The Man Behind The Curtain

In Relativity by Brian Koberlein2 Comments

Black Holes are where God divided by zero, so the saying goes. As short sayings go, that’s not a bad description of a black hole’s singularity, and it gives one a good idea why singularities are so problematic in physics.

In my last post I wrote about the cosmic censorship conjecture, and how it might be violated in hypothetical 5-dimensional black holes. I didn’t delve too deeply into the conjecture itself because there are actually multiple versions of the conjecture. The weak cosmic censorship conjecture is basically as I stated, that a singularity must be enclosed by an event horizon. It’s a bit more subtle than that, since an event horizon isn’t a local object in space, but rather a global structure in spacetime. So the formal definition is that a singularity can’t be seen by an observer sitting far away from a black hole (what is called null infinity). If you were close to a black hole you might catch a glimpse of a singularity, but the structure of spacetime is such that you couldn’t tell someone far from a black hole what you saw.

The upshot of all this is that anybody reasonably distant from a black hole could never see a singularity or interact with it. Since it can’t effect things outside a black hole, we don’t have to care about how it would affect things. As long as Pandora’s box stays closed, there’s no need to worry. As I mentioned in the earlier post, there are theoretical examples in general relativity where a singularity can be seen by a distant observer, but none of them seem likely to occur in a real situation.

The strong cosmic censorship conjecture takes a different approach. This is where the “dividing by zero” idea comes in. If you were to divide a number by zero, you might think the answer is infinity. After all, zero can go into a number like 1 an infinite number of times. But the actual answer is undefined. The formal reason has to do with the subtleties of mathematics, but for our purposes suppose we first divided by a very small number. For example, if we divide 1 by 0.01, the answer would be 100. If we divided 1 by 0.0001 we get 10,000. If we kept dividing by an ever smaller number, our answer would get bigger and bigger. This seems to say that 1/0 is infinity, but suppose instead we divided 1 by -0.01. In that case the answer would be -100. Dividing by -0.0001 we get -10,000. That would make 1/0 negative infinity. Starting with a small negative number or a small positive number gets us to the same 1/0, so is the answer positive or negative? The ratio 1/0 is meaningless without knowing how we approached zero.

A singularity is similar to this in that it is indeterministic. If all you have is a singularity, you have no idea how it became a singularity. Likewise if a singularity were to interact with other objects in the universe, the outcome would be unpredictable. So the strong cosmic censorship conjecture proposes that general relativity must be deterministic. As a result, singularities must be excluded from interaction with the rest of the universe.

It turns out there are solutions to Einstein’s field equations that satisfy the weak conjecture but not the strong conjecture, and vice versa. By itself general relativity is not bound by either censorship conjecture. But general relativity is also perfectly fine with warp drive and time travel as well, and they don’t seem to be physically possible for reasons beyond relativity. So it’s likely that some physical process prevents these kinds of weird singularities from occurring.

Then again, we’ve been wrong before.


  1. RichK: in General Relativity, a black hole has precisely three observable properties (potentially, externally), its mass, its ‘rotation’, and its electrical charge.

    Perhaps, one day, when there is a good theory of quantum gravity, it will be possible to show that a black hole has more observable properties …

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