Black Hole Puzzle

The puzzle of 101 starship captains

101 starship captains, bored with life in the Federation, decide to arrange their starships in a line, equally spaced, and let them fall straight into an enormous spherically symmetrical black hole—one right after the other. What does the 50th captain see, the moment their starship crosses the event horizon?

(Suppose there’s no accretion disk or other junk blocking the view.)

Background

A somewhat surprising fact is that the more massive a black hole is, the closer to flat is the spacetime geometry near the event horizon. This means an object freely falling into a larger black hole feels smaller tidal forces near the horizon. For example, we sometimes see stars getting ripped apart by tidal forces before they cross the horizon of large black holes. This happens for black holes lighter than 10⁸ solar masses. But a more massive black hole can swallow a Sun-sized star without ripping it apart before it crosses the horizon! It just falls through the horizon and disappears. The tidal forces increase as the star falls further in, and they must eventually disrupt the star. But because it’s behind the event horizon at that point, light can’t escape, so we never see this.

My puzzle is assuming a large enough black hole that the starships can fall through the horizon without getting stretched and broken.

The view from outside

Suppose you stay far from the black hole and watch the 101 starships fall in. What do you see?

You see these ships approach the black hole but never quite reach it. Instead, they seem to move slower and slower, and the light from them becomes increasingly redshifted. They fade from view as their light gets redshifted into the infrared and becomes weaker and weaker.

The experience of an infalling captain

This may fool you into thinking the ships don’t fall into the black hole. But the experience of the infalling captains is very different!

Each starship passes through the horizon and enters the black hole. While the spacetime curvature is small at the horizon, it quickly increases. Each captain dies as their ship and their body get torn apart by tidal forces. After a finite amount of time as ticked out by their own watch, each captain hits the singularity, where the curvature becomes infinite. At least, this is what general relativity predicts. In fact, general relativity probably breaks down before a singularity occurs! But this is not what concerns us here.

More relevant is that general relativity predicts that none of the captains ever see the singularity: it is not a region in space in front of them, it is always in their future, until the instant they meet it—at which point they are gone. (That is, general relativity has nothing more to say about them.)

For this it helps to look at a Penrose diagram of the black hole:

Time moves up the page and “away from the black hole” is to the right. Light always moves at 45 degree angles, as shown. The singularity is the black line segment at top. Thus, if you’re in the gray shaded region, you’re doomed to hit the singularity unless you move faster than light! But you’ll never see the singularity until you hit it, because there isn’t any 45 degree line from you going back in time that reaches the singularity.

The red line is the event horizon: this is the boundary of the gray shaded region. Once you cross this, you are doomed unless you can move faster than light.

What the 50th captain sees

As the 50th captain falls into the black hole, they see 49 starships in front of them and 49 starships behind them. This is true for all times: before they cross the horizon, when they cross it, and after they cross it.

They never see any starship hit the singularity—not even the 49 starships in front of them. That’s because the singularity is always in their future.

At the moment the 50th captain crosses the horizon, the 49 starships they see before them are also crossing the horizon (but not the 49 in back.)

The reason is that the event horizon is lightlike: light moves along its surface. You can see this in the diagram, since the horizon is drawn as a 45 degree line. Thus, the light of the 49 previous ships emitted as they cross the horizon moves tangent to the horizon, so the 50th captain sees that light exactly when they too cross the horizon!

It may help to imagine the 49th and 50th starships falling into the black hole:

Captain Alice falls in along the orange line, and Captain Bob falls in after her along the green line. This is an approximation: they actually fall in along a curve I’m too lazy to compute. But since ‘everything is linear to first order’, we can approximate this curve by a straight line if we’re only interested in what happens near when they cross the horizon.

Here the black line segments are rays of light emitted by Alice and seen by Bob:

These light rays move along 45 degree lines. In particular, you can see that when Alice crosses the horizon, she emits light that will be seen by Bob precisely when he crosses the horizon!

The redshift

Thus, as Captain Bob falls into the black hole, he will see Captain Alice in front of him for the rest of his short life. But she will be redshifted. How much?

Greg Egan calculated it, and here is his result:

Egan imagined Alice and Bob starting from rest at distances of 11 and 12 times the Schwarzschild radius of the black hole, respectively. (This sentence only makes sense if I tell you what coordinate system he was using: he used Schwarzchild coordinates, a commonly used coordinate system for nonrotating black holes.)

Then Egan graphed the frequency of light Bob sees divided by the frequency of light Alice emitted, as a function of time as ticked off by Bob’s watch. Thus, in the vertical axis, “1” means no redshift at all, and smaller numbers mean more redshift!

Alice as seen by Bob becomes more and more redshifted as time goes by. She becomes infinitely redshifted at the instant Bob hits the singularity. Nothing very special happens in this graph at the moment Bob crosses the horizon, though the light he sees then is from Alice crossing the horizon.

We could make a more fancy graph like this showing the redshift of all 49 ships in front of the 50th captain, and all 49 after the 50th captain. That might be worth doing with, say, 2 ships in front and 2 behind. But I will stop here for now, and let my more ambitious readers give it a try!