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

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.