Using the geodesic method and the perturbative approach, we study the deflection of light by time-periodic spherically symmetric gravitational fields. Assuming the weakness of the gravitational field, we derive general formulas that determine the deflection angle in the leading order approximation. The formulas are valid for both time-periodic and static metrics. Using these results, we calculate the deflection angle of a light ray passing through a spherically symmetric oscillating distribution of a self-gravitating scalar field with a logarithmic potential. It turned out that in this case the deflection angle does not depend on time in the leading order.
Gravitational lensing by the light sphere of compact objects like black holes and wormholes will give us information on the compact objects. In this paper, we provide an improved strong deflection limit analysis in a general asymptotically flat, static, spherically symmetric spacetime. The strong deflection limit analysis also works in ultrastatic spacetimes. As an example of an ultrastatic spacetime, we reexamine the deflection angle in the strong deflection limit in an Ellis wormhole spacetime. Using the strong deflection limit, we obtain the deflection angle analytically for the Reissner-No…
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…
Gravitational lensing inside and outside of a marginally unstable photon sphere in a general, static, spherically symmetric, and asymptotically-flat spacetime in strong deflection limits
Applying the perturbative approach to geodesic equations, we study motion of the test particles in time-dependent spherically symmetric spacetimes created by oscillating dark matter. Assuming the weakness of the gravitational field, we derive general formulas that describe infinite trajectories of the test particles and determine the total deflection angle in the leading order approximation. The obtained formulas are valid for both time-dependent and static matter configurations. Using these results, we calculate the deflection angle of a test particle passing through a spherically symmetric o…
In this paper, the perihelion precession and deflection of light have been investigated in the 4-dimensional general spherically symmetric spacetime, and the main equation is obtained. As the application of this main equation, the Reissner-Nordstorm-AdS solution and Clifton-Barrow solution in $f(R)$ gravity have been taken for the examples. We find that both the electric charge and $f(R)$ gravity can affect the perihelion precession and deflection of light, while the cosmological constant can only effect the perihelion precession. Moreover, we clarify a subtlety in the deflection of light in t…
We investigate the deflection angle in a strong deflection limit for a marginally unstable photon sphere in a general asymptotically flat, static and spherically symmetric spacetime under some assumptions to calculate observables. The deflection angle of a light ray reflected by the marginally unstable photon sphere diverges nonlogarithmically while the one reflected by a photon sphere diverges logarithmically. We apply our formula to a Reissner-Nordstrom spacetime and Hayward spacetime.
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…
Both null and timelike rays experience trajectory bending in a gravitational field. In this work, we systematically develop a perturbative method to compute the deflection angle of rays with general velocity $v$ in arbitrary static and spherically symmetric spacetimes and in equatorial plane of arbitrary static and axisymmetric spacetimes. We show that the expansion in the large closest approach $x_0$ limit depends on the asymptotic behavior of the metric functions only, and the generated integrand is always integrable, resulting in a deflection angle in a series form of either $x_0$ or $b$, t…
We firstly present a new asymptotical flat and spherically symmetric solution in the generalized Einstein-Cartan-Kibble-Sciama (ECKS) theory of gravity and then investigate the propagation of photon in this background. This solution possesses three independent parameters which affect sharply photon sphere, deflection angle of light ray and gravitational lensing. Since the condition of existence of horizons is not inconsistent with that of photon sphere, there exists a special case where there is horizon but no photon sphere in this spacetime. Especially, we find that in this special case, the …
The gravitational deflection of light ray is an important prediction of General Theory of Relativity. In this paper we develop analytical expression of the deflection of light ray without any weak field approximation due to a charged gravitational body represented by Reissner_Nordström (RN) and Janis-Newman-Winicour (JNW) space time geometry, using material medium approach. It is concluded that although both the geometries represent the charged, non-rotating, spherically symmetric gravitating body, but the effect of charge on the gravitational deflection is just opposite to each other. The gra…
This work investigates the influence of the Lorentz symmetry breaking in the bending angle of massive particles and light for bumblebee black hole solutions. The solutions analyzed break the Lorentz symmetry due to a non-zero vacuum expectation value of the bumblebee field. We use the Ishihara method, which allows us to study the bending angle of light for finite distances, and it is applicable to non-asymptotically flat spacetimes when considering the receiver viewpoint. In order to analyze the deflection of massive particles, we systematize the Ishihara method for its application in the Jaco…
The purpose of the current Letter is to give some relations between gravitational lensing in the strong-deflection limit and the frequencies of the quasinormal modes of spherically symmetric, asymptotically flat black holes. On the one side, the obtained relations can give a physical interpretation of the strong-deflection limit parameters. On the other side, they also give an alternative method for the measurement of the frequencies of the quasinormal modes of spherically symmetric, asymptotically flat black holes. They could be applied to the localization of the sources of gravitational wave…
The purpose of this paper is twofold. First, we introduce a geometric approach to study the circular orbit of a particle in static and spherically symmetric spacetime based on Jacobi metric. Second, we apply the circular orbit to study the weak gravitational deflection of null and time-like particles based on Gauss-Bonnet theorem. By this way, we obtain an expression of deflection angle and extend the study of deflection angle to asymptotically non-flat black hole spacetimes. Some black holes as lens are considered such as a static and spherically symmetric black hole in the conformal Weyl gra…
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 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…
We present a framework for studying gravitational lensing in spherically symmetric spacetimes using 1+1+2 covariant methods. A general formula for the deflection angle is derived and we show how this can be used to recover the standard result for the Schwarzschild spacetime.
We estimate the changes in solar system tests of general relativity when "dark energy" arises from a Weyl scaling invariant action, using the modified Schwarzschild-like spherically symmetric solution obtained in this case by Adler and Ramazano\vglu. We show that the standard results of general relativity are modified by amounts that are far below current experimental errors. However, because the metric is not conformally flat when the central mass $M$ vanishes, there is a contribution to light deflection proportional to the cosmological constant $Λ$, with opposite sign to the one proportional…
Nonlinear electrodynamic (NLED) theories are well-motivated for their extensions to classical electrodynamics in the strong field regime, and have been extensively investigated in seeking for regular black hole solutions. In this paper, we focus on two spherically symmetric and static black hole solutions based on two types of NLED models: the Euler-Heisenberg NLED model and the Bronnikov NLED model, and calculate the weak deflection angle of light by these two black holes with the help of the Gauss-Bonnet theorem. We investigate the effects of the one-loop corrections to quantum electrodynami…
In order to clarify effects of the finite distance from a lens object to a light source and a receiver, the gravitational deflection of light has been recently reexamined by using the Gauss-Bonnet (GB) theorem in differential geometry [Ishihara et al. 2016]. The purpose of the present paper is to give a short review of a series of works initiated by the above paper. First, we provide the definition of the gravitational deflection angle of light for the finite-distance source and receiver in a static, spherically symmetric and asymptotically flat spacetime. We discuss the geometrical invariance…
Static, spherically symmetric black holes and compact objects without an event horizon have unstable (stable) circular orbits of a light called photon (antiphoton) sphere. A Damour-Solodukhin wormhole has been suggested as a simple black hole mimicker and the difference of its metric tensors from a black hole is described by a dimensionless parameter $λ$. The wormhole with two flat regions has two photon spheres and an antiphoton sphere for $λ<\sqrt{2}/2$ and a photon sphere for $λ\geq \sqrt{2}/2$. When the parameter $λ$ is $\sqrt{2}/2$, the photon sphere is marginally unstable because of dege…
In this paper, the homotopy-perturbation method (HPM) is applied to obtain approximate analytical solutions for the gravitational deflection of light in General Relativity near Schwarzschild black hole surrounded by quintessence (Kiselev black hole). In order to demonstrate that HPM is able to yield acceptable solutions for the null-geodesics with easily computable terms, the HPM is tested for the simple examples of spherically symmetric spacetimes such as Schwarzschild and Reissner-Nordström black holes. After that, the null-geodesics of light passing the vicinity of Kiselev black hole are st…
We discuss a way to obtain information about higher dimensions from observations by studying a brane-based spherically symmetric solution. The three classic tests of General Relativity are analyzed in details: the perihelion shift of the planet Mercury, the deflection of light by the Sun, and the gravitational redshift of atomic spectral lines. The braneworld version of these tests exhibits an additional parameter $b$ related to the fifth-coordinate. This constant $b$ can be constrained by comparison with observational data for massive and massless particles.
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…
We are developing a general, unified, and rigorous analytical framework for using gravitational lensing by compact objects to test different theories of gravity beyond the weak-deflection limit. In this paper we present the formalism for computing corrections to lensing observables for static, spherically symmetric gravity theories in which the corrections to the weak-deflection limit can be expanded as a Taylor series in one parameter, namely the gravitational radius of the lens object. We take care to derive coordinate-independent expressions and compute quantities that are directly observab…
\begin{abstract} Using the Keeton and Petters approach, we determine the deflection angle. We also investigate the motion of photons around a parameterized black hole in the presence of non-magnetized cold plasma by using a new ray-tracing algorithm. In spherically symmetric spacetime, we examine the influence of the plasma by applying the Hamiltonian equation on the deflection angle as well as shadow. It is examine to derive the rays analytically from Hamilton's equation by separating the metric and the plasma frequency. We study that the presence of plasma affects the deflection angle as wel…
We investigate strong lensing by a spherically symmetric mass distribution in the framework of the Einstein-Straus solution with positive cosmological constant and concentrate on the case of a spatially closed Universe ($k=+1$). We develop a method based on integration of differential equations in order to make possible the computation of light deflection and time delay. By applying our results to the lensed quasar SDSS J1004+4112, we find that the bending of light and time delay depend on wether the present value of the scale factor $a_0$ is or is not much smaller than a value close to $9.1 \…
In this article, we demonstrate the weak gravitational lensing in the context of Bocharova-Bronnikov-Melnikov-Bekenstein (BBMB) black hole. To this desire, we derive the deflection angle of light in a plasma medium by BBMB black hole using the Gibbons and Werner method. First, we obtain the Gaussian optical curvature and implement the Gauss-Bonnet theorem to investigate the deflection angle for spherically symmetric spacetime of BBMB black hole. Moreover, we also analyze the graphical behavior of deflection angle by BBMB black hole in the presence of plasma medium.
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 this paper, we discuss the weak gravitational lensing in the context of stringy black holes. Initially, we examine the deflection angle of photon by charged stringy black hole. For this desire, we compute the Gaussian optical curvature and implement the Gauss-Bonnet theorem to investigate the deflection angle for spherically balanced spacetime of stringy black hole. We also analyze the influence of plasma medium in the weak gravitational lensing for stringy black hole. Moreover, the graphical impact of impact parameter $b$ , black hole charge $Q$ on deflection angle by charged stringy black…
We study motions of photons in an unmagnetized cold homogeneous plasma medium in the five-dimensional charged static squashed Kaluza-Klein black hole spacetime. In this case, a photon behaves as a massive particle in a four-dimensional spherically symmetric spacetime. We consider the light deflection by the squashed Kaluza-Klein black hole surrounded by the plasma in a weak-field limit. We derive corrections of the deflection angle to general relativity, which are related to the size of the extra dimension, the charge of the black hole and the ratio between the plasma and the photon frequencie…
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…
We study the light rays in a static and spherically symmetric gravitational field of null aether theory (NAT). To this end, we employ the Gauss-Bonnet theorem to compute the deflection angle by a NAT black hole in the weak limit approximation. Using the optical metrics of the NAT black hole, we first obtain the Gaussian curvature and then calculate the leading terms of the deflection angle. Our calculations show how gravitational lensing is affected by the NAT field. We also show once again that the bending of light stems from a global and topological effect.
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…
We examine in a semiclassical framework the deflection function of strong gravitational lensing, for static and spherically symmetric black holes, endowed with a photon sphere. From a first-order WKB analysis near the maximum of the Regge-Wheeler potential, we extract the real phase shifts from the S-matrix elements and then we derive the associated semiclassical deflection function, characterized by a logarithmic divergent behavior. More precisely, using the complex angular momentum techniques, we show that the Regge poles and the associated greybody factor residues, for a massless scalar fie…
The deflection and gravitational lensing of light and massive particles in arbitrary static, spherically symmetric and asymptotically (anti-)de Sitter spacetimes are considered in this work. We first proved that for spacetimes whose metric satisfying certain conditions, the deflection of null rays with fixed closest distance will not depend on the cosmological constant $Λ$, while that of timelike signals and the apparent angle in gravitational lensing still depend on $Λ$. A two-step perturbative method is then developed to compute the change of the angular coordinate and total travel time in t…
The gravitational field of supermassive black holes is able to strongly bend light rays emitted by nearby sources. When the deflection angle exceeds $π$, gravitational lensing can be analytically approximated by the so-called strong deflection limit. In this paper we remove the conventional assumption of sources very far from the black hole, considering the distance of the source as an additional parameter in the lensing problem to be treated exactly. We find expressions for critical curves, caustics and all lensing observables valid for any position of the source up to the horizon. After anal…
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 $ν$. The innermost bound and stable circular orbits are calculated and the effective potential is analysed. Consistent with numerical results in earlier literature, for $ν<1/2$, 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 spa…
We consider the classical theory of a gravitational field with spin 2 and non-vanishing (Pauli-Fierz) mass in flat spacetime, coupled to electromagnetism and point particles. We establish the law of light propagation and calculate the amount of deflection in the background of a spherically symmetric gravitational field. As the mass tends to zero, the deflection is shown to converge to 3/4 of the value predicted by the massless theory (linearized General Relativity), even though the spherically symmetric solution of the gravitational field equations has no regular limit. This confirms an old ar…
This paper explores ``black hole'' solutions of various Einstein-wave matter systems admitting an isometry of their domain of outer communications taking every point to its future. In the first two parts, it is shown that such solutions, assuming in addition that they are spherically symmetric and the matter has a certain structure, must be Schwarzschild or Reissner-Nordstrom. Non-trivial examples of matter for which the result applies are a wave map and a massive charged scalar field interacting with an electromagnetic field. The results thus generalize work of Bekenstein [1] and Heusler [12]…
Gravitational light deflection and shadows in a warm plasma
University of Haifa（海法大学）；Haifa Research Center for Theoretical Physics and Astrophysics, University of Haifa（海法大学海法理论物理与天体物理学研究中心）；University of Bremen（不来梅大学）
A perturbative method to compute the deflection angle of both timelike and null rays in arbitrary static and spherically symmetric spacetimes in the strong field limit is proposed. The result takes a quasi-series form of $(1-b_c/b)$ where $b$ is the impact parameter and $b_c$ is its critical value, with coefficients of the series explicitly given. This result also naturally takes into account the finite distance effect of both the source and detector, and allows to solve the apparent angles of the relativistic images in a more precise way. From this, the BH angular shadow size is expressed as …
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…
We extend to higher order a recently published method for calculating the deflection angle of light in a general static and spherically symmetric metric. Since the method is convergent we obtain very accurate analytical expressions that we compare with numerical results.
In this paper, we use a new asymptotically flat and spherically symmetric solution in the generalized Einstein-Cartan-Kibble-Sciama (ECKS) theory of gravity to study the weak gravitational lensing and its shadow cast. To this end, we first compute the weak deflection angle of generalized ECKS black hole using the Gauss-Bonnet theorem in plasma medium and in vacuum. Next by using the Newman-Janis algorithm without complexification, we derive the rotating generalized ECKS black hole and in the sequel study its shadow. Then, we discuss the effects of the ECKS parameter on the weak deflection angl…
