Black holes are complicated.
The Schwarzschild spacetime, which is the original basis for the theory of black holes, is more complicated than the simple form of the metric would have you think. The usual form of the metric is, to remind you, .
At the Schwarzschild radius, , this metric blows up (i.e., goes to infinity). That means, originally, the metric really only makes sense outside this radius, .
But last time, we talked about how this apparent singularity is not actually a problem. After a change of coordinates,^{1} the weirdness at the Schwarzschild radius goes away, and the spacetime is suddenly well behaved all the way down to .
Well, except that light can’t escape from inside this radius, due to the extreme bending of spacetime. That’s why we call the sphere at the event horizon, and the whole thing a black hole.
So, the original presentation of the Schwarzschild metric was problematic, and so it only made sense for , outside the black hole. But a coordinate change let the spacetime make sense inside the black hole as well, for . We were able to extend the spacetime.
So that leads to a question. Can we extend the spacetime even further?
It doesn’t seem like there’s anywhere to extend to, like there was before, when we extended into the interior of the black hole.
But it turns out, despite all appearances, we can extend the spacetime further, and discover another universe on the other side of the black hole.
But to understand that we need to talk about a special coordinate change for special relativity.
To remind you, Minkowski space, the spacetime of special relativity, can be interpreted as a solution of general relativity. The manifold is , with metric . However, if we change to spherical spacetime coordinates, the metric in these becomes . (Same spacetime and metric, different coordinates.)
Like we did last time, we’re going to mostly ignore the sphere part of the metric (the and parts), and focus on the and parts. And we’re going to do a weird kind of coordinate change. In fact, it’s weird enough that let’s just sketch a diagram for the Minkowski spacetime in these new coordinates, rather than write out a metric.
It’s kind of weird, but what we did is we compressed the infinite range of time and the infinite range of radii down to a compact area. For clarity, we also drew some coordinate lines based on the old coordinates.
One nice part about this diagram, sometimes called a Penrose diagram,^{2} is that light still travels at 45 degree angles.
Notice they end up at the (upper) right diagonal line. This line is called (future) null infinity. It’s called null, since light follows null paths, and infinity, since that’s where light “ends up” after traveling infinite distance. It’s infinitely far away, through represented as a line in our diagram.
On the other hand, if you are sitting still, no matter what radius you’re sitting at, you end up at the top point. This top point is called future timelike infinity. It’s “infinity” since it takes infinite time to get there, and timelike, since it’s where you end up if you travel along a timelike path (such as sitting still.) The bottom point is similarly past timelike infinity.^{3}
To summarize, the most important things to take away from this diagram is that the top and bottom points are infinitely far to the future or past, and the right sides represent infinitely far away.
These Penrose diagrams are kind of weird, but they are incredibly useful in understanding black holes.
So, what does the Penrose diagram for the Schwarzschild spacetime look like?
Due to the black hole, this diagram looks a bit different. The right part of the spacetime, approaching null infinity, though, works exactly the same way as in the diagram for special relativity.^{4} The top point is infinitely far into the future, while the right lines are infinitely far away.
On the left, the diagonal line cutting the spacetime in two is the event horizon of the black hole, at . Inside the black hole, the top line is the singularity at .
Light still travels at 45 degree angles in this diagram. But if that light was emitted inside the black hole, that means the light must fall into the singularity .
We saw this before, but it’s particularly easy to see in this diagram.
It’s also interesting to notice that the event horizon is also at 45 degrees. In other words, the event horizon (and thus the black hole) is expanding at the speed of light!
It’s not that the radius is changing. But if you shot a ray of light out from the black hole right on the event horizon, it would stay on the event horizon forever, travelling “outward.” And the path of the light in the spacetime would exactly be that line of the event horizon. So, in a very real sense, the event horizon is going outward at the speed of light. It’s just that the spacetime is so curved, light isn’t moving!
As one demonstration for why this diagram is useful for understanding the spacetime, let’s think about what happened when we pushed Hitler out the airlock into the black hole.
If Hitler is moving directly toward the black hole, his path in the spacetime might look like this.
Of course, we’ll be sitting at a safe distance, say at a fixed value.
What would we see as Hitler approaches the black hole?
The light reflecting off of his face would travel out to us along null geodesics, i.e., along 45 degree lines. Let’s sketch a few of these light paths.
Now, remember that the only infinities occur at the boundaries of the diagram.^{5} That means that Hitler, according to his own perception, falls into the black hole in some finite amount of time.
But notice that the light coming from him reaches us (sitting at a safe distance) at later and later times. (Remember that the top of the diagram, where our path ends, represents the infinite future. So the light rays look evenly spaced on the diagram, but the time between when each ray reaches us is longer and longer.) Even though some light is leaving Hitler shortly before he falls through the horizon, it doesn’t reach us until almost infinitely later!
In other words, Hitler experiences falling through the event horizon in finite time, but we will never actually see him fall in. We’ll just see him get closer and closer^{6}.
Pretty weird, huh?
So, this Penrose diagram, so far, is pretty cool.
But I promised that we could extend the spacetime, and I haven’t shown you how yet.
The idea how is simple enough. You can write down the metric in the coordinates of the Penrose diagram, though I’ll leave it to a footnote^{7}. The trick is that, while the spacetime that we have been using represents in these coordinates, the metric is perfectly nice and well-behaved for as well!
If we add that part of the spacetime to our Penrose diagram, we get something like this.
The diamond on the right, remember, is our universe. The diamond on the left is an alternate universe!
Anything could be going on in that other universe, but we could never find out.
See, the only way our universes connect is through the black hole.
Since we can only travel on paths at angles less than 45 degrees (timelike paths), the only way that our paths could cross with the path of an alien from the alternate universe is inside the black hole.
Certainly, inside the black hole, we could talk about the wonders of our respective universes^{8}, but only for a short time.
Then we all get eaten by the singularity.
What about the triangular region at the bottom?
The best way to think of that region is in analogy with the black hole region (the top triangular region.) The black hole is a region where light and matter and energy can go in, but nothing can ever come out.
The bottom region is the time-reversed version of this; light and matter and energy can leave, but it can never go back.
Since it’s the opposite of a black hole, we call it a white hole.
Unfortunately, this alternate universe is really just a mathematical construct.
Remember, the Schwarzschild spacetime is supposed to model the region outside of a star. If a star runs out of fuel, and is massive enough, it will collapse to a radius smaller than the Schwarzschild radius ^{9}, forming a black hole.
But the matter is still there, inside the black hole, at least until it falls into the singularity. (And who knows what happens then; probably something weird and quantum.) But that matter “censors” the alternate universe part of the solution from showing up.
The easiest way to see this is to draw another Penrose diagram.^{10}
The star stuff starts out bigger than the Schwarzschild radius. But as it collapses, it eventually falls within that radius.
But since the Schwarzschild solution is only valid outside the star, there isn’t any alternate dimension “beyond” the matter falling inward. There’s just the center of the matter.
Is it possible to extend the spacetime even farther?
Well, something we can’t do is extend past the singularity at . As we talked about last time, the curvature goes to infinity there, meaning there is infinite gravity. We shouldn’t be able to extend past that.^{11}
There doesn’t seem to be a way to extend the Schwarzschild spacetime any farther.
But if we add a little electric/magnetic charge (and so get a different black hole spacetime), it turns out we can extend past the equivalent of .
And it causes all sorts of problems, because it seems like it might be there, even for a collapsing star.
But that’s a story for another day.
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First post in this series: A Mathematical Intro to General Relativity, Part 1
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- Remember, a change of coordinates changes the presentation of the metric/spacetime, but doesn’t actually change the spacetime itself. ↩
- It’s also sometimes called the conformal compactification of the spacetime. ↩
- There’s also spacelike infinity, which is where things going faster than the speed of light (like tachyons) end up. ↩
- Though it’s not the same spacetime. Minkowski space (special relativity) has no curvature, but the Schwarzschild spacetime does. However, the Penrose diagrams look the same near null infinity. ↩
- Technically, only at the top and right parts of the diagram, as we’ll discuss a bit later. ↩
- However, he will get dimmer and dimmer, since the photons will be reaching us at a slower rate. Similarly, the light will also be getting “redshifted” (the wavelength will be increasing). Put together, that means that Hitler will become harder and harder to see, until it becomes impossible to detect him in practicality. ↩
- In the Penrose diagram coordinates, is the center point, where the event horizon intersects the lower diagonal line. In one version of these coordinates, the metric is , where is our old coordinate. ↩
- Mmmmm, bacon… ↩
- Which is about 3 kilometers for our sun. ↩
- This diagram isn’t of any particular solution, though you could come up with one that works pretty close to this way. ↩
- That argument shows that we can’t extend the spacetime past the singularity as a spacetime of general relativity. There’s a recent paper that shows you can’t even extend past that singularity as a manifold, much less as a solution to Einstein’s equations. ↩