I'm trying to understand something about time dilation on orbiting bodies in General Relativity. I was watching the Matt Parker video General Relativity: Top 05 mishaps and he seems to calculate the time dilation of the GPS satellite as ($r_s$ is the Schwartzchild Radius) :
$t_0 = t_f*\sqrt {1-r_s/r}$
The article in Wikipedia Gravitational Time Dilation under Circular Orbit is:
$t_0 = t_f*\sqrt {1-(3/2)*(r_s/r)}$
My logic says the orbiting satellite is in free-fall, so does not feel the effects of gravity
$t_0 = t_f$
