Black Hole Bombs and Spinning Space Time (script) (Patreon)

If there’s one thing cooler than a black hole it’s a rotating black hole. Why? Because we can use them as futuristic power generators, galactic-scale bombs, and portals to other universes.

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Two months after Einstein presented his complete general theory of relativity in 1915, a young german physicist named Karl Schwarzschild was the first to fully solve its equations for a realistic situation. Schwarzschild’s eponymous metric describes the warping of space and time around a spherically symmetric mass. And if that mass is sufficiently compressed, the metric predicts an event horizon - a spherical surface where the fabric of space cascades downwards at the speed of light, and where the flow of time halts from the perspective out the outside universe. It predicts the inescapable region of space that we now call the black hole.

Our observations of the universe have since told us that black holes are very real. We’ve seen the gravitational waves caused by their mergers, we’ve witnessed the havoc they wreak on their surroundings in distant quasars and in our own galaxy, and we’ve even taken an image of a black hole with the event horizon telescope image. But none of these real black holes are particularly well described by Schwarzschild’s solution. That’s because the Schwarzschild black hole has no rotation. But all real, astrophysical black holes are spinning.

According to the no-hair theorem, black holes can have three and only three properties: mass, electric charge, and spin. 

Every black hole must have mass. Compacting a lot of mass into a tiny region is what makes them black holes in the first place. Essentially no black holes have electric charge - if somehow one does acquire charge it would quickly lose it because it would repel like charges and attract opposite charges. But essentially all real black holes will be rotating. The spin of a black hole comes from the combined angular momentum of everything that went into forming it. That includes the rotation of the star’s core that collapsed into the black hole in the first place, and the rotation of any latter object sucked through the vent horizon. Even non-rotating objects will affect black hole spin if they fall in at some angle. Some black holes might not have MUCH spin - because angular momentum can cancel out if objects fall in with different spins or in different directions. But some rotation is always expected.

Despite the importance of spin in black holes, it took nearly half a century before Einstein’s equations were solved for the rotating case. That was by Roy Kerr in 1963, yielding the Kerr metric and describing the Kerr black hole, which has mass and rotation but no charge. So why did it take so long? Well, general relativity is hard, and the spherical symmetry in Schwarzschild’s solution eliminated a lot of complexity. Without that simplification the algebra was diabolical. But the final Kerr metric doesn’t look SO bad. I guess. It can be used to calculate the path of any body moving near or even within a Kerr black hole - or indeed any rotating mass.

Today we’re going to look at what the Kerr solution can tell us about the spacetime outside a rotating black hole. We’ll save the even weirder details of the Kerr black hole’s interior for another episode. For a preview check out our episode on time machines. Yup, the math says you can visit your past within a Kerr black hole. For the bit about visiting other universes you’ll just have to wait.

Let’s start by talking about what is actually rotating in a Kerr black hole. It’s tempting to just say that some physical thing deep beneath the event horizon is rotating. But if nothing can escape the event horizon, how can that internal rotation influence the outside? In fact, how can any effect of gravity extend from beneath the event horizon? In fact it sort of doesn’t. Both the gravitational field and its rotation can be thought of as properties of the spacetime itself.

Black holes are self-sustaining holes in the fabric of spacetime. Space at the event horizon cascades downwards, dragging more space behind it, sort of like how water drags itself near the edge of a waterfall. In a Kerr black hole, space above the event horizon is dragged around in a circle - so less waterfall and more whirlpool. Water spiraling down a drain in a flat sink doesn’t know about the hole - it only knows about the motion of the water around it. In fact it’s possible to construct a black hole in general relativity rotating or otherwise - without any mass. Warp spacetime so it looks like the exterior of a black hole, and that warping will persist. So what is rotating? Spacetime is rotating.

The flow of space around a rotating black hole is known as frame-dragging. We see it around any rotating mass. In frame dragging, any “freefall” trajectory - the path taken by an object moving freely in the gravitational field - is dragged in the direction of the object’s spin. Gravity Probe B measured the Earth’s frame dragging and it was exactly as Einstein’s theory predicted - incredibly weak in Earth’s case. But in the case of a Kerr black hole, this circular flow of spacetime changes everything.

Let’s approach our Kerr black hole. Contrary to the common misconception, long as you don’t get too close to the event horizon it’s possible to orbit a black hole in a perfectly stable way - For a non-rotating black hole you can execute a stable circular orbit as close as 3 times the radius of the event horizon - or 3 Schwarzschild radii. Any closer and no stable orbits exist - unless you’re firing your rockets like crazy, you must spiral either inwards or outwards. But for a rotating black hole, frame dragging gives a you little extra kick, and so stable orbits exist much closer to the event horizon. For a black hole rotating as fast as possible, stable orbits exist all the way down to the event horizon. As long as you’re traveling in the same direction as the black hole spin. If you’re orbiting in the opposite direction, there are no stable orbits within 9 Schwarzschild radii.

The name for the size of these innermost stable circular orbits is ... innermost stable circular orbit. Or ISCO. For a black hole that is currently feeding - perhaps devouring a companion star or, in the case of quasars, a bunch of its host galaxy’s gas - the ISCO is expected to eventually be detectable as a dark circle in the middle of the otherwise insanely bright accretion disk formed by infalling matter. That’s because any gas that gets that close will quickly be swallowed. To date ISCOs have not been directly detected - although there is tentative evidence in gravitational lensing studies of quasars.

So yeah, you can orbit “safely” pretty close to the Kerr black hole’s event horizon.  Just above the event horizon is a particularly bizarre region called the ergosphere. There, frame dragging carries space around the black hole at faster than the speed of light. That means everything - even light - must move in the direction of the black hole’s spin. 

The situation here is actually similar to the state below the event horizon where space moves downwards faster than light. In the math, that faster-than-light flow of space is represented in a particularly weird way - space and time switch places. In particular, the radial direction becomes time-like, so downward motion becomes as inevitably one-directional as time. Well, in the ergosphere the angular coordinate becomes time-like - it’s as difficult to resist orbiting the black hole than it is to travel backwards in time - which is to say it’s impossible. That same switch also allows us to extract energy from the ergosphere, as we’ll see.

The ergosphere extends all the way down to the event horizon - that’s where the downward flow of space also reaches the speed of light. 

The event horizon in the rotating case is not spherical - it’s squished at the poles, just like the rotating Earth. More spin equals more squished. The ergosphere has a similar shape - but also dips at the poles to touch the event horizon - it’s pumpkin shaped.

Now I know you’d just love to drop below the event horizon right now - but you’re going to have to wait. There’s still plenty to do in the ergosphere - like building a hyper-advanced black hole engine.

The great Roger Penrose figured this out in the early 70s. It goes like this: a massive object is dropped into the ergosphere of a kerr black hole on a carefully tuned trajectory. If the object is split into two pieces at exactly the right point, one half will go plummeting through the event horizon while the other is ejected from the ergosphere and escapes. In fact it escapes with more kinetic energy than it had coming in - up to 20% of the energy than was bound up in the mass of the half that was lost.

This energy is extracted from the rotational energy in the ergosphere, slowing the black hole’s spin. To get a little more technical - it works because the weird space-time flip in the Kerr metric of the ergosphere allows one half of the object to acquire negative energy, which is transferred to the black hole, while the other half gains the difference in energy as kinetic energy.

So there’s your black hole engine: maneuver rocks into a Kerr black hole, blow them apart at the right instant, and then catch the kinetic energy of the pieces that get ejected.

Sounds a little messy, but this can also be done with light. Light that is directed through the ergosphere in the direction of rotation will also extract rotational energy and emerge amplified in a process called superradiance. 

This way the rotational energy can be extracted with, in principle, 100% efficiency. Oh, and you can also build a black hole bomb this way - by surrounding the Kerr black hole with mirrors. Then just shine a flashlight at it and its photons pass through the ergosphere again and again, becoming exponentially amplified until ... boom.

The last process we’ll talk about is actually important in the real universe. It’s the Blandford-Znajek process. In this case you have a magnetic field produced by the flow of material around the black hole in an accretion disk. The flow of space in the ergosphere spins up the magnetic field into a gigantic particle accelerator. Charged particles are accelerated along those magnetic field and can radiate intense light. Ultimately, the energy of that light is extracted from the rotational energy of the black hole. It’s hypothesized that some jets observed from accreting black holes may be powered by this process.

Jets produced by fast-rotating black holes are also a contender for another astrophysical phenomenon - gamma ray bursts. When a truly gigantic star collapses at the end of its life, and if its core was rotating fast enough, that core will produce Kerr black hole that can suck infalling material into an accreting vortex and spit it back out in powerful jets, again powered by the black hole’s rotation. If we happen to be along the paths of one of these jets, relativistic effects massively magnify its brightness. We see these as gamma ray bursts.

Rotating black holes are very real and powerful players in the energetics of our universe - but they’re also very worrying to physicists, because they threaten several physics-breaking phenomena - time travel, universe-hopping, and naked singularities. As you’ll see when we drop below the event horizon into the deeper weirdness of the Kerr spacetime.