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After more than a century since its original formulation by A. Einstein, the theory of General Relativity (GR) has survived all the experimental tests with flying colors. While the so-called ``classical tests of GR'' provided early confirmations of the theory in our Solar System, observational pieces of evidence have grown in number and complexity over the years, providing experimental validations of the predictions from GR in plenty of different astrophysical scenarios [1]. The next major breakthrough in experimental gravitation is expected to come from the discovery of pulsars orbiting a SMBH. Thanks to their astounding intrinsic rotation stability, pulsars are considered among the best tools to probe gravitational fields . These systems, however, probe a comparatively weak gravitational field, with masses of the components of the system (M) on the order of units of solar masses and compactness of the order of one part over a million. However, a pulsar on a tight orbit (with orbital periods below 100 yrs) around a supermassive compact object like Sagittarius A* in our Galactic Center, would probe a totally different regime of gravity (see Figure), and analysis of the times-of-arrival (TOA) of the pulses emitted by such objects (the so-called pulsar timing analysis) would supersede all previous tests of GR in the strong field regime [2].
Figure: The gravitational potential for different astrophysical probes of gravity, as a function of the mass of the gravitational source. Pulsars at the Galactic Center would allow to fill the experimental gap between the S-stars analyses and horizon scales tests of gravity.
An overview on the photon propagation time
The problem of pulsar timing has been historically formulated for binary pulsar systems, in order to develop techniques able to fit the increasing amount of Time-Of-Arrival (TOA) data for such systems found in the local Universe. Our understanding and our ability to model the time delay of double pulsar systems heavily rely on the post-Newtonian approximation [3-7] by which the general relativistic motion of such a system is described with remarkable accuracy. At the post-Newtonian level, one can consider, perturbatively, all sorts of relativistic effects, both on the pulsar motion and on the propagation of light rays, up to any order of expansion, without having to solve the fully relativistic two-body problem. A comprehensive treatment of all possible sources of delay and their expression in the post-Newtonian approximation can be found in the pioneering work by [4], on whose basis all modern timing codes are formulated [8].
For our purpose, we will focus on the first-order post-Newtonian effects that, due to the perturbative nature of the post-Newtonian approximation, represent the dominant contribution to the timing delay. First of all, we need to distinguish the effects that change the position and the time at which photons are emitted by the pulsar with respect to the Newtonian case (e.g. the orbital precession and the Einstein delay), from those that directly alter the photon travel time (the Rømer delay and the Shapiro delay). More specifically, differently from Keplerian orbits, trajectories in GR (and eventually in modified theories of gravity) do not coincide with closed ellipses but suffer from pericenter advance that, on each orbital period, shifts the angular position of the pericenter by an angle
being M the mass of the central SMBH, a the semi-major axis of the pulsar's orbit and e its eccentricity. Clearly, this post-Keplerian effect changes the position in space from where the photon is emitted by the pulsar. Additionally, relativistic effects at first post-Newtonian order can also alter the time of the emission te as perceived by a distant observer. This shift is related to a slow-down of the pulsar's proper time with respect to the coordinate time measured by such an observer, due to a combination of special relativistic and gravitational time dilation [2]. These contributions sum up to the so-called Einstein delay whose amplitude, for a pulsar-SMBH system, is given by
where T is the pulsar's orbital period around the SMBH. This amplitude is modulated along the orbit according to the law
where u is a parameter corresponding to the orbital eccentric anomaly [4]. On the other hand, at first order, one can consider the propagation of photons on a straight line and compute their travel time from the emitter's position re to the observer's position ro as a linear sum of different effect:
Here, the first term represents the classical Rømer delay related to the photon propagation time across the pulsar's orbit and the second one is the Shapiro time delay related to the time dilation experienced by light rays when grazing the region where the central object curves space-time substantially, and are given by:
where
is the unit vector pointing from the emitter to the observer. It is worth noticing that both formulas for the Rømer and the Shapiro delay are computed by assuming that the photons propagate on a straight line. However, differently from the double-pulsar settings, where the mass of the two companions are generally comparable with each other (thus requiring solving a full two-body problem, which doesn't generally have a closed-form solution), in the case of a pulsar orbiting a SMBH the extreme mass ratio allows to effectively treat the pulsar as a test particle in the gravitational field of the massive object. This consideration opens to the possibility of approaching the delay problem with a completely analytical treatment, without resorting to approximations of any sort.
Bibliography
Will C. M., 2014, Living Reviews in Relativity, 17, 4
Liu K., Wex N., Kramer M., Cordes J. M., Lazio T. J. W., 2012, ApJ, 747, 1
Damour T., 1983, Phys. Rev. Lett., 51, 1019
Damour T., Deruelle N., 1986, Ann. Inst. Henri Poincaré Phys. Théor, 44, 263
Damour T., Schafer G., 1988, Nuovo Cimento B Serie, 101B, 127
Damour T., Gibbons G. W., Taylor J. H., 1988a, Phys. Rev. Lett., 61, 1151
Damour T., Gibbons G. W., Taylor J. H., 1988b, Phys. Rev. Lett., 61, 1151
Edwards R. T., Hobbs G. B., Manchester R. N., 2006, MNRAS, 372, 1549
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