The Effects of Finite Distance on the Gravitational Deflection Angle of Light

Abstract

In order to clarify the effects of the finite distance from a lens object to a light source and a receiver, the gravitational deflection of light has been recently reexamined by using the Gauss–Bonnet (GB) theorem in differential geometry (Ishihara et al. 2016). The purpose of the present paper is to give a short review of a series of works initiated by the above paper. First, we provide the definition of the gravitational deflection angle of light for the finite-distance source and receiver in a static, spherically symmetric and asymptotically flat spacetime. We discuss the geometrical invariance of the definition by using the GB theorem. The present definition is used to discuss finite-distance effects on the light deflection in Schwarzschild spacetime for both the cases of weak deflection and strong deflection. Next, we extend the definition to stationary and axisymmetric spacetimes. We compute finite-distance effects on the deflection angle of light for Kerr black holes and rotating Teo wormholes. Our results are consistent with the previous works if we take the infinite-distance limit. We briefly mention also the finite-distance effects on the light deflection by Sagittarius A * .

Finite-distance gravitational deflection of massive particles by a Kerr-like black hole in the bumblebee gravity model

“…In these studies, the possible astronomical application according to finite-distance correlations was considered. As well, Ono and Asada gave a comprehensive review on finite-distance deflection of light [68]. In addition, Arakida applied different definitions of the deflection angle to study the finite-distance deflection of light [69].…”

Weak Deflection Angle and Shadow by Tidal Charged Black Hole

“…After neglecting the plasma medium effect, we find the same angle as we find in the non-plasma case. Now if we neglect the plasma effect ( ω e ω ∞ → 0), then this deflection angle Equation (52) reduces into angle Equation (51). This shows the correctness of our angle in the presence of a plasma medium.…”

“…The weak deflection angle, using the GBT for various spacetimes, was studied by many physicists. For example, the deflection angle of light was studied for BHs and wormholes by the following authors [35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53]: Ovgun et al studied for different spacetimes, such as Schwarzschild-like spacetime, the bumblebee gravity model [54][55][56][57][58][59][60], and Javed et al studied the impact of various matter fields [61][62][63][64][65]. Next, Ishihara et al [66] showed that it is conceivable to calculate the weak deflection angle using the finite-distances method.…”

Deriving Weak Deflection Angle by Black Holes or Wormholes using Gauss-Bonnet Theorem

“…The surface that the bending of light occurs in is the key to compute the weak deflection angle; the rays of light are treated as space-like geodesics of the optical metric. Introducing the Gauss-Bonnet theorem, an approach that utilizes these attributes by relating the topology of the surface to its intrinsic geometry, it facilitates the bending angle to be invariant under coordinate transformations according to [61].…”

“…On the other hand, every observable object is at a finite redshift from the observer, us. Owing to this, the above equations need a tweak called finite-distance corrections: it is written as the difference between the deflection angles for the asymptotic case and the finite-distance case: [61] δ α = α − α∞ .…”