Light propagation and gravitational lensing on the Weyl-like spacetime   in scalar-tensor theories of gravity
  We study light propagation and gravitational lensing in scalar-tensor theories of gravity by using a static, axisymmetric exterior solution. The solution has asymptotic flatness properties and is reduced to Voorhees's one in the case of a constant scalar field. Our studies are done by using a technique of the conformal transformation such that their results are independent of details of scalar-tensor theories. For some specific cases, we analytically obtain a deflection angle of the light path and find that it can become negative. The appearance of a negative deflection angle indicates ``reflection'' of a light path, and we investigate under which conditions the light reflection occurs. As for the optical scalars, the Weyl source-term shows significantly different properties compared with that in the Schwarzschild spacetime. We therefore classify a space of the model parameters into four distinct regions on the basis of the qualitative properties of the Weyl source-term and find a close relationship between this classification and the occurrence of the light reflection. We finally solve the null geodesic equations and the optical scalar equations numerically. We find that a picture of the thin lens is applicable and give a simple analytic model for the optical scalars. As for the properties of gravitational lensing, the deflection angle and the image distortion rate are obtained as functions of the impact parameter. Again, we find a close relationship between their qualitative properties and the classification above. 
Shin-ei Tsuneishi, Kazuya Watanabe and Tooru Tsuchida