Gravitational lensing in Tangherlini spacetime in the weak gravitational field and the strong gravitational field

The gravitational lensing effects in the weak gravitational field by exotic lenses have been investigated intensively to find nonluminous exotic objects. Gravitational lensing based on 1/r^n fall-off metric, as a one-parameter model that can treat by hand both the Schwarzschild lens (n=1) and the Ellis wormhole (n=2) in the weak field, has been recently studied. Only for n=1 case, however, it has been explicitly shown that effects of relativistic lens images by the strong field on the light curve can be neglected. We discuss whether relativistic images by the strong field can be neglected for n>1 in the Tangherlini spacetime which is one of the simplest models for our purpose. We calculate the divergent part of the deflection angle for arbitrary n and the regular part for n=1, 2 and 4 in the strong field limit, the deflection angle for arbitrary n under the weak gravitational approximation. We also compare the radius of the Einstein ring with the radii of the relativistic Einstein rings for arbitrary n. We conclude that the images in the strong gravitational field have little effect on the total light curve and that the time-symmetric demagnification parts in the light curve will appear even after taking account of the images in the strong gravitational field for n>1.

Microlensed image centroid motions 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 $n eq 1$ for an extended spherical distribution of matter such as an isothermal sphere) in the weak-field approximation [Kitamura, Nakajima and Asada, PRD 87, 027501 (2013), Izumi et al. PRD 88 024049 (2013)]. 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 $n=1$ 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.

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

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.

Demagnifying gravitational lenses toward hunting a clue of exotic matter and energy

We examine a gravitational lens model inspired by modified gravity theories and exotic matter and energy. We study an asymptotically flat, static, and spherically symmetric spacetime that is modified in such a way that the spacetime metric depends on the inverse distance to the power of positive $n$ in the weak-field approximation. It is shown analytically and numerically that there is a lower limit on the source angular displacement from the lens object to get demagnification. Demagnifying gravitational lenses could appear, provided the source position $beta$ and the power $n$ satisfy $beta > 2/(n+1)$ in the units of the Einstein ring radius under a large-$n$ approximation. Unusually, the total amplification of the lensed images, though they are caused by the gravitational pull, could be less than unity. Therefore, time-symmetric demagnification parts in numerical light curves by gravitational microlensing (F.Abe, Astrophys. J. 725, 787, 2010) may be evidence of an Ellis wormhole (being an example of traversable wormholes), but they do not always prove it. Such a gravitational demagnification of the light might be used for hunting a clue of exotic matter and energy that are described by an equation of state more general than the Ellis wormhole case. Numerical calculations for the $n=3$ and 10 cases show maximally $sim 10$ and $sim 60$ percent depletion of the light, when the source position is $beta sim 1.1$ and $beta sim 0.7$, respectively.