Shadows and weak gravitational lensing for black holes within Einstein-Maxwell-scalar theory

In this study,we investigated the optical properties of charged black holes within the Einstein-Maxwellscalar(EMS)theory.We evaluated the shadow cast by these black holes and obtained analytical solutions for both the radius of the photon sphere and that of the shadow.We observed that black hole parametersγandβboth influence the shadow of black holes.We also found that the photon sphere and shadow radius increase as a consequence of the presence of the parameterγ.Interestingly,the shadow radius decreases first and then remains unchanged owing to the impact of the parameterβ.Finally,we analyzed the weak gravitational lensing and total magnification of lensed images around black holes.We found that the charge of the black holes and the parameterβboth have a significant impact,reducing the deflection angle.Similarly,the same behavior for the total magnification was observed,also as a result of the effect of the charge of the black holes and the parameterβ.

A note on the novel 4D Einstein–Gauss–Bonnet gravity

Recently, a novel 4 D Einstein–Gauss–Bonnet gravity has been proposed by Glavan and Lin(2020 Phys. Rev. Lett. 124 081301) by rescaling the coupling α→α(D-4) and taking the limit D→ 4 at the level of equations of motion. This prescription, though was shown to bring non-trivial effects for some spacetimes with particular symmetries, remains mysterious and calls for scrutiny. Indeed, there is no continuous way to take the limit D→4 in the higher Ddimensional equations of motion because the tensor indices depend on the spacetime dimension and behave discretely. On the other hand, if one works with 4 D spacetime indices the contribution corresponding to the Gauss–Bonnet term vanishes identically in the equations of motion. A necessary condition(but may not be sufficient) for this procedure to work is that there is an embedding of the 4 D spacetime into the higher D-dimensional spacetime so that the equations in the latter can be properly interpreted after taking the limit. In this note, working with2 D Einstein gravity, we show several subtleties when applying the method used in(2020 Phys.Rev. Lett. 124 081301).