Time delay of timelike particles in gravitational lensing of the Schwarzschild spacetime

Abstract

Time delay in Schwarzschild spacetime for null and timelike signals with arbitrary velocity v is studied. The total travel time t 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, ∆t v between signals with different velocities but coming from same side of the lens and ∆t p between signals from different sides of the lens, as well as the difference ∆t pv between two ∆t p 's are calculated. These time delays are applied to the gravitational-lensed supernova neutrinos and gravitational waves (GW). It is shown that the ∆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 ∆t pv 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 ∆t pv between GW and GRB can only reach 1.45 × 10 −5 second for very large source distance (2 × 10 4 [Mpc]) and source angle (10 [as]) if v GM = (1 − 3 × 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 ∆t pv is to be used to further constrain the GW velocity.

Constraining the spacetime spin using time delay in stationary axisymmetric spacetimes

“…Here we define t c−v as the deviation of the timelike signal's time delay from that of the null signal. t c−v has a particular advantage in GW/GRB dual lensing: because it is independent of the GW and GRB emission time difference and therefore can be used to constrain the GW speed [31,33].…”

“…Compared with time delay of light signals alone, the difference between time delays of light and neutrinos or light and GW signals can provide stronger constraints on the cosmology parameters [29][30][31]. In addition, the time delay of these signals can determine the properties of test particles like mass ordering of neutrinos and velocity of GW [31][32][33]. Although it is known that neutrinos as well as GWs in some gravitational theories beyond General Relativity have non-zero masses, most of the previous works on their time delay treated them as null signals [31,[34][35][36].…”

Time delay of photons coupled to Weyl tensor in a regular phantom black hole

“…In the present work, in order to obtain a more comprehensive picture of the photons coupled to the Weyl tensor in the regular phantom black hole, we investigate its time delay in both weak and strong deflection gravitational lensing, which is absent in the previous work [88]. These time-domain signals are important for determining the properties of black holes [89][90][91][92][93][94] and probing new physics [95][96][97][98][99][100][101][102].…”

Kerr black hole surrounded by a cloud of strings and its weak gravitational lensing in Rastall gravity

“…In addition, massive particle is an important class of element particles in our universe, and the analysis of the signatures of the gravitational lensing will be useful to understand the properties of these particles. Due to the importance of this problem, various works [60][61][62][63][64][65][66][67][68][69] have studied gravitational deflection of massive particles by different lens objects in differential gravity models. With GW method, Crisnejo et al studied not only the weak gravitational deflection of light in a plasma medium, but also the deflection of massive particles, in SSS/SAS spacetimes [70][71][72].…”