Particle acceleration for a stationary observer in Schwartzschild spacetime

We explore the acceleration of a massive particle in free fall in a stationary observer' s proper reference frame in the gravitational field of a spherical body. Comparisons of the results are made between Newtonian theory and General Relativity. It is seen that the acceleration has a velocity dependence in GR which only approaches the Newtonian value as the particle speed approaches 0 (and the particle is well outside the gravitational\ radius of the gravitating body). The GR result shows a critical vertical velocity at c/$\sqrt{2}$ where the vertical acceleration goes to 0. The GR result is found to be $a_x=\frac{2 M v_x v_z}{r ^2\sqrt{1-\frac{2 M}{r}}},\ a_y=\frac {2 M v_y v_z}{r ^2\sqrt{1-\frac{2 M}{r}}},\ a_z=-\frac{M \left (1-2 v_z^2\right)}{r ^2\sqrt{1-\frac{2 M}{r}}}$ which compares with the Newtonian result of $a_x=0,\ a_y=0,\ a_z=-\frac{M }{r ^2}$ (in units where c=1, G=1).