Negative time delay of light by a gravitational concave lens
  Gravitational lens models, some of which might act as if a concave lens, have been recently investigated by using a static and spherically symmetric modified spacetime metric that depends on the inverse distance to the $n$-th power [Kitamura, Nakajima and Asada, PRD 87, 027501 (2013)]. We reexamine the time delay of light in a gravitational concave lens as well as a gravitational convex one. The frequency shift due to the time delay is also investigated. We show that the sign of the time delay in the lens models is the same as that of the deflection angle of light. The size of the time delay decreases with increase in the parameter $n$. We discuss also possible parameter ranges that are relevant to pulsar timing measurements in our galaxy. 
Koki Nakajima, Koji Izumi, Hideki Asada
gr-qc astro-ph.CO