Timelike Trajectories Research Articles

Geodesics structure and deflection angle of electrically charged black holes in gravity with a background Kalb–Ramond field

This article investigates the geodesic structure and deflection angle of charged black holes in the presence of a nonzero vacuum expectation value background of the Kalb–Ramond field. Topics explored include null and timelike geodesics, energy extraction by collisions, and the motion of charged particles. The photon sphere radius is calculated and plotted to examine the effects of both the black hole charge (Q) and the Lorentz-violating parameter (b) on null geodesics. The effective potential for timelike geodesics is analyzed, and second-order analytical orbits are derived. We further show that the combined effects of Lorentz-violating parameter and electric charge can mimic a Kerr black hole spin parameter up to its maximum values, i.e., a/M∼1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a/M \\sim 1$$\\end{document} thus suggesting that the current precision of measurements of highly spinning black hole candidates may not rule out the effect of Lorentz-violating parameter. The center of mass energy of colliding particles is also considered, demonstrating a decrease with increasing Lorentz-violating parameter. Circular orbiting particles of charged particles are discovered, with the minimum radius for a stable circular orbit decreasing as both b and Q increase. Results show that this circular orbit is particularly sensitive to changes in the Lorentz-violating parameter. Additionally, a timelike particle trajectory is demonstrated as a consequence of the combined effects of parameters b and Q. Finally, the light deflection angle is analyzed using the weak field limit approach to determine the Lorentz-breaking effect, employing the Gauss–Bonnet theorem for computation. Findings are visualized with appropriate plots and thoroughly discussed.

Moving interfaces and two-dimensional black holes

Conformal field theories can exchange energy through a boundary interface. Imposing conformal boundary conditions for static interfaces implies energy conservation at the interface. Recently, the reflective and transmittive properties of such static conformal interfaces have been studied in two dimensions by scattering matter at the interface impurity. In this note, we generalize this to the case of dynamic interfaces. Motivated by the connections between the moving mirror and the black hole, we choose a particular profile for the dynamical interface. We show that a part of the total energy of each side will be lost in the interface. In other words, a time-dependent interface can accumulate or absorb energy. While, in general, the interface follows a time-like trajectory, one can take a particular limit of a profile parameter(β), such that the interface approaches a null line asymptotically(β → 0). In this limit, we show that for a class of boundary conditions, the interface behaves like a semipermeable membrane - it behaves like a (partially) reflecting mirror from one side and is (partially) transparent from the other side. We also consider another set of conformal boundary conditions for which, in the null line limit, the interface mimics the properties expected of a horizon. In this case, we devise a scattering experiment, where (zero-point subtracted) energy from one CFT is fully transmitted to the other CFT, while from the other CFT, energy can neither be transmitted nor reflected, i.e., it gets lost in the interface. This boundary condition is also responsible for the thermal energy spectrum which mimics Hawking radiation. This is analogous to the black hole where the horizon plays the role of a one-sided ‘membrane’, which accumulates all the interior degrees of freedom and radiates thermally in the presence of quantum fluctuation. Stimulated by this observation, we comment on some plausible construction of wormhole analogues.

Particle motion around traversable wormholes: Possibility of closed timelike geodesics

The present work investigates the general wormhole solution in Einstein gravity with an exponential shape function around an ultrastatic and a finite redshift geometry. The geodesic motion around the wormholes is studied in which the deflection angle of the orbiting photon sphere is found to be negative after a certain region, indicating the presence of repulsive effect of gravity in both the ultrastatic and finite redshift wormholes. Various unbounded and bounded timelike trajectories are presented on the wormhole embedding diagrams, in which some of the bound orbits involve intersection points that may lead to causality violating geodesics. Another class of closed timelike geodesics are obtained in the unstable circular trajectory that appeared at the wormhole throat. Finally, the trajectories are classified in terms of the family of CTG orbits.

Charged Particles Moving around a Spherically Symmetric Dilatonic Black Hole

For the static spherically symmetric dilatonic black hole described by the Gibbons–Maeda–Garfinkle–Horowitz–Strominger geometry, we analyze the timelike trajectories for electrically charged test particles. Both cases of an electric black hole and a magnetic one are considered. Finally, we are obtaining the solution to the Klein–Gordon equation in terms of Heun confluent functions and the corresponding energy spectrum. A special attention is given to the role of the dilaton parameter.

Geodesic analysis and steady accretion on a traversable wormhole

In this work, we analyze the behavior of light and matter as they pass near and through a traversable wormhole throat. In particular, we study the trajectories of massive and massless particles and the dust accretion around a traversable wormhole previously reported in Rueda et al. (2022). For massive particles, we integrate the trajectory equation for ingoing and outgoing geodesics and classify the orbits of particles scattered by the wormhole in accordance with their asymptotic behavior far from the throat. We represent all the time-like trajectories in an embedding surface where it is shown explicitly the trajectories of (i) particles that deviate from the throat and remain in the same universe, (ii) particles that traverse the wormhole to another universe, and (iii) particles that get trapped in the wormhole in unstable circular orbits. For the massless particles, we numerically integrate the trajectory equation to show the ray-tracing around the wormhole specifying the particles that traverse the wormhole and those that are only deviated by the throat. For the study of accretion, we consider the steady and spherically symmetric accretion of dust. Our results show that the wormhole parameters can significantly affect the behavior of light and matter near the wormhole. Some comparisons with the behavior of matter around black holes are made.

On the geometry and dynamics of relativistic particles in generalized Robertson–Walker spacetimes

In this work, we consider geometrical models describing spinning particles in the context of generalized Robertson–Walker spacetimes. We study spacelike and timelike trajectories for massive and massless particles governed by two different Lagrangians: one depending on the torsion and the other on the curvature of the trajectory. For both, we are able to relate the curvature and torsion of such trajectories with the curvature of the spatial fiber of the cosmological model. Moreover, several conserved quantities are described as well as their possible relations with physical parameters of the particle.

One-Parameter Hyperbolic Spatial Locomotions and Invariants of the Axode

In this paper, based on the E. Study map, direct appearances were sophisticated for one-parameter hyperbolic dual spherical locomotions and invariants of the axodes. With the suggested technique, the Disteli formulae for the axodes were acquired and the correlations through kinematic geometry of a timelike line trajectory were provided. Then, a ruled analogy of the curvature circle of a curve in planar locomotions was expanded into generic spatial locomotions. Lastly, we present new hyperbolic proofs for the Euler–Savary and Disteli formulae.

Optical quantum longitudinal conformable normalization energy of timelike spherical magnetic fibers

In this work, we calculate new concept for their Fermi–Walker conformable derivatives with spherical timelike magnetic fiber in spherical de-Sitter space [Formula: see text]. The description is advanced for timelike magnetic trajectory. First, we find conformable fractional derivatives of spherical magnetic Lorentz fields. Moreover, we compute the Fermi–Walker conformable derivatives of the normalization and recursion operators for these spherical vector fields. Furthermore, we give some characterizations of Lorentz fields of electromagnetic fields of associated with spherical magnetic conformable particle. Finally, we obtain the energy of the conformable derivative of the normalization function of the Lorentz forces and we have energy relations of some vector fields associated with spherical frame in the de-Sitter space [Formula: see text].

If you want to cross singularity, wrap it!

In two-dimensional string theory, a probe D0-brane does not see the black hole singularity due to a cancellation between its metric coupling and the dilaton coupling. A similar mechanism may work in the Schwarzschild black hole in large D dimensions by considering a suitable wrapped membrane. From the asymptotic observer, the wrapped membrane looks disappearing into nothing while the continuation of the time-like trajectory beyond the singularity suggests that it would reappear as an instantaneous space-like string stretching from the singularity. A null trajectory can be extended to a null trajectory beyond the singularity. Not only the effective particle but an effective string from the wrapped membrane can exhibit the same feature.

Revisiting timelike and null geodesics in the Schwarzschild spacetime: general expressions in terms of Weierstrass elliptic functions

The theory of Schwarzschild geodesics is revisited. Basing on a result by Weierstrass and Biermann, we derive a formula describing all non radial, timelike and null trajectories in terms of Weierstrass elliptic functions. Quite remarkably, a single formula works for an entire geodesic trajectory, even if it passes through turning points. Using this formula, we derive expressions for the proper and coordinate time along the geodesic.

Spacelike trajectories in BTZ spacetime and comparison with timelike and lightlike trajectories

Spacelike trajectories or the motions of tachyons in (2 + 1) dimensional BTZ spacetime are discussed. A study of the effective potential in this case has also been done and it is shown that unlike that for tardyons or photons it does not have any local or global extremum. It rather possesses an inflection point that depends, along with the blackhole parameters, on the integrals of motion. Tachyons are, however, found to be sensitive, like photons and tardyons, to the usual frame dragging effect due to the rotational feature of the BTZ spacetime and thus pure radial motion cannot persist. It is further found that circular orbits exist nowhere except at the horizon. That the timelike object shows the same behaviour in this case is a remarkable feature of (2 + 1) dimensional gravity. Moreover, the trapping of tachyons at the horizon is also indicated. Comparison of the spacelike trajectories with the timelike and the lightlike trajectories and the corresponding effective potentials have been done analytically and graphically.

Localized nonrelativistic quantum systems in curved spacetimes: A general characterization of particle detector models

In this manuscript we provide a consistent way of describing a localized non-relativistic quantum system undergoing a timelike trajectory in a background curved spacetime. Namely, using Fermi normal coordinates, we identify an inner product and canonically conjugate position and momentum operators defined in the rest space of the trajectory for each value of its proper time. This framework then naturally provides a recipe for mapping a quantum theory defined in a non-relativistic background to a theory around a timelike trajectory in curved spacetimes. This is done by reinterpreting the position and momentum operators and by introducing a local redshift factor to the Hamiltonian, which gives rise to new dynamics due to the curvature of spacetime and the acceleration of the trajectory. We then apply our formalism to particle detector models, that is, to the case where the non-relativistic quantum system is coupled to a quantum field in a curved background. This allows one to write a general definition for particle detector models which is able to recover the previous models in the literature. Our framework also allows one to estimate the regime of validity of these models, characterizing the situations where particle detectors can be used to accurately probe quantum fields.

Magnetic trajectories corresponding to Killing magnetic fields in a three-dimensional warped product

The aim of this paper is to investigate Killing magnetic trajectories of varying electrically charged particles in a three-dimensional warped product [Formula: see text] with positive warping function [Formula: see text], where [Formula: see text] is an open interval in [Formula: see text] equipped with an induced semi-Euclidean metric on [Formula: see text]. First, Killing vector fields on [Formula: see text] are characterized and it is observed that lifts to [Formula: see text] of Killing vector fields tangent to [Formula: see text] are also Killing on [Formula: see text]. Now, any Killing vector field on [Formula: see text] corresponds to a Killing magnetic field on [Formula: see text]. Magnetic trajectories (also known as magnetic curves) of charged particles which move under the influence of Lorentz force generated by Killing magnetic fields on [Formula: see text] are obtained in both Riemannian and Lorentzian cases. Moreover, some examples are exhibited with pictures determining Killing magnetic trajectories in hyperbolic [Formula: see text]-space [Formula: see text] modeled by the Riemannian warped product [Formula: see text]. Furthermore, some examples of spacelike, timelike and lightlike Killing magnetic trajectories are given with their possible graphs in the Lorentzian warped product [Formula: see text].

Phase space mixing in an external gravitational central potential

This article is devoted to the study of the dynamical behavior of a collisionless kinetic gas in d = 1, 2, 3 space dimensions which is trapped in a rotationally symmetric potential well. Although at the microscopic level the trajectories of individual gas particles are quasi-periodic and characterized by their d fundamental frequencies, at the macroscopic level the gas relaxes in time to a stationary state, provided the potential satisfies a certain non-degeneracy condition. In this article, we provide a mathematically precise formulation for this relaxation process which is due to phase space mixing. In particular, we prove that a physically relevant class of macroscopic observables computed from the one-particle distribution function, such as particle and energy densities, pressure and stress tensors, converge in time to the corresponding observables associated with an averaged distribution function. The latter can be determined from the initial datum and depends only on integrals of motion. Thus, the final state of the gas is described by an effective distribution function depending only on integrals of motion, which considerably reduces the degrees of freedom of the gas configuration. We discuss some applications to gravitational physics, including the propagation of a collisionless gas in typical potentials arising in stellar dynamics and the modeling of dark matter halos, and we also generalize our results to a relativistic gas whose individual particles follow bound timelike trajectories in the exterior region of a static, spherically symmetric black hole spacetime.

Particle creation by wormholes: A 1 + 1 model

The propagation of a free massless scalar field in a [Formula: see text]-dimensional Minkowski space modeling, a wormhole is considered. The wormhole model consists on two timelike trajectories, which represent the entrance and the exit of the wormhole, connected via some transfer function that specifies how incoming modes that reach the entrance are transferred to the exit. We find that particles and energy fluxes are generically produced except for transfer functions that represent global conformal transformations. We consider several examples involving exit trajectories which are asymptotically inertial, asymptotically null, and also involving a faster-than-light motion to illustrate the peculiarities of the emitted energy fluxes and quantum correlations.

Holographic Space-Time and Quantum Information

The formalism of Holographic Space-time (HST) is a translation of the principles of Lorentzian geometry into the language of quantum information. Intervals along time-like trajectories, and their associated causal diamonds, completely characterize a Lorentzian geometry. The Bekenstein-Hawking-'t Hooft-Jacobson-Fischler-Susskind-Bousso Covariant Entropy Principle, equates the logarithm of the dimension of the Hilbert space associated with a diamond to one quarter of the area of the diamond's holographic screen, measured in Planck units. The most convincing argument for this principle is Jacobson's derivation of Einstein's equations as the hydrodynamic expression of this entropy law. In that context, the null energy condition (NEC) is seen to be the analog of the local law of entropy increase. The quantum version of Einstein's relativity principle is a set of constraints on the mutual quantum information shared by causal diamonds along different time-like trajectories. The implementation of this constraint for trajectories in relative motion is the greatest unsolved problem in HST. The other key feature of HST is its claim that, the degrees of freedom localized in the bulk of a diamond are constrained states of variables defined on the holographic screen. This principle gives a simple explanation of otherwise puzzling features of BH entropy formulae, and resolves the firewall problem for black holes in Minkowski space. It motivates a covariant version of the CKN\cite{ckn} bound on the regime of validity of quantum field theory (QFT) and a detailed picture of the way in which QFT emerges as an approximation to the exact theory.

Gyroscope precession frequency analysis of a five-dimensional charged rotating Kaluza-Klein black hole

We study the spin precession frequency of a test gyroscope attached to a stationary observer in the five-dimensional rotating Kaluza-Klein black hole (RKKBH). We derive the conditions under which the test gyroscope moves along a timelike trajectory in this geometry, and the regions where the spin precession frequency diverges. The magnitude of the gyroscope precession frequency around the KK black hole diverges at two spatial locations outside the event horizon. However, in the static case, the behavior of the Lense-Thirring frequency of a gyroscope around the KK black hole is similar to the ordinary Schwarzschild black hole. Since a rotating Kaluza-Klein black hole is a generalization of the Kerr-Newman black hole, we present two mass-independent schemes to distinguish these two spacetimes.

Wave Equations with Moving Potentials

In this paper, we study the some reversed Strichartz estimates along general time-like trajectories for wave equations in $${\mathbb {R}}^{3}$$. Some applications of the reversed Strichartz estimates and the structure of wave operators to the wave equation with one potential are also discussed. These techniques are useful to analyze the stability problem of traveling solitons.

Kinematic Censorship as a Constraint on Allowed Scenarios of High-Energy Particle Collisions

In recent years, it was found that the energy $E_{c.m.}$ in the centre of mass frame of two colliding particles can be unbounded near black holes. If collision occurs exactly on the horizon, $E_{c.m.}$ is formally infinite. However, in any physically reasonable situation this is impossible. We collect different scenarios of such a kind and show why in every act of collision $E_{c.m.}$ is indeed finite (although it can be as large as one likes). The factors preventing infinite energy are diverse: the necessity of infinite proper time, infinite tidal forces, potential barrier, etc. This prompts us to formulate a general principle according to which the limits in which $E_{c.m.}$ becomes infinite are never achieved. We call this the kinematic censorship (KC). Although by itself the validity of KC is quite natural, its application allows one to forbid scenarios of collisions predicting infinite $E_{c.m.}$ without going into details. The KC is valid even in the test particle approximation, so explanation of why $E_{c.m.}$ cannot be infinite, does not require references (common in literature) to the non-linear regime, backreaction, etc. The KC remains valid not only for free moving particles but also if particles experience the action of a finite force. For an individual particle, we consider a light-like continuous limit of a time-like trajectory in which the effective mass turns into zero. We show that it cannot be accelerated to an infinite energy during a finite proper time under the action of such a force. As an example, we consider dynamics of a scalar particle interacting with a background scalar field.

The inverse kinematics of rolling contact of timelike curves lying on timelike surfaces

Rolling contact between two surfaces plays an important role in robotics and engineering such as spherical robots, single wheel robots, and multi-fingered robotic hands to drive a moving surface on a fixed surface. The rolling contact pairs have one, two, or three degrees of freedom (DOFs) consisting of angular velocity components. Rolling contact motion can be divided into two categories: spin-rolling motion and pure-rolling motion. Spin-rolling motion has three (DOFs), and pure-rolling motion has two (DOFs). Further, it is well known that the contact kinematics can be divided into two categories: forward kinematics and inverse kinematics. In this paper, we investigate the inverse kinematics of spin-rolling motion without sliding of one timelike surface on another timelike surface in the direction of timelike unit tangent vectors of their timelike trajectory curves by determining the desired motion and the coordinates of the contact point on each surface. We get three nonlinear algebraic equations as inputs by using curvature theory in Lorentzian geometry. These equations can be reduced as a univariate polynomial of degree six by applying the Darboux frame method. This polynomial enables us to obtain rapid and accurate numerical root approximations and to analyze the rolling rate as an output. Moreover, we obtain another outputs: the rolling direction and the compensatory spin rate.