Numerical Approaches to Spacetime Singularities

This Living Review updates a previous version which its itself an update of a review article. Numerical exploration of the properties of singularities could, in principle, yield detailed understanding of their nature in physically realistic cases. Examples of numerical investigations into the formation of naked singularities, critical behavior in collapse, passage through the Cauchy horizon, chaos of the Mixmaster singularity, and singularities in spatially inhomogeneous cosmologies are discussed.Comment: 51 pages, 6 figures may be found in online version: Living Rev. Relativity 2002-1 at www.livingreviews.or

The dynamics of precessing binary black holes using the post-Newtonian approximation

We investigate the (conservative) dynamics of binary black holes using the Hamiltonian formulation of the post-Newtonian (PN) equations of motion. The Hamiltonian we use includes spin-orbit coupling, spin-spin coupling, and mass monopole/spin-induced quadrupole interaction terms. In the case of both quasi-circular and eccentric orbits, we search for the presence of chaos (using the method of Lyapunov exponents) for a large variety of initial conditions. For quasi-circular orbits, we find no chaotic behavior for black holes with total mass 10 - 40 solar masses when initially at a separation corresponding to a Newtonian gravitational-wave frequency less than 150 Hz. Only for rather small initial radial distances, for which spin-spin induced oscillations in the radial separation are rather important, do we find chaotic solutions, and even then they are rare. Moreover, these chaotic quasi-circular orbits are of questionable astrophysical significance, since they originate from direct parametrization of the equations of motion rather than from widely separated binaries evolving to small separations under gravitational radiation reaction. In the case of highly eccentric orbits, which for ground-based interferometers are not astrophysically favored, we again find chaotic solutions, but only at pericenters so small that higher order PN corrections, especially higher spin PN corrections, should also be taken into account.Comment: 18 pages, 26 figure

Chaos embedded opposition based learning for gravitational search algorithm

Due to its robust search mechanism, Gravitational search algorithm (GSA) has achieved lots of popularity from different research communities. However, stagnation reduces its searchability towards global optima for rigid and complex multi-modal problems. This paper proposes a GSA variant that incorporates chaos-embedded opposition-based learning into the basic GSA for the stagnation-free search. Additionally, a sine-cosine based chaotic gravitational constant is introduced to balance the trade-off between exploration and exploitation capabilities more effectively. The proposed variant is tested over 23 classical benchmark problems, 15 test problems of CEC 2015 test suite, and 15 test problems of CEC 2014 test suite. Different graphical, as well as empirical analyses, reveal the superiority of the proposed algorithm over conventional meta-heuristics and most recent GSA variants.Comment: 33 pages, 5 Figure

Quantum origin of the early inflationary Universe

We give a detailed presentation of a recently proposed mechanism of generating the energy scale of inflation by loop effects in quantum cosmology. We discuss the quantum origin of the early inflationary Universe from the no-boundary and tunneling quantum states and present a universal effective action algorithm for the distribution function of chaotic inflationary cosmologies in both of these states. The energy scale of inflation is calculated by finding a sharp probability peak in this distribution function for a tunneling model driven by the inflaton field with large negative constant $\xi$ of non-minimal interaction. The sub-Planckian parameters of this peak (the mean value of the corresponding Hubble constant $H\simeq 10^{-5}m_P$, its quantum width $\Delta H/H\simeq 10^{-5}$ and the number of inflationary e-foldings $N\geq 60$) are found to be in good correspondence with the observational status of inflation theory, provided the coupling constants of the theory are constrained by a condition which is likely to be enforced by the (quasi) supersymmetric nature of the sub-Planckian particle physics model.Comment: 43 pages, LaTeX, figures not include

Gravitational Search and Harmony Search Algorithms for Solving the Chemical Kinetics Optimization Problems

The article is dedicated to the analysis of the global optimization algorithms application to the solution of inverse problems of chemical kinetics. Two heuristic algorithms are considered - the gravitational search algorithm and the harmony algorithm. The article describes the algorithms, as well as the application of these algorithms to the optimization of test functions. After that, these algorithms are used to search for the kinetic parameters of two chemical processes – propane pre-reforming on Ni-catalyst and catalytic isomerization of pentane-hexane fraction. For the first process both algorithms showed approximately the same solution, while for the second problem the gravitational search algorithm showed a smaller value of the minimizing function. Wherefore, it is concluded that on large-scale problems it is better to use the gravitational search algorithm rather than the harmony algorithm, while obtaining a smaller value of the minimizing function in a minimum time. On low-scale problems both algorithms showed approximately the same result, while demonstrating the coincidence of the calculated data with the experimental ones

Motion around a Monopole + Ring system: I. Stability of Equatorial Circular Orbits vs Regularity of Three-dimensional Motion

We study the motion of test particles around a center of attraction represented by a monopole (with and without spheroidal deformation) surrounded by a ring, given as a superposition of Morgan & Morgan discs. We deal with two kinds of bounded orbits: (i) Equatorial circular orbits and (ii) general three-dimensional orbits. The first case provides a method to perform a linear stability analysis of these structures by studying the behavior of vertical and epicyclic frequencies as functions of the mass ratio, the size of the ring and/or the quadrupolar deformation. In the second case, we study the influence of these parameters in the regularity or chaoticity of motion. We find that there is a close connection between linear stability (or unstability) of equatorial circular orbits and regularity (or chaoticity) of the three-dimensional motion.Comment: 13 pages, 17 figures, to appear in MNRA