Probing a Finslerian Schwarzschild black hole with the orbital precession of Sagittarius A*

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Orbits of Finslerian Schwarzschild black hole have been investigated in this paper. The Finslerian Schwarzschild black hole is a warp product spacetime where its two dimensional subspace possesses constant Finslerian curvature. Due to this special geometrical structure, four constants of motion have been obtained from geodesic equations of Finslerian Schwarzschild spacetime. Finslerian parameter $\ensuremath{\epsilon}$ which describes deviation between Finslerian Schwarzschild black hole and Schwarzschild black hole affects both orbital precession and orbital plane precession. Orbit of Finslerian Schwarzschild black hole will ergodically fill a toruslike region. It is shown that the orbital precession of Finslerian Schwarzschild black hole is a constant if we calculate the two nearby perihelion of orbit by arc length of two dimensional subspace of Finslerian Schwarzschild black hole. However, present astronomical observations calculate the two nearby perihelion of orbit by arc length of Riemannian 2-sphere. Under this viewpoint, the orbital precession of Finslerian Schwarzschild black hole will depend on the Finslerian parameter and orbital elements. Observations of orbits of several stars around Sagittarius A* by GRAVITY collaboration provide an approach to falsify Finslerian Schwarzschild black hole. We have used data of orbital precession of the star S2 as an anchor to constrain the Finslerian parameter. The value of $\ensuremath{\epsilon}$ is range from $\ensuremath{-}1.42\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$ to $1.66\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$. Then, we used the constrained $\ensuremath{\epsilon}$ to give prediction of orbital precession of other stars around Sagittarius A*. Due to the parity symmetry violation, it is shown in numerical results that both the direction of motion and the sign of $\ensuremath{\epsilon}$ will affect the orbital precession.