A numerical examination of certain null geodesies of a high angular momentum kerr field

An observer situated anywhere but in the equatorial plane of a high angular momentum Kerr field cannot see the ring singularity. In the visual field of such an observer, what demarcates his own universe from that through the ring? The projections onto a certain submanifold of the null geodesics which pass through a point on the symmetry axis of a specific Kerr field are examined numerically. All the distinct projections are obtained by varying one parameter, essentially the quadratic Killing tensor constant. Various interesting features of the geodesics emerge. Through the ring is a region in which there exist closed time-like curves and which can be used to construct closed time-like curves through any non-singular point of the manifold. Only geodesies of negative angular momentum can enter this region.     

NULL GEODESICS IN STATIONARY ERNST-WILD SPACE-TIME

Null geodesics in equatorial plane of stationary Ernst-Wild space-time representing a Kerr black hole immersed in axlsymmetric magnetic field are investigated.As the case of static Ernst space-time there are two types of geodesics which depend on a dimensionless parameter β of the magnetic field.Differing from the static Ernst space-time,however,the critical β_C now depends on a dimensionless parameter q of the angular momentum of the Kerr black hole,and the relation between β_C and q is qualitatively different in two cases:L>O and L<0 where L is a dimensionless parameter of angular momentum of test particles moving along nu11 geodesics.     

The collisional Penrose process

Shortly after the discovery of the Kerr metric in 1963, it was realized that a region existed outside of the black hole’s event horizon where no time-like observer could remain stationary. In 1969, Roger Penrose showed that particles within this ergosphere region could possess negative energy, as measured by an observer at infinity. When captured by the horizon, these negative energy particles essentially extract mass and angular momentum from the black hole. While the decay of a single particle within the ergosphere is not a particularly efficient means of energy extraction, the collision of multiple particles can reach arbitrarily high center-of-mass energy in the limit of extremal black hole spin. The resulting particles can escape with high efficiency, potentially serving as a probe of high-energy particle physics as well as general relativity. In this paper, we briefly review the history of the field and highlight a specific astrophysical application of the collisional Penrose process: the potential to enhance annihilation of dark matter particles in the vicinity of a supermassive black hole.     

Mean field theory and geodesics in general relativity

It is proven that all geodesics in a mean gravitational field can be interpreted locally as the averages of geodesics in the unaveraged field. The respective time-like, space-like or null character of averaged and unaveraged geodesics is discussed carefully. Finally, some important astrophysical and cosmological applications and consequences are investigated.     

Geometric analysis of particular compactly constructed time machine spacetimes

We formulate the concept of time machine structure for spacetimes exhibiting a compactly constructed region with closed timelike curves. After reviewing essential properties of the pseudo Schwarzschild spacetime introduced by Ori, we present an analysis of its geodesics analogous to the one conducted in the case of the Schwarzschild spacetime. We conclude that the pseudo Schwarzschild spacetime is geodesically incomplete and not extendible to a complete spacetime. We then introduce a rotating generalization of the pseudo Schwarzschild metric, which we call the pseudo Kerr spacetime. We establish its time machine structure and analyze its global properties.     

Geometrical locus of massive test particle orbits in the space of physical parameters in Kerr space–time

Gravitational radiation of binary systems can be studied by using the adiabatic approximation in General Relativity. In this approach a small astrophysical object follows a trajectory consisting of a chained series of bounded geodesics (orbits) in the outer region of a Kerr Black Hole, representing the space time created by a bigger object. In our paper, we study the entire class of orbits, both of constant radius (spherical orbits), as well as non-null eccentricity orbits, showing a number of properties on the physical parameters and trajectories. The main result is the determination of the geometrical locus of all the orbits in the space of physical parameters in Kerr space–time. This becomes a powerful tool to know if different orbits can be connected by a continuous change of their physical parameters. A discussion on the influence of different values of the angular momentum of the hole is given. Main results have been obtained by analytical methods.     

Gödel black hole,closed timelike horizon and the study of particle emissions

We show that a particle, with positive orbital angular momentum, following an outgoing null/timelike geodesic, shall never reach the closed timelike horizon present in the (4 + 1)-dimensional rotating Gödel black hole space–time. Therefore a large part of this space–time remains inaccessible to a large class of geodesic observers, depending on the conserved quantities associated with them. We discuss how this fact and the existence of the closed timelike curves present in the asymptotic region make the quantum field theoretic study of the Hawking radiation, where the asymptotic observer states are a pre-requisite, unclear. However, the semi classical approach provides an alternative to verify the Smarr formula derived recently for the rotating Gödel black hole. We present a systematic analysis of particle emissions, specifically for scalars, charged Dirac spinors and vectors, from this black hole via the semiclassical complex path method.     

Structure of geodesics in the regular Hayward black hole space-time

The regular Hayward model describes a non-singular black hole space-time. By analyzing the behaviors of effective potential and solving the equation of orbital motion, we investigate the time-like and null geodesics in the regular Hayward black hole space-time. Through detailed analyses of corresponding effective potentials for massive particles and photons, all possible orbits are numerically simulated. The results show that there may exist four orbital types in the time-like geodesics structure: planetary orbits, circular orbits, escape orbits and absorbing orbits. In addition, when \(\ell \), a convenient encoding of the central energy density \(3/8\pi \ell ^{2}\), is 0.6M, and b is 3.9512M as a specific value of angular momentum, escape orbits exist only under \(b>3.9512M\). The precession direction is also associated with values of b. With \(b=3.70M\) the bound orbits shift clockwise but counter-clockwise with \(b=5.00M\) in the regular Hayward black hole space-time. We also find that the structure of null geodesics is simpler than that of time-like geodesics. There only exist three kinds of orbits (unstable circle orbits, escape orbits and absorbing orbits).     

The Kerr space-time near the ring singularity

We investigate the geometry of the Kerr space-time near the ring singularity. A systematic study of the mathematical and physical structure of this region reveals that the singularity in the Kerr space-time is naturally understood in terms of a subset of the immersion of the set defined byr=0 (in Boyer-Lindquist coordinates) in the Kerr space-time. It is well known that the Kerr space-time is not a differentiable manifold (due to the curvature singularity) or a topological manifold, but a well defined topological space with a structure that is manifested by the constrast in taking limits along the hypersurface atr=0 and the equatorial plane which approach singularity. We find that the ring singularity is either an edge or a self-intersection of the topological space depending on which extension of the metric throughr=0 is implemented. A major result of this analysis is the extrapolation to the general accelerating case of Carter's proof that the only nonspacelike geodesics which can reach the ring singularity are restricted to the equatorial plane. For finite magnitudes of proper acceleration, it is shown that only lightlike trajectories that asymptotically approach the null generator of the ring singularity can reach it from above or below the equatorial plane.     

A Toy Model for Magnetic Field Configurations in Black Hole Accretion Discs

We discuss the feature of the magnetic field configuration arising from double counter oriented electric currentrings in the accretion disc around a Kerr black hole （BH）. We discuss the relevant physical quantities corresponding to this configuration： （1） the power and torque transferred by the large-scale magnetic field, （2） the angular momentum and energy fluxes transferred from the BIt to the inner disc, （3） the radiation flux from the disc. In addition, we discuss the possibility that the closed magnetic field anchored at the disc probably evolves to the open magnetic field, which is helpful to produce the jet from the disc.     

Creation of a time-machine in (2+1)-dimensional gravity

A non-stationary model of two concentric counter-rotating layers is constructed in (2+1)-dimensional Einstein theory. It is shown that the model which satisfies both the weak and the dominant energy conditions, may evolve from a state with no closed time-like curves to a state where such curves are present in a finite region of space. No singularities are formed in the process.     

一类类光测地线的加速度

类光测地线γ0(λ)与二维类空超曲面φ正交，λ为其仿射参量.假如在类光测地线γ0(λ)上存在一点r(r=γ0(λr))共轭于类空超曲面φ，则对于γ0(λ)上任一点q(q=γ0(λq))满足λq＞λr，一定能把γ0连续变形成一条从φ到q的类时曲线.当产生类时曲线的变分矢量场不是类光测地线上的广义Jacobi场时，这些类时曲线在趋于类光测地线时，它们的固有加速度趋于无穷大. 关键词： 类光测地线 共轭点 变分     

On causality violation on a Kerr-de Sitter spacetime

The causal properties of the family of Kerr-de Sitter spacetimes are analyzed and compared to those of the Kerr family. First, an inextendible Kerr-de Sitter spacetime is obtained by joining together Carter’s blocks, i.e. suitable four dimensional spacetime regions contained within Killing horizons or within a Killing horizon and an asymptotic de Sitter region. Based on this property, and leaving aside topological identifications, we show that the causal properties of a Kerr-de Sitter spacetime are determined by the causal properties of the individual Carter’s blocks viewed as spacetimes in their own right. We show that any Carter’s block is stably causal except for the blocks that contain the ring singularity. The latter are vicious sets, i.e. any two events within such block can be connected by a future (respectively past) directed timelike curve. This behavior is identical to the causal behavior of the Boyer–Lindquist blocks that contain the Kerr ring singularity. These blocks are also vicious as demonstrated long ago by Carter. On the other hand, while for the case of a naked Kerr singularity the entire spacetime is vicious and thus closed timelike curves pass through any event including events in the asymptotic region, for the case of a Kerr-de Sitter spacetime the cosmological horizons protect the asymptotic de Sitter region from a-causal influences. In that regard, a positive cosmological constant appears to improve the causal behavior of the underlying spacetime.     

A dynamical approach to symplectic and spectral invariants for billiards

The Lazutkin parameter for curves which are invariant under the billiard ball map is viewed symplectically in a way which makes it analogous to the sum of the values of a generating function over a closed orbit. This leads to relations among lengths of closed geodesics, lengths of invariant curves for the billiard map, rotation numbers, and the Lazutkin parameter. These relations establish the Birkhoff invariant and the expansion for the lengths of invariant curves in terms of the Lazutkin parameter as symplectic and spectral invariants (for the Dirichlet spectrum) and provide invariants which characterize a family of ellipses among smooth curves with positive curvature. Geodesic flow on a bounded planar region gives rise to several geometric objects among which are closed reflected geodesics and invariant curves-closed curves whose tangents are invariant under reflection at the boundary. On a bounded domain, the map that assigns to each geodesic segment its successor after reflection at the boundary is called the billiard ball map and its dual (in the cotangent bundle for the boundary) is called the boundary map.     

Constant curvature f(R) gravity minimally coupled with Yang–Mills field

In this work, we have studied accretion of the dark energies in new variable modified Chaplygin gas (NVMCG) and generalized cosmic Chaplygin gas (GCCG) models onto Schwarzschild and Kerr?CNewman black holes. We find the expression of the critical four velocity component which gradually decreases for the fluid flow towards the Schwarzschild as well as the Kerr?CNewman black hole. We also find the expression for the change of mass of the black hole in both cases. For the Kerr?CNewman black hole, which is rotating and charged, we calculate the specific angular momentum and total angular momentum. We showed that in both cases, due to accretion of dark energy, the mass of the black hole increases and angular momentum increases in the case of a Kerr?CNewman black hole.     

Bounded region with closed time-like curves in T-dual pp-waves backgrounds

In the 10-dimensional pp-waves background we have found using T-duality a 4-dimensional space-time in which closed time-like curves appear in the region bounded by two coaxial elliptic cylinders. This 4-dimensional space-time is similar to the Gödel-type space-time only in this region. Outside of this region the causal pathology does not appear.     

Gravi-inertial reference frames in Schwarzschild and Kerr spaces

Gravi-inertial reference frames — the analogue of inertial frames in two-dimensional space — are constructed in Schwarzschild and Kerr spaces. With their help the energy, momentum, and angular momentum of these systems are determined. A form is found for writing Kerr's solution in Bondi-Sachs coordinates in the slow-rotation approximation. A new solution of Einstein's equations is found in this approximation which describes the gravitational field of a rotating and radiating source.     

Influence of dark energy on time-like geodesic motion in Schwarzschild spacetime

In this paper we investigate the influence of the dark energy on the time-like geodesic motion of a particle in Schwarzschild spacetime by analysing the behaviour of the effective potential which appears in an equation of motion. For the non-radial time-like geodesics, we find a bound orbit when the particle energy is in an appropriate range, and also find another possible orbit, which is that the particle drops straightly into the singularity of a black hole or escapes to infinity. For the radial time-like geodesics, we find an unstable circular orbit when the particle energy is the critical value, in which case it is possible for the particle to escape to infinity.     

Minisuperspace model of Machian resolution of Problem of Time. I. Isotropic case

It is well known that the three parameters that characterize the Kerr black hole (mass, angular momentum and horizon area) satisfy several important inequalities. Remarkably, some of these inequalities remain valid also for dynamical black holes. This kind of inequalities play an important role in the characterization of the gravitational collapse. They are closed related with the cosmic censorship conjecture. In this article recent results in this subject are reviewed.