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 the incoming facilities (e.g., the Square Kilometre
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., $\lesssim1000$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
$\sim2.6$yr and an eccentricity of $0.3-0.9$ under timing precision of $1-5$ms,
within $\sim 8$yr the spin magnitude and the orientations of the GC MBH can be
constrained with $2\sigma$ error of $10^{-3}-10^{-2}$ and $10^{-1}-10^\circ$,
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 the distance to the Solar System of the MBH can be
constrained about $\sim10\mu$as, $\sim1\mu$as$/$yr, $\sim 10 M_\odot$ and
$\sim1$pc, respectively.
