Spherical photon orbits around a 5D Myers–Perry black hole

We study the motion of bound null geodesics with fixed coordinate radius around a five-dimensional rotating black hole. These spherical photon orbits are not confined to a plane, and can exhibit interesting quasiperiodic behaviour. We provide necessary conditions for the existence of these orbits, and explicitly compute the radii of circular orbits in the equatorial and polar planes. Finally, we plot representative examples of some of the types of possible orbits, commenting on their qualitative features.     

Circular orbits and relative strains in Schwarzschild space-time

The equation of the relative strain is analyzed in tetrad form with respect to a family of observers moving on spatially circular orbits, in the Schwarzschild space-time. We select a field of tetrads, which we term phase locking frames, and explicitly calculate how, in the equatorial plane, the orbital acceleration, its gradient and the Fermi drag add together to compensate the curvature and assure equilibrium among a set of comoving neighbouring particles. While equilibrium is achieved in the radial and azimuthal directions, in the direction orthogonal to the equatorial plane there is a residue of acceleration which pulls a particle towards that plane leading to a harmonic oscillation with a frequency equal to the proper frequency of the orbital revolution. This measurement, combined with those of the frequency shift of an incoming photon and the frequency of precession of the local compass of inertia, enables one to determine the relativistic ratio 2M/r, whereM is the gravitational mass of the source andr the coordinate radius of the circular orbits.     

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.     

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.     

Null geodesics in black hole metrics with non-zero cosmological constant

We study the radial motion along null geodesics in the Reissner-Nordström-de Sitter and Kerr-de Sitter space-times. We analyze the properties of the effective potential and we discuss circular orbits. We find that:

Photon motion in Kerr–de Sitter spacetimes

We study the general motion of photons in the Kerr–de Sitter black-hole and naked singularity spacetimes. The motion is governed by the impact parameters X, related to the axial symmetry of the spacetime, and q, related to its hidden symmetry. Appropriate ‘effective potentials’ governing the latitudinal and radial motion are introduced and their behavior is examined by the ‘Chinese boxes’ technique giving regions allowed for the motion in terms of the impact parameters. Restrictions on the impact parameters X and q are established in dependence on the spacetime parameters \(M, \Lambda , a\). The motion can be of orbital type (crossing the equatorial plane, \(q>0\)) and vortical type (tied above or below the equatorial plane, \(q<0\)). It is shown that for negative values of q, the reality conditions imposed on the latitudinal motion yield stronger constraints on the parameter X than that following from the reality condition of the radial motion, excluding the existence of vortical motion of constant radius. The properties of the spherical photon orbits of the orbital type are determined and used along with the properties of the effective potentials as criteria of classification of the KdS spacetimes according to the properties of the motion of the photon.     

Numerical calculation of bound geodesics in the Kerr metric

Timelike geodesics, especially bound orbits in the equatorial plane (?=π/2) and spherical orbits (r=const), are calculated numerically. We plot the orbits using the Kerr-Schild coordinate system. The periastron advance and the dragging of nodes have the same values in any coordinate system and can be directly measured by an observer at infinity.     

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.     

Equatorial circular orbits in Kerr–anti-de Sitter spacetimes

Equatorial circular orbits of test particles in the Kerr–anti-de Sitter black-hole and naked-singularity spacetimes are analyzed and their properties like the existence, orientation and stability are discussed. Due to the attractive cosmological constant ( $\varLambda <0$ ), all particles moving along equatorial orbits are still bound in the gravitational field of the central object. In general, there are two families of equatorial circular orbits. Particles moving along minus-family orbits possess negative angular momentum and, thus, they are counterrotating from the point of view of the locally non-rotating frames (LNRF). Particles moving along plus-family orbits possess, in most cases, positive angular momentum and belong to corotating particles from the point of view of the LNRF. Nevertheless, in stationary regions inside black holes and also near naked singularities with appropriately chosen value of the cosmological constant and rotational parameter $a<1.299$ , there are also counterrotating plus-family circular orbits. Moreover, in spacetimes with $a<1.089$ , some of these orbits are characterized by negative specific energy, indicating the bounding energy of the particle, moving along such an orbit, higher than its rest energy. In black-hole spacetimes, all such orbits are radially unstable, but in naked-singularity spacetimes, stable counterrotating orbits with negative specific energy exist.     

Equatorial Circular Orbits of Neutral Test Particlesin Weyl Spacetimes

A general stability study of equatorial circular orbits in static axially symmetric gravitating systems is presented. We investigate the motion of neutral test particles in circular geodesics such as the marginally stable orbit, the marginally bounded orbit, and the photon orbit are analyzed. We find general expressions for the radius, specific energy, specific angular momentum, and the radius of the marginally stable orbit, both for null and timelike circular geodesics. Different solutions were expressed in different coordinates systems: cylindrical coordinates, oblate spheroidal coordinates, and prolate spheroidal coordinates are considered. We show that all null circular trajectories are unstable, and that there aren’t marginally stable null geodesics, whereas for timelike geodesics the motion can be unbounded, bounded, or circular.     

Charged particle trajectories in a magnetic field on a curved space-time

In this paper we have studied the motion of charged particles in a dipole magnetic field on the Schwarzscbild background geometry. A detailed analysis has been made in the equatorial plane through the study of the effective potential curves. In the case of positive canonical angular momentum the effective potential has two maxima and two minima giving rise to a well-defined potential well rear the event horizon. This feature of the effective potential categorises the particle orbits into four classes, depending on their energies. (i) Particles, coming from infinity with energy less than the absolute maximum ofV _eff, would scatter away after being turned away by the magnetic field. (ii) Whereas those with energies higher than this would go into the central star seeing no barrier. (iii) Particles initially located within the potential well are naturally trapped, and they execute Larmor motion in bound gyrating orbits. (iv) and those with initial positions corresponding to the extrema ofV _eff follow circular orbits which are stable for non-relativistic particles and unstable for relativistic ones. We have also considered the case of negative canonical angular momentum and found that no trapping in bound orbits occur for this case. In the case when particles are not confined to the equatorial plane we have found that the particles execute oscillatory motion between two mirror points if the magnetic field is sufficiently high, but would continuously fall towards the event horizon otherwise. An erratum to this article is available at .     

Models of spherical shells as sources of Majumdar–Papapetrou type spacetimes

By starting with a seed Newtonian potential–density pair we construct relativistic thick spherical shell models for a Majumdar–Papapetrou type conformastatic spacetime. As a simple example, we considerer a family of Plummer–Hernquist type relativistic spherical shells. As a second application, these structures are then used to model a system composite by a dust disk and a halo of matter. We study the equatorial circular motion of test particles around such configurations. Also the stability of the orbits is analyzed for radial perturbation using an extension of the Rayleigh criterion. The models considered satisfying all the energy conditions.     

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.     

MALBEC: a new CUDA-C ray-tracer in general relativity

A new CUDA-C code for tracing orbits around non-charged black holes is presented. This code, named MALBEC, take advantage of the graphic processing units and the CUDA platform for tracking null and timelike test particles in Schwarzschild and Kerr. Also, a new general set of equations that describe the closed circular orbits of any timelike test particle in the equatorial plane is derived. These equations are extremely important in order to compare the analytical behavior of the orbits with the numerical results and verify the correct implementation of the Runge–Kutta algorithm in MALBEC. Finally, other numerical tests are performed, demonstrating that MALBEC is able to reproduce some well-known results in these metrics in a faster and more efficient way than a conventional CPU implementation.     

Orbital and epicyclic motion of charged test particles around non-rotating Einstein-Æther black holes

The study of charged test particle dynamics in the combined black hole gravitational field and magnetic field around it could provide important theoretical insight into astrophysical processes around such compact object. We have explored the orbital and epicyclic motion of charged test particles in the background of non-rotating Einstein-Æther black holes in the presence of external uniform magnetic field. We numerically integrate the equations of motion and analyze the trajectories of the charged test particles. We examined the stability of circular orbits using effective potential technique and study the characteristics of innermost stable circular orbits. We analyze the key features of quasi-harmonic oscillations of charged test particles nearby the stable circular orbits in an equatorial plane of the black hole, and investigate the radial profiles of the frequencies of latitudinal as well as radial harmonic oscillations in dependence on the strength of magnetic field, mass of the black hole and dimensionless coupling constants of the theory. We demonstrate that the magnetic field and dimensionless parameters of the theory have strong influence on charged particle motion around Einstein-Æther black holes.     

Stability of Circular Orbits around a Tidal Charged Black Hole

We study the effects of the tidal charge on the equatorial circular motion of neutral test particles near a tidal charged black hole. This analysis investigates stable as well as unstable circular orbits in all possible configurations of nonextremal and extremal cases. It is found that a negative tidal charge will increase the energy and angular momentum of a neutral test particle moving around a black hole. We obtain a continuous region of stability for both extremal and nonextremal cases. We conclude that the region of stability as well as radius of last stable circular orbit shows increasing behavior for Q < 0.     

Massive test particle motion in 5-dimensional electromagnetic-free Kaluza-Klein theory

A class of static, vacuum solutions of (free-electromagnetic) Kaluza-Klein equations with three-dimensional spherical symmetry is studied. In order to explore the dynamic in such spacetimes, geodesic equations are obtained and the effective potential for massive test particles is analyzed. Particular attention is devoted to the properties of the four-dimensional counterpart of these solutions in their Schwarzschild limit. A modification of the circular stable orbits compared with the Schwarzschild case is investigated.     

Gyroscopic Precession and Centrifugal Force in the Ernst Spacetime

The phenomenon of gyroscopic precession in the Ernst spacetime is studied within the framework of the Frenet-Serret formalism. General formulae are obtained for circular orbits. At the same time general relativistic analogues of inertial forces such as gravitational and centrifugal forces are also investigated in the Ernst spacetime. Reversal of gyroscopic precession as well as centrifugal force is considered at the circular photon orbits. These phenomena are examined in the Melvin universe as a special case of the Ernst spacetime by setting the mass parameter equal to zero.     

Existence and stability of circular orbits in static and axisymmetric spacetimes

The existence and stability of timelike and null circular orbits (COs) in the equatorial plane of general static and axisymmetric (SAS) spacetime are investigated in this work. Using the fixed point approach, we first obtained a necessary and sufficient condition for the non-existence of timelike COs. It is then proven that there will always exist timelike COs at large \(\rho \) in an asymptotically flat SAS spacetime with a positive ADM mass and moreover, these timelike COs are stable. Some other sufficient conditions on the stability of timelike COs are also solved. We then found the necessary and sufficient condition on the existence of null COs. It is generally shown that the existence of timelike COs in SAS spacetime does not imply the existence of null COs, and vice-versa, regardless whether the spacetime is asymptotically flat or the ADM mass is positive or not. These results are then used to show the existence of timelike COs and their stability in an SAS Einstein-Yang-Mills-Dilaton spacetimes whose metric is not completely known. We also used the theorems to deduce the existence of timelike and null COs in some known SAS spacetimes.     

Inspection methods for spacecrafts with nuclear power plants

Inspection methods for space objects (SOs) using an active apparatus with onboard detecting equipment are considered. A method based on sequential maneuvering of a spacecraft inspector in the equatorial plane to the points of convergence with various inspected objects in crossed orbits is proposed.