So, I may have watched one too many episodes of PBS SpaceTime. But my fascination with black holes goes back a while, and I’ve always thought that the idea of singularities is weird.
Supposedly, the Penrose Singularity Theorem proves that there are singularities in black holes, because everything inside must travel towards the center at the speed of light. But the more I think about it, the more I’m convinced that that’s just what it looks like from the outside.
Of course, you can’t see what’s going on inside a black hole, so discussing what it “looks like from the outside” is pretty pointless. It’s far more interesting to consider what they look like from the inside. Of course, nobody knows for sure, and it’s entirely possible that black holes are made up of unicorn and rainbows, but let’s assume that physics work more or less the same inside black holes as outside for now.
So, let’s start with making a black hole. To do that, we’re going to use a kugelblitz. Kugelblitz is german for ball lightning, but in this context, it specifically means making a black hole out of light. I’m going to basically steal the setup from the PBS SpaceTime kugelblitz challenge episode episode…
Imagine a super-advanced spacefaring race who have decided humanity needs to be removed from the universe. Maybe the talked to hitlers ghost, saw EP8, or maybe it’s some sort of dark-forest scenario. Whatever the reason, they line up their armada around the solar system, and they all fire a laser pulse towards earth. The total energy of this pulse is a hundred thousand times larger than the total mass-energy of the sun, easily enough to create a black hole.
Once this laser pulse gets within one light-second of earth, an event horizon forms, creating a black hole, and one second later, the earth is completely destroyed. The interesting part here is that for a whole second, earth doesn’t even know that there is anything wrong. No hints even. Since nothing travels faster than the speed of light, no information about what’s going on can reach earth faster than the light that destroys earth. (Unless it gets stopped by rainbows and unicorns of course!)
Since earth is destroyed, it’s still a little difficult to reason about what’s happening inside the black hole, so let us change the kugelblitz scenario a bit. Let’s say that our space-faring aliens are also pacifists who don’t want to actually kill us. They just don’t want to share a universe with us. So, they very carefully aim their lasers to miss earth by one half light-second. Let’s also assume that as the lasers pass earth, there are so many of them, and so perfectly space that they form a perfect sphere of photons around the earth.
So now what happens? Does earth get crushed into a singularity? Are we still doomed? How can we find out?
The answer should lie in the path of the light. The path light takes is always the shortest, straightest path possible, and only curves in spacetime can change that. (Or interaction with matter.)
First of all, we’ll need to invoke Newton’s shell theorem. This theorem says that a perfect shell of matter or energy will not affect anything inside it, regardless of location. This theorem should save earth from getting torn apart by 100000 sun-masses passing earth, since we conveniently specified that the lasers would form a perfect shell.
So, now, we can take a stroll on earth and watch the lasers pass by. Assuming we survive, we can then watch the lazers travel off into space. Except, maybe not. Since the lasers got close enough to earth to create a black hole, the lasers can’t leave. They also cannot change direction, and they can’t slow down.
The classical interpretation gives us two different, and conflicting answers. A photon that is one nm inside the perfect shell would feel no gravity from the shell, and would thus continue straight, while the Penrose singularity theorem tells us that the whole sphere is going to be pulled into the center and form a singularity. Ok, so maybe the answers aren’t actually conflicting, but getting two different answers for two photons that only nanometers apart seems like an unlikely amount of space-time curvature, especially since we can make that curvature arbitrarily large by making the distance between the photon and the shell arbitrarily small.
For now, let’s assume the earth in this scenario survives for a bit and see if that leads to something more sensible. Let’s say you happened to spend the night camping while all this is happening, what would you see? Well, the answer would have to be something that would let light from the rest of the universe still reach us, while light / radio / whatever from us cannot reach them. The only thing that would make sense to me is that everything in the universe would appear to move away from us, faster than the speed of light. We could still see the sun, as it’s racing away from us, but we could never send a signal to it, because it would never catch up with it.
From our perspective, space would stretch, creating a space between us and the rest of the universe faster than anything could ever catch up with it. Newtons shell theorem says that the space inside the sphere of photons should not stretch though. That means that space stretching should be most intense at the sphere, and outside the one light-second swartchild radius, it should be less than light-speed.
But this observation seems to make our space-time curvature problem worse. A hypothetical particle outside the sphere should move away from us faster than the speed of light, which means that a photon that was on nm outside of the shell would be pulled away from us rather than get pulled in towards us.
Wouldn’t it be neat if the pull of gravity and the stretching of space would just cancel each other out though? That would resolve most of these weird problems.
My guess is that, it depends on the observer what happens.
From the inside, the space stretching would appear appear larger than the force of gravity. The lasers would appear to move away from earth. From the outside, the space stretching appears to be taking things into the center, rather than out, so the lasers all get smushed into the center, and if you’re somewhere in the between, the lasers might appear to stay in orbit forever.
Now, I really wish I had the math to actually work out if any of this is accurate.
As far as I can tell, the Penrose Singularity Theorem isn’t exactly violated by this idea, it’s just that there the singularity is observer-dependent. And there is no need for a singularity if space on the inside of the black hole expands faster than the speed of light.
IF any of this is remotely right, it would seem that this sort of thing would happen all the time. Because any time a black hole is formed, there is some internal sphere where the inside mass isn’t large enough to form a black hole. Inside that sphere, the space-time curvature would be less than light speed, and so there would be no singularity there. Again, someone with far more math than I would have to figure out if this reasoning is enough to not have to worry about singularities or not…
Anyways, I just had to write all this gobblygook down before my head explodes.