标题： 强场极限下零和类时信号的时间延迟及其简单解释 显示英文标题

标题： Time delay in the strong field limit for null and timelike signals and its simple interpretation

摘要： 引力透镜效应不仅发生在零质信号上，还在某些超越广义相对论的理论中发生在类时信号上，例如中微子和大质量引力波。 在本工作中，我们研究了在一般静态且球对称时空中，由具有任意渐近速度 $v$ 的信号形成的不同相对论图像之间的时间延迟。 使用一种微扰方法来计算强场极限下的总旅行时间，发现它是一个小参数 $a=1-b_c/b$ 的准级数，其中 $b$ 是偏转参数， $b_c$ 是其临界值。 该级数的系数完全由粒子球 $r_c$ 附近度规函数的行为决定，且该级数中只有第一项包含弱对数发散。 在领先非平凡阶次的时间延迟 $\Delta t_{n,m}$ 被证明等于粒子球周长除以局部信号速度，并乘以绕数和红移因子。 假设Sgr A*超大质量黑洞是一个Hayward黑洞，我们能够验证总时间的准级数形式，并揭示时空参数$l$、信号速度$v$以及源/探测器坐标差$\Delta\phi_{sd}$对时间延迟的影响。 发现当$l$从0增加到其临界值$l_c$时，$r_c$和$\Delta t_{n,m}$都减小。 $\Delta t_{n+1,n}$随$l$从 0 到$l_c$的变化可能高达$7.2\times 10^1$[s]，其测量值可以用来限制$l$的值。 而对于超相对论中微子或引力波，$\Delta t_{n,m}$的变化太小而无法分辨。 $\Delta t_{n,-n}$对$\Delta \phi_{sd}$的依赖表明，为了在时间上分辨来自透镜两侧的两组图像，$|\Delta \phi_{sd}-\pi|$必须大于某个值。 显示英文摘要

摘要： Gravitational lensing can happen not only for null signal but also timelike signals such as neutrinos and massive gravitational waves in some theories beyond GR. In this work we study the time delay between different relativistic images formed by signals with arbitrary asymptotic velocity $v$ in general static and spherically symmetric spacetimes. A perturbative method is used to calculate the total travel time in the strong field limit, which is found to be in quasi-series of the small parameter $a=1-b_c/b$ where $b$ is the impact parameter and $b_c$ is its critical value. The coefficients of the series are completely fixed by the behaviour of the metric functions near the particle sphere $r_c$ and only the first term of the series contains a weak logarithmic divergence. The time delay $\Delta t_{n,m}$ to the leading non-trivial order was shown to equal the particle sphere circumference divided by the local signal velocity and multiplied by the winding number and the redshift factor. By assuming the Sgr A* supermassive black hole is a Hayward one, we were able to validate the quasi-series form of the total time, and reveal the effects of the spacetime parameter $l$, the signal velocity $v$ and the source/detector coordinate difference $\Delta\phi_{sd}$ on the time delay. It is found that as $l$ increase from 0 to its critical value $l_c$, both $r_c$ and $\Delta t_{n,m}$ decrease. The variation of $\Delta t_{n+1,n}$ for $l$ from 0 to $l_c$ can be as large as $7.2\times 10^1$ [s], whose measurement then can be used to constrain the value of $l$. While for ultra-relativistic neutrino or gravitational wave, the variation of $\Delta t_{n,m}$ is too small to be resolved. The dependence of $\Delta t_{n,-n}$ on $\Delta \phi_{sd}$ shows that to temporally resolve the two sequences of images from opposite sides of the lens, $|\Delta \phi_{sd}-\pi|$ has to be larger than certain value.