Constraining the spacetime spin using time delay in stationary axisymmetric spacetimes

Total travel time $t$ and time delay $\Delta t$ 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 $r_s$ and lens-detector distances $r_d$, signal velocity $v$, and asymptotic expansion coefficients of the metric functions. The time delay $\Delta t$ to the leading order(s) were shown to be determined by the spacetime mass $M$, spin angular momentum $a$ and post-Newtonian parameter $\gamma$, and kinematic variables $r_s,~r_d,~v$ and source angular position $\beta$. When $\beta\ll \sqrt{aM}/r_{s,d}$, $\Delta t$ 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 $\beta\lesssim 1.5\times 10^{-5}$ [$^{\prime\prime}$], then $\Delta t$ will be able to reach the $\mathcal{O}(1)$ second level, which is well within the time resolution of current GRB, gravitational wave and neutrino observatories. Therefore measuring $\Delta t$ in GL of these signals will allow us to constrain the spin of the Sgr A*.