# A few questions related to frame dragging

I am trying to get my head around a few concepts related to frame dragging and related physics. In regards to black holes that have no charge and all their mass is tied up in rotational kinetic energy then the ergosphere is a maximum size. By definition, as I see it defined, the outer boundary of the ergosphere is the point at which a photon becomes stationary with respect to outside observation if it is orbiting retrograde. The radius at the equator of this maximal case is $2GM$ making it twice the inner event horizon radius if I am understanding correctly. The innermost stable orbit for massive particles is at GM in a pro-grade direction. If any of this is incorrect please let me know.

So my questions are:

1. Pretty much by definition it is impossible for a massive particle to go slower than the speed of light relative to outside observers if it is within the ergosphere - true? What is the limit of this speed relative to outside observers? I'm looking for the limit before it passes into the inner horizon.

2. Since the frame dragging extends outward (presumably to infinity with extremely negligible effect) what would happen in the case where you had a ring of orbiting black holes equally spaced with spin aligned such that the axis of rotation was tangential to the circle? Without touching event horizons it would appear to be able to pull massive particles through the center of the circle in a straight line rather than a curved path as above - correct? And if the holes were close enough would that allow the ergospheres to merge (temporarily as I'd assume everything would merge eventually) would that allow faster than light travel for massive particles in a straight line? I'm neglecting that the orbits of the holes are unstable and would radiate gravitational waves and that they are massive enough so tidal forces are 'small' compared to the particle or mass of particles.

3. If the holes above are arranged as stated would this extend the ergosphere inward toward the center if they are closely spaced at some distance? I've seen cases for general non-maximal spin cases of two holes and am unable to determine how this would play out.

Thanks for taking the time to read and if I put too much in a single question I apologize but didn't want to spam several closely related questions.

• Pretty much by definition it is impossible for a massive particle to go slower than the speed of light relative to outside observers if it is within the ergosphere - true? No, GR doesn't have a well-defined notion of the speed of an object relative to a distant object. SR's prohibition on superluminal speeds is purely local.
– user4552
Commented Sep 10, 2013 at 15:14