Constraining the spacetime spin using time delay in stationary axisymmetric spacetimes

Abstract

Abstract 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*.