Timelike Particles Research Articles

Escape, bound and capture geodesics in local static coordinates in Schwarzschild spacetime

The classical geodesics of timelike particles in Schwarzschild spacetime is analyzed according to the particle starting radius $r$, velocity $v$ and angle $\alpha$ against the radial outward direction in the reference system of an local static observer. The region of escape, bound and capture orbits in the parameter space of $(r,~v,~\alpha)$ are solved using the three cases of the effective potential. It is found that generally for radius smaller than $4M$ or velocity larger than $c/\sqrt{2}$ there will be no bound orbits. While for fixed radius larger than $4M$ (or velocity smaller than $c/\sqrt{2}$), as velocity (or radius) increase from zero (or $2M$), the particle is always captured until a critical value $v_{\mathrm{crit1}}$ (or $r_{\mathrm{crit1}}$) when the bound orbit start to appear around $\alpha=\pi/2$ between a double-napped cone structure. As the velocity (or radius) increases to another critical value $v_{\mathrm{crit2}}$ (or $r_{\mathrm{crit2}}$) then the bound directions and escape directions in the outward cone become escape directions, leaving only the inward cone separating the capture and bound directions. The angle of this cone will increase to its asymptotic value as velocity (or radius) increases to its asymptotic value. The implication of these results in shadow of black holes formed by massive particles, in black hole accretion and in spacecraft navigation is briefly discussed.

Geometric phase for timelike spherical normal magnetic charged particles optical ferromagnetic model

We introduce the theory of optical spherical Heisenberg ferromagnetic spin of timelike spherical normal magnetic flows of particles by the spherical frame in de Sitter space. Also, the concept of timelike spherical normal magnetic particles is investigated, which may have evolution equations. Afterward, we reveal new relationships with some integrability conditions for timelike spherical normal magnetic flows in de-Sitter space. In addition, we obtain total phases for spherical normal magnetic flows. We also acquire perturbed solutions of the nonlinear Schrödinger's equation that governs the propagation of solitons in de-Sitter space . Finally, we provide some numerical simulations to supplement the analytical outcomes.

Time delay of timelike particles in gravitational lensing of the Schwarzschild spacetime

Time delay in Schwarzschild spacetime for null and timelike signals with arbitrary velocity $v$ is studied. The total travel time $t_\mathrm{if}$ is evaluated both exactly and approximately in the weak field limit, with the result given as functions of signal velocity, source-lens and lens-observer distances, angular position of the source and lens mass. Two time delays, $\Delta t_v$ between signals with different velocities but coming from same side of the lens and $\Delta t_\mathrm{p}$ between signals from different sides of the lens, as well as the difference $\Delta t_{\mathrm{p}v}$ between two $\Delta t_\mathrm{p}$'s are calculated. These time delays are applied to the gravitational-lensed supernova neutrinos and gravitational waves (GW). It is shown that the $\Delta t_v$ between different mass eigenstates of supernova neutrinos can be related to the mass square difference of these eigenstates and therefore could potentially be used to discriminate neutrino mass orderings, while the difference $\Delta t_{\mathrm{p}v}$ between neutrino and optical signals can be correlated with the absolute mass of neutrinos. The formula for time delay in a general lens mass profile is derived and the result is applied to the singular isothermal sphere case. For GWs, it is found that the difference $\Delta t_{\mathrm{p}v}$ between GW and GRB can only reach $1.45\times 10^{-5}$ second for very large source distance ($2\times 10^4$ [Mpc]) and source angle (10 [as]) if $v_{GM}=(1-3\times 10^{-15})c$. This time difference is at least three order smaller than uncertainties in time measurement of the recently observed GW/GRB signals and thus calls for improvement if $\Delta t_{\mathrm{p}v}$ is to be used to further constrain the GW velocity.

Traversable wormholes in Einsteinian cubic gravity

In this work, we investigate wormhole configurations described by a constant redshift function in Einstein-Cubic gravity ( ECG ). We derive analytical wormhole geometries by assuming a particular equation of state ( EoS ) and investigate the possibility that these solutions satisfy the standard energy conditions. We introduce exact asymptotically flat and anti-de Sitter (AdS) spacetimes that admit traversable wormholes. These solutions are obtained by imposing suitable values for the parameters of the theory so that the resulted geometries satisfy the weak energy condition ( WEC ) in the vicinity of the throat, due to the presence of higher-order curvature terms. Moreover, we find that AdS solutions satisfy the WEC throughout the spacetime. A description of the geodesic motion of time-like and null particles is presented for the obtained wormhole solutions. Also, using gravitational lensing effects, observational features of the wormhole structure are discussed.

Can massless wormholes mimic a Schwarzschild black hole in the strong field lensing?

Recent trend of research indicates that not only massive but also massless (asymptotic Newtonian mass zero) wormholes can reproduce post-merger initial ring-down gravitational waves characteristic of black hole horizon. In the massless case, it is the non-zero charge of other fields, equivalent to what we call here the "Wheelerian mass", that is responsible for mimicking ring-down quasi-normal modes. In this paper, we enquire whether the same Wheelerian mass can reproduce black hole observables also in an altogether different experiment, viz., the strong field lensing. We examine two classes of massless wormholes, one in the Einstein-Maxwell-Dilaton (EMD) theory and the other in the Einstein-Minimally-coupled-Scalar field (EMS) theory. The observables such as the radius of the shadow, image separation and magnification of the corresponding Wheelerian masses are compared with those of a black hole (idealized SgrA* chosen for illustration) assuming that the three types of lenses share the same minimum impact parameter and distance from the observer. It turns out that, while the massless EMS\ wormholes can closely mimic the black hole in terms of strong field lensing observables, the EMD wormholes show considerable differences due to the presence of dilatonic charge. The conclusion is that masslessless alone is enough to closely mimic Schwarzschild black hole strong lensing observables in the EMS theory but not in the other, where extra parameters also influence those observables. The motion of timelike particles is briefly discussed for completeness.

Circular orbit of a test particle and phase transition of a black hole

The radius of the circular orbit for the time-like or light-like test particle in a background of general spherically symmetric spacetime is viewed as a characterized quantity for the thermodynamic phase transition of the corresponding black hole. We generally show that the phase transition information of a black hole can be reflected by its surrounding particle's circular orbit.

Gravitational lensing of massive particles in Reissner–Nordström black hole spacetime

In this work we study the deflection angle and gravitational lensing of both lightlike and timelike neutral rays in Reissner–Nordström (RN) black hole spacetimes. The exact deflection angle is found as an elliptical function of the impact parameter b and velocity of the ray, and the charge Q of the spacetime. In obtaining this angle, we found the critical impact parameter and radius of particle sphere that are also dependent on and Q. In general, both the increase of velocity and charge reduces the as well as . To study the effect of and Q on the deflection angle , its weak and strong deflection limits, relativistic and non-relativistic limits, and small charge and extremal RN limits are analyzed carefully. It is found that both the increase of velocity and charge reduces the deflection angle. For weak deflection, the velocity and charge corrections appear respectively in the and orders. For strong deflections, these two corrections appear in the same order. The apparent angles and magnifications of weak and strong regular lensing, and retro-lensing are studied for both lightlike and timelike rays. In general, in all cases the increase of velocity or charge will decrease the apparent angle of any order. We show that velocity correction is much larger than that of charge in the weak lensing case, while their effects in the strong regular lensing and retro-lensing are comparable. It is further shown that the apparent angle and magnification in strong regular lensing and retro-lensing can be effectively unified. These observables at different orders in these two kinds of lensing are staggered: the apparent angles can be ordered in a staggered way and the magnifications form two staggered geometric series. Finally, we argue that the correction of and Q on the apparent angle can be correlated to mass or mass hierarchy of timelike particles with certain energy. In addition, the effects of and Q on shadow size of black holes are discussed.

GRay2: A General Purpose Geodesic Integrator for Kerr Spacetimes

Abstract Fast and accurate integration of geodesics in Kerr spacetimes is an important tool in modeling the orbits of stars and the transport of radiation in the vicinities of black holes. Most existing integration algorithms employ Boyer–Lindquist (BL) coordinates, which have coordinate singularities at the event horizon and along the poles. Handling the singularities requires special numerical treatment in these regions, often slows down the calculations, and may lead to inaccurate geodesics. We present here a new general-purpose geodesic integrator, GRay2, that overcomes these issues by employing the Cartesian form of Kerr–Schild (KS) coordinates. By performing particular mathematical manipulations of the geodesic equations and several optimizations, we develop an implementation of the Cartesian KS coordinates that outperforms calculations that use the seemingly simpler equations in BL coordinates. We also employ the OpenCL framework, which allows GRay2 to run on multicore CPUs as well as on a wide range of graphics processing units hardware accelerators, making the algorithm more versatile. We report numerous convergence tests and benchmark results for GRay2 for both time-like (particle) and null (photon) geodesics.

Photoproduction of Triplets on Free Electrons and the Search for the Dark Photon

We have investigated the photoproduction of triplets on free electrons, γe– → e+e–e–, in which dark photon A′ can be formed as an intermediate state with subsequent decay into an e+e– pair. This effect appears as a result of the so-called kinetic mixing and is characterized by the small parameter describing the intensity of the interaction of the dark photon with charged Standard Model (SM) particles in terms of electric charge e. The search for a manifestation of A′ in this process is advantageous since the background to the A′ signal is purely electrodynamic and, hence, can be calculated with the required accuracy. We calculate this background taking into account the identity of final electrons. As regards A′, its contribution is taken into account only in Compton-type diagrams (of virtual Compton scattering) in which a virtual dark photon is a time-like particle and its propagator has the form of a Breit–Wigner resonance. It is only in the vicinity of resonance that A′ can be manifested. We calculate the invariant mass distribution of both e+e– pairs formed in the process and analyze the kinematic region in which relatively small squares of momenta transferred from the target electron to the formed electrons are excluded. In these conditions, the contribution to the differential cross section from Compton-type diagrams is not suppressed relative to the contribution of the remaining (Borsellino) diagrams. A number of limitations on parameter depending on the dark photon mass and the statistics (number) of events are obtained for a special method of gathering events in which the invariant mass of one e+e– pair remains fixed and of the other pair is scanned.

Space-like Particle Production: An Interpretation Based on the Majorana Equation

This study reconsiders the decay of an ordinary particle in bradyons, tachyons and luxons in the field of the relativistic quantum mechanics. Lemke already investigated this from the perspective of covariant kinematics. Since the decay involves both spacelike and timelike particles, the study uses the Majorana equation for particles with an arbitrary spin. The equation describes the tachyonic and bradyonic realms of massive particles, and approaches the problem of how spacelike particles might develop. This method confirms the kinematic constraints that Lemke theory provided and proves that some possible decays are more favourable than others are.

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.

Periodic orbits around a spherically symmetric naked singularity

The motion of time-like test particles in the Fisher/Janis-Newman-Winicour (F/JNW) spacetime is studied with the Hamiltonian formulation of the geodesic equations. The spacetime is characterised by its mass parameter $r_g$ and scalar field parameter $\nu$. The innermost bound and stable circular orbits are calculated and the effective potential is analysed. Consistent with numerical results in earlier literature, for $\nu<1/2$, particles with non-zero angular momentum encounter an infinite potential barrier, preventing them from reaching the naked singularity at $r=r_g$. Periodic orbits in the spacetime are also obtained. Compared to the periodic orbits around the Schwarzschild black hole, it is found that typically lower energies are required for the same orbits in the F/JNW spacetime.

Equatorial geodesics of dyonic Kerr-Newman black hole pierced by a cosmic string

This paper is devoted to study the circular geodesics of the dyonic Kerr-Newman black hole with a cosmic string passing through it. We investigate circular geodesics of null and timelike particle. In this context, we find the circular photon orbit as well as the innermost stable circular orbit. The angular velocity and time period for the timelike particle are calculated. The effect of electric and magnetic charge as well as of the cosmic string parameter on the effective potential is analyzed numerically. Finally, we discuss the role of these parameters on the energy extraction by the Penrose process. We conclude that the string parameter does not affect the gain energy of the particle but it decreases with respect to charge.

Chaotic motion of particles in the accelerating and rotating black holes spacetime

We have investigated the motion of timelike particles along geodesic in the background of accelerating and rotating black hole spacetime. We confirmed that the chaos exists in the geodesic motion of the particles by Poincar\'e sections, the power spectrum, the fast Lyapunov exponent indicator and the bifurcation diagram. Moreover, we probe the effects of the acceleration and rotation parameters on the chaotic behavior of a timelike geodesic particle in the black hole spacetime. Our results show that the acceleration brings richer physics for the geodesic motion of particles.

Motion of charged particles around a magnetized/electrified black hole

Geodesic equations of timelike and null charged particles in the Ernst metric are studied. We consider two distinct forms of the Ernst solution where the Maxwell potential represents either a uniform electric or magnetic field. Circular orbits in various configurations are considered, as well as their perturbations and stability. We find that the electric field strength must be below a certain charge-dependent critical value for these orbits to be stable. The case of the magnetic Ernst metric contains a limit which reduces to the Melvin magnetic universe. In this case the equations of motion are solved to reveal cycloidlike or trochoidlike motion, similar to those found by Frolov and Shoom around black holes immersed in test magnetic fields.

Bianchi Type-I Cosmological Models for Inextensible Flows of Biharmonic Particles by Using Curvature Tensor Field in Spacetime

In this article, we study flows of biharmonic particles of a new spacetime using Bianchi type-I (B-I) cosmological model. We give a geometrical description of timelike biharmonic particle in spacetime. Moreover, we obtain evolution of curvatures of this particles.

Constant Energy of Time Involute Particles of Biharmonic Particles in Bianchi Type-I Cosmological Model Spacetime

In this paper, we study energy of time involute particles of biharmonic particles in Bianchi type-I (B-I) cosmological model spacetime. We give a geometrical description of energy for a Frenet vector fields of timelike biharmonic particle. Finally, using the Frenet frame of the given particle, we obtain different cases for this particles and give important characterizations about them in Bianchi type-I (B-I) cosmological model spacetime.

Examination of a simple example of gravitational wave memory

We examine a simple example of gravitational wave memory due to the decay of a point particle into two point particles. In the case where one of the decay products is null, there are two types of memory: a null memory due to the null particle and an ordinary memory due to the recoiling timelike particle. In the case where both decay products are timelike, there is only ordinary memory. However, this ordinary memory can mimic the null memory in the limit where one of the decay products has a large velocity.

Bianchi Type-I Cosmological Models for Biharmonic Particles and its Transformations in Spacetime

In this article, we study timelike biharmonic particles in Bianchi type-I (B-I) cosmological model spacetime. We give a geometrical description of timelike biharmonic particle in spacetime. Moreover, we obtain transformations this particles. Some physical and geometric behaviour of these particles are also discussed in Bianchi type-I (B-I) cosmological model spacetime.

Electromagnetic Fields on Time-Involute Particles Around Biharmonic Particles and its Lorentz Transformations in Heisenberg Spacetime

In this paper, we obtain the new parametric representation for a time-involute particles in Heisenberg spacetime \( \mathcal {H}_{1}^{4}\). By using the Frenet frame, we derive the necessary and sufficient conditions to construct a biharmonic particle Heisenberg spacetime \(\mathcal {H}_{1}^{4}\). We give a geometrical description of time-involute particles around timelike biharmonic particle in \(\mathcal {H} _{1}^{4}\). Moreover, we obtain Lorentz transformations this particles. Finally, we give the relationship of electromagnetic fields on Heisenberg spacetime.