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تسجيل مستخدم جديدOn the applicability of constrained symplectic integrators in general relativity
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J. Frauendiener
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The purpose of this note is to point out that a naive application of symplectic integration schemes for Hamiltonian systems with constraints such as SHAKE or RATTLE which preserve holonomic constraints encounters difficulties when applied to the numerical treatment of the equations of general relativity.
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Symplectic integrators that preserve the geometric structure of Hamiltonian flows and do not exhibit secular growth in energy errors are suitable for the long-term integration of N-body Hamiltonian systems in the solar system. However, the constructi
on of explicit symplectic integrators is frequently difficult in general relativity because all variables are inseparable. Moreover, even if two analytically integrable splitting parts exist in a relativistic Hamiltonian, all analytical solutions are not explicit functions of proper time. Naturally, implicit symplectic integrators, such as the midpoint rule, are applicable to this case. In general, these integrators are numerically more expensive to solve than same-order explicit symplectic algorithms. To address this issue, we split the Hamiltonian of Schwarzschild spacetime geometry into four integrable parts with analytical solutions as explicit functions of proper time. In this manner, second- and fourth-order explicit symplectic integrators can be easily made available. The new algorithms are also useful for modeling the chaotic motion of charged particles around a black hole with an external magnetic field. They demonstrate excellent long-term performance in maintaining bounded Hamiltonian errors and saving computational cost when appropriate proper time steps are adopted.
In previous papers, explicit symplectic integrators were designed for nonrotating black holes, such as a Schwarzschild black hole. However, they fail to work in the Kerr spacetime because not all variables can be separable, or not all splitting parts
have analytical solutions as explicit functions of proper time. To cope with this difficulty, we introduce a time transformation function to the Hamiltonian of Kerr geometry so as to obtain a time-transformed Hamiltonian consisting of five splitting parts, whose analytical solutions are explicit functions of the new coordinate time. The chosen time transformation function can cause time steps to be adaptive, but it is mainly used to implement the desired splitting of the time transformed Hamiltonian. In this manner, new explicit symplectic algorithms are easily available. Unlike Runge Kutta integrators, the newly proposed algorithms exhibit good long term behavior in the conservation of Hamiltonian quantities when appropriate fixed coordinate time steps are considered. They are better than same order implicit and explicit mixed symplectic algorithms and extended phase space explicit symplectic like methods in computational efficiency. The proposed idea on the construction of explicit symplectic integrators is suitable for not only the Kerr metric but also many other relativistic problems, such as a Kerr black hole immersed in a magnetic field, a Kerr Newman black hole with an external magnetic field, axially symmetric core shell systems, and five dimensional black ring metrics.
In a previous paper, second- and fourth-order explicit symplectic integrators were designed for a Hamiltonian of the Schwarzschild black hole. Following this work, we continue to trace the possibility of the construction of explicit symplectic integr
ators for a Hamiltonian of charged particles moving around a Reissner-Nordstrom black hole with an external magnetic field. Such explicit symplectic methods are still available when the Hamiltonian is separated into five independently integrable parts with analytical solutions as explicit functions of proper time. Numerical tests show that the proposed algorithms share the desirable properties in their long-term stability, precision and efficiency for appropriate choices of step sizes. For the applicability of one of the new algorithms, the effects of the black holes charge, the Coulomb part of the electromagnetic potential and the magnetic parameter on the dynamical behavior are surveyed. Under some circumstances, the extent of chaos gets strong with an increase of the magnetic parameter from a global phase-space structure. No the variation of the black holes charge but the variation of the Coulomb part is considerably sensitive to affect the regular and chaotic dynamics of particles orbits. A positive Coulomb part is easier to induce chaos than a negative one.
We give a possible splitting method to a Hamiltonian for the description of charged particles moving around the Reissner-Nordstrom-(anti)-de Sitter black hole with an external magnetic field. This Hamiltonian can be separated into six analytical solv
able pieces, whose solutions are explicit functions of proper time. In this case, second- and fourth-order explicit symplectic integrators are easily available. They exhibit excellent long-term behavior in maintaining the boundness of Hamiltonian errors regardless of ordered or chaotic orbits if appropriate step-sizes are chosen. Under some circumstances, an increase of positive cosmological constant gives rise to strengthening the extent of chaos from the global phase space; namely, chaos of charged particles occurs easily for the accelerated expansion of the universe. However, an increase of the magnitude of negative cosmological constant does not. The different contributions on chaos are because the cosmological constant acts as a repulsive force in the Reissner-Nordstrom-de Sitter black hole, but an attractive force in the Reissner-Nordstrom-anti-de Sitter black hole.
The Hamiltonian system of general relativity and its quantization without any matter or gauge fields are discussed on the basis of the symplectic geometrical theory. A symplectic geometry of classical general relativity is constructed using a general
ized phase space for pure gravity. Prequantization of the symplectic manifold is performed according to the standard procedure of geometrical quantization. Quantum vacuum solutions are chosen from among the classical solutions under the Einstein-Brillouin-Keller quantization condition. A topological correction of quantum solutions, namely the Maslov index, is realized using a prequantization bundle. In addition, a possible mass spectrum of Schwarzschild black holes is discussed.
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