Extremal Black Holes (Script) (Patreon)

Black hole singularities break physics - fortunately, the universe seems to conspire to protect itself from their causality-destroying madness. At least, so says the cosmic censorship hypothesis. Only problem is many physicists think it might be wrong, and that naked singularities may exist after all.

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Strange things happen inside black holes. Space and time switch roles, pathways open up to other universes, and in some cases time travel becomes possible. Causality breaks down, and so the supreme sensibleness of physics is badly threatened. And surrounding all of this weirdness is the event horizon: the surface around the central singularity where the inward flow of space reaches the speed of light, and time freezes from the perspective of the outside universe. In the most real possible sense, the interior of the black hole is its own separate spacetime, excised from our universe. 

The event horizon sounds dire, but it’s a blessing. Without it, the infinitely dense singularity and its surrounding madness is exposed to the outside universe. We would have what we call a naked singularity. This would wreak havoc on our understanding of causality, and our precious laws of physics would become unhinged. Given the dire consequences, Roger Penrose proposed the Cosmic Censorship Hypothesis - which basically states that every gravitational singularity must be surrounded by an event horizon. In an upcoming episode we’re going to look deeper at why many physicists believe the cosmic censorship hypothesis must be true - even without knowing exactly WHY it’s true. And we’ll see why other physicists have begun to doubt it’s validity.

But today we’re going to look at the astrophysics and see how exactly a naked singularity might be formed in the first place, and how the universe seems to work very hard to protect itself from them. So let’s talk about how to dissolve an event horizon, although I would ask you to please not try this at home, for the sake of all of physics.

According to the so-called no-hair theorem, black holes can have only three properties - mass, electric charge, and spin. Mass is what makes a black hole a black hole, and so the simplest black holes have only this property. These are Schwarzschild black holes, and with only mass that means they also only have inward-pulling gravity. With nothing to counter the inward flow of space, an event horizon is inevitable. In recent episodes we also explored the rotating, or Kerr black hole. Inside, we found that the rapid rotation has spun the point-like singularity into a ring of infinite density.  The time machine is on the other side of that ring, by the way.

The same rotation that f orms the ring also drags the fabric of space into a vortex which counters the inward pull due to the singularity’s mass. The result is that we get this region around the singularity where the faster-than-light inward flow of space is halted. It’s like the eye of the storm, and it’s separated from the surrounding cascade by the inner event horizon. As the spin of a Kerr black hole increases, the spacetime waterfall is beaten back, and so the inner horizon grows. At a certain very high rotation rate, the inner and outer horizons merge - which means they both vanish. There’s no longer a region where the inward flow of space exceeds light speed. Instead, there’s a smooth run of normal - albeit rapidly spinning space all the way down to the singularity ring. The NAKED singularity ring.

There’s a similar situation with the charged black hole - which, absent rotation, is described by the Reisner-Nordstrom metric. The presence of electric charge at the central singularity - which point-like in this case - results in a negative pressure that again resists the inward pull of gravity. Reisner-Nordstrom black holes also have an inner horizon, interior to which space and time seem normal-ish. The more electric charge you drop into a black hole, the larger its inner horizon becomes. And just as with the rotating black hole, at some point the inner and outer horizons become one and vanish and you’re left with a naked singularity.

At the tipping point - when the inner and outer horizons are right next to each other - you have what we call an extremal black hole - a black hole with the maximum amount of spin or charge while still having an event horizon. In both cases, the amount of angular momentum or charge you can fit into a black hole before it becomes extremal depends on the mass. More mass means more inward gravity, and so the black hole can hold more spin and charge before going extremal.

So that’s how you make an extremal black hole - or even a naked singularity. Just add enough spin or charge to an existing black hole.

And actually there’s another way to do it in the case of the charged black hole. Normal black holes leak their mass away by emitting Hawking radiation. That radiation cian be any type of elementary particle - but in the case of the most massive black holes, it’s mostly just photons. That’s because the more massive the black hole the lower the temperature of the radiation. In very massive black holes the Hawking radiation has trouble mustering the energy for anything but weak photons. So that means a massive CHARGED black hole will slowly leak away its mass while retaining its charge. The outer event horizon will shrink until it meets the inner horizon, and again you have an extremal black hole. This can’t happen with rotating black holes because they leak away their angular momentum as well as their mass.

Once you have an extremal black hole by whatever method, it lasts for a very, very long time - if not forever. Hawking radiation is a direct result of there being an event horizon. So naked singularities don’t Hawking-radiate, and extremal black holes radiate only very slowly. That leaves us with a strange situation - in the far distant future, even if all particles in the universe decay, we may be left with only radiation and these naked, charged singularities.  So, there we have a few recipes for theoretical disaster. At first glance, it appears that extremal black holes are certainly possible. So why shouldn’t it be possible to add a little more spin or charge to produce true naked singularities?

The cosmic censorship hypothesis tells us that something will always stop a gravitational singularity being stripped of its event horizon - but it doesn’t tell us the physical mechanism. Let’s look at what we know.

Rotating black holes gain their angular momentum from things they swallow. Stuff doesn’t normally fall straight into a black hole - it spirals in as its orbit decays. It’s that orbital angular momentum that is fed to the black hole. But in order for an object orbiting a black hole to fall into it, it actually has to lose at least some of that angular momentum. Otherwise it would just keep orbiting forever. For example, if there’s a disk of gas surrounding the black hole like in a quasar, then the gas only spirals inwards because angular momentum is carried outwards by friction. By the time the gas reaches the black hole it has lost much of the angular momentum it started with.

The faster a black hole is rotating, the more angular momentum that gas has to lose in order to fall in. That’s because space gets dragged around the rotating black hole, giving the gas a sort of boost so it can still orbit even with very little of its own angular momentum. In the case of an extremal Kerr black hole - one that’s rotating nearly fast enough to lose its event horizon - the gas near the event horizon orbits entirely riding on the carousel of frame-dragged state, and has no angular momentum of its own. Therefore just on the verge of becoming extremal, the black hole can’t gain spin from accretion anymore. More generally, there is no trajectory into an extremal black hole that can add angular momentum from the trajectory or the “orbit” itself.

An alternative is to throw in something that’s actually spinning itself - so it has intrinsic angular momentum. It’s not yet clear whether we can avoid the naked singularity in this case. At any rate, our observations of gravitational waves from colliding black holes and various other methods for estimate black hole spin has not yet reveals a single black hole with a spin high enough to be super-extremal.

For charged black holes the situation is in some ways easier, but has its own weirdness. We don’t actually expect real, astrophysical black holes to retain any significant change. A charged black hole in the vicinity of any matter would repel like charges and attract and swallow opposite charges, and so quickly neutralize. But imagine we create a charged black hole and isolate it from all other matter. Then surely we can just throw charged particles into the black hole. We have to be careful, because those particles increase the mass of the black hole as well as the charge - and if the mass increases too much it won’t go extremal. But electrons have very tiny masses for comparatively large charge - just factoring the electrons mass, it should be easy to send a black hole over the extremal limit by feeding it a stream of electrons.

But here we get to something that seems like a bit too neat to be a coincidence. As Einstein taught us, mass and energy are equivalent. And there’s an enormous amount of energy in the electric field of all those electrons that you smooshed together into the black hole. In fact the field itself will always generate enough mass to prevent the black hole from losing its event horizon.

Once again, the universe appears to have a mechanism to avoid the naked singularity. But there still isn’t an underlying understanding of why - or even if - cosmic censorship must be maintained. In fact, if you define the cosmic censorship hypothesis in more technical terms, it seems that physics should allow its violation. And that would be a big problem for physics. To really understand that problem, and the implications of the cosmic censorship hypothesis being true or not, you’ll have to wait for an upcoming deeper dive to witness the horrors of the cosmicly uncensored spacetime.