# How long do you have to rescue someone from a black hole?

Forever, and if you hold tight for a bit, I’ll explain why.

I’ve been teaching a grad course in general relativity this term, and I’ve been having the time of my life. While my own work in gravitational lensing uses general relativity, I hadn’t taught a course in it before, and so I hadn’t delved as deeply into some problems as I might have liked. It’s now finals week, and while I won’t reveal my hand on what’s going on the final, I can tell you about a problem or two that won’t be on the exam.

In particular, let me relate a question put to me by an io9 reader named Doug:

We were watching a Science Channel show about black holes last night. At one point the narrator said that as an object approaches the horizon, it’s image appears to slow down. Slow down to the point that, to an observer far away, the object appears to freeze and stay at the horizon forever.

My son had a good question. As he put it, wouldn’t the horizon apparently be full of junk? If the hole eats gas that gets hot, wouldn’t the glow of the gas get stuck at the horizon and appear to glow forever? We’re not talking about Hawking Radiation here, just glow or images from various things that fall into the hole. Shouldn’t the Black Hole, from an observer’s point of view from far away, look like a ball of “junk”; realistically, shouldn’t it look like a glowing ball from the frozen image of hot glowing gas that falls into the black hole?

That is an excellent question. I covered some of the related material in an old io9 article, but you and your son already understand the crux of it: while the infalling observer (or detritus or whatever) measures a finite amount of infall time, an observer far away will measure the infall to take literally forever. However, there are two things that prevent the event horizon of the black hole from looking like a cosmic junkyard, either locally or far away.

1. Space is warped. I’m going to put this in somewhat mathematical terms here, but hopefully it will make some sense. Consider that near the surface of the earth, if there are two objects — me on the surface at a radius of 6400km and a low-lying satellite at a radius of 6401km, there is a distance of 1km between us. Of course! Nothing could be simpler! But the problem is that curved space — and space is _very_ curved near the event horizon of a black hole — doesn’t work that way. Here’s the equation, and I’m afraid you’ll have to trust me. The distance between two points, one at radius, $r$, and one at radius $r+dr$ is (and I’m terribly sorry about this):

$Delta l=frac{Delta r}{sqrt{1-frac{R_s}{r}}}$
where $R_s$ is known as the Schwarzschild radius, and represents the surface of the black hole. In case you’re wondering, the value of the Schwarzschild radius is:
$R_s=frac{2GM}{c^2}$
where $M$ is the mass of the gravitational source. As a point of reference, if you plug in the mass of the sun, you get a Schwarzschild radius of about 3km. For the earth, it’s a paltry 9 centimeters. Since anyone standing on the surface of the earth is much, much further away than 9 centimeters from the center, you can basically ignore this effect.

But near the surface of a black hole r is almost equal to $R_s$, so the denominator in the “metric” (the expression for $Delta l$) gets HUGE. In other words, though lots of things are “very near” the event horizon, that doesn’t mean that they’re very near to each other as seen locally.

If you want to ignore the equations and get an intuitive idea of what I mean, take a look at the “rubber sheet” analogy of the bending of spacetime near a black hole:

As you get close to the event horizon, you need to travel more and more distance “up” the sheet in order to move even the tiniest distance in radius.

2. Redshifts get enormous as you get close to the event horizon. Normally, if I fire a green laser toward you, you’d see green light. However, relativity has another effect: if you emit light in a deep potential well, the photons lose energy and become redshifted. You might see red instead of green. If the effect is strong enough, you’d see infrared — or rather, since your eyes aren’t sensitive to infrared, you wouldn’t see anything.

Even though it takes forever for objects to fall below the event horizon, it doesn’t take them forever to disappear. Since this post is tagged as “technical” I’ll give you one more equation. To a person hanging out near a black hole (things are subtly, but not qualitatively different to a person falling in), for every time, $Delta tau$ that passes by their watch, an amount:
$Delta t=frac{Delta tau}{sqrt{1-frac{R_s}{r}}}$
passes for someone far away. Again, as $r$ gets very close to the event horizon of the black hole, this factor becomes huge

Likewise, the light from them gets more and more redshifted, so they become invisible to us (and to any other detector). We get a double whammy, actually, since not only does every photon lose energy, but because time runs slower near the black hole, the apparent rate of emitted photons slows down to a trickle as well.

Incidentally, all of this weirdness has an even stranger implication that just recently occurred to me independent of Doug’s email.

Suppose you had a friend who fell into a black hole without a jetpack, and suppose (for whatever reason) he/she wouldn’t be killed by the tidal forces when he/she fell in. This could happen if the black hole was massive enough — more than about a hundred thousand times the mass of the sun. If you wanted to stage a rescue mission, you literally have all the time in the universe, so long as your rockets are powerful enough.

The only problem, however, is that the longer you wait, the closer he’ll get to the black hole, and the faster you’re going to have to turn your ship around after you do so. At some point, while he may not die falling into the black hole, he will die from the rescue itself.

Despite the fact that you could be rescued, none of this is meant to be an inducement to actually jump into a black hole.

-Dave

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### 6 Responses to How long do you have to rescue someone from a black hole?

1. Don says:

If time is slowed down in the black-hole, shouldn’t the one who fall in the black-hole felt like a few seconds while his friend rescuing him would gone through years instead of the other way round?

• Joshua Knox says:

your semi right because the man out side would only see the man going in get slower and every thing else would be normal, the rest is right and just to be clear he would be torn to shreds before any thing could reach him.

• Joshua Knox says:

Oh and so you know I’m 11 and this smart. But do your research you will find I’m right.

2. dave says:

Indeed. I fixed the error. And thanks.

3. anna nomis says:

so a black hole’s gravity is so strong that it creates a ripple in space-time, literally slowing down space? or is it not the gravity that does that? black holes are confusing, yet awesome, yet not awesome because they can kill you.

4. anna nomis says:

p.s. like my name?