Strong Deflection Limit Research Articles

Strong deflection limit analysis of black hole lensing in inhomogeneous plasma

Strong deflection limit analysis of black hole lensing in inhomogeneous plasma

Analytical perturbations of relativistic images in Kerr space-time

Light rays passing very close to black holes may wind several times before escaping. For any given electromagnetic source around the black hole, a distant observer would thus observe two infinite sequences of images on either side of the black hole. These images are generated by light rays performing an increasing numbers of loops. The strong deflection limit provides a simple analytic formalism to describe such higher order images for spherically symmetric metrics, while for axially symmetric black holes one typically resorts to numerical approaches. Here we present the leading order perturbation to higher order images when the black hole spin is turned on. We show that the images slide around the black hole shadow as an effect of space-time dragging. We derive analytical formulae for their shifts and the perturbation of their time delays. We also discuss how such simple analytical formulae for images by Kerr black holes can be of great help in many applications.

Deflection of light by a Reissner–Nordström black hole and Painlevé VI equation

We consider the bending angle of the trajectory of a photon incident from and deflected to infinity around a Reissner–Nordström black hole. We treat the bending angle as a function of the squared reciprocal of the impact parameter and the squared electric charge of the background normalized by the mass of the black hole. It is shown that the bending angle satisfies a system of two inhomogeneous linear partial differential equations with polynomial coefficients. This system can be understood as an isomonodromic deformation of the inhomogeneous Picard–Fuchs equation satisfied by the bending angle in the Schwarzschild spacetime, where the deformation parameter is identified as the background electric charge. Furthermore, the integrability condition for these equations is found to be a specific type of the Painlevé VI equation that allows an algebraic solution. We solve the differential equations both at the weak and strong deflection limits. In the weak deflection limit, the bending angle is expressed as a power series expansion in terms of the squared reciprocal of the impact parameter and we obtain the explicit full-order expression for the coefficients. In the strong deflection limit, we obtain the asymptotic form of the bending angle that consists of the divergent logarithmic term and the finite O(1) term supplemented by linear recurrence relations which enable us to straightforwardly derive higher order coefficients. In deriving these results, the isomonodromic property of the differential equations plays an important role. Lastly, we briefly discuss the applicability of our method to other types of spacetimes such as a spinning black hole.

Gravitational lensing by a stable rotating regular black hole

Recent observational data from the Event Horizon Telescope (EHT) collaboration provide convincing realistic evidence for the existence of black hole rotation. From a phenomenological perspective, a recently proposed stable rotating regular (SRR) black hole circumvents the theoretical flaws of the Kerr solution. For the purpose of obtaining observational signatures of this black hole, we study its gravitational lensing effect. In the strong deflection limit, we calculate the deflection angle of light on the equatorial plane, the radius of the photon sphere, and other observables. The observables include the relativistic image position, separation, magnification, and time delays between different images. Then, by modeling M87* and Sgr A* as the SRR black hole, we compute their observables and evaluate the deviation of the observables from the Kerr case. In the weak deflection limit, we calculate the light deflection angle on the equatorial plane of M87* and Sgr A* via the Gauss-Bonnet theorem (GBT). With the growth of deviation parameter e, the gravitational lensing effect in the weak deflection limit intensifies monotonically, and the gravitational lensing effect in the strong deflection limit changes dramatically only at high spins. Our research may contribute to distinguish between SRR black holes from Kerr black holes under higher-precision astronomical observations.

Effect of electric interaction on the deflection and gravitational lensing in the strong field limit

The deflection angle Delta phi 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 naturally contains the finite distance effect and takes a quasi-power series form with a logarithmic divergence at the leading order. The coefficients of the series contain both the gravitational and electric contributions. Using the Reissner–Nordström spacetime as an example, we found that an electric repulsion (or attraction) tends to decrease (or increase) the critical impact parameter b_c. If the repulsion is strong enough, then b_c can shrink to zero and the critical particle sphere r_{0c} will disappear. These results are applied to the gravitational lensing of charge signal, from which we solved the image positions, their magnifications and time delays. It is found that in general, the electric repulsion (or attraction) will decrease (or increase) the image apparent angles, the black hole shadow sizes as well as their magnifications but increase (or decrease) the time delays.

Deflection and gravitational lensing with finite distance effect in the strong deflection limit in stationary and axisymmetric spacetimes

We study the deflection and gravitational lensing (GL) of both timelike and null signals in the equatorial plane of arbitrary stationary and axisymmetric spacetimes in the strong deflection limit. Our approach employs a perturbative method to show that both the deflection angle and the total travel time take quasi-series forms ∑ n=0 [Cn ln (1-bc/b) + Dn ] (1-bc/b) n , with the coefficients Cn and Dn incorporating the signal velocity and finite distance effect of the source and detector. This new deflection angle allows us to establish an accurate GL equation from which the apparent angles of the relativistic images and their time delays are found. These results are applied to the Kerr and the rotating Kalb-Ramond (KR) spacetimes to investigate the effect of the spacetime spin in both spacetimes, and the effective charge parameter and a transition parameter in the rotating KR spacetime on various observables. Moreover, using our approach, the effect of the signal velocity and the source angular position on these variables is also studied.

Gravitational lensing by using the 0th order of affine perturbation series of the deflection angle of a ray near a photon sphere

The 0th order of affine perturbation series of the deflection angle of a ray near a photon sphere is more accurate than a deflection angle in a strong deflection limit, which is used often, because the later has hidden error terms. We investigate gravitational lensing by using 0th order affine perturbation series of the deflection angle in a general asymptotically-flat, static, and spherical symmetric spacetime with the photon sphere. We apply our formula to Schwarzschild black hole, Reissner–Nordström black hole, and Ellis–Bronnikov wormhole spacetimes as examples. By comparing observables by using the deflection angles, we show that we can ignore the effect of the hidden error terms in the the deflection angle in the strong deflection limit on the observables in a usual lens configuration with the photon sphere since the hidden error terms are tiny. On the other hand, in a retro lensing configuration, the deflection angle in the strong-deflection-limit analysis have error of several percent and the 0th order of affine perturbation series of the deflection angle has almost half of the error. Thus, in the retro lensing configuration, we should use the 0th order of affine perturbation series of the deflection angle rather than the deflection angle in the strong-deflection-limit analysis. The 0th order of affine perturbation series of the deflection angle can give a brighter magnification by a dozen percent than the one by using the deflection angle in the strong-deflection-limit analysis.

Affine perturbation series of the deflection angle of a ray near the photon sphere of a Reissner-Nordström black hole

We investigate the affine perturbation series of the deflection angle of a ray near the photon sphere of an extreme Reissner-Nordstr\"{o}m black hole. We compare the 0th and 1st orders of the affine perturbation series with the deflection angle in a strong deflection limit. We conclude that the 0th order of the affine perturbation series is more accurate than the deflection angle in the strong deflection limit and that we should improve a strong-deflection-limit analysis by using the 0th order of the affine perturbation series.

Shape of higher-order images of equatorial emission rings around a Schwarzschild black hole: Analytical description with polar curves

Higher-order photon rings can be expected to be detected in a more detailed image of the black hole found in future observations. These rings are lensed images of the luminous matter surrounding the black hole and are formed by photons that loop around it. In this paper we have succeeded to derive an analytical expression for the shape of the higher-order rings in the form that is most convenient for application: the explicit equation of the curve in polar coordinates. The formula describes the apparent shape of the higher-order image of the circular ring with the given radius around Schwarzschild black hole as viewed by distant observer with an arbitrary inclination. For the derivation, the strong deflection limit of the gravitational deflection is used. Our formula is a simple and efficient alternative to the numerical calculation of ray trajectories, with the main application to studying the shape of $n=2$ and $n=3$ photon rings.

Erratum: Nonlogarithmic divergence of a deflection angle by a marginally unstable photon sphere of the Damour-Solodukhin wormhole in a strong deflection limit [Phys. Rev. D 101 , 104021 (2020)

Received 22 August 2022DOI:https://doi.org/10.1103/PhysRevD.106.049901© 2022 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasClassical black holesGeneral relativityGravitation, Cosmology & Astrophysics

Analytical study of higher-order ring images of the accretion disk around a black hole

Gravitational lensing of a light source by a black hole leads to the appearance of higher-order images produced by photons that loop around the black hole before reaching the observer. Higher-order images were widely investigated numerically and analytically, in particular using the so-called strong deflection limit of gravitational deflection. After recent observations of the black hole image, attention has been drawn to higher-order rings, which are lensed images of the accreting matter of the black hole environment and can appear near the boundary of the black hole shadow. In this article, we use the strong deflection limit technique to investigate higher-order ring images of the luminous accretion disk around a Schwarzschild black hole. We consider a thin disk given by the inner and outer radii and an observer located far from the black hole on the axis of symmetry. For this configuration, it becomes possible to find the angular radii, thicknesses, and solid angles of higher-order rings in the form of compact analytical expressions. We show that the size of the rings is mainly determined by the position of the inner boundary of the accretion disk, which makes it possible to use them to distinguish between different accretion models. Possible generalizations of our model to nonsymmetric images can help to make the estimation of black hole angular momentum. We also present the analytical estimation of fluxes from higher-order images. Our method makes it easy to investigate $n=2$ and $n=3$ higher-order rings, the possible observation of which in future projects is currently being discussed.

Gravitational lensing by a Bronnikov-Kim wormhole under a weak-field approximation and in a strong deflection limit

We consider gravitational lensing under a weak-field approximation and in a strong deflection limit by a Bronnikov-Kim wormhole with the same metric as the one of a wormhole which has been suggested in Einstein-Dirac-Maxwell theory. The metric approaches into the metric of an extreme charged Reissner-Nordstr\"om black hole in a black hole limit and it becomes the metric of a spatial Schwarzschild wormhole in an ultrastatic limit. In both of the black hole limit and the ultrastatic limit, the coefficient of a divergent term and the constant term of the deflection angle of a light in the strong deflection limit can be obtained exactly without expanding parameters of the spacetime. Interestingly, in the both limits to the black hole and the ultrastatic wormhole, we obtain exactly the same coefficient and constant term in the strong deflection limit.

Deflection Angle in the Strong Deflection Limit for the Photons Coupled to Weyl Tensor in a Conformal Gravity Black Hole

Deflection Angle in the Strong Deflection Limit for the Photons Coupled to Weyl Tensor in a Conformal Gravity Black Hole

Gravitational time delay effects by Kerr and Kerr-Newman black holes in strong field limits

We study the time delay between two relativistic images due to strong gravitational lensing of the light rays caused by the Kerr and Kerr-Newman black holes. The trajectories of the light rays are restricted on the equatorial plane. Using the known form of the deflection angle in the strong deflection limit (SDL) allows us to analytically develop the formalism for the travel time of the light from the distant source winding around the black hole several times and reaching the observer. We find that the black hole with higher mass or with spin of the extreme black hole potentially have higher time delay. The effect of the charge of the black hole enhances the time delay between the images lying on the opposite side of the optical axis resulting from the light rays when one light ray is in the direct orbit and the other is in the retrograde orbit. In contrary, when both light rays travel along either direct or retrograde orbits giving the images on the same side of the optical axis, the charge effect reduces the time delay between them. We then examine the time delay observations due to the galactic and supermassive black holes respectively

Decoding a black hole metric from the interferometric pattern of the relativistic images of a compact source

Photons emitted by light sources in the neighbourhood of a black hole can wind several times around it before fleeing towards the observer. For spherically symmetric black holes, two infinite sequences of images are created for any given source, asymptotically approaching the shadow border with decreasing magnitude. These sequences are reflected by a characteristic staircase structure in the complex visibility function that may be used to decode the properties of the black hole metric. Recalling the formalism of gravitational lensing in the strong deflection limit, we derive analytical formulae for the height, the width and the periodicities of the steps in the visibility as functions of the black hole parameters for the case of a single compact and distant source. With respect to diffuse emission by the whole accretion flow, this ideal framework provides clean insight and model-independent information on the metric. These basic formulae can then be used to build visibilities for more complicated sources and track the changes induced by alternative metrics and ultimately test General Relativity. As simple examples, we include visibilities for Reissner-Nordström and Janis-Newman-Winicour metrics.

Strong gravitational lensing by Kerr and Kerr-Newman black holes

We study the strong gravitational lensing due to the Kerr black holes with angular momentum $a$ and the Kerr-Newman black holes with additional charge $Q$. We first derive the analytical expressions of the deflection angles of light rays that particularly diverge as they travel near the photon sphere. In this strong deflection limit, the light rays can circle around the black hole multiple times before reaching the observer, giving relativistic images. The obtained analytical expressions are then applied to compute the angular positions and the magnifications of relativistic images due to the supermassive galactic black holes. In this work, we focus on the outermost image with reference to the optical axis. We find that its angular separation from the one closest to the optical axis and the magnification increase in angular momentum $a$ of the black holes for light rays with direct orbits. Additionally, the effects of the charge $Q$ of black holes also increase the angular separation of the outermost image from the others and the magnification for both direct and retrograde orbits. The potentially increasing observability of the relativistic images from the effects of angular momentum and charge of the black holes will be discussed.

Iterative solutions for the gravitational lens equation in the strong deflection limit

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 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 ${\mathrm{A}}^{*}$ 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.

Gravitational lensing in the Simpson-Visser black-bounce spacetime in a strong deflection limit

A Simpson-Visser spacetime has two nonnegative parameters $a$ and $m$ and its metric is correspond with (i) a Schwarzschild metric for $a=0$ and $m\neq0$, (ii) a regular black hole metric for $a<2m$, (iii) a one-way traversable wormhole metric for $a=2m$, (vi) a two-way traversable wormhole metric for $a>2m$, and (v) an Ellis-Bronnikov wormhole metric for $a\neq0$ and $m=0$. The spacetime is one of the most useful spacetimes for the purpose of comprehensively understanding gravitational lensing of light rays reflected by a photon sphere of black holes and wormholes. We have investigated gravitational lensing in the Simpson-Visser spacetime in a strong deflection limit in all the nonnegative parameters of $a$ and $m$. In a case of $a=3m$, two photon spheres and an antiphoton sphere at the throat degenerate into a marginally unstable photon sphere. The deflection angle of the light rays reflected by the marginally unstable photon sphere at the throat diverges nonlogarithmically in the strong deflection limit.

Deflection angle of a light ray reflected by a general marginally unstable photon sphere in a strong deflection limit

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.

Strong gravitational lensing by a strongly naked null singularity

We study strong gravitational lensing in a static, spherically symmetric, naked singularity spacetime, without a photon sphere. The nature of the singularity is found to be lightlike. We discuss the characteristic lensing features of this naked singularity in the strong deflection limit. In spite of the absence of a photon sphere in this spacetime, the bending angle of light diverges, as it approaches the singularity. However, unlike black holes, it is found that the nature of this divergence is nonlogarithmic, and we derive an analytic formula for the same. Moreover, the relativistic rings produced due to strong lensing by the singularity are found to be well separated from each other, making them easy to resolve and possibly detect. These features are expected to be important in the study of strong lensing by ultracompact objects, especially ones without event horizons.