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I understand that the event horizon of a black hole forms at the radius from the singularity where the escape velocity is c. But it's also true that you don't have to go escape velocity to escape an object if you can maintain some kind of thrust. You could escape the earth at 1 km/h if you could maintain the proper amount of thrust for enough time. So if you pass just beneath the event horizon, shouldn't you be able to thrust your way back out, despite the >c escape velocity? Or does this restriction have to do solely with relativistic effects (time stopping for outside observers at the event horizon)?

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The thrust is simply helping you accelerate to a point where you reach the escape velocity. No amount of thrust would be sufficient to let you reach the speed of light. –  Omar Dec 15 '11 at 16:10
    
@Omar: Yes, but in pure Newtonian physics you could escape without ever reaching the surface escape velocity. It's impossible only because of relativistic effects. –  Keith Thompson Dec 15 '11 at 19:22
    
@KeithThompson You are right. In Newtonian physics one needs to get far enough away to a point where he escape velocity is low enough. So no surface escape velocity required. For a BH, however, one needs to be moving at the speed of light just to hover above the event horizon. So in theory if you could reach $c$, you could remain on top of the event horizon. Of course, once inside, you have no chance because of the space-time distortion effects. –  Omar Dec 15 '11 at 20:31
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up vote 17 down vote accepted

@Florin is absolutely right, but sometimes a picture is worth a thousand words. This website has multiple pictures and explanations about how the future light cone starts to point only to the inside of the black hole once you pass the event horizon.

Here is one of the images:

enter image description here

Time is vertical, the cylinder represents the event horizon and the cones are the future light cones for the observer as they fall into the black hole.

Note that even if the observer could instantaneously accelerate to almost the speed of light they would still be confined to the future light cone. So once the event horizon is crossed they cannot escape.

UPDATE: As @Florin says in a comment, rotating black holes are even stranger than than the non rotating Schwarzchild black hole described above. In particular there is a region outside of the event horizon, called the ergosphere where space-time itself is dragged around the black hole at faster than the speed of light (relative to distant stars). It is possible to enter the ergosphere and still escape to infinity. In fact some of the black holes rotational energy can be extracted in this region and may be source of energy for gamma ray bursts.

I'd also like to point out that there is no local experiment that can be performed to determine when the observer has crossed the event horizon. It will "feel" perfectly normal - it is only the future destination of light rays that changes when the horizon is crossed and that cannot be determined locally. In fact the "normal Newtonian" concept of the "force" of gravity doesn't have to be particularly strong at the horizon. A more massive black hole has a lower event horizon "gravity force". A hand waving way of seeing this is that the Newtonian force is proportional to $\frac{1}{R^2}$ but the event horizon radius is proportional to the mass, $M$; so the surface force is proportional to $\frac{1}{M}$. See this Wikipedia article for a more rigorous discussion of what it might even mean to talk about the concept of "force" in General Relativity.

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Excellent example. The problem is not how fast you're moving. The problem is that the "up" direction does not exist anymore; any direction you look, you look down. Spacetime is tremendously mangled, tied into a knot on itself. You can only choose between shorter or longer ways down. BTW, even though all the gauges go to infinity at the center, I believe the time to hit the singularity is actually finite. So your fate is sealed no matter what. –  Florin Andrei Dec 16 '11 at 20:52
    
BTW, for completion - if this is a rotating black hole, I believe I've heard recently that stable orbits inside the event horizon are actually possible, with some people speculating that planets may exist, even harboring life - if the black hole is big enough. So, in other words, if the black hole is rotating, not all directions point down; some directions point "around". But that's all; there's still no "up", or at least no "all the way up". But this whole concept is very very far out. –  Florin Andrei Dec 16 '11 at 20:57
    
"I'd also like to point out that there is no local experiment that can be performed to determine when the observer has crossed the event horizon. It will "feel" perfectly normal" - well, at least upon approaching the horizon the radius of the BH will shrink so that it will finally evaporate when you touch the horizon. You will experience the plank temperature and infinite power flow at that moment. –  Anixx Jul 28 '12 at 7:18
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I wish those who popularize science would stop talking about the speed of light. It just confuses people.

Forget about speed. Think in terms of energy. You'd need an infinite amount of energy to get out of it. Infinite. Meaning, it's like dealing with the Mafia: no matter how much you're spending, it's still not enough.

Think in terms of topology. From the inside of the event horizon, there are no possible trajectories to the outside. This is a purely geometric issue; spacetime is so mangled that any trajectory you could conceivably draw, starting at your current position, goes only to the inside of the event horizon.

Moreover, any trajectory inside the event horizon is only pointing down. That's right, everywhere you look, you look down towards the center.

The 3 spacelike dimensions and 1 timelike dimension of normal space-time now become 3 timelike and 1 spacelike dimension. All those 3 timelike dimensions terminate at the central singularity. Which means, once inside, you can only go down.

That's how seriously broken spacetime is. It's not a matter of speed, heck even energy doesn't matter, it's that space and time themselves are tied into a big screwball knot that makes no sense and is seemingly designed to suck you in.

Physicists refer to this as the collapse of space-time geometry.

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I don't think it is bad to think in terms of escape velocity. The reason you would need an infinite amount of energy is because you would need to reach $c$. I agree with space-time distortion taking place, but for simple terms thinking of escape velocity is fine. –  Omar Dec 15 '11 at 16:09
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Most people automatically think "so, go faster than speed of light and you'll escape". Which is BS because "speed of light" in the local frame really means infinite speed (it's limited only in a remote frame). This is what most people don't understand because nobody cares to explain it to them - that "speed of light" and infinite speed are the same thing, in the frame of reference of the moving object. Pop-sci articles just treat everything from the p.o.v. of the remote frame, which is nonsense from a relativistic standpoint - the choice of frame is crucial. –  Florin Andrei Dec 15 '11 at 16:59
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What I'm saying is this: In the frame of the moving object, you can cover arbitrarily large distances in arbitrarily short durations; effectively, relativistic effects combine to allow you "infinite" speed - in this frame only. However, in the frame of the external observer, the moving object appears limited to speed of light. Pop-sci only presents things from the external frame, neglecting to show how in the moving object's frame you can go anywhere in as short time as you want. Faster than c is impossible because c in external frame is infinite effective speed in the moving frame. –  Florin Andrei Dec 15 '11 at 19:06
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E.g., you could go from here to the Andromeda galaxy in a second - in the rocket's frame; effectively a huge speed (due to space contraction, etc). However, the trip would still take 2.5 million years in the Earth's frame. Speed of light is never a limit for the rocket itself (and I'm talking here about effective speed - I know c remains the same in any frame). It's only a limit for those left behind. –  Florin Andrei Dec 15 '11 at 19:08
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I'm afraid that is not the case. There are no special frames of reference. Laws of Physics hold in all frames of reference therefore you cannot go faster than the speed of light in any frame of reference. Time dilation and length contraction effects will mean that both observers, travelling and stationary, will measure the same speed. –  Omar Dec 15 '11 at 19:34
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I completely agree with @FrankH's answer. I just want to emphasize the underlying misconception many authors give when talking about escape velocities from black holes. One can have a "black hole" of sorts in Newtonian physics, sans GR, and indeed this has been known since the 18th century (see the Wikipedia page, for instance). All you do is find the radius outside a mass M where the escape velocity is equal to the speed of light.

Unfortunately, this radius corresponds exactly to the Schwarzschild radius, $2GM/c^2$, leading many to think the Newtonian arguments apply somewhat. One key difference is what is meant by "escape." The escape velocity is the speed you'd have to go to head off to infinity with no further thrust. With any less velocity, you could move a certain nonzero distance away from the surface of a Newtonian black hole before being pulled back in. Given a source of thrust, the argument in the original question shows that you could always escape such an object. With GR, things are different, and you are never allowed to go any amount above the event horizon, no matter how much rocket fuel you have on hand.

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It is actually possible to get out of the horizon. To get out one only have to reach speed higher than c.

But no object that bears information can reach that speed, this would violate casualty. So only things that bear no information can get out from inside the horizon.

One such thing is the Hawking radiation (not only photons but also particles) which can be viewed as quantum tunneling out of the BH. Note that in all cases of tunelling (say at nucleus decay) the emitted particle reaches higher than c speed. This does not violate casualty because the process is probablistic and as such cannot be used for iunformation transfer.

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When scientists say the escape speed of a black hole is greater than the speed of light, they mean that if you leave the surface of black hole at a speed which is near the speed of light, and during the trip you do not get extra boost of speed, then eventually you will fall back. That is absolutely true. But if you get a speed boost during the trip, you can maintain your speed to be near the speed of light, then eventually, you will move to a orbit whose escape speed is lower than the speed of light, then you can escape the black hole. Thus the myth of "nothing can escape black hole" is wrong.

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I'd like to downvote but don't want to lose points. Anyway, this answer is completely wrong. –  JLA May 10 '13 at 5:27
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I'll sacrifice my rep. This is so wrong. It would be true classically only. But classically the speed of light isn't a limit which still makes this somewhat wrong. –  Brandon Enright May 10 '13 at 6:14
    
I did not say the speed of spaceship is faster than speed of light. What I said is that spaceship maintain the speed to be near the speed of light, e.g, 99% of speed of light. This speed is achievable. By this speed, spaceship should be able to reach the orbit whose escape speed is less than the speed of light. –  Larry May 10 '13 at 6:42
    
There is no such point inside a black hole where you would need to travel at less than c to escape, so you're answer is still completely wrong. Once you enter a black hole, you can't even turn around, much less escape. –  JLA May 10 '13 at 7:02
    
Jla, you just repeated your answer multiple times without detailed explanation. –  Larry May 10 '13 at 7:11
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protected by Qmechanic May 10 '13 at 6:50

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