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*.

研究了任意速度的零时空和时空信号在静止轴对称（SAS）时空的赤道平面中重力透镜（GL）图像之间的总传播时间t和时间延迟\（\ Delta t \）。使用微弱场限中的摄动方法，将一般SAS时空中的t表示为冲击参数b的准系列，其系数涉及源透镜距离\（r_s \）和透镜检测器距离\（r_d \），信号速度v和度量函数的渐近展开系数。时间延迟\（\ Delta t \）到前导阶数由时空质量M，自旋角动量a和后牛顿参数\（\ gamma \）以及运动学变量\（r_s，〜r_d，〜v \）和源确定角位置\（\ beta \）。当\（\ beta \ ll \ sqrt {aM} / r_ {s，d} \）时，\（\ Delta t \）由线性旋转a的贡献决定。将Sgr A *超大质量黑洞建模为Kerr–Newman黑洞，我们证明只要\（\ beta \ lesssim 1.5 \乘以10 ^ {-5} \） [ \（^ {\ prime \ prime} \ ） ]，那么\（\ Delta t \）将能够到达\（\ mathcal {O}（1）\）第二层，这正好在当前GRB，重力波和中微子观测站的时间分辨率之内。因此，在这些信号的GL中测量\（\ Delta t \）将使我们能够约束Sgr A *的自旋。