  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 (SBH) 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. Finally, we check the effect of the
non-commutativity on the shadow radius of the SBH, where we show the similarity
between the non-commutativity and the black hole mass.
