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Register a new userIterative solutions for the gravitational lens equation in the strong deflection limit
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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 limit in terms of a small offset angle. By using the ratio of the lens mass to the lens distance, we discuss a slightly different method for iterative solutions and the behavior of the convergence. Finite distance effects begin at the third order in the iterative method. The iterative solutions in the strong deflection limit are estimated for Sgr $A^{*}$ and M87. These results suggest that only the linear order solution can be relevant with current observations, while the finite distance effects at the third order may be negligible in the Schwarzschild lens model for these astronomical objects.
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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 to discuss also possible observational implications. The proposed corrections for Sgr A$^{ast}$ for instance are able to amount to $sim 10^{-5}$ arcseconds for some parameter range, which may be within the capability of near-future astronomy, while also the correction for the Sun in the weak field limit is $sim 10^{-5}$ arcseconds.
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 a simple formula containing metric functions and particle/photon sphere radius. The magnification of the relativistic images were shown to diverge at different values of the source-detector angular coordinate difference, depending on the relation between the source and detector distance from the lens. To verify all these results, we then applied them to the Hayward BH spacetime, concentrating on the effects of its charge parameter $l$ and the asymptotic velocity $v$ of the signal. The BH shadow size were found to decrease slightly as $l$ increase to its critical value, and increase as $v$ decreases from light speed. For the deflection angle and the magnification of the images however, both the increase of $l$ and decrease of $v$ will increase their values.
In this paper we apply the strong deflection limit approach to investigate the gravitational lensing phenomena beyond general relativity. This is accomplished by considering the lensing effects related to black hole solutions that emerge out of the domain of Einstein gravity, namely, the ones acquired from the method of geometric deformation and the Casadio-Fabbri-Mazzacurati brane-world black holes. The lensing observables, for those brane-world black hole metrics, are compared with the standard ones for the Schwarzschild case. We prove that brane-world black holes could have significantly different observational signatures, compared to the Schwarzschild black hole, with terms containing the post-Newtonian parameter, for the case of the Casadio-Fabbri-Mazzacurati, and terms with variable brane-world tension, for the method of geometric deformation.
The gravitational field of supermassive black holes is able to strongly bend light rays emitted by nearby sources. When the deflection angle exceeds $pi$, 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 analyzing the spherically symmetric case we focus on the Kerr black hole, for which we present an analytical 3-dimensional description of the higher order caustic tubes.
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 convenient for the purpose of doing an iterative analysis. As an explicit example of an asymptotically nonflat spacetime, we consider a static and spherically symmetric solution in Weyl conformal gravity, especially a case that $gamma$ parameter in the Weyl gravity model is of the order of the inverse of the present Hubble radius. For this case, we examine iterative solutions for the finite-distance lens equation up to the third order. The effect of the Weyl gravity on the lensed image position begins at the third order and it is linear in the impact parameter of light. The deviation of the lensed image position from the general relativistic one is $sim 10^{-2}$ microarcsecond for the lens and source with a separation angle of $sim 1$ arcminute, where we consider a cluster of galaxies with $10^{14} M_{odot}$ at $sim 1$ Gpc for instance. The deviation becomes $sim 10^{-1}$ microarcseconds, even if the separation angle is $sim 10$ arcminutes. Therefore, effects of the Weyl gravity model are negligible in current and near-future observations of gravitational lensing. On the other hand, the general relativistic corrections at the third order $sim 0.1$ milliarcseconds can be relevant with VLBI observations.
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