In this week's "Ask a Physicist," we tackle a general relativistic paradox: If time slows down near the event horizon of a black hole, how does anything ever fall in?
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Today's question comes to us from David Sirola who asks:
If a black hole warps space-time to such a degree to slow and stop time, how can anything ever disappear past the event horizon (or whatever point t=0)? It would seem to me in my superficial understanding, that ultimately, after all the fun at the outer edges of the hole, that nothing ever really happens, since happens implies a time/cause/effect relationship.
What am I missing here?
Let's get one thing out of the way from the outset: Black holes are awesome. They are the only major disturbance of space-time which have the advantage of actually being known to exist. Almost every large galaxy, including our own, seems to have a supermassive black hole at the center.
And black holes are ridiculously simple objects — or at least the non-rotating ones are, which are the only ones I'm going to talk about here. They basically consist of an infinitely compact "singularity" at the center and an outer boundary known as an "event horizon" from which nothing can escape (and here's where I'm supposed to use an ominously spooky voice) not even light. These guys are tiny, astronomically speaking. Were our sun to become a black hole, it would be smaller in radius than the city of Philadelphia. Even the 3 million solar mass black hole at the center of the Milky Way could comfortably fit inside the orbit of Mercury.
Okay, you probably knew all of that. I still need to dispel a few myths before we get into the hardcore space-warping.
- Black holes don't suck.
- Suppose the sun were to suddenly turn into a black hole. Would you notice? Sure you would. The sun would blink out of existence and you'd quickly freeze to death. But in your dying moments, you'd no doubt be struck by the fact that J.J. Abrams lied to you. Rather than get pulled into the black-hole sun, the earth would just keep orbiting that seemingly empty point in the sky, exactly as it always had. Only icier.
- You can't actually see them.
- Black holes are called that because they don't give off any light. I don't want to get into a nerd-fight here, partly because my mom says I'm not allowed, but mostly because things will go more smoothly if I anticipate a few objections. Somebody is likely to point out that we do, indeed, "see" black holes in the form of quasars in other galaxies. But this isn't quite right. What you're really seeing is hot, glowing gas falling onto the black hole or even larger glowing gas clouds surrounding the whole shebang. And by the way, with the exception of giant radio jets, we can't even generally resolve these clouds. When you see detailed accretion disks in news stories about black holes, that's somebody using MS Paint or whatever they use these days to make artist's conceptions.
- Let me further anticipate a black-belt level nerd who might introduce an even better possibility: Hawking Radiation. This is one of the coolest ideas in astrophysics, and one that most physicists believe, even though we've never observed it. Near the event horizon of black holes, particles and antiparticles are constantly being created in pairs. Every now and again, one of the particles escapes and creates some radiation (and takes with it some of the mass-energy of the black hole). But here's the deal: Hawking radiation is far too dim, and far too long of wavelength to ever be seen directly.
Now that we've got the lay of the land, we can get into the question of what's so special about the event horizon.
One of the major predictions of general relativity is that time runs slower near massive bodies than far away. On earth, we don't notice this effect, since the effect is only about 1 part in a billion. However, if you were of a stout enough constitution (we're talking, like 18+2, or something like that) you could hang out on a neutron star where the effect is more like 20% or more. Hang out for a few years, and even more time will have passed far away. What you've done here is built a (pretty crappy) time machine into the future. Also, this would be a one-way trip.
Black holes do it one better, and at the event horizon the time distortion effect literally becomes infinite. There's the paradox. If time slows down infinitely near the surface, presumably it takes stuff longer and longer to get closer and closer to the event horizon. How does anything ever actually fall in?
Let's imagine you have a friend who you didn't mind sacrificing for science. Suppose he decided to jump feet first into a black hole. What do you see when he crosses the event horizon?
Well, first off, you're not going to see him cross the event horizon at all because he's going to be torn to shreds by tidal forces long before-hand. He's also going be squeezed by the strong gravitational forces until he's ripped apart atom by atom. To those of you who either whispered under your breath (or more likely squealed with delight), "Spaghettification," good for you!
It's unfortunate for your friend, however, as he will most certainly not survive the ordeal. There is a glimmer of good news, however. A friend and former professor of mine, Rich Gott did an interesting calculation in which he found that regardless of the size of the black hole, it would take approximately one tenth of a second between the moment when you first felt mildly uncomfortable to the time when you are ripped atom from atom. Incidentally, if you'd like to read more about what's in store when falling in, you should check out Neil Tyson's discussion of the subject or my own.
But let's forget about this unpleasantness. Even if your friend could somehow survive the process, you couldn't see him fall in because eventually his signal is going to disappear. If your clock is running slow, this means that everything you could possibly use to measure time — including the frequency of light, will also appear to run slow. Light emitted from your friend's ship, for example, becomes longer and longer in wavelength as he approaches the event horizon until you can't see him at all. Even if you were looking at him with a radio detector, eventually his signal would be too low a frequency for you to see him.
Of course, from his perspective, it happens the other way around. Photons (and other particles) coming from the outside will appear to have a much higher frequency and much higher energies than they would otherwise. Even if he could somehow have survived the spaghettification, the high energy particles would rip him apart. This is a common theme, and one of the big puzzles of black holes. After all, if everything — keys and chairs and friends and particles lose their identity when they fall in — where does that information go?
And there would be a lot of particles, too. After all, just as you see your friend running slow, he sees you running fast. Indeed, someone dangling near the edge of a black hole would see the rest of the universe infinitely sped up. He could literally see the entire future of the universe.
Sort of. This only works if we can dangle someone just outside the black hole without them falling in. Supposing they were actually falling, they'd cross the event horizon and barely notice it (except for the dying part). From the perspective of people (and particles) inside the black hole there is no paradox. Everything falls in in a perfectly reasonable amount of time.
How reasonable? Well, I suppose I'd better answer the original question. Let's see you dropped your friend into a black hole the mass of the sun, and let him go at the same distance the earth currently is from the sun. It takes a surprisingly long time to fall in (from his perspective), a bit over 2 months. Of course, except for the last second or so, this is pretty uneventful. In fact, up until the last minute or so, your friend isn't even traveling an appreciable fraction of the speed of light and is so far outside the event horizon that you two could have a perfectly normal, nearly time-synchronized, conversation.
But after your friend falls in, and he tries to tell you how long it took, he's just SOL. Remember, nothing can escape, not even light. But of course, you knew that already.
Dave Goldberg is the author, with Jeff Blomquist, of "A User's Guide to the Universe: Surviving the Perils of Black Holes, Time Paradoxes, and Quantum Uncertainty." (follow us on facebook or twitter.) He is an Associate Professor of Physics at Drexel University. Feel free to send email to firstname.lastname@example.org with any questions about the universe.