scholarly journals Chaos in Relativity and Cosmology

1999 ◽  
Vol 172 ◽  
pp. 1-16
Author(s):  
G. Contopoulos ◽  
N. Voglis ◽  
C. Efthymiopoulos
Keyword(s):  
Black Holes ◽  
General Solution ◽  
Periodic Orbits ◽  
Rest Mass ◽  
Scattering Problem ◽  
Period Doubling ◽  
Characteristic Number ◽  
Type I ◽  
Chaotic Scattering ◽  
Mixmaster Model

AbstractChaos appears in various problems of Relativity and Cosmology. Here we discuss (a) the Mixmaster Universe model, and (b) the motions around two fixed black holes, (a) The Mixmaster equations have a general solution (i.e. a solution depending on 6 arbitrary constants) of Painlevé type, but there is a second general solution which is not Painlevé. Thus the system does not pass the Painlevé test, and cannot be integrable. The Mixmaster model is not ergodic and does not have any periodic orbits. This is due to the fact that the sum of the three variables of the system (α + β + γ) has only one maximum for τ = τm and decreases continuously for larger and for smaller τ. The various Kasner periods increase exponentially for large τ. Thus the Lyapunov Characteristic Number (LCN) is zero. The “finite time LCN” is positive for finite τ and tends to zero when τ → ∞. Chaos is introduced mainly near the maximum of (α + β + γ). No appreciable chaos is introduced at the successive Kasner periods, or eras. We conclude that in the Belinskii-Khalatnikov time, τ, the Mixmaster model has the basic characteristics of a chaotic scattering problem, (b) In the case of two fixed black holes M1 and M2 the orbits of photons are separated into three types: orbits falling into M1 (type I), or M2 (type II), or escaping to infinity (type III). Chaos appears because between any two orbits of different types there are orbits of the third type. This is a typical chaotic scattering problem. The various types of orbits are separated by orbits asymptotic to 3 simple unstable orbits. In the case of particles of nonzero rest mass we have intervals where some periodic orbits are stable. Near such orbits we have order. The transition from order to chaos is made through an infinite sequence of period doubling bifurcations. The bifurcation ratio is the same as in classical conservative systems.

Periodic orbits and chaos around two black holes

1990 ◽  
Vol 431 (1881) ◽  
pp. 183-202 ◽  
Keyword(s):  
Black Holes ◽  
Periodic Orbits ◽  
Initial Conditions ◽  
Classical Problem ◽  
Period Doubling ◽  
Satellite Orbits ◽  
Unstable Periodic Orbits ◽  
Newtonian Case ◽  
Stable Periodic ◽  
Escaping Orbits

We calculate orbits of photons and particles in the relativistic problem of two extreme Reissner-Nordström black holes (fixed). In the case of photons there are three types of non-periodic orbits, namely orbits falling into the black holes M 1 and M 2 (types I and II), and orbits escaping to infinity (III). The various types of orbits are separated by orbits asymptotic to the three main types of (unstable) periodic orbits: ( a ) around M 1 , ( b ) around M 2 and ( c ) around both M 1 and M 2 . Between two non-periodic orbits of two different types there are orbits of the third type. The initial conditions of the three types of orbits form three Cantor sets, and this fact is a manifestation of chaos. The role of higher-order periodic orbits is explained. In the case of particles the situation is similar in the parabolic and hyperbolic cases. However, in the elliptic case the orbits are contained inside a curve of zero velocity, and there are no escaping orbits. Instead we have orbits trapped around stable periodic orbits (IV) and stochastic orbits not falling on the black holes M 1 and M 2 (V). The stable periodic orbits become unstable by period doubling, and infinite period doublings lead to chaos. The newtonian limit is an integrable problem (essentially the same as the classical problem of two fixed centres). The periodic orbits of type ( c ) are topologically similar to the corresponding relativistic orbits, but they differ considerably numerically. We prove that in the newtonian case there are no satellite orbits around M 1 or M 2 (of type ( a ) or ( b )). The post-newtonian case is non-integrable. In this case there are in general orbits of all three types ( a ), ( b ) and ( c ).

Chaotic scattering on a double well: Periodic orbits, symbolic dynamics, and scaling

1993 ◽  
Vol 3 (4) ◽  
pp. 475-485 ◽  
Author(s):  
Vincent Daniels ◽  
Michel Vallières ◽  
Jian‐Min Yuan
Keyword(s):  
Periodic Orbits ◽  
Symbolic Dynamics ◽  
Chaotic Scattering

QUANTIZATION OF ENTROPY SPECTRA OF BLACK HOLES

2013 ◽  
Vol 22 (02) ◽  
pp. 1330001 ◽  
Author(s):  
YONGJOON KWON ◽  
SOONKEON NAM
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Correspondence Principle ◽  
Scattering Problem ◽  
Classical System ◽  
Quasinormal Modes ◽  
Entropy Spectrum ◽  
Total Transmission ◽  
Transmission Modes ◽  
Characteristic Modes

From the quasinormal modes (QNM) of black holes, we obtain the quantizations of the entropy and horizon area of black holes via Bohr–Sommerfeld quantization, based on Bohr's correspondence principle. For this, we identify the appropriate action variable of the classical system corresponding to a black hole. By considering the BTZ black holes in topologically massive gravity as well as Einstein gravity, it is found that the spectra of not the horizon areas but the entropies of black holes are equally spaced. We also propose that other characteristic modes of black holes, which are non-QNM or holographic QNM, can be used in quantization of entropy spectra just like QNM. From these modes, it is found that only the entropy spectrum of the warped AdS3 black hole is equally spaced as well. Furthermore, by considering a scattering problem in a black hole, we propose that the total transmission modes and total reflection modes of black holes can be regarded as characteristic modes of black holes and result in the equally spaced entropy of the Kerr and Reissner–Nordström black holes. Finally, we conclude that there is a universal behavior that the entropy spectra of various black holes are equally spaced.

scholarly journals Chaotic scattering around black holes

1997 ◽  
Vol 224 (4-5) ◽  
pp. 234-238 ◽  
Author(s):  
Juan M. Aguirregabiria
Keyword(s):  
Black Holes ◽  
Chaotic Scattering

Bound orbits around modified Hayward black holes

2021 ◽  
Author(s):  
Bo Gao ◽  
Xue-Mei Deng
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Angular Momentum ◽  
Gravitational Field ◽  
Periodic Orbits ◽  
Kerr Black Hole ◽  
Periodic Oscillations ◽  
Test Particles ◽  
Spin Parameter ◽  
Bound Orbits

The neutral time-like particle’s bound orbits around modified Hayward black holes have been investigated. We find that both in the marginally bound orbits (MBO) and the innermost stable circular orbits (ISCO), the test particle’s radius and its angular momentum are all more sensitive to one of the parameters [Formula: see text]. Especially, modified Hayward black holes with [Formula: see text] could mimic the same ISCO radius around the Kerr black hole with the spin parameter up to [Formula: see text]. Small [Formula: see text] could mimic the ISCO of small-spinning test particles around Schwarzschild black holes. Meanwhile, rational (periodic) orbits around modified Hayward black holes have also been studied. The epicyclic frequencies of the quasi-circular motion around modified Hayward black holes are calculated and discussed with respect to the observed Quasi-periodic oscillations (QPOs) frequencies. Our results show that rational orbits around modified Hayward black holes have different values of the energy from the ones of Schwarzschild black holes. The epicyclic frequencies in modified Hayward black holes have different frequencies from Schwarzschild and Kerr ones. These might provide hints for distinguishing modified Hayward black holes from Schwarzschild and Kerr ones by using the dynamics of time-like particles around the strong gravitational field.

scholarly journals Periodic Orbits Around Kerr Sen Black Holes

2019 ◽  
Vol 71 (12) ◽  
pp. 1461 ◽  
Author(s):  
Chang-Qing Liu ◽  
Chi-Kun Ding ◽  
Ji-Liang Jing
Keyword(s):  
Black Holes ◽  
Periodic Orbits

scholarly journals GENERICITY ASPECTS IN GRAVITATIONAL COLLAPSE TO BLACK HOLES AND NAKED SINGULARITIES

2012 ◽  
Vol 21 (08) ◽  
pp. 1250066 ◽  
Author(s):  
PANKAJ S. JOSHI ◽  
DANIELE MALAFARINA ◽  
RAVINDRA V. SARAYKAR
Keyword(s):  
Black Holes ◽  
Initial Data ◽  
Gravitational Collapse ◽  
Naked Singularity ◽  
Matter Field ◽  
Einstein Equations ◽  
Type I ◽  
Naked Singularities ◽  
Physical Conditions ◽  
Perfect Fluids

Here we investigate the genericity and stability aspects for naked singularities and black holes that arise as the final states for a complete gravitational collapse of a spherical massive matter cloud. The form of the matter considered is a general Type I matter field, which includes most of the physically reasonable matter fields such as dust, perfect fluids and such other physically interesting forms of matter widely used in gravitation theory. Here, we first study in some detail the effects of small pressure perturbations in an otherwise pressure-free collapse scenario, and examine how a collapse evolution that was going to the black hole endstate would be modified and go to a naked singularity, once small pressures are introduced in the initial data. This allows us to understand the distribution of black holes and naked singularities in the initial data space. Collapse is examined in terms of the evolutions allowed by Einstein equations, under suitable physical conditions and as evolving from a regular initial data. We then show that both black holes and naked singularities are generic outcomes of a complete collapse, when genericity is defined in a suitable sense in an appropriate space.

scholarly journals Effects of Chaos in the Galactic Halo

1996 ◽  
Vol 169 ◽  
pp. 419-421
Author(s):  
W.J. Schuster ◽  
C. Allen
Keyword(s):  
Periodic Orbits ◽  
Vertical Structure ◽  
Galactic Halo ◽  
Galactic Center ◽  
Galactic Plane ◽  
Orbital Energy ◽  
Chaotic Scattering ◽  
Halo Stars ◽  
Chaotic Orbits ◽  
Families Of Periodic Orbits

To study the way in which the principal periodic orbits in a Galactic potential determine orbital structure, horizontal and vertical surfaces of section, i.e. (dR/dt, R) and (dz/dt, z), are being used to explore the potential of Allen & Santillán (1991) and to investigate possible vertical structure in the Galactic halo. The chaotic “scattering” process due to the nearly spherical mass distribution close to the Galactic center in conjunction with the confinement of the chaotic orbits produces a vertical segregation of both chaotic and non-chaotic orbits in the halo. Certain zmax, zmin are preferred by the chaotic orbits over others as a result of the conservation of the total orbital energy and of the interaction and confinement of the chaotic orbits by the principal families of periodic orbits (Figure 1). Some of these periodic orbits have been identified. Correlations between the structure found in the observed W distribution and that of the numerically determined zmax, zmin histograms are shown for our sample of 280 halo stars (Schuster et al. 1993). W is the star's velocity perpendicular to the Galactic plane and zmax, zmin the maximum distances above or below the Galactic plane, respectively, reached by the star in the course of its orbit. This vertical structure may explain certain puzzling observations of the galactic halo, such as conflicting c/a values for the shape of the halo, and unusual velocity dispersions and/or distributions near the Galactic poles. These results are in good agreement with Hartwick's (1987) two component model for the halo.

CHAOTIC BEHAVIOR INDUCED BY THERMAL MODULATION IN A MODEL OF THE CONVECTIVE FLOW OF A FLUID MIXTURE IN A POROUS MEDIUM

1999 ◽  
Vol 09 (02) ◽  
pp. 383-396 ◽  
Author(s):  
J.-M. MALASOMA ◽  
P. WERNY ◽  
C.-H. LAMARQUE
Keyword(s):  
Porous Medium ◽  
Convective Flow ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Chaotic Attractors ◽  
Global Behavior ◽  
Type I ◽  
Fluid Mixture ◽  
Numerical Investigations ◽  
Period Doubling Bifurcations

Numerical investigations of the global behavior of a model of the convective flow of a binary mixture in a porous medium are reported. We find a complex behavior characterized by the presence of coexisting periodic, quasiperiodic and chaotic attractors. Bifurcations of periodic solutions and routes to chaos via type-I intermittency and period-doubling bifurcations are described. Boundary crises and band merging crises have also been observed.

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