Are rotating black holes producing a frame dragging effect inside the event horizon? Are rotating black holes producing a frame dragging effect inside the event horizon? Is that effect moving space inside the event horizon at speeds far greater than the speed of light?
 A: 
Krešimir Bradvica asked: "Are rotating black holes producing a frame dragging effect inside the event horizon? Is that effect moving space inside the event horizon at speeds far greater than the speed of light?"

Yes, for example here is an equatorial orbit of the third kind inside the inner horizon at $\rm r=0.5 \ G M/c^2$ inside a black hole with spin $\rm a=0.9 \ G M^2/c$ where the frame dragging velocity is $\rm 14.6969 \ c$ in the prograde, i.e. counterclockwise direction.
The small dot followed by the dashed trail is the ZAMO moving with the frame dragging velocity, while the red dot followed by the solid trail is the free falling test particle in orbit, who orbits with $\rm v=0.693 \ c$ in the retrograde, i.e clockwise direction (relative to a local ZAMO).
The coordinates are Boyer Lindquist in the cartesian projection, but since the $\rm r$ of the trajectory is constant the trajectory also looks the same in Doran Raindrop and Kerr Schild coordinates:

A photon or observer that locally orbits in the clockwise direction relative to a ZAMO (so the orbit has negative axial angular momentum, in this example $\rm L_z=-1.9933 \ G M m/c$) still moves counterclockwise relative to the fixed stars and sees the sky rotate accordingly since the frame dragging velocity there is much higher than the speed of light (the frame dragging velocity gets smaller than $\rm c$ inside the inner ergosphere, which is the central red region, again).

Krešimir Bradvica asked: "Is that effect moving space inside the event horizon at speeds far greater than the speed of light?"

That effects becomes larger than $\rm c$ already inside the outer ergosphere (which is the pumpkin shaped region around the ellipsoid horizon) though, so that also happens outside of the horizon. Inside the horizon the radial infall velocity gets larger than $\rm c$, but the transverse frame dragging velocity goes to $\rm c$ much sooner.
The frame dragging velocity relative to a static observer (at rest with respect to the fixed stars) is
$${\rm v_{fd}}=\sqrt{g_{\rm t \phi} \ g^{\rm t \phi}}=\sqrt{1-g_{\rm t t} \ g^{\rm t t}}=\omega \sqrt{-g_{\phi \phi} \ g^{\rm t t}}$$
(in natural units where $\text{G=M=c=1}$), with the delayed angular velocity of the frame dragging
$$\omega={\rm d \phi/d t} = -g_{\rm t \phi}/g_{\phi \phi}$$
so the plot for the frame dragging velocity in the equatorial plane of our BH with $\rm a=0.9$ is

where our ZAMO is at the second vertical gridline at $\rm r=0.5 \to R=1.02956$ where the horizontal gridline is at the aforementioned $14.6969 \ \rm c$. Between the third and forth vertical gridline (the region inside the outer, but outside the inner horizon) the $\rm v_{fd}$ is imaginary since no ZAMO can stay at a fixed radial distance there. Between the ring at $\rm 0 \leq R<a$ in the equatorial plane, $\rm v_{fd}=0$. The plot for $\omega$ is

The vertical gridlines are

*

*the ring singularity and inner ergosphere at $\rm R = \sqrt{0^2+a^2} = 0.9$

*the position of our test particle and ZAMO at $\rm R = \sqrt{0.5^2+a^2} = 1.02956$

*the inner cauchy horizon at $\rm R = \sqrt{{r_{H}^{-}}^2+a^2} = 1.06218$

*the outer event horizon at $\rm R = \sqrt{{r_{H}^{+}}^2+a^2} = 1.69463$

*the outer ergosphere at $\rm R = \sqrt{{r_{E}^{+}}^2+a^2} = 2.19317$
