During the investigation grade work titled Study of geodesics in the spacetime of black holes type Sagittarius A, it was determined the trajectories of motion known as \textit{geodesics} around a system composed by two supermassive black holes (SMB) whose physical properties are the same of the black hole Sagittarius A located at the center of the Milky Way; for this investigation, the Roy Kerr's solution to Einstein's field equations was used, and it was decomposed as a linear superposition of the Schwarzschild metric tensor $g_{schw}$ and a perturbation tensor $g_{pert}$; the superposition of these two tensors was called general metric of the system $g_{\mu\nu}$; as the general metric of the system is a perturbation model instead of an exact solution of the Einstein field equations, the spacetime intervals were determined for which the general metric of the system is a solution that correctly describes the topology of spacetime.\

Different simulations were created to analyze the behavior of the Ricci tensor associated to $g_{\mu\nu}$, the numerical value of the Ricci tensor oscillated around $10^{-6}m^{-2}$ and $10^{-31}m^{-2}$ converging to a Minkowski spacetime; at the same time, the scalar curvature of Ricci was computed obtaining a value of $-1.97\cdot 10^{-34}$ whose negative value confirms that any general manifold higher to 3 dimensions admits a negative scalar curvature, based on the general metric of the system, the postulates of general theory of relativity were employed to determine the differential equations that govern the geodesic curves, the symmetries of the general metric tensor allowed to determine conserved quantities as the energy $E$, and linearize constants of motion as the angular momentum $L$ that reduced the structure of the differential equations to models that were solved by numerical methods as the finite difference method or analytically applying approximations to the same differential equations

The solution obtained of the differential equations describes the orbit of a supermassive star B0-2 V, known as S2; the trajectory $r(\phi)$ is composed by the superposition of 3 orders of approximation, $r^{(0)}(\phi)$ is a function that describes an elliptical orbit that converges to the solution obtained by Newton’s equations in his theory of planetary motion, from this solution the angular momentum can be parameterized around black holes in terms of the semimajor axis and the eccentricity of the ellipse; the first order of approximation $ r^{(1)}(\phi)$ is a solution which describes the precession of elliptical orbits, in this case, it is determined a precession of $75$ $ arcsec/year$; finally, $ r^{(2)}(\phi)$ allowed us to determine the average energy mass per orbit that the S2 star would have employing an approximation to the gravitational energy constructed from the symmetries of the solution, and whose value oscillates around $-2.9 \cdot 10^{13} $$m^2/s^2$.

General Theory of Relativity