Constraining the spacetime spin using time delay in stationary axisymmetric spacetimes

TL;DR

This paper develops a perturbative method to relate gravitational lensing time delays in stationary axisymmetric spacetimes to the spacetime's mass and spin, enabling potential measurements of black hole spin via time delay observations.

Contribution

It introduces a quasi-series expansion for travel time and shows how time delays depend on spin and mass, providing a new way to constrain black hole spin through lensing observations.

Findings

Time delay $\Delta t$ is primarily influenced by the black hole's spin and mass.

For Sgr A*, $\Delta t$ can reach seconds, detectable by current observatories.

Measuring $\Delta t$ can constrain the spin of Sgr A*.

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 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*.