Eur. Phys. J. C (2020) 80: 932
https://doi.org/10.1140/epjc/s10052-020-08496-5

Regular Article - Theoretical Physics

Constraining the spacetime spin using time delay in stationary axisymmetric spacetimes

¹ School of Physics and Technology, Wuhan University, 430072, Wuhan, China
² MOE Key Laboratory of Artificial Micro- and Nano-structures, Center for Astrophysics, School of Physics and Technology, Wuhan University, 430072, Wuhan, China

^b junjijia@whu.edu.cn

Received: 19 June 2020
Accepted: 25 September 2020
Published online: 9 October 2020

Abstract

Total travel time t and time delay $\mathrm{\Delta }t$ 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 $\mathrm{\Delta }t$ 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 $\beta \ll \sqrt{\mathit{aM}}/r_{s,d}$, $\mathrm{\Delta }t$ 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 $\beta \lesssim 1.5\times {10}^{-5}$ [${}^{″}$], then $\mathrm{\Delta }t$ 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 $\mathrm{\Delta }t$ in GL of these signals will allow us to constrain the spin of the Sgr A*.