Investigating Graviton Mass Effects on Black Hole Lensing in dRGT Massive Gravity

TL;DR

The paper investigates gravitational lensing by non-asymptotically flat black holes in the dRGT massive gravity framework with a nonzero graviton mass $m_g$. It applies the Gauss-Bonnet theorem to compute the weak-field light deflection for a finite-distance source and observer, deriving an explicit expression for the deflection angle $\hat{\alpha}$ that includes contributions from mass $M$, charge $Q$, cosmological constant $\Lambda$, and massive-gravity parameters $\gamma$ and $\zeta$, up to second order in the small parameter $\varepsilon$. The angular radius of the Einstein ring $\Theta_E$ is then obtained from the lens equation, with analytic reductions to the asymptotically flat RN/Schwarzschild limits, showing explicit dependence on the graviton mass via $\zeta$. The results indicate that a nonzero $m_g$ leaves distinctive imprints on lensing observables and can be constrained by current bounds from gravitational-wave events and Solar System tests, highlighting the potential for lensing as a probe of massive gravity.

Abstract

In this paper, we delve into the gravitational lensing and photon trajectories in the vicinity of non-asymptotically flat black hole spacetimes within the framework of dRGT massive gravity, incorporating a non-zero graviton mass. We assume that both the observer and the light source are located at a finite distance from the lens object, and calculate the gravitational deflection angle of light ray by such a black hole based on Gauss-Bonnet theorem. Furthermore, we derive the angular radius of Einstein rings associated with black holes in dRGT massive gravity, thereby facilitating a comprehensive discussion on the ramifications of the graviton mass. Notably, our findings reveal pronounced impacts of the graviton mass on both light deflection angles and Einstein ring characteristics, underscoring its significance in dRGT massive gravity and enhancing the detectability of black holes in gravitational lensing observations, thereby opening new avenues for future research.

Figures (1)

• Figure 1: Weak field limit deflection angle of black hole in dRGT massive gravity. (a) black curve represents Schwartzschild-AdS black hole and red curve represents RN-AdS black hole with $\gamma=0, \zeta=0$ ( i.e. $\Lambda=-m_g^2<0$); (b) blue curve plotted with example values of $\alpha, \beta$ for $\Lambda<0, \gamma<0, \zeta>0$ and red curve plotted for $\Lambda<0, \gamma<0, \zeta<0$; (c) plotted with example values of $\alpha, \beta$ for $\Lambda=0, \gamma=0, \zeta>0$ and (d) plotted for $\Lambda>0, \gamma>0, \zeta>0$. Here we take $b=1, M/b=0.2, u_S=1/25, u_R=1/30$.