  We develop the kinematics in Matrix Gravity, which is a modified theory of
gravity obtained by a non-commutative deformation of General Relativity. In
this model the usual interpretation of gravity as Riemannian geometry is
replaced by a new kind of geometry, which is equivalent to a collection of
Finsler geometries with several Finsler metrics depending both on the position
and on the velocity. As a result the Riemannian geodesic flow is replaced by a
collection of Finsler flows. This naturally leads to a model in which a
particle is described by several mass parameters. If these mass parameters are
different then the equivalence principle is violated. In the non-relativistic
limit this also leads to corrections to the Newton's gravitational potential.
We find the first and second order corrections to the usual Riemannian geodesic
flow and evaluate the anomalous nongeodesic acceleration in a particular case
of static spherically symmetric background.
