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Time delay in Schwarzschild spacetime for null and timelike signals witharbitrary velocity $v$ is studied. The total travel time $t_\mathrm{if}$ isevaluated both exactly and approximately in the weak field limit, with theresult given as functions of signal velocity, source-lens and lens-observerdistances, angular position of the source and lens mass. Two time delays,$\Delta t_v$ between signals with different velocities but coming from sameside of the lens and $\Delta t_\mathrm{p}$ between signals from different sidesof 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 thegravitational-lensed supernova neutrinos and gravitational waves (GW). It isshown that the $\Delta t_v$ between different mass eigenstates of supernovaneutrinos can be related to the mass square difference of these eigenstates andtherefore could potentially be used to discriminate neutrino mass orderings,while the difference $\Delta t_{\mathrm{p}v}$ between neutrino and opticalsignals can be correlated with the absolute mass of neutrinos. The formula fortime delay in a general lens mass profile is derived and the result is appliedto the singular isothermal sphere case. For GWs, it is found that thedifference $\Delta t_{\mathrm{p}v}$ 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 timedifference is at least three order smaller than uncertainties in timemeasurement of the recently observed GW/GRB signals and thus calls forimprovement if $\Delta t_{\mathrm{p}v}$ is to be used to further constrain theGW velocity.

Time delay of timelike particles in gravitational lensing of Schwarzschild spacetime