Constraining the spacetime spin using time delay in stationary axisymmetric spacetimes

Abstract

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 distancesr_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 1.5times 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*.