The gravitational deflection angle of light for an observer and source at finite distance from a lens object has been studied by Ishihara et al. [Phys. Rev. D, 94, 084015 (2016)], based on the Gauss-Bonnet theorem with using the optical metric. Their approach to finite-distance cases is limited within an asymptotically flat spacetime. By making several assumptions, we give an interpretation of their definition from the observer's viewpoint: The observer assumes the direction of a hypothetical light emission at the observer position and makes a comparison between the fiducial emission direction and the direction along the real light ray. The angle between the two directions at the observer location can be interpreted as the deflection angle by Ishihara et al. The present interpretation does not require the asymptotic flatness. Motivated by this, we avoid such asymptotic regions to discuss another integral form of the deflection angle of light. This form makes it clear that the proposed deflection angle can be used not only for asymptotically flat spacetimes but also for asymptotically nonflat ones. We examine the proposed deflection angle in two models for the latter case; Kottler (Schwarzschild-de Sitter) solution in general relativity and a spherical solution in Weyl conformal gravity. Effects of finite distance on the light deflection in Weyl conformal gravity result in an extra term in the deflection angle, which may be marginally observable in a certain parameter region. On the other hand, those in Kottler spacetime are beyond reach of the current technology.
By using a method improved with a generalized optical metric, the deflection of light for an observer and source at finite distance from a lens object in a stationary, axisymmetric and asymptotically flat spacetime has been recently discussed [Ono, Ishihara, Asada, Phys. Rev. D 96, 104037 (2017)]. By using this method, in the weak field approximation, we study the deflection angle of light for an observer and source at finite distance from a rotating Teo wormhole, especially by taking account of the contribution from the geodesic curvature of the light ray in a space associated with the genera…
In this paper we present an analytical treatment of gravitational lensing by Kerr black holes in the limit of very large deflection angles, restricting to observers in the equatorial plane. We accomplish our objective starting from the Schwarzschild black hole and adding corrections up to second order in the black hole spin. This is sufficient to provide a full description of all caustics and the inversion of lens mapping for sources near them. On the basis of these formulae we argue that relativistic images of Low Mass X-ray Binaries around Sgr A* are very likely to be seen by future X-ray in…
By using a method improved with a generalized optical metric, the deflection of light for an observer and source at finite distance from a lens object in a stationary, axisymmetric and asymptotically flat spacetime has been recently discussed [Ono, Ishihara, Asada, Phys. Rev. D {\bf 96}, 104037 (2017)]. In this paper, we study a possible extension of this method to an asymptotically nonflat spacetime. We discuss a rotating global monopole. Our result of the deflection angle of light is compared with a recent work on the same spacetime but limited within the asymptotic source and observer [Jusu…
We generalize our previous work on gravitational lensing by a Kerr black hole in the strong deflection limit, removing the restriction to observers on the equatorial plane. Starting from the Schwarzschild solution and adding corrections up to the second order in the black hole spin, we perform a complete analytical study of the lens equation for relativistic images created by photons passing very close to a Kerr black hole. We find out that, to the lowest order, all observables (including shape and shift of the black hole shadow, caustic drift and size, images position and magnification) depen…
The deflection angle of a light ray can be arbitrarily large near a light sphere. The time-symmetrical shape of light curves of a pair of light rays reflected by a light sphere of a lens object does not depend on the details of the lens object. We consider retrolensing light curves of sunlight with deflection angles $π$ and $3π$ by an Ellis wormhole, which is the simplest Morris-Thorne wormhole. If an Ellis wormhole with a throat parameter $a=10^{11}$ km is $100$ pc away from an observer and if the Ellis wormhole, the observer, and the sun are aligned perfectly in this order, the apparent magn…
We will mainly discuss the measurable angle (local angle) of the light ray $ψ_P$ at the position of the observer $P$ instead of the total deflection angle (global angle) $α$ in Kerr spacetime. We will investigate not only the effect of the gravito-magnetic field or frame dragging but also the contribution of the motion of the observer with a coordinate radial velocity $v^r$ and a coordinate transverse velocity $bv^ϕ$ ($b$ is the impact parameter and $v^ϕ$ is a coordinate angular velocity) which are converted from the components of the 4-velocity of the observer $u^r$ and $u^ϕ$, respectively. B…
We discuss the deflection of light and Shapiro delay under the influence of gravity as described by Schwarzschild metric. We obtain an exact expression based on the coordinate velocity, as first set forth by Einstein, and present a discussion on the effect of velocity anisotropy. We conclude that the anisotropy in the coordinate velocity, as the velocity apparent to a distant observer, gives rise to a third order error in the deflection angle, so that the practical astronomical observations from gravitational lensing data remain inconclusive on the anisotropy. However, measurement of Shapiro d…
After discussing some subtle issues concerning the computation of deflection angle, a general but simple expression of bending angle of light rays in weak deflection limit has been presented for a general static and spherically symmetric space-time. In this context the importance of proper choice of polar axis has been highlighted. Applying the prescribed method the explicit expression of bending angle up to an accuracy of second order in mass is obtained for the Schwarzschild geometry without restricting the source and the observer to be at infinite distance away from the lens.
In Newtonian gravity (NG) it is known that the gravitational field anywhere inside a spherically symmetric distribution of mass is determined only by the enclosed mass. This is also widely believed to be true in general relativity (GR), and the Birkhoff theorem is often invoked to support this analogy between NG and GR. Here we show that such an understanding of the Birkhoff theorem is incorrect and leads to erroneous calculations of light deflection and delay time through matter. The correct metric, matching continuously to the location of an external observer, is determined both by the enclo…
We derive angle differences between two null geodesics, propagating from light sources to an observer, on the de Sitter spacetime with multi-lensing objects. Assuming the lensing objects are mass monopoles on the de Sitter background, we derive the metric tensor by solving the Einstein equation perturbatively. On that spacetime, we solve a null geodesic parametrized by the coordinate time. Using the null geodesics, we define the angle differences in a coordinate invariant way. We take in the relativistic effects up to the first order of perturbation and clarify the magnitude of approximation e…
Various authors have investigated the problem of light deflection by radially moving gravitational lenses - with inconsistent results. In this paper we generalize the calculations for arbitrary lens velocities and show that, to first order in the lens velocity v, the deflection angles scales with 1-v. We discuss the seeming inconsistency of relativistic light deflection with the classical picture of moving test particles by generalizing the lens effect to test particles of arbitrary velocity, including light as a limiting case. We show that the effect of radial motion of the lens is very diffe…
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…
Using a perturbative technique, in this work we study the deflection of null and timelike signals in the extended Einstein-Maxwell spacetime, the Born-Infeld gravity and the charged Ellis-Bronnikov (CEB) spacetime in the weak field limit. The deflection angles are found to take a (quasi-)series form of the impact parameter, and automatically takes into account the finite distance effect of the source and observer. The method is also applied to find the deflections in CEB spacetime with arbitrary dimension. It's shown that to the leading non-trivial order, the deflection in some $n$-dimensional…
We show that in the weak field limit the light deflection alone cannot distinguish between different $R + F[g(\square)R]$ models of gravity, where $F$ and $g$ are arbitrary functions. This does not imply, however, that in all these theories an observer will see the same deflection angle. Owed to the need to calibrate the Newton constant, the deflection angle may be model-dependent after all necessary types of measurements are taken into account.
As an extension of our previous paper, instead of the total deflection angle $α$, we will mainly focus on discussing the measurable angle of the light ray $ψ$ at the position of the observer in Kerr--de Sitter spacetime which includes the cosmological constant $Λ$. We will investigate the contributions of the radial and transverse motions of the observer which are connected with the radial velocity $v^r$ and transverse velocity $bv^ϕ$ ($b$ is the impact parameter) as well as the spin parameter $a$ of the central object which induces the gravitomagnetic field or frame dragging and the cosmologi…
Two exact lens equations have been recently shown to be equivalent to each other, being consistent with the gravitational deflection angle of light from a source to an observer, both of which can be within a finite distance from a lens object [Phys. Rev. D 102, 064060 (2020)]. We examine methods for iterative solutions of the gravitational lens equations in the strong deflection limit. It has been so far unclear whether a convergent series expansion can be provided by the gravitational lens approach based on the geometrical optics for obtaining approximate solutions in the strong deflection li…
In this work are reviewed several aspects of gravitational lensing produced by astrophysical bodies that strongly curve the spacetime in their vicinity. When an object with a photon sphere (e.g. a black hole) is interposed between a source and an observer, in addition to primary and secondary images due to small deflections of the light rays, there are two infinite sets of images called relativistic, produced by light passing close to the photon sphere, which undergoes a large deflection. The positions and magnifications of relativistic images can be obtained numerically or analytically, in th…
Debate persists as to whether the cosmological constant $Λ$ can directly modify the power of a gravitational lens. With the aim of reestablishing a consensus on this issue, I conduct a comprehensive analysis of gravitational lensing in the Schwarzschild--de Sitter spacetime, wherein the effects of $Λ$ should be most apparent. The effective lensing law is found to be in precise agreement with the $Λ=0$ result: $α_\mathrm{eff} = 4m/b_\mathrm{eff}+15πm^2/4b_\mathrm{eff}^2 +O(m^3/b_\mathrm{eff}^3)$, where the effective bending angle $α_\mathrm{eff}$ and impact parameter $b_\mathrm{eff}$ are define…
We present a simple prescription for the rotation of polarization produced by a relativistically moving gravitational lens, applicable to arbitrary deflection angle and arbitrary velocity of the lens. When geometric optics is applicable, two independent components contribute to the total rotation of polarization: (i) in the frame of the lens the polarization vector experiences minimal rotation defined by the deflection angle (as measured by a set of remote observers, or no rotation if defined in terms of parallel-propagated tetrad); (ii) the effect of the motion of the lens on the polarization…
A generalized lens equation for weak gravitational fields in Schwarzschild metric and valid for finite distances of source and observer from the light deflecting body is suggested. The magnitude of neglected terms in the generalized lens equation is estimated to be smaller than or equal to 15 Pi/4 (m/d')^2, where m is the Schwarzschild radius of massive body and d' is Chandrasekhar's impact parameter. The main applications of this generalized lens equation are extreme astrometrical configurations, where 'Standard post-Newtonian approach' as well as 'Classical lens equation' cannot be applied. …
We investigate gravitational lensing by a primary photon sphere which is a sphere filled with unstable circular light orbits, and by a secondary photon sphere on a wormhole throat in a black-bounce spacetime which is suggested in [F. S. N. Lobo, M. E. Rodrigues, M. V. d. S. Silva, A. Simpson, and M. Visser, Phys. Rev. D 103, 084052 (2021)] in strong deflection limits. There is an antiphoton sphere between the primary photon sphere and the secondary photon sphere. If a light source and an observer are on the same side of the wormhole throat, in addition to an infinite number of images slightly …
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 to spinning signals with arbitrary asymptotic velocity and takes into account the finite distance effect of the source and the observer. The deflection angle $Δφ$ and total travel time $Δt$ are expressed as (quasi-)power series whose coefficients are …
The presence of dark matter around a black hole remarkably affects its spacetime. We consider the effects of dark matter on the shadow of a new solution to the Einstein equations that describes a rotating black hole in the background of perfect dark matter fluid (PFDM), along with its extension to nonzero cosmological constant $Λ$. Working in Boyer-Lindquist coordinates, we consider the effects of the PFDM parameter $α$ on the shadow cast by a black hole with respect to an observer at position $(r_o,θ_o)$. By applying the Gauss-Bonnet theorem to the optical geometry we find that notable distor…
The trajectory deflection and gravitational-electromagnetic dual lensing (GEL) of charged signal in general charged static and spherically symmetric spacetimes are considered in this work. We showed that the perturbative approach previously developed for neutral particles can be extended to the electromagnetic interaction case. The deflection angle still takes a (quasi-)series form and the finite distance effect of both the source and observer can be taken into account. Comparing to pure gravitational case, the apparent angles of the images in the GEL, their magnifications and time delay all r…
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…
Light deflection in the post-linear gravitational field of two bounded point-like masses is treated. Both the light source and the observer are assumed to be located at infinity in an asymptotically flat space. The equations of light propagation are explicitly integrated to the second order in $G/c^2$. Some of the integrals are evaluated by making use of an expansion in powers of the ratio of the relative separation distance to the impact parameter $(r_{12}/ξ)$. A discussion of which orders must be retained to be consistent with the expansion in terms of $G/c^2$ is given. It is shown that the …
Motivated by recent work on the Modified Maxwell (ModMax) black holes [Phys.Lett.B 10.1016/j.physletb.2020.136011], which are invariant in duality rotations and conformal transformations founded in [ Phys.Rev.D 10.1103/PhysRevD.102.121703], we probe its effects on the shadow cast, weak field gravitational lensing, and neutrino propagation in its vicinity. Using the EHT data for the shadow diameter of Sgr. A* and M87*, and LIGO/VIRGO experiments for the dyonic ModMax black hole perturbations, we find constraints for ModMax parameters such as $Q_\text{m}$ and the screening factor $γ$. We also an…
Assuming a static and spherically symmetric spacetime, we propose a novel concept of the total deflection angle of a light ray. The concept is defined by the difference between the sum of internal angles of two triangles; one of the triangles lies on curved spacetime distorted by a gravitating body and the other on its background. The triangle required to define the total deflection angle can be realized by setting three laser-beam baselines as in planned space missions such as LATOR, ASTROD-GW, and LISA. Accordingly, the new total deflection angle is, in principle, measurable by gauging the i…
The deflection angle of Kerr-MOG black holes is studied for different values of the parameter in modified gravity (MOG). To this end, we employ the Gauss-Bonnet theorem, which was first studied by Gibbons and Werner and then extended by Ono, Ishihara and Asada, who use a generalized optical metric where the deflection of light for an observer and source at finite distance. By using this method, we study the weak gravitational lensing by Kerr-MOG black hole. Our computations show that with an increase in the MOG parameter ($α$), the deflection angle becomes significantly larger than that of Ker…
Future space astrometry missions are planned to measure positions and/or parallaxes of celestial objects with an accuracy of the order of the microarcsecond. At such a level of accuracy, it will be indispensable to take into account the influence of the mass multipole structure of the giant planets on the bending of light rays. Within the parametrized post-Newtonian formalism, we present an algorithmic procedure enabling to determine explicitly this influence on a light ray connecting two points located at a finite distance. Then we specialize our formulae in the cases where 1) the light sourc…
This paper uses the Schwarzschild metric to derive an effective refractive index and acceleration vector that account for relativistic deflection of light rays, in an otherwise classical kinematic framework. The new refractive index and the known path equation are integrated to give accurate results for travel time and deflection angle, respectively. A new formula for coordinate acceleration is derived which describes the path of a massless test particle in the vicinity of a spherically symmetric mass density distribution. A standard ray-shooting technique is used to compare the deflection ang…
The general expression of the angular distance between two point sources as measured by an arbitrary observer is given. The modelling presented here is rigorous, covariant and valid in any space-time. The sources of light may be located at a finite distance from the observer. The aberration and the gravitational deflection of light are treated in a unified way. Assuming the gravitational field to be weak, an explicit expansion of the angular separation within the post-post-Minkowskian approximation is carried out. The angular separation within the post-Newtonian approximation truncated at the …
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…
We discuss, without assuming asymptotic flatness, a gravitational lens for an observer and source that are within a finite distance from a lens object. The proposed lens equation is consistent with the deflection angle of light that is defined for nonasymptotic observer and source by Takizawa et al. [Phys. Rev. D 101, 104032 (2020)] based on the Gauss-Bonnet theorem with using the optical metric. This lens equation, though it is shown to be equivalent to the Bozza lens equation[Phys. Rev. D 78, 103005 (2008)], is linear in the deflection angle. Therefore, the proposed equation is more convenie…
Weak field deflection angle and analytical parameter estimation of the Lorentz-violating Bumblebee parameter through the black hole shadow using EHT data
In this paper, we investigate the influence of the plasma surrounding the gravitational lens on the effect of microlensing. In presence of plasma around the lens, the deflection angle is determined by both the gravitational field of the lens and the chromatic refraction in the inhomogeneous plasma. We calculate microlensing light curves numerically for point-mass lens surrounded by power-law density distribution of plasma. A variety of possible curves is revealed, depending on the plasma density and frequency of observations. In the case of significant influence of plasma, the shape of microle…
Experimental verification of the existence of gravimagnetic fields generated by currents of matter is important for a complete understanding and formulation of gravitational physics. Although the rotational `intrinsic' gravimagnetic field has been extensively studied and is now being measured by the Gravity Probe B, the `extrinsic' gravimagnetic field generated by the translational current of matter is less well studied. The present paper uses the post-Newtonian parametrized Einstein and light geodesics equations to show that the `extrinsic' gravimagnetic field generated by the translational c…
In this paper we study the light bending caused by a slowly rotating source in the context of quadratic theories of gravity, in which the Einstein--Hilbert action is extended by additional terms quadratic in the curvature tensors. The deflection angle is computed employing the method based on the Gauss--Bonnet theorem and working in the approximation of a weak lens; also, we assume that the source and observer are at an infinite distance. The formalism presented is very general and applies to any spacetime metric in the limit of weak gravitational field and slow rotation. We find the explicit …
In this paper, we study the deflection angle for wormhole-like static aether solution by using Gibbons and Werner technique in non-plasma, plasma, and dark matter mediums. For this purpose, we use optical spacetime geometry to calculate the Gaussian optical curvature, then implement the Gauss-Bonnet theorem in weak field limits. Moreover, we compute the deflection angle by using a technique known as Keeton and Petters technique. Furthermore, we analyze the graphical behavior of the bending angle $ψ$ with respect to the impact parameter $b$, mass $m$ as an integration constant, and parameter $q…
In 2007 Rindler and Ishak showed that, contrary to previous claims, the value of the cosmological constant does have an effect on light deflection by a gravitating object in an expanding universe, modeled by a Schwarzschild-de~Sitter spacetime. In this paper we consider light bending in the more general situation of a gravitating object in a cosmological background with varying expansion rate $H(t)$. We calculate numerically the null geodesics representing light rays deflected by a black hole in an accelerating Friedmann-Lemaître-Robertson-Walker universe, modeled by a McVittie metric. Keeping…
We discuss boson stars as possible gravitational lenses and study the lensing effect by these objects made of scalar particles. The mass and the size of a boson star may vary from an individual Newtonian object similar to the Sun to the general relativistic size and mass of a galaxy close to its Schwarzschild radius. We assume boson stars to be transparent which allows the light to pass through them though the light is gravitationally deflected. We assume boson stars of the mass $M = 10^{10}M_\odot$ to be on non-cosmological distance from the observer. We discuss the lens equation for these st…
Applying the Gibbons-Werner method to bound orbits of massive particles in stationary spacetimes
Hunan University of Science and Technology（湖南科技大学）；Key Laboratory of Intelligent Sensors and Advanced Sensing Materials of Hunan Province, Hunan University of Science and Technology（湖南省智能传感与先进材料重点实验室，湖南科技大学）
