Gravitational lens without asymptotic flatness: Its application to the Weyl gravity. (arXiv:2006.00682v2 [gr-qc] UPDATED)
<a href="http://arxiv.org/find/gr-qc/1/au:+Takizawa_K/0/1/0/all/0/1">Keita Takizawa</a>, <a href="http://arxiv.org/find/gr-qc/1/au:+Ono_T/0/1/0/all/0/1">Toshiaki Ono</a>, <a href="http://arxiv.org/find/gr-qc/1/au:+Asada_H/0/1/0/all/0/1">Hideki Asada</a>
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.
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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