Mapping the Multiverse (Script) (Patreon)

This is a map of the multiverse. Or in physics-ese, it’s the maximally extended Penrose diagram of a Kerr spacetime. And in english: when you solve Einstein’s equations of general relativity for a rotating black hole, the universe does not come to an abrupt halt at the bottom of the gravitational pit. Instead, a path can be traced out again but you do not end up in the universe that you started in. Like I said, it’s a map of the multiverse.

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In order to learn some multiverse cartography, and to decide whether the strange new regions on this map are real, we’re going to need to continue a journey that we began a little while ago when we explored the region just above a rotating black hole. Back then we stopped short of dropping below the event horizon - the boundary of no return. Today we’re going to take that plunge.

A black hole results when enough mass is concentrated in a small enough space that the gravitational field becomes too strong for even light to escape. The inescapable surface surrounding a black hole is called the event horizon. If that mass is NOT rotating and does not have any electric charge, the result is a Schwarzschild black hole, which is about as simple as these bizarre objects can get. Add a little spin or charge and things get even weirder. We’re going to focus on rotating black holes here, but a lot of this also applies to charged black holes, as we’ll see.

A rotating black hole is described mathematically by the Kerr metric. It describes the way spacetime warps and flows in the vicinity of a spinning mass. We sometimes call a rotating black hole with no electric charge a Kerr black hole.

Let’s think about what that rotation does in familiar terms. Think of the worst spinny amusement park rides, or an astronaut high-g training machine - you feel yourself flung outwards by centrifugal force. Well, in a Kerr black hole that rotation results in an outward pressure that partially counteracts the inward flow due to gravity. This leads to weird effects. Above the event horizon is the ergosphere - a place where the fabric of space itself is whipped into a vortex so fast that not even light can resist its motion. The event horizon itself is distorted - wider at the equator than the poles, just like the rotating earth. We saw all that in a previous episode. Now let’s peek below the event horizon to see how weird things really get.

Before we do that, let’s remember how to read our multiverse map - our Penrose, or Carter-Penrose diagram - maybe it’ll help us find our way out again. A couple of things to remember about this map. Time flows up - for the most part - and one dimension of space is left or right. Light always travels a 45 degree path, and normal objects can only travel paths steeper than that. Your so-called “future light cone” encompasses the parts of the multiverse you could possibly get to at less than light speed. Your past light cone shows you the parts of the multiverse that you can see, because light has had time to reach your position from those parts. 

For most of the rest of the episode we’ll be using a couple of videos side by side. I just finished making this one. You can either remake this or use it directly. We’ll need some “explorer” to be making the journey indicated by the dashed line. That explorer could either be a rocketship or a cartoon monkey (with a jetpack ideally). Whichever you prefer. It should trace out the dashed lines as it moves.

Different sections of the path will correspond to different paragraphs below, which I’ll indicate there. See the figures below. Track segments through the linked movie are colour coded on the diagram below, and then the corresponding tracks shown on the Penrose diagram. All this will be done full screen, with the black hole interior and the Penrose diagram side by side. There may be a third video showing the view of the falling observer - this will be generated by someone else and just needs to be embedded.

As we take this journey we’re also going to plot our path on the Penrose diagram. 

When we first enter the Kerr black hole, we find it’s just like every non-rotating Schwarzschild black hole we’ve ever visited. Down becomes the future, because falling space drags us there faster than the speed of light. Space and time switch places. In a regular black hole we’d be crushed by the central singularity pretty quickly. 

But now as we fall the outward pressure due to the black hole’s rotation starts to win against that inward flow. In fact the flow eventually slows to less than the speed of light. This leads to a second event horizon called the inner horizon. Now, some pretty terrible things happen here, and you don’t survive the crossing. But we’re going to pretend that you’re Deadpool or something you get across intact-ish - what do you see? On the other side, space and time back in their proper order. You’re now free to NOT fall, so you can jet around and explore the interior.

Below you is the black hole singularity - but it looks odd. The singular, central point has been spun up into a ring of infinite density. It’s surrounded by a second ergosphere, where again the rotational flow of space exceeds light speed. So what next? First probably take a ride in the ergosphere, because why not? When you’re dizzy enough you can try to turn around and go back the way you came through the inner horizon, or you can plunge through the singularity ring. In either case, you’re in for a weird time.

To understand what happens here, we have to think like a general relativist. In order to create these maps of spacetime, physicists use the equations of general relativity to trace what we call geodesics - these are the paths that objects take when not experiencing any force. For example, the path a ball takes if I drop or throw it - after it leaves my hand. A nice way to map the gravitational field is according to the geodesics of objects in freefall that start motionless relative to the gravitational body. For example, if you drop a trillion balls from space they’ll trace the downward cascade of space - and that flow would be pretty spherically symmetric. Same with the interior of a Schwarzschild black hole -  every ball, every geodesic reaches the same point and, very importantly, ends there.

But within a Kerr black hole, the singularity is a ring, and the trillion balls would fall towards the disk bounded by that ring. Only the ones exactly on the equator hit the ring and, those geodesics do end - same infinite squish as in a Schwarzschild black hole. But those that approach the disk experience an overwhelming anti-gravitational force from the spinning singularity. Our balls, and the geodesics they follow, rebound and travel back out.

Before we take that outward ride, let’s push a little harder and try to get through the ring. With a lot of speed - and I mean a LOT - it’s possible to overcome the anti-gravitational field in the middle of the ring and punch through. But you don’t find yourself on the other side of the same region. The ring is like a portal to a new, very different region of spacetime - and that’s because the geodesics passing through one side do not map to those passing through the other. Through the portal things get very weird. For one thing, the ring singularity becomes entirely repulsive - as though it had negative mass. Also, there are no horizons above you - there’s a straight path to the outside universe - but it’s certainly not our universe. For one thing, the lack of event horizons above you means the ring is a naked singularity, as well as being repulsive. Surrounding the ring on this side is a toroidal region in which it’s possible to accelerate to infinite speed. And as some of you probably remember - faster than light travel means time travel. There are trajectories in this torus that lead you back to your starting location - in space AND time. So-called closed-timelike curves. This is the Carter time machine, after the aussie physicist Brandon Carter who did much of the early exploration of the Kerr spacetime.

This time travel and naked singularity stuff is good reason to think this part of the mathematical structure of the Kerr black hole is NOT real. We’ll come back to how it can be avoided in the math - for now, let’s whip once around back to our own past and hop back through the ring.

At this point we can follow an outgoing geodesic across the inner horizon. However you’re in for a surprise. That outgoing flow continues above the inner horizon, and will eventually eject you past the outer horizon. That’s a little confusing because black holes are supposed to be inescapable. So what’s with this conveyor belt of space taking you out of the black hole? Don’t you have the same conveyor belt pulling you in over that same region? Well you sort of through the same space, but not through the same time. The black hole is in your past. You’d need to travel back in time-slash-faster than light to get back there. Extending the Kerr metric from ring singularity into the future leads to something else. It leads to what looks like a white hole. Now we saw that in the Schwarzschild solution, a white hole exists in the past of a purely hypothetical eternal non-rotating black hole.

The Kerr black hole also has a white hole in its past, but it has one in the future too. And that’s where the outgoing geodesics go, carrying you with them. But not only is the black hole in your past - the entire universe that you came from is in your past. You caught a glimpse of the entire history of that universe the moment you crossed into the white hole. That white hole will eject you as forcibly as the black hole pulled you down. Eject you where? This has to be a new universe, because it’s definitely not the one you came from. It’s not bizarro negative universe either - here, the laws of physics are the same as where you started - at least as far as general relativity is concerned.

If you try to head back to the white hole, you won’t find it - you’ll only find a new black hole that lies in its future. And if you enter that you’ll be able to take the entire fair ground ride once again, skipping ahead to yet another universe, and so on ad infinitum.

This may all sound fun, but unfortunately there’s the inconvenient fact of your utter obliteration before you ever cross the inner event horizon in the first place. The complete Penrose diagram for the Kerr spacetime has not one, but two inner event horizons leading to two parallel wormholes. You reach one if you embrace your fall and head down, the other if you try to resist and exit the black hole. These correspond to the region below the event horizon but they’re causally disconnected, with the difference being that time flows forward if you go in one way and backwards if the other. So you have this messed up situation where the direction of the flow of time actually clashes at the inner horizon. This would be okay-ish if the black hole was empty, but it’s not okay if there’s even the tiniest bit of matter or radiation.

If there’s any stuff at all in the black hole then the forward flow of time carries a current of positive energy, while the backwards carries negative energy. When they meet they don’t cancel out - instead they stream past each other producing enormous pressure. That pressure produces its own gravitational effect on par with the black hole itself, accelerating the streams further. The result is an exponential runaway effect. So if you do get to the inner horizon you’re in a bath of energy on par with the Big Bang. Good luck with that. Besides, this so-called mass inflation means that the inner structure of the Kerr black hole is catastrophically unstable, and the whole thing collapses, shutting off any potential magical portals to time machines or new universes - probably before they ever exist.

This whole counter-streaming instability thing was figured out by Roger Penrose in the context of charged, non-rotating black holes. But it almost certainly applies to Kerr black holes also because the two are very similar. In electrically charged, or Reissner-Nordström black holes the electromagnetic field within causes massive tension, or negative pressure that produces an antigravitational effect. That resists the inward flow just like rotation does in a Kerr black hole. You have the same inner horizon and chain of wormhole-connected universes. They just don’t have the ring singularity leading to the bizarro-verse, so Kerr is more fun than Reissner-Nordström.

So there you have it. Now you know everything you need in order to travel the multiverse and travel through time. Just a little wishful thinking, a rotating black hole, and don’t forget to slip your map in your back pocket. Never travel the multiverse without a Carter-Penrose diagram of the maximally extended Kerr spacetime.