Weak Gravitational lensing by phantom black holes and phantom wormholes using the Gauss–Bonnet theorem

In this paper, we study the deflection of light by a class of phantom black hole and wormhole solutions in the weak limit approximation. More specifically, in the first part of this work we study the deflection of light by Garfinkle–Horowitz–Ströminger black hole and Einstein–Maxwell anti-dilaton black hole using the optical geometry and the Gauss–Bonnet theorem. Our calculation shows that gravitational lensing is affected by the phantom scalar field (phantom dilaton). In the second part of this work, we explore the deflection of light by a class of asymptotically flat phantom wormholes. In particular we have used three types of wormholes: wormhole with a bounded/unbounded mass function, and a wormhole with a vanishing redshift function. We show that the particular choice of the shape function and mass function plays a crucial role in the final expression for the deflection angle of light. In the third part of the paper we verify our findings with the help of standard geodesics equations. Finally, in the fourth part of this paper we consider the problem for the observational relevance of our results studying the creation of the weak field Einstein rings.