ergosphere of black holes is considered an indirect rotational quality, its just the effect of the massive rotation speed morphing the area of space outside the event horizon.
(2) If they rotate in opposite directions, do the ergospheres add up, shooting the particle outwards?
(1) If they rotate in the same direction, do the ergospheres cancel each other out, not moving the particle at all?
Until the particle is not under any influence of a
ergophere rotation, the objects pulled into a black hole is due to its
gravitational pull (which extends way further than the ergosphere).
From what I suspect, the particle will stay in the middle for both situations until the above statement is true,
assuming the particle can sustain pull forces from both the black holes combined (realistically not possible), the black hole have the same mass and are equidistant from the particles (as you have mentioned) and they have the same gravitational pull.
Now, once the black the particle enters the ergospheres of the black holes (assuming the both the black hole move at the exact same speed and the particle enters the ergospheres at the same time):
In condition (1):
The particle will stay constant / still. This is because, referencing the image above the opposite rotational forces of black holes
B on the particle, will cancel out and with the balanced gravitational forces, the particle simply stays still.
In condition (2):
The object will be ejected out in the direction relative to the added rotational direction of the black holes. Each force has two components; horizontal and vertical,
Fx and Fy as seen in the image above. Due to the direction of
x component of the rotations of the black holes, they will
cancel out, leaving only the
y. Since the
y component is in the same direction for both black holes and at a equal magnitude (quantity), its only obvious that the particle should move straight in the
y and relative to the rotation direction of the black holes (downwards in the image) and it will eventually be ejected.
Other possibilities include:
The particles rotates around the two black holes, following the path above
Experimental / Real Life:
To stay true to reality, the above theory will not work due to a simple reason: When the black holes come near enough one another, the will
orbit each other, which gradually
decays, leading them to
collide and become a bigger black hole, which in-turn leads to two scenarios:
either the force and factors affecting the particle cause it to be
ejected out into space,
your particle gets
sucked into the black hole :(.
So why did I do the geometrical theory above?
- Its always fun to play around with geometry and theory and what it would be like excluding any other variables and problems.
- If I just said the black hole ate up the particle, does not really seem to explain a lot + plus its cooler to explain geometry.
Hope this helps :) !!