Probing the Spinning of the Massive Black Hole in the Galactic Center via Pulsar Timing: A Full Relativistic Treatment

Pulsars around the massive black hole (MBH) in the Galactic center (GC) are expected to be revealed by upcoming facilities (e.g., the Square Kilometer Array). Under a full relativistic framework with the pulsar approximated as a test particle, we investigate the constraints on the spinning of the MBH by monitoring the timing of surrounding pulsars. For GC pulsars orbiting closely around the MBH (e.g., less than or similar to 1000 au), we find that full relativistic treatment in modeling accurately their timing signals can be necessary, as the relativistic signals are orders of magnitude larger than the time-of-arrival measurement accuracies. Although usually there are near degeneracies among MBH spin parameters, the constraints on the spinning of the MBH are still very tight. By continuously monitoring a normal pulsar in orbits with a period of similar to 2.6 yr and an eccentricity of 0.3-0.9 under a timing precision of 1-5 ms, within similar to 8 yr the spin magnitude and the orientations of the GC MBH can be constrained with 2 sigma error of 10(-3)-10(-2)and 10(-1)-10(0), respectively. Even for pulsars in orbits similar to the detected star S2/S0-2 or S0-102, we find that the spinning of the MBH can still be constrained within 4-8 yr, with the most significant constraints provided near the pericenter passage. If the proper motion of the pulsars with astrometric accuracy of 10 mu as can also be collected along with the timing measurement, then the position, velocity, mass, and distance to the solar system of the MBH can be constrained to similar to 10 mu as, similar to 1 mu as yr(-1), similar to 10 M-circle dot, and similar to 1 pc, respectively.