https://doi.org/10.1140/epjc/s10052-021-09659-8
Regular Article - Theoretical Physics
Time delay in the strong field limit for null and timelike signals and its simple interpretation
MOE Key Laboratory of Artificial Micro- and Nano-structures, School of Physics and Technology, Wuhan University, 430072, Wuhan, China
Received:
11
May
2021
Accepted:
16
September
2021
Published online:
10
October
2021
Gravitational lensing can happen not only for null signals 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 a quasi-power series of the small parameter 
where b is the impact parameter and 
is its critical value. The coefficients of the series are completely fixed by the behaviour of the metric functions near the particle sphere 
and only the first term of the series contains a weak logarithmic divergence. The time delay 
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 
on the time delay. It is found that as l increases from 0 to its critical value 
, both and decrease. The variation of 
for l from 0 to can be as large as 
[s], whose measurement then can be used to constrain the value of l. While for ultra-relativistic neutrino or gravitational wave, the variation of is too small to be resolved. The dependence of 
on shows that to temporally resolve the two sequences of images from opposite sides of the lens, 
has to be larger than a certain value, or equivalently if is small, the time resolution of the observatories has to be good.
© The Author(s) 2021
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