Observations of gravitational lenses in strong gravitational fields give us a clue to understanding dark compact objects. In this paper, we extend a method to obtain a deflection angle in a strong deflection limit provided by Bozza [Phys. Rev. D 66, 103001 (2002)] to apply to ultrastatic spacetimes. We also discuss on the order of an error term in the deflection angle. Using the improved method, we consider gravitational lensing by an Ellis wormhole, which is an ultrastatic wormhole of the Morris-Thorne class.

Strong deflection limit analysis and gravitational lensing of an Ellis wormhole

Continuing work initiated in an earlier publication [A. Ishihara, Y. Suzuki, T. Ono, T. Kitamura, and H. Asada, Phys. Rev. D 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 into account 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 ${\mathrm{A}}^{*}$ for instance are able to amount to $\ensuremath{\sim}{10}^{\ensuremath{-}5}\text{ }\text{ }\mathrm{arcseconds}$ 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 $\ensuremath{\sim}{10}^{\ensuremath{-}5}\text{ }\text{ }\mathrm{arcseconds}$.

Finite-distance corrections to the gravitational bending angle of light in the strong deflection limit

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)]. 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 [Jusufi et al., Phys. Rev. D 95, 104012 (2017)], in which they employ another approach proposed by Werner with using the Nazim's osculating Riemannian construction method via the Randers-Finsler metric. We show that the two different methods give the same result in the asymptotically far limit. We obtain also the corrections to the deflection angle due to the finite distance from the rotating global monopole. Near-future observations of Sgr A * will be able to put a bound on the global monopole parameter $\ensuremath{\beta}$ as $1\ensuremath{-}\ensuremath{\beta}l{10}^{\ensuremath{-}3}$ for a rotating global monopole model, which is interpreted as the bound on the deficit angle $\ensuremath{\delta}l8\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$ [rad].

Deflection angle of light for an observer and source at finite distance from a rotating global monopole

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-Nordstr\"om spacetime. The point of the improvement is the definition of a standard variable in the strong deflection limit analysis. We show that the choice of the variable is as important as the choice of the coordinates and we conclude that one should choose a proper variable for a given spacetime.

Deflection angle in the strong deflection limit in a general asymptotically flat, static, spherically symmetric spacetime

Gravitational lensing is a good probe into the topological structure of dark gravitating celestial objects. In this paper, we investigate the light curve of a light ray that passes through the throat of an Ellis wormhole, the simplest example of traversable wormholes. The method developed here is also applicable to other traversable wormholes. To study whether the light curve of a light ray that passes through a wormhole throat is distinguishable from that which does not, we also calculate light curves without the passage of a throat for an Ellis wormhole, a Schwarzschild black hole, and an ultrastatic wormhole with the spatial geometry identical to that of the Schwarzschild black hole in the following two cases: (i) ``microlensing,'' where the source, lens, and observer are almost aligned in this order and the light ray starts at the source, refracts in the weak gravitational field of the lens with a small deflection angle, and reaches the observer; and (ii) ``retrolensing,'' where the source, observer, and lens are almost aligned in this order, and the light ray starts at the source, refracts in the vicinity of the light sphere of the lens with a deflection angle very close to $\ensuremath{\pi}$, and reaches the observer. We find that the light curve of the light ray that passes through the throat of the Ellis wormhole is clearly distinguishable from that by the microlensing but not from that by the retrolensing. This is because the light curve of a light ray that passes by a light sphere of a lens with a large deflection angle has common characters, irrespective of the details of the lens object. This implies that the light curves of the light rays that pass through the throat of more general traversable wormholes are qualitatively the same as that of the Ellis wormhole.

Light curves of light rays passing through a wormhole

Gravitational lens models with negative convergence (surface mass density projected onto the lens plane) inspired by modified gravity theories, exotic matter and energy have been recently discussed in such a way that a static and spherically-symmetric modified spacetime metric depends on the inverse distance to the power of positive n (n=1 for Schwarzschild metric, n=2 for Ellis wormhole) in the weak-field approximation [Kitamura, Nakajima and Asada, PRD 87, 027501 (2013)], and it has been shown that demagnification of images could occur for n > 1 lens models associated with exotic matter (and energy), though they cause the gravitational pull on light rays. The present paper considers gravitational lensing shear by the demagnifying lens models and other models such as negative-mass compact objects causing the gravitational repulsion on light rays like a concave lens. It is shown that images by the lens models for the gravitational pull are tangentially elongated, whereas those by the repulsive ones are radially distorted. This feature of lensed image shapes may be used for searching (or constraining) localized exotic matter or energy with gravitational lensing surveys. It is suggested also that an underdense region such as a cosmic void might produce radially elongated images of background galaxies rather than tangential ones.

Gravitational lensing shear by an exotic lens object with negative convergence or negative mass

Gravitational lens models with negative convergence inspired by modified gravity theories, exotic matter, and energy have been recently examined in such a way that a static and spherically symmetric modified spacetime metric depends on the inverse distance to the $n$th power ($n=1$ for Schwarzschild metric, $n=2$ for Ellis wormhole, and $n\ensuremath{\ne}1$ for an extended spherical distribution of matter such as an isothermal sphere) in the weak-field approximation. Some of the models act as if a convex lens, whereas the others are repulsive on light rays like a concave lens. The present paper considers microlensed image centroid motions by the exotic lens models. Numerical calculations show that, for large $n$ cases in the convex-type models, the centroid shift from the source position might move on a multiply connected curve like a bow tie, while it is known to move on an ellipse for the $n=1$ case and to move on an oval curve for $n=2$. The distinctive feature of the microlensed image centroid may be used for searching (or constraining) localized exotic matter or energy with astrometric observations. It is shown also that the centroid shift trajectory for concave-type repulsive models might be elongated vertically to the source motion direction like a prolate spheroid, whereas that for convex-type models such as the Schwarzschild one is tangentially elongated like an oblate spheroid.

Chisaki Hagiwara

Papers

Gravitational lensing shear by an exotic lens object with negative convergence or negative mass

Microlensed image centroid motions by an exotic lens object with negative convergence or negative mass