Gravitational lens without asymptotic flatness: Its application to Weyl gravity

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 the 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 y 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 similar to 10(-2) microarcseconds for the lens and source with a separation angle of similar to 1 arcminute, where we consider a cluster of galaxies with 10(14) M-circle dot at similar to 1 Gpc for instance. The deviation becomes similar to 10(-1) microarcseconds, even if the separation angle is similar to 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 similar to 0.1 milfiarcseconds can be relevant with VLBI observations.