreting mass is injected inward in the vicinity of the central $z=0$ plane, where also radiation is assumed to be created. The rest of the disk remains approximately stationary. Only conservation laws are employed and the gas-radiation interaction in the bulk of the disk is described in the thin-gas approximation. We demonstrate that this scheme is numerically viable and yields a structure of the bulk that is influenced by the radiation and (indirectly) by the prescribed mass accretion rate. The obtained disk configurations are typical for environments in Active Galactic Nuclei (AGN), with the central mass of the order of $10^7 M_{\astrosun}$ to $10^8 M_{\astrosun}$, quasi-Keplerian rotation curves, disk masses ranging from about $10^6 M_{\astrosun}$ to $10^7 M_{\astrosun}$, and the luminosity ranging from $10^6 L_{\astrosun}$ to $10^9 L_{\astrosun}$. These luminosities are much lower than the corresponding Eddington limit.

astro-ph.SR↗

arXiv2012

Virial theorem for radiating accretion discs

A continuum version of the virial theorem is derived for a radiating self-gravitating accretion disc around a compact object. The central object is point-like, but we can avoid the regularization of its gravitational potential. This is achieved by applying a modified Pohozaev-Rellich identity to the gravitational potential of the disk only. The theorem holds for general stationary configurations, including discontinuous flows (shock waves, contact discontinuities). It is used to test numerical solutions of a model of self-gravitating radiative accretion discs. The presented virial theorem should be useful in the analysis of those (possibly radiating) hydrodynamical systems in astrophysics where the central mass and the mass of the fluid are comparable and none of them can be neglected.

astro-ph.SR↗