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![The apparent angle (5.9) as a function ofˆjofˆofˆj using the Sun as the lens. Physical value of r d = 1 [AU], ˆ a = 0.218 [71] are used. For other parameters, we assume r s = 8.1 [kpc] and ϕ 0 = 1.04θ .](profile/Fan_Gaofeng/publication/363811697/figure/fig2/AS:11431281087285394@1664530500236/The-apparent-angle-59-as-a-function-ofjofofj-using-the-Sun-as-the-lens-Physical.png)
The apparent angle (5.9) as a function ofˆjofˆofˆj using the Sun as the lens. Physical value of r d = 1 [AU], ˆ a = 0.218 [71] are used. For other parameters, we assume r s = 8.1 [kpc] and ϕ 0 = 1.04θ .
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Spin of a test particle is a fundamental property that can affect its motion in a gravitational field. In this work we consider the effect of particle spin on its deflection angle and gravitational lensing in the equatorial plane of arbitrary stationary and axisymmetric spacetimes. To do this we developed a perturbative method that can be applied t...
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... this case, two of the images with s o = + will be very weakly lensed so that their apparent angles will be within the solar radius θ and not observable. In figure 5 therefore we only plot the apparent angles θ K,−s j for the same range ofˆjofˆofˆj as in figure 4 for a fixed ϕ 0 = 1.04θ . It is seen that similar to the case in figure 4 (b), whenˆjwhenˆwhenˆj 10 3 , the apparent angles of two spin directions start to deviate from each other. ...
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The deflection angle $$\Delta \phi $$ Δ ϕ of charged signals in general charged spacetime in the strong deflection limit is analyzed in this work using a perturbative method generalized from the neutral signal case. The solved $$\Delta \phi $$ Δ ϕ naturally contains the finite distance effect and takes a quasi-power series form with a logarithmic d...
Citations
... (3.1) for the reduced system. Finally, it would be interesting to extend the results to the spacetimes with non-zero torsion [59] or apply to deflections and lensing of the massive particles [60]. ...
We analyze the motion of the spinning body (in the pole-dipole approximation) in the gravitational and electromagnetic fields described by the Mathisson-Papapetrou-Dixon-Souriau equations. First, we define a novel spin supplementary condition for the electromagnetic interactions which generalizes the one proposed by Ohashi-Kyrian-Semer\'ak for gravity. As a result, we get the whole family of charged spinning particle models in the curved spacetime with remarkably simple dynamics (momentum and velocity are parallel). Applying the reparametrization procedure, for a specific dipole moment, we obtain equations of motion with constant mass and gyromagnetic factor. Next, we show that these equations follows from an effective Hamiltonian formalism, previously interpreted as a classical model of the charged Dirac particle.
... Ladino and Larrañaga study the motion of spinning test particles around an improved rotating black hole [45], where using the Mathisson-Papapetrou*** -Dixon (MPD) equations and the Tulczyjew spin supplementary condition, the authors investigated the equatorial circular orbits, finding that the event horizon and the radius of the Innermost Stable Circular Orbit (ISCO) for the quantum-improved rotating black hole are smaller than Schwarzschild and Kerr classi-cal solutions. The dynamics of spinning test particles have called the attention of the community, and several works consider different spacetime backgrounds [45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60][61][62][63]. In this manuscript, we consider the motion of charged spinning test particle motion around quantum-improved charged black holes; we also study the dynamics of magnetized test particles. ...
In the present work, we aimed to investigate the dynamics of spinning charged and magnetized test particles around both electrically and magnetically charged quantum-improved black holes. We derive the equations of motion for charged spinning test particles using the Mathisson-Papapetrou-Dixon ***equations with the Lorentz coupling term. The radius of innermost stable circular orbits (ISCOs), specific angular momentum, and energy for charged spinless, uncharged spinning, and charged spinning test particles around the charged and non-charged quantum-improved black holes are analyzed separately. We found that the quantum parameter increases the maximum spin value, $$s_\textrm{max}$$ s max , which leads to the nonphysical motion (superluminal motion) of the charged spinning test particle. In contrast, the black hole charge decreases its value. We also found that, in contrast to the Reissner Nordström black hole, spinning charged test particles in the quantum-improved charged black hole have higher $$s_\textrm{max}$$ s max ; moreover, positively charged spinning particles can have higher values of $$s_\textrm{max}$$ s max near the extreme black hole cases when compared with uncharged spinning particles. Finally, we investigate the magnetized test particle’s dynamics in the spacetime of a quantum-improved magnetically charged black hole in Quantum Einstein Gravity using the Hamilton–Jacobi equation. We show that the presence of $$\omega $$ ω increases the maximum value of the effective potential and decreases the minimum energy and angular momentum of magnetized particles at their circular orbits. We found an upper constraint in the black hole charge at the ISCO.
... Different techniques, such as the Gauss-Bonnet theorem-based geometrical method [18,19] and perturbative methods [26,41,46], have been used to study deflections of not only null but timeline signals. Various effects, such as the finite distance effect of the source and detectors [9][10][11], as well as those of the spacetime parameters (spin, charges [16], magnetic field [14] etc.) and signal properties (spin [15] and charge [12,45]), have been extensively investigated in recent years. Observationally, however Kerr BH is still the most simple and natural BH considered by astronomers [24,25], and even the Kerr-Newmann (KN) spacetime is not as popular due to its nonzero charge. ...
... and the s l in front of χ is valid in the weak deflection limit. These relations can be used to replace the L and K in the integrals (15) and (16) when we carry out these integrations later. Among the six coordinates (r s , θ s , ϕ s ) and (r d , θ d , ϕ d ), we will assume that the r s , r d and θ s are known. ...
... The key to carry out the integrations of Eqs. (15) and (16) is to find a proper way to expand the integrands, after which the integral can always be carried out to any desired accuracy. The weak deflection assumption provides a natural choice of such an expansion parameter, that is, the ratio M/r 0 between the source mass and the closest radius. ...
This paper investigates the off-equatorial plane deflection and gravitational lensing of both null and timeline signals in Kerr spacetime in the weak deflection limit, with the finite distance effect of the source and detector taken into account. The deflection in both the Boyer-Linquidist $\phi$ and $\theta$ directions are computed as power series of $M/r_0$ and $r_0/r_{s,d}$, where $M,~r_{s,d}$ are the spacetime mass and source and detector radius respectively, and $r_0$ is the minimal radius of the trajectory. The coefficients of these series are simple trigonometric functions of $\theta_\text{e}$, the extreme value of $\theta$ coordinate of the trajectory. A set of exact gravitational lensing equations is used to solve $r_0,~\theta_\text{e}$ for a given deflection $\delta\theta$ and $\delta\phi$ of the source, and two images are always obtained. The apparent angles and their magnifications of these images are solved and their dependence on various parameters, especially spacetime spin $\hat{a}$ are analyzed in great detail. It is found that generally there exist two critical spacetime spin which separate the signals reaching the detector from different sides of the spin axis and the images to appear either in the first or the second, and the third and the fourth quadrant of the celestial plane. Three potential applications of these results are discussed.
... If we had instead taken them to be anti-parallel, we would have obtained a plus sign in the last term in the numerator of (3.116). This implies that the sign in the last term in the numerator of (3.116) is such that for parallel (antiparallel) orbital angular momentum and spin the deflection angle decreases (increases) in agreement with expectations [212]. We remark that, in view of the bound on the spin, a ≤ Gm, and of the PM limit Gm ≪ b, the parameter a/b is always very small. ...
Motivated by conceptual problems in quantum theories of gravity, the gravitational eikonal approach, inspired by its electromagnetic predecessor, has been successfully applied to the transplanckian energy collisions of elementary particles and strings since the late eighties, and to string-brane collisions in the past decade. After the direct detection of gravitational waves from black-hole mergers, most of the attention has shifted towards adapting these methods to the physics of black-hole encounters. For such systems, the eikonal exponentiation provides an amplitude-based approach to calculate classical gravitational observables, thus complementing more traditional analytic methods such as the Post-Newtonian expansion, the worldline formalism, or the Effective-One-Body approach. In this review we summarize the main ideas and techniques behind the gravitational eikonal formalism. We discuss how it can be applied in various different physical setups involving particles, strings and branes and then we mainly concentrate on the most recent developments, focusing on massive scalars minimally coupled to gravity, for which we aim at being as self-contained and comprehensive as possible.
... J. M. Ladino and E. A. Larrañaga study the motion of spinning test particles around an improved rotating black hole [45], where using the Mathisson-Papapetrou-Dixon (MPD) equations and the Tulczyjew spin supplementary condition, the authors investigated the equatorial circular orbits, finding that the event horizon and the radius of the Innermost Stable Circular Orbit (ISCO) for the quantum-improved rotating black hole are smaller than Schwarzschild and Kerr classical solutions. The dynamics of spinning test particles have called the attention of the community, and several works consider different spacetime backgrounds [45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60][61][62]. In this manuscript, we consider the motion of charged spinning test particle motion around quantum-improved charged black holes; we also study the dynamics of magnetized test particles. ...
In the present work, we aimed to investigate the dynamics of spinning charged and magnetized test particles around both electrically and magnetically charged quantum-improved black holes. We derive the equations of motion for charged spinning test particles using the Mathisson-Papapetrou-Dixon equations with the Lorentz coupling term. The radius of innermost stable circular orbits (ISCOs), specific angular momentum, and energy for charged spinless, uncharged spinning, and charged spinning test particles around the charged and non-charged quantum-improved black holes are analyzed separately. We found that the quantum parameter increases the maximum spin value, $s_\text{max}$, which leads to the nonphysical motion (superluminal motion) of the charged spinning test particle, whereas the black hole charge decreases its value. We also found that, in contrast to the Reissner Nordstr\"om black hole, spinning charged test particles in the quantum-improved charged black hole have higher $s_\text{max}$; moreover, positively charged spinning particles can have higher values of $s_\text{max}$ near the extreme black hole cases when compared with uncharged spinning particles. Finally, we investigate the magnetized test particle's dynamics around a quantum-improved magnetically charged black hole in Quantum Einstein Gravity using the Hamilton-Jacobi equation. We show that the presence of $\omega$ increases the maximum value of the effective potential and decreases the minimum energy and angular momentum of magnetized particles at their circular orbits. We found an upper constraint in the black hole charge at the ISCO.
... When F M is equipped with the Poisson bracket, the manifold M n is called the Poisson manifold. Comparing (36) and (37) with (11) and (12), we see that the infinite-dimensional vector space F M is equipped with the structure of a Lie algebra. Exercise 2.1. ...
... Proof. First, we note that (40) implies (36). Second, the mapping (39), being combination of derivatives, is bilinear and automatically obeys the Leibnitz rule. ...
This work contains a brief and elementary exposition of the foundations of Poisson and symplectic geometries, with an emphasis on applications for Hamiltonian systems with second-class constraints. In particular, we clarify the geometric meaning of the Dirac bracket on a symplectic manifold and provide a proof of the Jacobi identity on a Poisson manifold. A number of applications of the Dirac bracket are described: applications for the proof of the compatibility of a system consisting of differential and algebraic equations, as well as applications for the problem of reduction of a Hamiltonian system with known integrals of motion.
... When F M is equipped with the Poisson bracket, the manifold M n is called the Poisson manifold. Comparing (36) and (37) with (11) and (12), we see that the infinite-dimensional vector space F M is equipped with the structure of a Lie algebra. Exercise 2.1. ...
... Proof. First, we note that (40) implies (36). Second, the mapping (39), being combination of derivatives, is bilinear and automatically obeys the Leibnitz rule. ...
This work contains a brief and elementary exposition of the foundations of Poisson and symplectic geometries, with an emphasis on applications for Hamiltonian systems with second-class constraints. In particular, we clarify the geometric meaning of the Dirac bracket on a symplectic manifold and provide a proof of the Jacobi identity on a Poisson manifold. A number of applications of the Dirac bracket are described: applications for the proof of the compatibility of a system consisting of differential and algebraic equations, as well as applications for the problem of reduction of a Hamiltonian system with known integrals of motion.
This paper studies the deflection of charged particles in a dipole magnetic field in Schwarzschild spacetime background in the weak field approximation. To calculate the deflection angle, we use Jacobi metric and Gauss-Bonnet theorem. Since the corresponding Jacobi metric is a Finsler metric of Randers type, we use both the osculating Riemannian metric method and generalized Jacobi metric method. The deflection angle up to fourth order is obtained and the effect of the magnetic field is discussed. It is found that the magnetic dipole will increase (or decrease) the deflection angle of a positively charged signal when its rotation angular momentum is parallel (or antiparallel) to the magnetic field. It is argued that the difference in the deflection angles of different rotation directions can be viewed as a Finslerian effect of the nonreversibility of the Finsler metric. The similarity of the deflection angle in this case with that for the Kerr spacetime allows us to directly use the gravitational lensing results in the latter case. The dependence of the apparent angles on the magnetic field suggests that by measuring these angles the magnetic dipole might be constrained.
