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Register a new userConstruction of second-order six-dimensional Hamiltonian-conserving scheme
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It is shown analytically that the energy-conserving implicit nonsymplectic scheme of Bacchini, Ripperda, Chen and Sironi provides a first-order accuracy to numerical solutions of a six-dimensional conservative Hamiltonian system. Because of this, a new second-order energy-conserving implicit scheme is proposed. Numerical simulations of Galactic model hosting a BL Lacertae object and magnetized rotating black hole background support these analytical results. The new method with appropriate time steps is used to explore the effects of varying the parameters on the presence of chaos in the two physical models. Chaos easily occurs in the Galactic model as the mass of the nucleus, the internal perturbation parameter, and the anisotropy of the potential of the elliptical galaxy increase. The dynamics of charged particles around the magnetized Kerr spacetime is easily chaotic for larger energies of the particles, smaller initial angular momenta of the particles, and stronger magnetic fields. The chaotic properties are not necessarily weakened when the black hole spin increases. The new method can be used for any six-dimensional Hamiltonian problems, including globally hyperbolic spacetimes with readily available (3+1) split coordinates.
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In this paper, an implicit nonsymplectic exact energy-preserving integrator is specifically designed for a ten-dimensional phase-space conservative Hamiltonian system with five degrees of freedom. It is based on a suitable discretization-averaging of the Hamiltonian gradient, with a second-order accuracy to numerical solutions. A one-dimensional disordered discrete nonlinear Schr{o}dinger equation and a post-Newtonian Hamiltonian system of spinning compact binaries are taken as our two examples. We demonstrate numerically that the proposed algorithm exhibits good long-term performance in the preservation of energy, if roundoff errors are neglected. This result is independent of time steps, initial orbital eccentricities, and regular and chaotic orbital dynamical behavior. In particular, the application of appropriately large time steps to the new algorithm is helpful in reducing time-consuming and roundoff errors. This new method, combined with fast Lyapunov indicators, is well suited related to chaos in the two example problems. It is found that chaos in the former system is mainly responsible for one of the parameters. In the latter problem, a combination of small initial separations and high initial eccentricities can easily induce chaos.
Working directly with a general Hamiltonian for the spacetime metric with the $3+1$ decomposition and keeping only the spatial covariance, we investigate the possibility of reducing the number of degrees of freedom by introducing an auxiliary constraint. The auxiliary constraint is considered as part of the definition of the theory. Through a general Hamiltonian analysis, we find the conditions for the Hamiltonian as well as for the auxiliary constraint, under which the theory propagates two tensorial degrees of freedom only. The class of theories satisfying these conditions can be viewed as a new construction for the type-II minimally modified gravity theories, which propagate the same degrees of freedom of but are not equivalent to general relativity in the vacuum. We also illustrate our formalism by a concrete example, and derive the dispersion relation for the gravitational waves, which can be constrained by observations.
We give an explicit relation, up to second-order terms, between scalar-field fluctuations defined on spatially-flat slices and the curvature perturbation on uniform-density slices. This expression is a necessary ingredient for calculating observable quantities at second-order and beyond in multiple-field inflation. We show that traditional cosmological perturbation theory and the `separate universe approach yield equivalent expressions for superhorizon wavenumbers, and in particular that all nonlocal terms can be eliminated from the perturbation-theory expressions.
We introduce a new kludge scheme to model the dynamics of generic extreme mass-ratio inspirals (stellar compact objects spiraling into a spinning supermassive black hole) and to produce the gravitational waveforms that describe the gravitational-wave emission of these systems. This scheme combines tools from different techniques in General Relativity: It uses a multipolar, post-Minkowskian (MPM) expansion for the far-zone metric perturbation (which provides the gravitational waveforms, here taken up to mass hexadecapole and current octopole order) and for the local prescription of the self-force (since we are lacking a general prescription for it); a post-Newtonian expansion for the computation of the multipole moments in terms of the trajectories; and a BH perturbation theory expansion when treating the trajectories as a sequence of self-adjusting Kerr geodesics. The orbital evolution is thus equivalent to solving the geodesic equations with time-dependent orbital elements, as dictated by the MPM radiation-reaction prescription. To complete the scheme, both the orbital evolution and wave generation require to map the Boyer-Lindquist coordinates of the orbits to the harmonic coordinates in which the different MPM quantities have been derived, a mapping that we provide explicitly in this paper. This new kludge scheme is thus a combination of approximations that can be used to model generic inspirals of systems with extreme mass ratios to systems with more moderate mass ratios, and hence can provide valuable information for future space-based gravitational-wave observatories like LISA and even for advanced ground detectors. Finally, due to the local character in time of our MPM self-force, this scheme can be used to perform studies of the possible appearance of transient resonances in generic inspirals.
The static, i.e., linear momentum independent, part of the next-to-leading order (NLO) gravitational spin(1)-spin(1) interaction Hamiltonian within the post-Newtonian (PN) approximation is calculated from a 3-dim. covariant ansatz for the Hamilton constraint. All coefficients in this ansatz can be uniquely fixed for black holes. The resulting Hamiltonian fits into the canonical formalism of Arnowitt, Deser, and Misner (ADM) and is given in their transverse-traceless (ADMTT) gauge. This completes the recent result for the momentum dependent part of the NLO spin(1)-spin(1) ADM Hamiltonian for binary black holes (BBH). Thus, all PN NLO effects up to quadratic order in spin for BBH are now given in Hamiltonian form in the ADMTT gauge. The equations of motion resulting from this Hamiltonian are an important step toward more accurate calculations of templates for gravitational waves.
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