Narrow Results

Let D_F = {(z₀, z) ∈ ℂⁿ | |z₀|² < b, ||z||² < F(|z₀|²)} be a strongly pseudoconvex Hartogs domain endowed with the Kähler metric g_F associated to the Kähler form .

This paper contains several results on the Riemannian geometry of these domains. These are summarized in Theorems 1.1–1.3. In the first one we prove that if D_F admits a non-special geodesic (see definition below) through the origin whose trace is a straight line then D_F is holomorphically isometric to an open subset of the complex hyperbolic space. In the second theorem we prove that all the geodesics through the origin of D_F do not self-intersect, we find necessary and sufficient conditions on F for D_F to be geodesically complete and we prove that D_F is locally irreducible as a Riemannian manifold. Finally, in Theorem 1.3, we compare the Bergman metric g_B and the metric g_F in a bounded Hartogs domain and we prove that if g_B is a multiple of g_F, namely g_B = λ g_F, for some λ ∈ ℝ⁺, then D_F is holomorphically isometric to an open subset of the complex hyperbolic space.

In this work we show that for the geodesic spray S of a Finsler function F, the most natural projective deformation leads to a non-Finsler metrizable spray, for almost every value of λ ∈ ℝ. This result shows how rigid is the metrizablility property with respect to certain reparameterizations of the geodesics. As a consequence, we obtain that the projective class of an arbitrary spray contains infinitely many sprays that are not Finsler metrizable.

A new definition of a strong curvature singularity is proposed. This definition is motivated by the definitions given by Tipler and Królak, but is significantly different and more general. All causal geodesics terminating at these new singularities, which we call generalized strong curvature singularities, are classified into three possible types; the classification is based on certain relations between the causal structure and the curvature strength of the singularities. A cosmic censorship theorem is formulated and proved which shows that only one class of generalized strong curvature singularities, corresponding to a single type of geodesics according to our classification, can be naked. Implications of this result for the cosmic censorship hypothesis are indicated.

Considering the definition of inertial forces acting on a test particle, following non-circular geodesics, in static and stationary space–times, we show that the centrifugal force reversal occurs only in the case of particles following prograde orbits around black holes. We first rewrite the covariant expressions for the acceleration components in terms of the lapse function, shift vector and the three-metric γ_ij, using the ADM 3 + 1 splitting and use these, for different cases as given by pure radial motion, pure azimuthal motion and the general non-circular motion. It is found that the reversal occurs between 2m and 3m, only when the azimuthal angular velocity of the particle supersedes the radial velocity, which indeed depends upon the physical parameters E, ℓ and the Kerr parameter a.

In this paper we study the geodesic structure of the Schwarzschild black hole in rainbow gravity analyzing the behavior of null and time-like geodesic. We find that the structure of the geodesics essentially does not change when the semiclassical effects are included. However, we can distinguish different scenarios if we take into account the effects of rainbow gravity. Depending on the type of rainbow functions under consideration, inertial and external observers see very different situations in radial and non-radial motion of a test particle.

We study the time-like geodesics and geodesic deviation for a two-dimensional (2D) stringy black hole (BH) spacetime in Schwarzschild gauge. We have analyzed the properties of effective potential along with the structure of the possible orbits for test particles with different settings of BH parameters. The exactly solvable geodesic deviation equation is used to obtain corresponding deviation vector. The nature of deviation and tidal force is also examined in view of the behavior of corresponding deviation vector. The results are also compared with an another 2D stringy BH spacetime.

In this paper, by using the effective potential for the photons, we analyze the null geodesics and all kinds of orbits corresponding to the energy levels for the Reissner–Nordström black hole surrounded by quintessence with and compare our results with those obtained for the Schwarzschild black hole surrounded by quintessence matter. We also investigate the circular orbits and calculate the angle of deflection of the photons.

In this paper, we study the geodesic structure for a geometry described by a spherically symmetric four-dimensional (4D) solution embedded in a five-dimensional (5D) space known as a brane-based spherically symmetric solution. Mainly, we have found that the extra dimension contributes to the existence of bounded orbits for the photons, such as planetary and circular stable orbits that have not been observed for other geometries.

The geodesics equations on de Sitter (dS) and anti-de Sitter (AdS) spacetimes of any dimensions, are the starting point for deriving the general form of the Boltzmann equation in terms of conserved quantities. The simple equation for the non-equilibrium Marle and Anderson–Witting models are derived and the distributions of the Boltzmann–Marle model on these manifolds are written down first in terms of conserved quantities and then as functions of canonical variables.

In this paper, we explore the geodesics motion of neutral test particles and the process of energy extraction from a regular rotating Hayward black hole (BH). We analyze the effect of spin as well as deviation parameter $g$, on ergoregion, event horizon and static limit of the said BH. By making use of geodesic equations on the equatorial plane, we determine the innermost stable circular and photon orbits. Moreover, we investigate the effective potentials and effective force to have information on motion and the stability of circular orbits. On studying the negative energy states, we figure out the energy limits of Penrose mechanism. Using Penrose mechanism, we found expression for the efficiency of energy extraction and observed that both spin and deviation parameters contribute to the efficiency of energy extraction. Finally, the obtained results are compared with that acquired from Kerr and braneworld Kerr BHs.

In this paper, the different properties of generalized Vaidya spacetime are considered. We define the location of horizons. We show that the apparent horizon can contain the event horizon. The locations of all types of horizons are compared with the ones in the usual Vaidya spacetime. We investigate the time-like geodesics in this spacetime. New corrections to Schwarzschild and Vaidya cases appear and we give conditions when these corrections are not negligible. Also, we consider the conformal Killing vector and transform the metric to conformally static coordinates. We introduce a new constant of motion along null and time-like geodesics, which is generated by a homothetic Killing vector. The conformally static coordinates allow diagonalizing of the generalized Vaidya spacetime. The surface gravity has been calculated for the dust and stiff fluid cases.

Some black hole-cosmic string models such as Reissner–Nordström, RN–de Sitter, Kerr–Newman and multi-black holes with cosmic string are given. Energy and angular momentum of a timelike particle in circular orbits in multi-black hole space–time are calculated. The geodesic equations for the timelike particles for the far region of the multi-black hole sources are calculated and small oscillations around the circular orbit obtained. It is seen that the particle's orbit precesses like the Lens–Thirring effect.

Two particle collisions close to the horizon of the rotating nonextremal black hole are analyzed. It is shown that high energy of the order of the Grand Unification scale in the centre of mass of colliding particles can be obtained when there is a multiple collision – the particle from the accretion disc gets the critical momentum in first collision with the other particle close to the horizon and then there is a second collision of the critical particle with the ordinary one. High energy occurs due to a great relative velocity of two particles and a large Lorentz factor. The dependence of the relative velocity on the distance to horizon is analyzed, the time of movement from the point in the accretion disc to the point of scattering with large energy as well as the time of back movement to the Earth are calculated. It is shown that they have reasonable order.

Three mechanisms of getting high energies in particle collisions in the ergosphere of the rotating black holes are considered. The consequences of these mechanisms for observation of ultra high energy cosmic rays particles on the Earth as result of conversion of superheavy dark matter particles into ordinary particles are discussed.

In this paper, we have explored the behavior of dynamics of neutral test particles near the brane-world black hole (BH). We have discussed the stability/instability near the circular orbits with the help of extremal and nonextremal BH. In this consideration, we investigate the last stable circular orbit $(r_{\mathrm{\text{LSCO}}})$ that decreases with the increase of black hole parameters and concluded that the circular orbits are stable at $r>r_{\mathrm{\text{LSCO}}}$. The effective potential vanishes at horizon in extremal BH. In case of nonextremal BH, the angular momentum decreases, whereas energy increases with the increase in dark parameter $\beta$. Moreover, we find the conditions for the naked singularity and also express the effective potential that increases with the increase of angular momentum. Finally, the center-of-mass energy (CME) is constructed by the collision of particles in the vicinity of brane-world black hole and observed that the CME decreases with the increase of black hole parameters.

Using the observational properties of Einstein’s gravitational field it is shown that a minimum of four non-coplanar mass probes are necessary for a Michelson and Morley interferometer to detect gravitational waves within the context of General Relativity. With fewer probes, some alternative theories of gravitation can also explain the observations. The conversion of the existing gravitational wave detectors to four probes is also suggested.

The paper deals with static traversable wormhole having shape function a polynomial of the radial coordinate of degree 2. Embedding of the wormholes in space–time is examined both analytically and graphically. Also both timelike and null geodesics are studied in this wormhole geometry.

A review is given of some aspects of the Riemannian geometry of quantum computation in which the quantum evolution is represented in the tangent space manifold of the special unitary unimodular group SU(2ⁿ) for n qubits. The Riemannian right-invariant metric, connection, curvature, geodesic equation for minimal complexity quantum circuits, Jacobi equation and the lifted Jacobi equation for varying penalty parameter are reviewed. Sharpened tools for calculating the geodesic derivative are presented. The geodesic derivative may facilitate the numerical investigation of conjugate points and the global characteristics of geodesic paths in the group manifold, the determination of optimal quantum circuits for carrying out a quantum computation, and the determination of the complexity of particular quantum algorithms.

In the present paper, we propose a novel generalization of the celebrated MMP algorithm in order to find the wavefront propagation and the cut-locus on a convex polyhedron with an emphasis on actual implementation for instantaneous visualization and numerical computation.

In this paper, we consider enumeration of geodesics on a polyhedron, where a geodesic means locally-shortest path between two points. Particularly, we consider the following preprocessing problem: given a point $s$ on a polyhedral surface and a positive real number $r$, to build a data structure that enables, for any point $t$ on the surface, to enumerate all geodesics from $s$ to $t$ whose length is less than $r$. First, we present a naive algorithm by removing the trimming process from the MMP algorithm (1987). Next, we present an improved algorithm which is practically more efficient on a non-convex polyhedron, in terms of preprocessing time and memory consumption. Moreover, we introduce a single-pair geodesic graph to succinctly encode a result of geodesic query. Lastly, we compare these naive and improved algorithms by some computer experiments.