Time delay of timelike particles in gravitational lensing of Schwarzschild spacetime

Abstract

Time delay in Schwarzschild spacetime for null and timelike signals with arbitrary velocity $v$ is studied. The total travel time $t_\mathrm{if}$ is evaluated both exactly and approximately in the weak field limit, with the result given as functions of signal velocity, source-lens and lens-observer distances, angular position of the source and lens mass. Two time delays, $\Delta t_v$ between signals with different velocities but coming from same side of the lens and $\Delta t_\mathrm{p}$ between signals from different sides of the lens, as well as the difference $\Delta t_{\mathrm{p}v}$ between two $\Delta t_\mathrm{p}$'s are calculated. These time delays are applied to the gravitational-lensed supernova neutrinos and gravitational waves (GW). It is shown that the $\Delta t_v$ between different mass eigenstates of supernova neutrinos can be related to the mass square difference of these eigenstates and therefore could potentially be used to discriminate neutrino mass orderings, while the difference $\Delta t_{\mathrm{p}v}$ between neutrino and optical signals can be correlated with the absolute mass of neutrinos. The formula for time delay in a general lens mass profile is derived and the result is applied to the singular isothermal sphere case. For GWs, it is found that the difference $\Delta t_{\mathrm{p}v}$ between GW and GRB can only reach $1.45\times 10^{-5}$ second for very large source distance ($2\times 10^4$ [Mpc]) and source angle (10 [as]) if $v_{GM}=(1-3\times 10^{-15})c$. This time difference is at least three order smaller than uncertainties in time measurement of the recently observed GW/GRB signals and thus calls for improvement if $\Delta t_{\mathrm{p}v}$ is to be used to further constrain the GW velocity.