Deflection and gravitational lensing of null and timelike signals in the Kiselev black hole spacetime in the weak field limit

Abstract

In this work we study the deflection and gravitational lensing of null and timelike signals in the Kiselev spacetime in the weak field limit, to investigate the effects of the equation of state parameter $\omega$ and the matter amount parameter $\alpha$. In doing this, we extend a perturbative method previously developed for asymptotically flat spacetimes whose metric functions have integer-power asymptotic expansions to the case that may or may not be asymptotically flat but with non-integer power expansions. It is found that in the asymptotically flat case ($-1/3<\omega<0$) the deflection angles are expressable as quasi-power series of the dimensionless quantities $M/b,~b/r_{s,d}$ and $\alpha/M^{1+3\omega}$ where $M,~b,~r_{s,d}$ are respectively the lens mass, impact parameter and source/detector radius. A similar series exist for the non-asymptotically flat case of ($-1<\omega<-1/3$), but with the closest radius $r_0$ replacing $b$. In the asymptotically flat (or non-flat) case, the increase of $\alpha$ or decrease of $\omega$ will increase (or increase) the deflection angle. Since the obtained deflection angles naturally take into account the finite distance effect of the source and the detector, we can establish an exact gravitational lensing equation, from which the apparent angles of the images and their magnifications are solved. It is found that generally for asymptotically flat case, increasing $\alpha$ or decreasing $\omega$ will increase the apparent angles of the images. While for non-asymptotically flat case, increasing $\alpha$ or $\omega$ will both lead to smaller apparent angles.

Deflection in higher dimensional spacetime and asymptotically non-flat spacetimes

“…For example, typical asymptotically de-Sitter spacetimes are also asymptotically non-flat because of the extra Λr 2 term in the metric functions A(r) and B(r). However such spacetimes still have a deflection angle at lowest order to be π [51][52][53]. In other words, asymptotic non-flatness is only a necessary but not sufficient condition for the bending angle at the lowest order to be non-π.…”

“…Here δ is the angle by the three velocity v itself and its ϕ-component v ϕ , and d OL is the distance from the observer to the lens. Using (46) for v ϕ , equation (52) for γ and d OL = √ c 0 r,…”

On dark energy effects on the accretion physics around a Kiselev spinning black hole

“…Hence, for a broad analysis of the Kiselev metric and for this choice of metric parameter we refer to Appendix A and for example, to 49,[51][52][53][54][55][56][57][58][59][60][61][62]. In Appendix A we proceed with the analysis of the spacetime properties within this choice of the α parameter and spin a > 0 by considering the ergosufaces and detailing the conditions for the horizons existence, then examining the extreme BH Kiselev spacetimes.…”

“…While rotating BHs and "quintessence" (the Kiselev solutions) were studied in depth in [47], the rotating Kiselev BHs in f (R, T ) gravity has been also explored, with the study of Penrose diagrams, in [48]. In [49] deflection and gravitational lensing of null and timelike signals in the weak field limit of the Kiselev BH spacetime have been explored with perturbative methods. Analysis in [50] discusses properties of the total stress energy for the generalized Kiselev BHs.…”