  The goal of this paper is to investigate the detection by GRAVITY of
different relativistic effects affecting the astrometric and/or spectroscopic
observations of S2 such as the transverse Doppler shift, the gravitational
redshift, the pericenter advance and higher-order general relativistic (GR)
effects, in particular the Lense-Thirring effect due to the angular momentum of
the black hole. We showed that the combination of S2 observations obtained with
the GRAVITY instrument and the spectrograph SINFONI (Spectrograph for INtegral
Field Observations in the Near Infrared) also installed at the VLT (Very Large
Telescope) will lead to the detection of various relativistic effects. Such
detections will be possible with S2 monitorings obtained within a few months or
years, depending on the effect. Strong constraints on the angular momentum of
Sgr~A* (e.g., at $1\sigma~=~0.1$) with the S2 star will be possible with a
simple stellar-orbit model without using a ray-tracing code but with
approximating the gravitational lensing effect. However, long monitorings are
necessary, and we thus must rely on the discovery of closer-in stars near
Sgr~A* if we want to efficiently constrain the black hole parameters with
stellar orbits in a short time, or monitor the flares if they orbit around the
black hole.
