Total travel time t and time delay $$\Delta t$$ Δ t between images of gravitational lensing (GL) in the equatorial plane of stationary axisymmetric (SAS) spacetimes for null and timelike signals with arbitrary velocity are studied. Using a perturbative method in the weak field limit, t in general SAS spacetimes is expressed as a quasi-series of the impact parameter b with coefficients involving the source-lens distance $$r_s$$ r s and lens-detector distances$$r_d$$ r d , signal velocity v, and asymptotic expansion coefficients of the metric functions. The time delay $$\Delta t$$ Δ t to the leading order(s) were shown to be determined by the spacetime mass M, spin angular momentum a and post-Newtonian parameter $$\gamma $$ γ , and kinematic variables $$r_s,~r_d,~v$$ r s , r d , v and source angular position $$\beta $$ β . When $$\beta \ll \sqrt{aM}/r_{s,d}$$ β ≪ aM / r s , d , $$\Delta t$$ Δ t is dominated by the contribution linear to spin a. Modeling the Sgr A* supermassive black hole as a Kerr–Newman black hole, we show that as long as $$\beta \lesssim 1.5\times 10^{-5}$$ β ≲ 1.5 × 10 - 5 [$$^{\prime \prime }$$ ″ ], then $$\Delta t$$ Δ t will be able to reach the $$\mathcal {O}(1)$$ O ( 1 ) second level, which is well within the time resolution of current GRB, gravitational wave and neutrino observatories. Therefore measuring $$\Delta t$$ Δ t in GL of these signals will allow us to constrain the spin of the Sgr A*.

Time delay in Schwarzschild spacetime for null and timelike signals with arbitrary velocity v is studied. The total travel time t if is evaluated both exactly and approximately in the weak field limit, with the result given as functions of signal velocity, source-lens and lens-observer distances, angular position of the source and lens mass. Two time delays, ∆t v between signals with different velocities but coming from same side of the lens and ∆t p between signals from different sides of the lens, as well as the difference ∆t pv between two ∆t p 's are calculated. These time delays are applied to the gravitational-lensed supernova neutrinos and gravitational waves (GW). It is shown that the ∆t v between different mass eigenstates of supernova neutrinos can be related to the mass square difference of these eigenstates and therefore could potentially be used to discriminate neutrino mass orderings, while the difference ∆t pv between neutrino and optical signals can be correlated with the absolute mass of neutrinos. The formula for time delay in a general lens mass profile is derived and the result is applied to the singular isothermal sphere case. For GWs, it is found that the difference ∆t pv between GW and GRB can only reach 1.45 × 10 −5 second for very large source distance (2 × 10 4 [Mpc]) and source angle (10 [as]) if v GM = (1 − 3 × 10 −15 )c. This time difference is at least three order smaller than uncertainties in time measurement of the recently observed GW/GRB signals and thus calls for improvement if ∆t pv is to be used to further constrain the GW velocity.

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.