Recently, a non-trivial 4D Einstein-Gauss-Bonnet (EGB) theory of gravity, by rescaling the GB coupling parameter as α/(D−4), was formulated in [1], which bypasses Lovelock's theorem and avoids Ostrogradsky instability. The theory admits a static spherically symmetric black hole, unlike 5D EGB or general relativity counterpart, which can have both Cauchy and event horizons. We generalize previous work, on gravitational lensing by a Schwarzschild black hole, in the strong and weak deflection limits to the 4D EGB black holes to calculate the deflection coefficients ā and b̄, while former increases and later decrease with increasing α. We also find that the deflection angle αD, angular position θ∞ and um decreases, but angular separation s increases with α. The effect of the GB coupling parameter α on positions and magnification of the source relativistic images is discussed in the context of SgrA* and M87* black holes. A brief description of the weak gravitational lensing using the Gauss-Bonnet theorem is presented.

Among the higher curvature gravities, the most extensively studied theory is the so-called Einstein–Gauss–Bonnet (EGB) gravity, whose Lagrangian contains Einstein term with the GB combination of quadratic curvature terms, and the GB term yields nontrivial gravitational dynamics in $$ D\ge 5$$ D ≥ 5 . Recently there has been a surge of interest in regularizing, a $$ D \rightarrow 4 $$ D → 4 limit of, the EGB gravity, and the resulting regularized 4D EGB gravity valid in 4D. We consider gravitational lensing by Charged black holes in the 4D EGB gravity theory to calculate the light deflection coefficients in strong-field limits $$\bar{a}$$ a ¯ and $$\bar{b}$$ b ¯ , while former increases with increasing GB parameter $$\alpha $$ α and charge q, later decrease. We also find a decrease in the deflection angle $$\alpha _D$$ α D , angular position $$\theta _{\infty }$$ θ ∞ decreases more slowly and impact parameter for photon orbits $$u_{m}$$ u m more quickly, but angular separation s increases more rapidly with $$\alpha $$ α and charge q. We compare our results with those for analogous black holes in General Relativity (GR) and also the formalism is applied to discuss the astrophysical consequences in the case of the supermassive black holes Sgr A* and M87*.

The recent time witnessed a surge of interest in strong gravitational lensing by black holes is due to the Event Horizon Telescope (EHT) results, which suggest comparing the black hole lensing in both general relativity and heterotic string theory. That may help us to assess the phenomenological differences between these models. Motivated by this, we consider gravitational lensing by the nonsingular Kerr-Sen black holes, which encompass Kerr black holes as a particular case, to calculate the light deflection coefficients p and q in strong-field limits, while the former increases with increasing parameters k and charge b, later decrease. We also find a decrease in the light deflection angle αD, angular position θ∞ decreases more slowly and impact parameter for photon orbits um more quickly, but angular separation s increases more rapidly with parameters b and k. We compare our results with those for Kerr black holes, and also the formalism is applied to discuss the astrophysical consequences in the case of the supermassive black holes NGC 4649, NGC 1332, Sgr A* and M87*. In turn, we also investigate the shadows of the nonsingular Kerr-Sen black holes and show that they are smaller and more distorted than the corresponding Kerr black holes and nonsingular Kerr black holes shadows. The inferred circularity deviation Δ C≤ 0.10, for the M87* black hole shadow, put constraints on the nonsingular Kerr-Sen black hole parameters (a, k) and (a, b). The maximum shadow angular diameter for b=0.30M and k=0.30M are, respectively, θd=35.3461 μas and θd=35.3355 μas. We also estimate the parameters associated with nonsingular Kerr-Sen black holes using the shadow observables.

We investigate strong field gravitational lensing by rotating Simpson-Visser black hole, which has an additional parameter (0 ≤ l/2M ≤ 1), apart from mass (M ) and rotation parameter (a). A rotating Simpson-Visser metric correspond to (i) a Schwarzschild metric for l/2M = a/2M = 0 and M = 0, (ii) a Kerr metric for l/2M = 0, |a/2M | < 0.5 and M = 0 (iii) a rotating regular black hole metric for |a/2M | < 0.5, M = 0 and l/2M in the range 0 < l/2M < 0.5 + (0.5) 2 − (a/2M ) 2 , and (iv) a traversable wormhole for a |a/2M | > 0.5 and l/2M = 0. We find a decrease in the deflection angle α D and also in the ratio of the flux of the first image and all other images r mag . On the other hand, angular position θ 1 increases more slowly and photon sphere radius x m decreases more quickly, but angular separation s increases more rapidly, and their behaviour is similar to that of the Kerr black hole. The formalism is applied to discuss the astrophysical consequences in the supermassive black holes NGC 4649, NGC 1332, Sgr A* and M87* and find that the rotating Simpson-Visser black holes can be quantitatively distinguished from the Kerr black hole via gravitational lensing effects. We find that the deviation of the lensing observables ∆θ 1 and ∆s of rotating Simpson Visser black holes from Kerr black hole for 0 < l/2M < 0.6 (a/2M = 0.45), for supermassive black holes Sgr A* and M87, respectively, are in the range 0.0422-0.11658 µas and 0.031709-0.08758 µas while |∆r mag | is in the range 0.2037-0.95668. It is difficult to distinguish the two black holes because the departure are in O(µas), which are unlikely to get resolved by the current EHT observations, and one has to wait for future observations by ngEHT can pin down the exact constraint. We also derive a two-dimensional lens equation and formula for deflection angle in the strong field limit by focusing on trajectories close to the equatorial plane.

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