From the Schwarzschild metric we obtain the higher-order terms (up to 20-th order) for the deflection of light around a massive object using the Lindstedt-Poincaré method to solve the equation of motion of a photon around the stellar object. Additionally, we obtain diagonal Padé approximants from the perturbation expansion, and we show how these are a better fit for the numerical data. Furthermore, we use these approximants in ray-tracing algorithms to model the bending of light around the massive object.
Expression for second post-Newtonian level gravitational deflection angle of massive particles is obtained in a model independent framework. Several of its important implications including the possibility of testing gravitational theories at that level are discussed.
In this paper, the gravitational deflection of relativistic massive particles up to the second post-Minkowskian order by static and spherically symmetric wormholes is investigated in the weak-field limit. These wormholes include the Janis-Newman-Winicour wormhole, a class of zero Ricci scalar scalar-tensor wormholes, and a class of charged Einstein-Maxwell-dilaton wormholes. With the Jacobi metric approach, the Gauss-Bonnet theorem is employed to study the gravitational deflection. In this scheme, the deflection angle as a topological effect is considered. Moreover, we analyze the influence of…
A comparison between the deflection angles of massive and massless particles in the Shchwarzschild space-time and their consequences on black hole shadows
In this paper the problem of gravitational deflection of relativistic charged massive particles in slowly rotating Kerr-Newman black holes is considered. Toward this purpose we have used two methods: Firstly, we have applied the Gauss-Bonnet theorem (GBT) and the optical geometry to evaluate the deflection angle of charged particles. Secondly, we have presented a detailed analysis of the deflection angle by means of the Hamilton-Jacobi equation recovering the same invariant result for the deflection angle in leading order terms. The crucial point behind the first method is to use the correspon…
We consider the problem of gravitational deflection of a propagating relativistic massive particles by rotating black holes (Kerr black holes) and rotating wormholes (Teo wormholes) in the weak limit approximation. In particular we have introduced an alternative way to calculate the deflection angle for massive particles based on the refractive index of the optical media and the Gauss-Bonnet theorem applied to the isotropic optical metrics. The refractive index governing the propagation of massive particles is calculated by considering those particles as a de Broglie wave packets. Finally appl…
Based on the Jacobi metric method, this paper studies the deflection of a charged massive particle by a novel four-dimensional charged Einstein-Gauss-Bonnet black hole. We focus on the weak field approximation and consider the deflection angle with finite distance effects. To this end, we use a geometric and topological method, which is to apply the Gauss-Bonnet theorem to the Jacobi space to calculate the deflection angle. We find that the deflection angle contains a pure gravitational contribution $δ_g$, a pure electrostatic $δ_c$ and a gravitational-electrostatic coupling term $δ_{gc}$. We …
In this paper, we study the weak gravitational deflection angle of relativistic massive particles by the Kerr-like black hole in the bumblebee gravity model. In particular, we focus on weak field limits and calculate the deflection angle for a receiver and source at a finite distance from the lens. To this end, we use the Gauss-Bonnet theorem of a two-dimensional surface defined by a generalized Jacobi metric. The spacetime is asymptotically non-flat due to the existence of a bumblebee vector field. Thus the deflection angle is modified and can be divided into three parts: the surface integral…
The purpose of this article is twofold. First, we extend the results presented in [Gabriel Crisnejo and Emanuel Gallo, Phys.Rev.D 97, 124016 (2018)] to stationary spacetimes. Specifically, we show that the Gauss-Bonnet theorem can be applied to describe the deflection angle of light rays in plasma media in stationary spacetimes. Second, by using a correspondence between the motion of light rays in a cold non magnetized plasma and relativistic test massive particles we show that this technique is not only powerful to obtain the leading order behavior of the deflection angle of massive/massless …
We apply the Gauss-Bonnet theorem to the study of light rays in a plasma medium in a static and spherically symmetric gravitational field and also to the study of timelike geodesics followed for test massive particles in a spacetime with the same symmetries. The possibility of using the theorem follows from a correspondence between timelike curves followed by light rays in a plasma medium and spatial geodesics in an associated Riemannian optical metric. A similar correspondence follows for massive particles. For some examples and applications, we compute the deflection angle in weak gravitatio…
Strong deflection of massive particles in spherically symmetric spacetimes
Università di Salerno（萨莱诺大学）；Istituto Nazionale di Fisica Nucleare, Sezione di Napoli（意大利国家核物理研究所那不勒斯分部）；University of Bremen（不来梅大学）；Goethe Universität（法兰克福歌德大学）
We study the gravitational deflection of relativistic massive particles by Janis-Newman-Winicour (JNW) spacetimes (also known as a rotating source with a surface-like naked singularity), and a rotating Kerr-like wormholes. Based on the recent article [K. Jusufi, Phys. Rev. D 98, 064017 (2018)], we extend some of these results by exploring the effects of naked singularity and Kerr-like objects on the deflection of particles. We start by introducing coordinate transformation leading to an isotropic line element which gives the refraction index of light for the corresponding optical medias. On th…
In this work we extend the approach used in [Emanuel Gallo and Osvaldo M. Moreschi, Phys. Rev. D 83, 12 083007 (2011)] to the study of weak gravitational lensing in a plasma medium. First, we present expressions for the deflection angle and optical scalars in terms of the components of the energy-momentum tensor for spherically symmetric lenses surrounded by a cold non-magnetized plasma. Second, we show that the same expressions can be deduced using the Gauss-Bonnet theorem. Finally, we establish a correspondence between the spatial orbits of photons in a non-homogeneous plasma and the non-geo…
In this paper, we study the weak gravitational deflection of relativistic massive particles for a receiver and source at finite distance from the lens in stationary, axisymmetric and asymptotically flat spacetimes. For this purpose, we extend the generalized optical metric method to the generalized Jacobi metric method by using the Jacobi-Maupertuis Randers-Finsler metric. More specifically,we apply the Gauss-Bonnet theorem to the generalized Jacobi metric space and then obtain an expression for calculating the deflection angle, which is related to Gaussian curvature of generalized optical met…
In this paper, we investigate the deflection of a charged particle moving in the equatorial plane of Kerr-Newman spacetime, focusing on weak field limit. To this end, we use the Jacobi geometry, which can be described in three equivalent forms, namely Randers-Finsler metric, Zermelo navigation problem, and $(n+1)$-dimensional stationtary spacetime picture. Based on Randers data and Gauss-Bonnet theorem, we utilize osculating Riemannian manifold method and the generalized Jacobi metric method to study the deflection angle, respectively. In the $(n+1)$-dimensional spacetime picture, the motion o…
Deflection angles of massive test particles moving along an unbound trajectory in the Schwarzschild metric are considered for the case of large deflection. We analytically consider the strong deflection limit, which is opposite to the commonly applied small deflection approximation and corresponds to the situation when a massive particle moves from infinity, makes several revolutions around a central object and goes to infinity. For this purpose we rewrite an integral expression for the deflection angle as an explicit function of the parameters determining the trajectory and expand it. Remarka…
The exact metric of a Schwarzschild black hole in the true radiation gauge was recently reported. In this work, we base on this gravity and calculate the gravitational deflection of relativistic massive particles up to the fourth post-Minkowskian order. It is found that the result is consistent with the previous formulations for both the case of dropping the fourth-order contribution and the case of light deflection. Our result might be helpful for future high-accuracy observations.
The aim of the present work is twofold: first, we present general remarks about the application of recent procedures to compute the deflection angle in spherically symmetric and asymptotically flat spacetimes, taking into account finite distance corrections based on the Gauss-Bonnet theorem. Second, and as the main part of our work, we apply this powerful technique to compute corrections to the deflection angle produced by astrophysical configurations in the weak gravitational regime when a plasma medium is taken into account. For applications, we use these methods to introduce new general for…
Nonlinear electrodynamics is known as the generalizations of Maxwell electrodynamics at strong fields and presents interesting features such as curing the classical divergences present in the linear theory when coupled to general relativity. In this paper, we consider the asymptotically flat Reissner-Nordström black hole solution with higher-order magnetic correction in Einstein-nonlinear-Maxwell fields. We study the effect of the magnetic charge parameters on the black hole, viz. weak deflection angle of photons and massive particles using Gauss-bonnet theorem. Moreover, we apply the Keeton-P…
Weak gravitational lensing by black holes and wormholes in the context of massive gravity (Bebronne and Tinyakov 2009) theory is studied. The particular solution examined is characterized by two integration constants, the mass $M$ and an extra parameter $S$ namely `scalar charge'. These black hole reduce to the standard Schwarzschild black hole solutions when the scalar charge is zero and the mass is positive. In addition, a parameter $λ$ in the metric characterizes so-called 'hair'. The geodesic equations are used to examine the behavior of the deflection angle in four relevant cases of the p…
In this paper we investigate the strong gravitational lensing with Gauss-Bonnet gravity. By considering the logarithmic term for deflection angle, we obtain the deflection angle $\hatα$ and corresponding parameters $\bar{a}$ and $\bar{b}$. Finally, by using the relations between the observables and lensing parameters, we estimate the observables such as, $θ_{\infty}$, $s$ and $r_m$, for the relativistic images in the Gauss-Bonnet background.
We consider the parametrized post-Newtonian (PPN) limit of ghost-free massive bimetric gravity with two mutually non-interacting matter sectors coupled to the two metrics. Making use of a gauge-invariant differential decomposition of the metric perturbations, we solve the field equations up to the linear PPN order for a static, point-like mass source. From the result we derive the PPN parameter $γ$ for spherically symmetric systems, which describes the gravitational deflection of light by visible matter. By a comparison to its value measured in the solar system we obtain bounds on the paramete…
The gravitational deflection of test particles including light, due to a radially moving Kerr-Newman black hole with an arbitrary constant velocity being perpendicular to its angular momentum, is investigated. In harmonic coordinates, we derive the second post-Minkowskian equations of motion for test particles, and solve them by high-accuracy numerical calculations. We then concentrate on discussing the kinematical corrections caused by the motion of the gravitational source to the second-order deflection. The analytical formula of light deflection angle up to second order by the moving lens i…
We provide a discussion on a light ray in a charged black hole solution in massive gravity. To serve the purpose, we exploit the optical geometry of the black hole solution and find the Gaussian curvature in weak gravitational lensing. Furthermore, we discuss the deflection angle of the light ray in both plasma and non-plasma mediums using the Gauss-Bonnet theorem on the black hole. We also analyze the Regge--Wheeler equation and derive rigorous bounds on the greybody factors of linearly charged massive BTZ black hole. We also study the shadow or silhouette generated by charged massive BTZ bla…
Both massless light ray and objects with nonzero mass experience trajectory bending in a gravitational field. In this work the bending of trajectories of massive objects in a Schwarzschild spacetime and the corresponding gravitational lensing (GL) effects are studied. A particle sphere for Schwarzschild black hole (BH) is found with its radius a simple function of the particle velocity and proportional to the BH mass. A single master formula for both the massless and massive particle bending angle is found, in the form of an elliptic function depending only on the velocity and impact parameter…
Continuing work initiated in an earlier publication [Ishihara, Suzuki, Ono, Kitamura, Asada, Phys. Rev. D {\bf 94}, 084015 (2016) ], we discuss a method of calculating the bending angle of light in a static, spherically symmetric and asymptotically flat spacetime, especially by taking account of the finite distance from a lens object to a light source and a receiver. For this purpose, we use the Gauss-Bonnet theorem to define the bending angle of light, such that the definition can be valid also in the strong deflection limit. Finally, this method is applied to Schwarzschild spacetime in order…
In this work we study the deflection angle $Δφ$ and gravitational lensing of both lightlike and timelike neutral rays in Reissner-Nordström (RN) spacetimes. The exact deflection angle is found as an elliptical function of the impact parameter $b$ and velocity $v$ of the ray, and the charge $Q$ of the spacetime. In obtaining this angle, we found the critical impact parameter $b_c$ and radius of particle sphere $r_c$ that are also dependent on $v$ and $Q$. In general, both the increase of velocity and charge reduces the $b_c$ as well as $r_c$. To study the effect of $v$ and $Q$ on the deflection…
In this work, we calculate the quantum correction effects on the deflection of light in the spacetime geometry of a quantum improved Kerr black hole pierced by an infinitely long cosmic string. More precisely, we calculate the deflection angle by applying the Gauss--Bonnet theorem (GBT) to the osculating optical geometries related to the quantum improved rotating black hole in the weak limit approximation. We find that the deflection angle of light is affected by the quantum effects as well as the global topology due to the presence of the cosmic string. Besides, we have managed to find the sa…
In this letter we apply the Gauss--Bonnet theorem (GB) to calculate the deflection angle by a quantum corrected Schwarzschild black hole in the weak limit approximation. In particular we calculate the light deflection by two types of quantum corrected black holes: the renormalization group improved Schwarzschild solution and the quantum corrected Schwarzschild solution in Bohmian quantum mechanics. We start from the corresponding optical metrics to use then the GB theorem and calculate the Gaussian curvature in both cases. We calculate the leading terms of the deflection angle and show that qu…
We derive the second-order post-Minkowskian solution for the small-deflection motion of test particles in the external field of the Kerr-Newman black hole via an iterative method. The analytical results are exhibited in the coordinate system constituted by the particles' initial velocity unit vector, impact vector, and their cross-product. The achieved formulas explicitly give the dependences of the particles' trajectory and velocity on the time once their initial position and velocity are specified, and can be applied not only to a massive particle, but also to a photon as well.
Deflection of light due to massive objects was predicted by Einstein in his General Theory of Relativity. This deflection of light has been calculated by many researchers in past, for spherically symmetric objects. But, in reality, most of these gravitating objects are not spherical instead they are ellipsoidal ( oblate) in shape. The objective of the present work is to study theoretically the effect of this ellipticity on the trajectory of a light ray. Here, we obtain a converging series expression for the deflection of a light ray due to an ellipsoidal gravitating object, characterised by an…
We investigate the Lorentz symmetry breaking effects (LSBE) on the deflection of light by a rotating cosmic string spacetime in the weak limit approximation. We first calculate the deflection angle by a static cosmic string for a fixed spacelike 4-vector case (FSL) with the corresponding effective-string optical metric using the Gauss-Bonnet theorem (GBT). Then, we focus on a more general scenario, namely we calculate the deflection angle by a rotating cosmic string applying the GBT to Randers effective-string metric. We obtain a significant modification in the deflection angle because of the …
From the Reissner-Nordstrom metric we obtain the higher-order terms for the deflection of light around a massive-charged black hole using the Lindstedt-Poincaré method to solve the equation of motion of a photon around the compact object. The corrections are performed developing the expansion in terms of $ε=\frac{r_{c}}{b}$ and $\fracδ{b}= \frac{Q^{2}}{6 πε_{0}Mc^{2} b}$. Additionally, from the perturbation expansion, we obtain multivariate Padé approximants. We also show how these are a better fit for the numerical data than the original formal Taylor series.
We compute the deflection angle to order (m/r_0)^2 and m/r_0*Lambda r_0^2 for a light ray traveling in a flat LambdaCDM cosmology which encounters a completely condensed mass region. We use a Swiss cheese model for the inhomogeneities and find that the most significant correction to the Einstein angle occurs not because of the non-linear terms but instead occurs because the condensed mass is embedded in a background cosmology.The Swiss cheese model predicts a decrease in the deflection angle of ~2% for weakly lensed galaxies behind the rich cluster A1689, and that the reduction can be as large…
We have derived the relationship between the radial proper distance, h, and the polar angle, phi, for a light ray that is emitted and travels in the neighborhood of the Earth's surface. General relativity predicts that, even locally, the equation which relates these two physical magnitudes differs from the one stated by the principle of equivalence. More precisely, we have proved that, in the weak field limit, the local physical trajectory, h=h(phi), is the one that would correspond to a massive Newtonian particle in a field two times greater. Therefore, and contrary to the generally accepted …
We study the null geodesics in a static circularly symmetric (SCS) black hole spacetime which is a solution in the $(2+1)$-dimensional massive gravity proposed by Bergshoeff, Hohm and Townsend (BHT massive gravity). We obtain analytic solutions for the null geodesic equation in the SCS black hole background and find the explicit form of deflection angles. We see that for various values of the impact parameter, the deflection angle can be positive, negative or even zero in this black hole spacetime. The negative deflection angle indicates the repulsive behavior of the gravity which comes from t…
We study the deflection of light by a magnetic dipole field in the generalized Born-Infeld electrodynamics. Using the effective index of refraction and the trajectory equation based on geometric optics, we compute the weak bending angle of light passing on the equator of the magnetic dipole. In the limit where the classical Born-Infeld parameter is infinite, the bending angle agree with the one computed from the Euler-Heisenberg Lagrangian. We also compute the bending angle using the geodesic equation of the effective metric induced by a massive object with magnetic dipole. In the massless lim…
The Weyl gravity appears to be a very peculiar theory. The contribution of the Weyl linear parameter to the effective geodesic potential is opposite for massive and nonmassive geodesics. However, photon geodesics do not depend on the unknown conformal factor, unlike massive geodesics. Hence light deflection offers an interesting test of the Weyl theory. In order to investigate light deflection in the setting of Weyl gravity, we first distinguish between a weak field and a strong field approximation. Indeed, the Weyl gravity does not turn off asymptotically and becomes even stronger at larger d…
A theory is derived relating the configuration of the cores of active galaxies, specifically candidates for presumed super-massive black hole binaries (SMBHBs), to time-dependent changes in images of those galaxies. Three deflection quantities, resulting from the monopole term, mass quadrupole term, and spin dipole term of the core, are examined. The resulting observational technique is applied to the galaxy 3C66B. This technique is found to under idealized circumstances surpass the technique proposed by Jenet et al. in accuracy for constraining the mass of SMBHB candidates, but is exceeded in…
Poincaré Gauge's theory of gravity is the most noteworthy alternative extension of general relativity that has a correspondence between spin and spacetime geometry. In this paper, we use Reissner-Nordstrom-de Sitter and anti-de Sitter solutions, where torsion $τ$ is added as an independent field, to analyze the weak deflection angles $\hatα$ of massive and null particles in finite distance regime. We then apply $\hatα$ to determine the Einstein ring formation in M87* and Sgr. A* and determine that relative to Earth's location from these black holes, massive torsion effects can provide consider…
In this letter, we use a recent wormhole solution known as a ringhole [Gonzalez-Diaz, Phys.\ Rev.\ D {\bf 54}, 6122, 1996] to determine the surface topology and the deflection angle of light in the weak limit approximation using the Gauss-Bonnet theorem (GBT). We apply the GBT and show that the surface topology at the wormhole throat is indeed a torus by computing the Euler characteristic number. As a special case of the ringhole solution, one can find the Ellis wormhole which has the surface topology of a 2-sphere at the wormhole throat. The most interesting results of this paper concerns the…
Deflection of Light due to Kerr Sen Black Hole in Heterotic String Theory using Material Medium Approach
National Institute of Technology, Agartala（阿加塔拉国家理工学院）；The Institute of Mathematical Sciences（数学科学研究所）；Hemvati Nandan Bahuguna Garhwal Central University（赫姆瓦蒂·南丹·巴胡加纳加尔瓦尔中央大学）；Center for Space Research, North-West University（西北大学空间研究中心）；Assam University（阿萨姆大学）
We investigate the shadow, deflection angle, and the greybody bounding of a Schwarzschild-AdS/dS black hole in scale-dependent gravity. We used the EHT data to constrain the parameter $ε$. We have found that within the $1σ$ level of uncertainty, $εM_0$ ranges at $10^{-11} - 10^{-16}$ orders of magnitude for Sgr. A*, and $10^{-11} - 10^{-17}$ for M87*. Using these parameters, we explored how the shadow radius behaves as perceived by a static observer. Using the known parameters for Sgr. A* and M87*, we found different shadow cast behavior near the cosmological horizon even if the same scaling p…
Using self-force methods, we consider the hyperbolic-type scattering of a pointlike particle carrying a scalar charge $Q$ off a Schwarzschild black hole. For given initial velocity and impact parameter, back-reaction from the scalar field modifies the scattering angle by an amount $\propto\! Q^2$, which we calculate numerically for a large sample of orbits (neglecting the gravitational self-force). Our results probe both strong-field and field-weak scenarios, and in the latter case we find a good agreement with post-Minkowskian expressions. The scalar-field self-force has a component tangent t…
The influence of the medium on the gravitational deflection of light rays is widely discussed in literature for the simplest non-trivial case: cold non-magnetized plasma. In this article, we generalize these studies to the case of an arbitrary transparent dispersive medium with a given refractive index. We calculate the deflection angle of light ray moving in a general spherically symmetric metric in the presence of medium with the spherically symmetric refractive index. The equation for the radius of circular light orbits is also derived. We discuss in detail the properties of these results a…
