The propagation delay in the timing of a pulsar orbiting a supermassive black hole

Abstract

The observation of a pulsar closely orbiting the galactic center supermassive black hole would open the window for an accurate determination of the black hole parameters and for new tests of General Relativity. An important relativistic effect which has to be taken into account in the timing model is the propagation delay of the pulses in the gravitational field of the black hole. Due to the extreme mass ratio of the pulsar and the supermassive back hole we use the test particle limit to derive an exact analytical formula for the propagation delay in a Schwarzschild spacetime. We then compare this result to the propagation delays derived in the usually employed post-Newtonian approximation, in particular to the Shapiro delay up to the second post-Newtonian order. For edge-on orbits we also consider modifications of the Shapiro delay which take the lensing effects into account. Our results are then used to assess the accuracy of the different orders of the post-Newtonian approximation of the propagation delay. This comparison indicates that for (nearly) edge-on orbits the new exact delay formula should be used.

Relativistic propagation and frame dragging time delay in the timing of a pulsar orbiting the supermassive black hole SgrA*

“…Firstly, we test our main formula (24) by choosing 𝑎 = 10 −10 . For this value of the rotation 𝑎 the frame dragging effects are negligible, and we were able to reproduce the results from Hackmann & Dhani (2019). For illustrative purposes, and to lay a ground for the discussions below, we show a single result in figure 3.…”

“…Note that we could as well have chosen another invariant characteristic of the pulsar orbit. However, already in Hackmann & Dhani (2019) it was pointed out that the circumference of a circle works well also in the comparison to post-Newtonian approaches, in contrast to other choices as, say, an identification of the proper orbital period.…”

“…In this section, we compare our derived exact formula for the propagation time delay in a Kerr spacetime with both the formula derived for Schwarzschild spacetime in Hackmann & Dhani (2019) and the post-Newtonian approximations reviewed in section 4.…”

“…In this approximation the gravitational field of the pulsar is neglected, as compared to that of the black hole, and we may treat the motion of the pulsar and its radiation as geodesics in the gravitational field of the spinning black hole. In this approach, Hackmann & Dhani (2019) investigated the Shapiro delay as the dominant nontrivial relativistic effect on the radio pulses within the setting of a Schwarzschild spacetime that represents the non-spinning black hole case. In the present work, we generalize this previous treatment to include in addition to the relativistic Shapiro delay also the delay induced by the rotation, the frame dragging time delay, for the case of a black hole characterized by an additional parameter, the spin, within the frame of the Kerr spacetime.…”

Observables from spherically symmetric modified dispersion relations

“…The Shapiro delay is the difference between the time of flight of a light-signal in the presence and the absence of a gravitating central object. The comparison of the derived Shapiro delay with the experimental measurements leads to test of the gravitational field in the weak [62], for example with lunar laser ranging [63] as well as the strong gravity regime with pulsars orbiting black holes [64].…”