Equivalence between two charged black holes in dynamics of orbits outside the event horizons

Using the FermiDirac distribution function, Balart and Vagenas gave a charged spherically symmetric regular black hole, which is a solution of Einstein field equations coupled to a nonlinear electrodynamics. In fact, the regular black hole is a Reissner-Nordstrom (RN) black hole when a metric function is given a Taylor expansion to first order approximations. It does not have a curvature singularity at the origin,but the RN black hole does. Both black hole metrics have horizons and similar asymptotic behaviors, and satisfy the weak energy conditions everywhere. They are almost the same in photon effective potentials, photon circular orbits and photon spheres outside the event horizons. There are relatively minor differences between effective potentials, stable circular orbits and innermost stable circular orbits of charged particles outside the event horizons of the two black holes immersed in external magnetic fields. Although the twomagnetized black holes allow different construction methods of explicit symplectic integrators, they exhibit approximately consistent results in the regular and chaotic dynamics of charged particles outside the event horizons. Chaos gets strong as the magnetic field parameter or the magnitude of negative Coulomb parameter increases, but becomes weak when the black hole charge or the positive Coulomb parameter increases. A variation of dynamical properties is not sensitive dependence on an appropriate increase of the black hole charge. The basic equivalence between the two black hole gravitational systems in the dynamics of orbits outside the event horizons is due to the two metric functions having an extremely small difference. This implies that the RN black hole is reasonably replaced by the regular black hole without curvature singularity in many situations.Comment: 18 pages, 12 figure

A Note on the Construction of Explicit Symplectic Integrators for Schwarzschild Spacetimes

In recent publications, the construction of explicit symplectic integrators for Schwarzschild- and Kerr-type spacetimes is based on splitting and composition methods for numerical integrations of Hamiltonians or time-transformed Hamiltonians associated with these spacetimes. Such splittings are not unique but have various options. A Hamiltonian describing the motion of charged particles around the Schwarzschild black hole with an external magnetic field can be separated into three, four, and five explicitly integrable parts. It is shown through numerical tests of regular and chaotic orbits that the three-part splitting method is the best of the three Hamiltonian splitting methods in accuracy. In the three-part splitting, optimized fourth-order partitioned Runge–Kutta and Runge–Kutta–Nyström explicit symplectic integrators exhibit the best accuracies. In fact, they are several orders of magnitude better than the fourth-order Yoshida algorithms for appropriate time steps. The first two algorithms have a small additional computational cost compared with the latter ones. Optimized sixth-order partitioned Runge–Kutta and Runge–Kutta–Nyström explicit symplectic integrators have no dramatic advantages over the optimized fourth-order ones in accuracy during long-term integrations due to roundoff errors. The idea of finding the integrators with the best performance is also suitable for Hamiltonians or time-transformed Hamiltonians of other curved spacetimes including Kerr-type spacetimes. When the numbers of explicitly integrable splitting sub-Hamiltonians are as small as possible, such splitting Hamiltonian methods would bring better accuracies. In this case, the optimized fourth-order partitioned Runge–Kutta and Runge–Kutta–Nyström methods are worth recommending

Charged Particle Motions near Non-Schwarzschild Black Holes with External Magnetic Fields in Modified Theories of Gravity

A small deformation to the Schwarzschild metric controlled by four free parameters could be referred to as a nonspinning black hole solution in alternative theories of gravity. Since such a non-Schwarzschild metric can be changed into a Kerr-like black hole metric via a complex coordinate transformation, the recently proposed time-transformed, explicit symplectic integrators for the Kerr-type spacetimes are suitable for a Hamiltonian system describing the motion of charged particles around the non-Schwarzschild black hole surrounded with an external magnetic field. The obtained explicit symplectic methods are based on a time-transformed Hamiltonian split into seven parts, whose analytical solutions are explicit functions of new coordinate time. Numerical tests show that such explicit symplectic integrators for intermediate time steps perform well long-term when stabilizing Hamiltonian errors, regardless of regular or chaotic orbits. One of the explicit symplectic integrators with the techniques of Poincaré sections and fast Lyapunov indicators is applied to investigate the effects of the parameters, including the four free deformation parameters, on the orbital dynamical behavior. From the global phase-space structure, chaotic properties are typically strengthened under some circumstances, as the magnitude of the magnetic parameter or any one of the negative deformation parameters increases. However, they are weakened when the angular momentum or any one of the positive deformation parameters increases

Chaos in a Magnetized Modified Gravity Schwarzschild Spacetime

Based on the scalar–tensor–vector modified gravitational theory, a modified gravity Schwarzschild black hole solution has been given in the existing literature. Such a black hole spacetime is obtained through the inclusion of a modified gravity coupling parameter, which corresponds to the modified gravitational constant and the black hole charge. In this sense, the modified gravity parameter acts as not only an enhanced gravitational effect but also a gravitational repulsive force contribution to a test particle moving around the black hole. Because the modified Schwarzschild spacetime is static spherical symmetric, it is integrable. However, the spherical symmetry and the integrability are destroyed when the black hole is immersed in an external asymptotic uniform magnetic field and the particle is charged. Although the magnetized modified Schwarzschild spacetime is nonintegrable and inseparable, it allows for the application of explicit symplectic integrators when its Hamiltonian is split into five explicitly integrable parts. Taking one of the proposed explicit symplectic integrators and the techniques of Poincaré sections and fast Lyapunov indicators as numerical tools, we show that the charged particle can have chaotic motions under some circumstances. Chaos is strengthened with an increase of the modified gravity parameter from the global phase space structures. There are similar results when the magnetic field parameter and the particle energy increase. However, an increase of the particle angular momentum weakens the strength of chaos

Shanghai cognitive intervention of mild cognitive impairment for delaying progress with longitudinal evaluation-a prospective, randomized controlled study (SIMPLE): rationale, design, and methodology

Abstract Background Mild cognitive impairment is an early stage of Alzheimer’s disease. Increasing evidence has indicated that cognitive training could improve cognitive abilities of MCI patients in multiple cognitive domains, making it a promising therapeutic approach for MCI. However, the effect of long-time training has not been widely explored. It is also necessary to evaluate the extent how it could reduce the convertion rate from MCI to AD. Methods/design The SIMPLE study is a multicenter, randomized, single-blind prospective clinical trial assessing the effects of computerized cognitive training on different cognitive domains in MCI patients. It is carried out in 7 centers in China. The study population includes patients aged 50–85, and they are randomly allocated to the training or control group. The primary outcome is to compare the conversion rate of MCI within 36-month follow-up. Structural and functional MRI will be used to interpret the effect of cognitive training. The cognitive training comprises a variety of games related with cognitive domains such as attention, memory, visualspatial ability and executive function. We cautiously set 50% reduction in the rate of conversion as estimated effect. With 80–90% statistical power and 12% as the overall probability of conversion within the study period, 600–800 patients are finally required in the study. The first patent has been recruited in April 2017. Discussion Previous studies suggested the benefit of cognitive training for MCI, but neither long-time nor Chinese culture were investigated. The SIMPLE designs and utilizes an improved computerized cognitive training approach and assesses its effects on MCI progress. In addition, neural activities explaining the effects on cognition function changes will be revealed, which could in turn to imply more useful therapeutic approaches. Trial registration ClinicalTrials.gov Identifier: NCT03119051

Additional file 1: of Shanghai cognitive intervention of mild cognitive impairment for delaying progress with longitudinal evaluation-a prospective, randomized controlled study (SIMPLE): rationale, design, and methodology

Description of neuropsychological battery. Here is the description of neuropsychological assessments used in in the SIMPLE study, including their versions, scores and clinical significance. (DOCX 35 kb