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I think it is safe to assume that humans are a long way off from having either the technology to compress a normal-sized sphere of uranium below its Schwarzschild radius, or the technology to create a sphere of uranium with a mass comparable to that of a star, so this is deep into idle-curiosity-land, but supposing that we "called into existence" a sphere of uranium whose mass and density was sufficient for both the effects of nuclear fission and gravitational collapse to be relevant,

does the sphere of uranium explode due to nuclear fission, or collapse into a black hole, first?

Of course, perhaps there is a combination of effects - maybe the sphere starts to undergo fission, but forms a black hole before any explosion can take place.

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  • $\begingroup$ The bomb used on Hiroshima (called "Little Boy") was a critical Uranium mass assembly device. It used on order of 64 kg of 80% U-235, so in the most naive interpretation this question is a no-brainer. $\endgroup$ – dmckee Jun 25 '13 at 20:19
  • $\begingroup$ @dmckee: But it didn't create conditions comparable to those where the uranium might have formed a black hole. I feel my question is fairly clear about specifying that I am imagining a situation where gravitational collapse is a relevant effect, in addition to nuclear fission. $\endgroup$ – Zev Chonoles Jun 25 '13 at 20:22
  • $\begingroup$ And you're going to get that much uranium together without it first exploding, how exactly? Whatever initial configuration you envision it will have to more through a phase where you begin to increase the density. Now it is not as easy as saying "as soon as 65 kg have the natural density it goes off" because criticality is notoriously dependent of the details of geometry, but you have to keep compacting it over orders of magnitude, so your done for long before you get any kind of collapse. $\endgroup$ – dmckee Jun 25 '13 at 20:31
  • $\begingroup$ @dmckee: Well, that is why I asked that the sphere be "called into existence", in a completely non-realistic way. But in comparision, I have certainly heard famous thought experiments about, for example, the Sun instantaneously disappearing - this seems like an acceptable fiction to consider. I completely agree, that it seems realistically impossible to get enough fissile material together in any way without it blowing up first. $\endgroup$ – Zev Chonoles Jun 25 '13 at 20:39
  • $\begingroup$ Which either (1) violates the conservation of energy or (2) requires a pre-existing configuration of some kind of energy satisfying the requirement to be a black hole, so it is already is a black hole. $\endgroup$ – dmckee Jun 25 '13 at 20:50
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I think it would be very hard to provide a rigorous answer to your question, but i can think of hand waving arguments suggesting it would do both. The outer portion of the sphere would explode while the centre would form a black hole.

My argument is that the mass of uranium required to form a black hole is around the same as the mass of a star big enough to form a black hole - give or take a few orders of magnitude. If you could magically conjure into existance a star sized lump of uranium then the neutron capture efficiency for all but the outermost layers would be 100% and the uranium nuclei would all fission within a fairly short time.

Suppose we take our sphere of uranium to have the mass of the Sun, about $2 \times 10^{33}$g, then this is $8.5 \times 10^{30}$ moles of uranium 235 or $5.1 \times 10^{54}$ atoms. Fission of $^{235}$U releases $3.2 \times 10^{-11}$J per nucleus, so the total energy released would be $1.6 \times 10^{44}$ joules. This is remarkably close to the energy released by a supernova, which is in the range $10^{44}$ to $10^{46}$J.

So the fissioning of the uranium would produce something very like a supernova, and we know that supernova can form black holes because the centre is compressed by the explosion. Whether this would happen with the uranium sphere is anyone's guess, because of course the details of the explosion would be quite different in the two cases. However I think there is a sporting chance that, like a supernova, you could end up with most of the sphere being blown away but a black hole remnant remaining.

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