When black holes collide

I thought I'd give the people who show up from time to time to complain about my occasional political posts something new to grumble about, so today we're talking physics. (Not so strange given how I view roleplaying, which is that it's about everything.) And don't panic. I'm going to do all this in terms a child could understand, honest.

I had a question about black holes that used to puzzle me as a teenager:

Say you’ve got a black hole of 4 million solar masses, so roughly 50 AU across. (I’m taking a really big one so that its tidal forces don’t turn everything that falls in into spaghetti.) An astronaut wearing a wristwatch falls towards it. What do we see? My understanding is that we’d see the astronaut slow down as he or she approached the event horizon, finally appearing to come to rest on the event horizon, very red-shifted and the watch display apparently having stopped. (From the astronaut’s point of view he or she accelerates towards the event horizon, the entire universe around them is blue shifted, and their watch continues to tell the time accurately right up to the moment they go through the event horizon – and maybe after as well? But that’s a whole other question.)

OK, now what if instead of an astronaut we drop another supermassive black hole into the first one? As the event horizons touch, do they freeze in place there like two balls glued together?

1. If not, then in watching the rate of merging of the horizons we’d be getting information out about how fast the two singularities are moving together. From our perspective we’d suppose them to move together infinitely slowly, but in any case no such information can escape the black hole.
2. But if they do just stay frozen like two balls touching at a point, then the universe should be festooned with very oddly-shaped black holes, apparently made up of lots of aggregated black spheres stuck one onto another like a 3D Mandelbrot set.

I don’t think that’s what the maths says. All black holes are supposed to be featureless and (nearly) spherical. So do they go from the two-balls-touching state to the one-big-ball state in a quantum leap – to coin a phrase?

For half a century I kept asking that question but never got a satisfactory answer. One astrophysicist did tell me that I was making the mistake of using second-order geometry when I needed to consider fourth-order geometry, but I live in this universe not the maths one. I wanted an explanation I could chew.

And eventually (there might be a life lesson in this) I figured out the answer for myself. I could have done that decades ago, too, if I'd just stopped to think that I'd packed a huge assumption into the original question when I envisaged them as both remaining spherical up to the point of contact. So here's that answer -- and look, Ma, no maths!

Imagine two supermassive black holes of equal size in otherwise locally empty space. At a wide separation they are spherical (if they're not rotating). The size is defined by the Schwarzschild radius, the point at which a particle cannot escape the gravity well of the black hole.

As the holes get closer together, consider two particles, one on a line between the two black holes and just within hole #1’s Schwarzschild radius, the other on the same line but the far side of hole #1 and just outside its original Schwarzschild radius.

As the gravity of hole #2 begins to have a significant effect, the particle between the two gets a gravitational boost that would allow it to escape the original radius. Conversely, the particle on the far side now has to escape not only hole #1’s gravity but the additional pull of hole #2.

So the effect as the holes approach each other is that the event horizons bulge on the far side and flatten on the near side. The extent of the event horizon becomes less on the near side and greater on the far side so that they resemble two distorted lenticular blobs that will (as they get closer and closer) asymptotically adopt the shape of hemispheres that will then merge into one new larger sphere.

If the physics of spacetime curvature allowed that process to be entirely seamless then there would be no release of energy, but of course the two merging holes don’t precisely resemble sections of a greater sphere even at the moment of contact – at that point, there’s an anomalous dimple in the curve around the great circle bisecting the new event horizon. Hence the associated burp of around 5% of the mass, and the new horizon of the combined hole will only extend as far as the inner part of that dimple.

I’m just guessing that you could compute the proportion of energy released purely from the geometry, mind you – I retain more faith in maths than I do mathematical ability, these days. And of course, an observer at infinity would say there was only one black hole there in the first place. But that's a detail.