Lyapunov exponents and geodesic stability of Schwarzschild black hole in the non-commutative gauge theory of gravity

In this paper, we study the stability of geodesic motion for both massive and massless particles using Lyapunov exponents in the non-commutative (NC) Schwarzschild black hole (BH) via the gauge theory of gravity. As a first step, we investigate the both time-like and null radial motion of particles, the mean result in NC geometry shows that the particles take infinity proper time to reach the NC singularity (infinite time affine parameter framework for photons). The proper/coordinate time of Lyapunov exponents and their ratio of time-like geodesic for the circular motion of this black hole shows a new behavior, which describes a new range of stable circular orbits between unstable ones. Then we analyze the circular motion of photons, where the result shows a new photon sphere near the event horizon which is not allowed in the commutative case, and the Lyapunov exponent is expressed in this geometry, where this confirms the instability of the outer photon sphere and the stability of the inner one. Moreover, we studied the effect of noncommutativity on the black hole shadow radius, We found a similarity between the non-commutativity and the mass of a black hole. Then we using experimental data from the event horizon telescope, we show that a noncommutativity parameter of the order of $\Theta^{\text{Phy}}\sim 10^{-32}m$.