Cluster of Black Holes around a Central Point

Under the circumstances you describe, my immediate reaction is that it would not be possible. The issue here is that the cluster would be fairly unstable. The black holes would all be mutually attracted to each other, and would soon coalesce into one large black hole - taking the Imperial research center with it.

For this to somehow work, the black holes would have to be in some odd stable orbits around each other. They would have a common barycenter - in this case, the Imperial research center - and would continue circling around it. The tricky part here would be to have them orbit in three dimensions - that is, to not simply have all their orbits in the same plane, but to have orbits at odd angles to each other. Also, there would undoubtedly be gaps between the black holes, which would be undesirable.

The whole setup would seem to be rather unstable, and so I'm inclined to say that this would be impossible.

Let's say that the cluster and the station popped into existence one day. I know, it violates so many laws of physics, but I want to delve into the station's hypothetical downfall, so reality will just have to wait. Anyway, so we have a cluster of black holes in stable orbits. What will happen?

Certain systems, such as binary neutron stars, emit radiation in the form of gravitational waves. A system composed of two black holes orbiting each other should do the same. We can calculate the rate at which their orbits will decay with the formula $$\frac{dr}{dt}=-\frac{64}{5}\frac{G^3}{c^5}\frac{(m_1m_2)(m_1+m_2)}{r^3}$$ where $m_1$ and $m_2$ are the masses and $r$ is the distance between the two black holes. If you imagine the black holes each have a mass of about five solar masses, and are separated by about one kilometer, you can easily do the math and figure out the rate at which the orbit will decay.

Alternatively, we can integrate like so: $$\int_{r_0}^r r'^3dr'=-\int_0^t \frac{64}{5}\frac{G^3}{c^5}(m_1m_2)(m_1+m_2)dt'$$ to find the relationship between the distance between the black holes and time, and calculate how long it will take for the two black holes to meet. This gives you the solution $$r(t)=\sqrt[4]{r_0^4-4\frac{64}{5}\frac{G^3}{c^5}(m_1m_2)(m_1+m_2)t}$$ where $r_0=r(0)$ is the initial radius. The time to coalescence is then $$\tau=r_0^4\left(4\frac{64}{5}\frac{G^3}{c^5}(m_1m_2)(m_1+m_2)\right)^{-1}$$

I'm not sure how this would all work with a system of $n$ black holes, but I would think they would still emit gravitational radiation. Perhaps you could use a modified formula (though I don't know for sure).

Finally, the Imperial research station would have to be in the exact center of the cluster. If it was off by just a bit, it would be pulled towards one side and consumed by a black hole. Perhaps the workers could somehow adjust the station's position, but they would have to be careful to keep it in equilibrium.

posted over 9 years ago

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