Deflection of light by black holes and massless wormholes in massive gravity

TL;DR

This paper investigates weak gravitational lensing caused by black holes and wormholes within massive gravity theory, deriving new analytical deflection angles and analyzing time delay effects in these exotic spacetimes.

Contribution

It introduces a new method to analyze light deflection in massive gravity, including a massless wormhole solution, and generalizes deflection angles for different parameter cases.

Findings

Derived new analytical deflection angles for various parameter cases.

Established a method to analyze time delay in black hole and wormhole spacetimes.

Found a massless wormhole solution with scalar charge in massive gravity.

Abstract

Weak gravitational lensing by black holes and wormholes in the context of massive gravity (Bebronne and Tinyakov 2009) theory is studied. The particular solution examined is characterized by two integration constants, the mass $M$ and an extra parameter $S$ namely `scalar charge'. These black hole reduce to the standard Schwarzschild black hole solutions when the scalar charge is zero and the mass is positive. In addition, a parameter $\lambda$ in the metric characterizes so-called 'hair'. The geodesic equations are used to examine the behavior of the deflection angle in four relevant cases of the parameter $\lambda$. Then, by introducing a simple coordinate transformation $r^\lambda=S+v^2$ into the black hole metric, we were able to find a massless wormhole solution of Einstein-Rosen (ER) \cite{Einstein} type with scalar charge $S$. The programme is then repeated in terms of the Gauss--Bonnet theorem in the weak field limit after a method is established to deal with the angle of deflection using different domains of integration depending on the parameter $\lambda$. In particular, we have found new analytical results corresponding to four special cases which generalize the well known deflection angles reported in the literature. Finally, we have established the time delay problem in the spacetime of black holes and wormholes, respectively.