Conceptually, what is the difference between these two frame-dragging equations, describing the rate of angular velocity of the space around a symmetrically spherical mass such as a black hole and why would you use one over the other?
1: $\boldsymbol{\Omega} = \frac{r_s\alpha c}{r^3 + \alpha^2 (r + r_s)}$
where $\alpha$ is related to the angular momentum: $\alpha = \frac{J}{Mc}$
$r$ is the position vector above the rotating mass and $r_s$ is the Schwarzschild radius: $r_s = \frac{2GM}{c^2}$
2: $\boldsymbol{\Omega} = \frac{GI}{c^2r^3}\left( \frac{3(\boldsymbol{\omega}\cdot \boldsymbol{r})\boldsymbol{r}}{r^2}-\boldsymbol{\omega} \right)$
where $r$ is the position vector above the rotating mass, $I$ is the moment of inertia of the rotating mass, and $ω$ is the angular velocity of the rotating mass.
