spacetime - A Sphere of Black Holes

Imagine a sphere of black holes surrounding a piece of space. Will this piece be separated from the rest of normal spacetime (at least for some time, till these black holes finally attracted themselves).

So, seen from the outside, we have a black hole, but with a non-singular interior.

Is this possible?

Answer

The radius of the event horizon of a black hole of mass $m$ is given by:

$$r_s = \frac{2GM}{c^2} \tag{1}$$

Let's consider your idea of taking $n$ black holes of mass $M$ and arranging them into a sphere. The total mass is $nM$, and the radius of the event horizon corresponding to this mass is:

$$R_s = n\frac{2GM}{c^2} \tag{2}$$

Now let's see how closely we have to pack our black holes to get them to form a spherical surface with their event horizons overlapping. The cross sectional area of a single black hole is $\pi r_s^2$, and since we have $n$ of them their total cross sectional area is just $n \pi r_s^2$. the surface area of a sphere of radius $R$ is $4\pi R^2$, and we can get a rough idea of $R$ by just setting the areas equal:

$$4\pi R^2 = n \pi r_s^2$$

Giving us:

$$R = \frac{\sqrt{n}}{2}r_s$$

Use equation (1) to substitute for $r_s$ and we find that the radius of our sphere of packed black holes is:

$$R = \frac{\sqrt{n}}{2}\frac{2GM}{c^2} \tag{3}$$

Bit if you compare equations (2) and (3) you find that $R < R_s$ because $\sqrt{n}/2 < n$. That means when you try and construct the sphere of black holes that you imagine you won't be able to do it. An event horizon will form before you can get the individual black holes to overlap. You won't be able to construct the black shell that you want and it's impossible to trap a normal bit of space inside a shell of black holes.

However there is a situation a bit like the one you're thinking about, and it's called the Reissner–Nordström metric. A normal black hole has just the single event horizon, but if you electrically charge the black hole you get a geometry with two event horizons, and inner one and an outer one. When you cross the outer horizon you enter a region of spacetime where time and space are switched round, just as in an uncharged black hole, and you can't resist falling inwards towards the second horizon. However when you cross the second horizon you're back into normal space. You can choose a trajectory that misses the singularity and travels outwards through both horizons so you re-emerge from the black hole. If you're interested I go into this further in my answer to Entering a black hole, jumping into another universe---with questions.

As for what the spacetime inside the second horizon looks like, well it's just spacetime. It's highly curved spacetime, but there's nothing extraordinary about it.