Gyroscopic Precession and Inertial Forces in Axially Symmetric Stationary Spacetimes

We study the phenomenon of gyroscopic precession and the analogues of inertial forces within the framework of general relativity. Covariant connections between the two are established for circular orbits in stationary spacetimes with axial symmetry. Specializing to static spacetimes, we prove that gyroscopic precession and centrifugal force both reverse at the photon orbits. Simultaneous non-reversal of these in the case of stationary spacetimes is discussed. Further insight is gained in the case of static spacetime by considering the phenomena in a spacetime conformal to the original one. Gravi-electric and gravi-magnetic fields are studied and their relation to inertial forces is established.     

Sen时空中的陀螺进动效应

采用Frenet-Serret形式研究了Sen时空中的陀螺进动效应,得出了沿圆轨迹以任意常数角速 度运动的轨道陀螺进动的一般公式,当陀螺在赤道平面上沿短程线运动时,由一般公式出发 可得到精确的轨道周期进动角公式.作为特例,研究了静态dilaton时空中的陀螺进动效应. 将所得结果分别与Kerr-Newman和Reissner-Nordstrm时空中的情形相比较,发现dilaton 耦合使陀螺进动速率减慢,轨道周期进动角减小. 关键词： Sen时空 Frenet-Serret标架 Fermi-Walker移动     

Cosmological black holes: A black hole in the Einstein-de Sitter universe

As an example of a dynamical cosmological black hole, a spacetime that describes an expanding black hole in the asymptotic background of the Einstein-de Sitter universe is constructed. The black hole is primordial in the sense that it forms ab initio with the big bang singularity and its expanding event horizon is represented by a conformal Killing horizon. The metric representing the black hole spacetime is obtained by applying a time dependent conformal transformation on the Schwarzschild metric, such that the result is an exact solution with a matter content described by a two-fluid source. Physical quantities such as the surface gravity and other effects like perihelion precession, light bending and circular orbits are studied in this spacetime and compared to their counterparts in the gravitational field of the isolated Schwarzschild black hole. No changes in the structure of null geodesics are recorded, but significant differences are obtained for timelike geodesics, particularly an increase in the perihelion precession and the non-existence of circular timelike orbits. The solution is expressed in the Newman-Penrose formalism.     

Circular orbits in extremal Reissner-Nordstrom spacetime

Circular null geodesic orbits, in extremal Reissner-Nordstrom spacetime, are examined with regard to their stability, and compared with similar orbits in the near-extremal situation. Extremization of the effective potential for null circular orbits shows the existence of a stable circular geodesic in the extremal spacetime, precisely on the event horizon which coincides with the null geodesic generator. Such a null orbit on the horizon is also indicated by the global minimum of the effective potential for circular timelike orbits. This type of geodesic is of course absent in the corresponding near-extremal spacetime, as we show here, testifying to differences between the extremal limit of a generic RN spacetime and the exactly extremal geometry.     

Circular orbits in cosmic string and Schwarzschild–AdS spacetime with Fermi–Walker transport

In this paper we discuss the Fermi–Walker transport of vectors along orbits in cosmic string and Schwarzschild–AdS spacetimes. We analyze the influence of acceleration on these holonomies. An effect similar to Thomas precession is observed within the process of Fermi–Walker transport along these circular orbits which are studied in the limit of vanishing cosmological constant in Schwarzschild–AdS case; also we obtain Fermi–Walker transport in a Schwarzschild background. In the case of a Schwarzschild spacetime, we analyze the quantized band holonomy invariance. In the limit of zero acceleration we recover the well-known results for holonomy matrix obtained by parallel transport in all these spacetimes.     

Equatorial Plane Circular Orbits in the Taub-NUT Spacetime

Accelerated circular orbits in the equatorial plane of the Taub-NUT spacetime are analyzed to investigate the effects of its gravitomagnetic monopole source. The effect of a small gravitomagnetic monopole on these orbits is compared to the corresponding orbits pushed slightly off the equatorial plane in the absence of the monopole.     

A tapered beam finite element for rotor dynamics analysis

The stiffness, mass and gyroscopic matrices of a rotating beam element are developed, a cubic function being used for the transverse displacement. Shear deflection is included by use of end nodal variables of shear strain, along with transverse displacement and cross-section rotation; rotatory inertia effects are included in the energy functional to provide a Timoshenko beam formulation. The gyroscopic effects for small perturbations are linearized as a skew symmetric damping matrix. The formulation is implemented by numerical integration for a linearly tapered circular beam. A technique of reduction of the shear nodal variable prior to global assembly is shown to provide little loss in accuracy with reduced system bandwidth. Numerical comparisons for three previously published beam models are included, with results presented for the case of forward and reverse precession to verify the gyroscopic effects. The utility of the element in a general program for rotor dynamics analysis is identified.     

Stability of Circular Orbits in General Relativity: a Phase Space Analysis

Phase space method provides a novel way for deducing qualitative features of nonlinear differential equations without actually solving them. The method is applied here for analyzing stability of circular orbits of test particles in various physically interesting environments. The approach is shown to work in a revealing way in Schwarzschild spacetime. All relevant conclusions about circular orbits in the Schwarzschild-de Sitter spacetime are shown to be remarkably encoded in a single parameter. The analysis in the rotating Kerr black hole readily exposes information as to how stability depends on the ratio of source rotation to particle angular momentum. As a wider application, it is exemplified how the analysis reveals useful information when applied to motion in a refractive medium, for instance, that of optical black holes.     

Geodesic motions of test particles in a relativistic core–shell spacetime

In this paper, we discuss the geodesic motions of test particles in the intermediate vacuum between a monopolar core and an exterior shell of dipoles, quadrupoles and octopoles. The radii of the innermost stable circular orbits at the equatorial plane depend only on the quadrupoles. A given oblate quadrupolar leads to the existence of two innermost stable circular orbits, and their radii are larger than in the Schwarzschild spacetime. However, a given prolate quadrupolar corresponds to only one innermost stable circular orbit, and its radius is smaller than in the Schwarzschild spacetime. As to the general geodesic orbits, one of the recently developed extended phase space fourth order explicit symplectic-like methods is efficiently applicable to them although the Hamiltonian of the relativistic core–shell system is not separable. With the aid of both this fast integrator without secular growth in the energy errors and gauge invariant chaotic indicators, the effect of these shell multipoles on the geodesic dynamics of order and chaos is estimated numerically.     

The properties and geodesics related to the NUT--Taub-like spacetime

Some properties related to the NUT--Taub-like spacetime, such as the surface of infinite red-shift, horizon, singularity and the area of the NUT--Taub-like black hole are discussed. Furthermore, the geodesics in the NUT--Taub-like spacetime are obtained in some special cases. Specifically, the circular orbits for a massive particle are derived, which can reduce to the cases of the Schwarzschild spacetime and the NUT--Taub spacetime when m^*=0 and m^*\ll M, respectively.     

Spin precession of a charged particle in a uniform magnetic field on a static space-time

We investigate the ratio of spin precession frequency to orbital frequency for a spinning charged particle confined to circular orbit in the equatorial plane of a compact object, with a uniform magnetic field, as described by the Wald and the Ernst potentials. In order to see the difference in behaviours for particles with differentg values we consider the cases of electron and proton separately.     

Geodesic Structure of a Non-linear Magnetic Charged Black Hole Surrounded by Quintessence

This paper aims to investigate the geodesic motion in the spacetime of a non-linear magnetic charged black hole surrounded by quintessence. By varying the Lagrangian corresponding to the metric, the orbital motion equation has been obtained. The effects of the magnetic charge $Q$, positive normalization factor $C$, angular momentum $b$, and energy $E$ on time-like and null geodesic motion are discussed from three aspects: orbital stability, orbital types, and circular orbits. By comparing the effects of the above parameters $C$, $b$ on the effective potential, it is found that quintessence has an impact on the types and stability of orbits. In addition, for time-like orbital motion, when $3.443113\leq b\leq6.392\,578$ (for fixed $C=0.0002$, $M = 1$, $Q=0.7$), there are bound orbits, and within this range, the stable circular orbits exist, and the radii of the innermost and outermost stable circular orbit are $r=5.912\,654$ and $r=56.745\,933$, respectively. For null orbital motion, the orbital types have only unstable circular orbit which occur at $r=2.951\,072$ ($E^{2}=E_{2}^{2}=0.4$), absorb orbits and escape orbits, but no stable circular orbits, and bound orbits.     

Relativistic precession of a gyroscope. I. Circular orbits

The relativistic precession of the rotation axis of a spherical gyroscope is treated in the framework of monadic specification of the frame of reference. It is found that the precession in the comoving frame compensates the rotation of the frame of reference itself. An exact expression is derived for the angular velocity of precession for motion of a gyroscope in circular epiequatorial orbits in the Kerr field. The results are compared with the approximate expression obtained by Schiff. Numerical examples are considered.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 64–68, March, 1981.     

The Larmor nutation

The classical non-relativistic problem of the motion of a charged particle in an external central force field and a weak uniform magnetic field is revisited to show that the motion of the kinetic angular momentumL = r × p of the particle, in the so-called Larmor approximation, is not a simple precession but is actually a composite motion involving precession as well as a high frequency nutation. The precession-nutation motion ofL is discussed in the Larmor approximation when the Larmor-frame-orbit of the charged particle is an ellipse (or a circle) for the case of the two central forces namely the Coulomb and the Hooke-law-force, which are the only two central forces known to permit closed orbits.     

On properties of vacuum axially symmetric spacetime of gravitomagnetic monopole in cylindrical coordinates

We investigate general relativistic effects associated with the gravitomagnetic monopole moment of a gravitational source through the analysis of the motion of test particles and electromagnetic fields distribution in the spacetime around the nonrotating cylindrical NUT source. We consider the circular motion of test particles in the NUT spacetime, their characteristics and the dependence of the effective potential on the radial coordinate for the different values of the NUT parameter and orbital momentum of test particles. It is shown that the bounds of stability for circular orbits are displaced toward the event horizon with the growth of the monopole moment of the NUT object. In addition, we obtain exact analytical solutions of the Maxwell equations for magnetized and charged cylindrical NUT stars.     

Circular Orbits in Stationary Axisymmetric Spacetimes

Several types of characteristics of spatially circular timelike trajectories in stationary axisymmetric spacetimes are related in a simple and covariant manner. The relations allow us to establish straightforward links between different phenomena often studied on circular orbits: mechanics of a single test particle, precession of gyroscopes with respect to important vectors defined along the orbit, geometrical parameters (curvatures) of the trajectory provided by the Frenet-Serret formalism, and geometrical properties (vorticity and shear) of the whole circular congruence.     

Position determination and strong field parallax effects for photon emitters in the Schwarzschild spacetime

Position determination of photon emitters and associated strong field parallax effects are investigated using relativistic optics when the photon orbits are confined to the equatorial plane of the Schwarzschild spacetime. We assume the emitter is at a fixed space position and the receiver moves along a circular geodesic orbit. This study requires solving the inverse problem of determining the (spatial) intersection point of two null geodesic initial data problems, serving as a simplified model for applications in relativistic astrometry as well as in radar and satellite communications.     

Particle dynamics around a dyonic charged black hole

In this article, we study the circular motion of particles and the well-known Penrose mechanism around a Kerr-Newman-Kasuya black hole spacetime. The inner and outer horizons, as well as ergosurfaces of the said black hole, are briefly examined under the effect of spin and dyonic charge. Moreover, by limiting our exploration to the equatorial plane, we discuss the characteristics of circular geodesics and investigate both photons, as well as marginally stable circular orbits. It is noted that black hole charge diminishing the radii of photon and marginally stable circular orbits. To investigate the nature of particle dynamics, we studied the effective potential and Lyapunov exponent. While inspecting the process of energy extraction, we derived the Wald inequality, which can help us to locate the energy limits of the Penrose process. Furthermore, we have found expressions for the negative energy states and the efficiency of energy extraction. The obtained result illustrates that both black hole rotation and dyonic charge contributes to the efficiency of energy extraction.     

Physical frames along circular orbits in stationary axisymmetric spacetimes

Three natural classes of orthonormal frames, namely Frenet-Serret, Fermi–Walker and parallel transported frames, exist along any timelike world line in spacetime. Their relationships are investigated for timelike circular orbits in stationary axisymmetric spacetimes, and illustrated for black hole spacetimes. Dedicated to Bahram Mashhoon for his 60th birthday.