Stellar rotation of S301 as a macroscopic gyroscope to test general relativity

Stellar trajectories around the Galactic Center provide a testing environment for general relativity. The intrinsic rotation of these stars evolves under covariant transport in curved spacetime and classical Newtonian quadrupole torques. We analyze the recently observed S301 S-star to quantify the relativistic precession of its rotational axis. Its 8.7-year period and eccentricity of $e=0.982$ localize geodetic precession and Newtonian quadrupole torques to a step function at periapsis. We incorporate first-order post-Newtonian corrections into the orbital kinematics to calculate the spatial trajectory. Sampling an isotropic distribution of initial orientations and viewing geometries over a 40-year period across a grid of equatorial velocities and rotational ellipticities, we calculate the statistical likelihood of an absolute shift in the projected rotational line broadening, $\mathrm{\mid }\mathrm{\Delta }v\sin i\mathrm{\mid }$. The relativistic geodetic shift scales linearly with $v_{\mathrm{r}\mathrm{o}\mathrm{t}}$ and the classical quadrupole shift is independent of rotation speed, scaling with $q$. The absolute maximum velocity shift saturates at $46.1\text{kms}$ for oblate stars. The absolute median shifts, driven by geodetic precession, range from $3\text{kms}$ to $6.3\text{kms}$. We calculate the time-domain observable $\mathrm{\mid }\mathrm{\Delta }v\sin i\mathrm{\mid }$ to provide a target for infrared spectrographs testing the Schwarzschild metric around Sgr~A${}^{\ast }$. The spin of S301 acts as a flying gyroscope whose drift, if measured, can test Einstein’s theory in a regime that has not previously been accessible.