Black Hole Thought Experiment

[69dodge]

It is true that nothing special happens to the spacetime near the event horizon of a large black hole. However the horizon is not really a stationary surface in the usual sense - it's a little better to think of it as a surface accelerating out away from the hole with infinite acceleration. Because the spacetime as a whole is curved, the horizon remains in one place as viewed by distant observers. But locally, it's equivalent to a surface undergoing infinite acceleration.*

*Think of a river as it approaches a waterfall. The velocity of the water gets larger and larger as you approach the falls, and at some point it's equal to the speed a lightfish could swim in still water. A lightfish at that point would have to propel herself (accelerate) upstream as fast as she could just to remain stationary. That's the lightfish horizon - any lightfish that floats past it will go over the falls. The main difference with a light horizon is that because the speed of light is the maximum speed attainable even after an infinite amount of acceleration, a black hole horizon requires infinite acceleration to remain stationary at.

I try to think of it that way, but I keep getting velocity and acceleration confused.

Does light accelerate infinitely hard as it travels, or does it just travel leisurely at constant speed?

 

[sol invictus]

I try to think of it that way, but I keep getting velocity and acceleration confused.

Does light accelerate infinitely hard as it travels, or does it just travel leisurely at constant speed?

It's more or less the same thing.

Imagine a surface accelerating at a very large (but finite) rate. Send a light beam chasing after it. The beam will eventually overtake the surface as long as it started close enough to it, but it will take a long time for that to happen. Since the surfaces parallel to but outside of a black hole horizon behave that way (they accelerate out), you can use this to understand why light takes a very long time to escape from near the horizon.

Now imagine the acceleration of the surface goes to infinity. Light can never catch it, unless it starts right on top of it - in which case it travels along with it - because the surface reaches the speed of light immediately.

So the infinitely accelerating surface moves just like light does, apart from one thing - if it accelerates infinitely forever, there's an instant where it's at rest as it reverses direction. But the two halves of its motion are lightlike.

 

[sol invictus]

Is there any reason the space craft couldn't be orbitting ome meter above the black hole (ignoring the issue of what it might run into)?

Yes, there is. The stable orbit nearest to the singularity is 1.5 times as far from it as the event horizon is.

Could somebody provide a quantitative estimate of the variation in gravity across one meter for something that was very close to the event horizon of a large black hole. Presumably the bigger the black hole the smaller the gravity gradient, but I sit back and wait for edification on that.

It depends not just on the size of the hole, but on the closeness of the object to the event horizon. In the limit of a very large hole the acceleration is simply proportional to the inverse distance to the horizon (as measured in the standard Schwarzschild metric). The proportionality constant is c^2 by dimensional analysis, so the force on a point particle of mass m a distance d from the horizon is m c^2/d times a constant of order 1. Note that there is no dependence on G, the gravitational constant - that's for the reason I was trying to explain above.

Assuming for a second that the black hole is large enough the gravity gradient is small enough across a one meter space craft that things aren't completely shredded then my guess is that from the point of view of the guy in the space craft he can put his arm out and pull it back. But from the point of view of a distant observer pulling the arm back takes an infinite amount of time.

No, because the force on his arm goes to infinity when it gets to the horizon.

OK, so there is an infinite force gradient across the event horizon if you are accelerating away from it (essentially the amount of the acceleration required), and it will procure the proffered pfinger. If you are going with the (gravity) flow, you don't see/feel it until you get much closer to the singularity and he spaghettification starts in. Fair?

Yes.

BTW, I believe that spaghettification from tidal forces has no direct relationship to the event horizon. For small black holes it happens outside the horizon, for large ones inside.

For freely falling observers, correct.

I don't know where I read the stuff about radiation inside the event horizon piling up as it attempted to exit outwards. Sorry.

It wasn't completely wrong - no need to apologize.

Please note that I am not a physicist and could be wrong, but I do not see any reason why you would automatically lose your finger, provided that the black hole is large enough for the gravitational gradient to be small. Think of a galaxy sized black hole.

The size of the hole only matters if it's small compared to the object we're discussing. For large holes, where the horizon can be approximated as a flat surface when the object is close, my analysis above applies.

Can you explain this to me? I understand that if your head is past the EH and you decide to wiggle your toes, you won't be able to because the original nerve signal is also past the EH and therefore can't escape.

However, I don't understand how that would change once your entire body was past the EH. Surely as long as your head stayed closer to the singularity than your feet, the nerve signal would never be able to reach your feet.

Someone holding themselves at fixed distance above the horizon (head first) is like standing in a very powerful wind blowing up from your feet to your head. When the wind gets to the speed of sound, no matter how loudly you shout the sound will never reach your feet.

But if you let go and fly along with that wind, you can talk to your feet to your heart's content.

 

[bellonax]

As you fall head-first, your feet catch up to where your head was when it sent the nerve signal, and then they even pass that position. So there's no reason why they couldn't get the signal. If you like: the signal doesn't reach them, they reach the signal.

But I still don't get this. The nerve impulse has to travel to your feet, right? I thought the idea was that, once you were past the Event Horizon, you cannot move away from the black hole, nor stay stationary. So surely any nerve impulses will never get to any parts of your body because they can't actually travel away from your head, nor stay where your head was until your feet catch up? They're being dragged towards the BH too?

Am I being a complete moron here?

Someone holding themselves at fixed distance above the horizon (head first) is like standing in a very powerful wind blowing up from your feet to your head. When the wind gets to the speed of sound, no matter how loudly you shout the sound will never reach your feet.

But if you let go and fly along with that wind, you can talk to your feet to your heart's content.

Sorry, I'm afraid I don't get this analogy.

 

[69dodge]

But I still don't get this. The nerve impulse has to travel to your feet, right?

It doesn't have to travel to where your feet were when your brain sent it. All that's required is that, at some point in time, it and your feet be in the same place.

I thought the idea was that, once you were past the Event Horizon, you cannot move away from the black hole, nor stay stationary.

Yes, that's right.

So surely any nerve impulses will never get to any parts of your body because they can't actually travel away from your head, nor stay where your head was until your feet catch up? They're being dragged towards the BH too?

Yes, nerve impulses are being dragged towards the black hole too, and they do fall in, but the rest of you falls in even faster. So the impulses and your feet still meet.

 

[davefoc]

The point of this question is, I think, whether there is something special about the event horizon that the guy in the space craft is going to sense.

My sense of it is that the answer would be no.

Assuming a spaceship with the power and strength to support the guy near the event horizon, the guy inside the space ship would have been squished into a slab of some kind homogeneous lump, maybe just pure quarks.

So talking about what happens when the guy sticks is hand into the event horizon is a bit problematic when under no conceivable circumstances could the experiment be done and doing the experiment somewhat above the event horizon or somewhat inside the event horizon would produce the same result.

 

[drkitten]

But I still don't get this. The nerve impulse has to travel to your feet, right? I thought the idea was that, once you were past the Event Horizon, you cannot move away from the black hole, nor stay stationary

My understanding is that this is not correct. I may be misremembering my physics, but as I remember it, a black hole is defined as having an escape velocity exceed the speed of light, which means that it takes an infinite amount of energy to get infinitely far away from the black hole.

That does not mean that it takes an infinite amount of energy to get six inches away from the black hole.

Nothing keeps you from "standing" on the "surface" of a black hole and jumping, but you'll always fall back to the surface.

My understanding is that you could even have a black hole with the surface gravity (and tidal forces at the surface) being reasonable/survivable if the black hole were large enough (and thus the singularity were far enough away). From the standpoint of a bunch of cavemen living on such a black hole, it wouldn't look much different than living on earth. You throw stuff up, it falls down.

 

[Nancarrow]

My understanding is that this is not correct. I may be misremembering my physics, but as I remember it, a black hole is defined as having an escape velocity exceed the speed of light, which means that it takes an infinite amount of energy to get infinitely far away from the black hole.

That does not mean that it takes an infinite amount of energy to get six inches away from the black hole.

Nothing keeps you from "standing" on the "surface" of a black hole and jumping, but you'll always fall back to the surface.

My understanding is that you could even have a black hole with the surface gravity (and tidal forces at the surface) being reasonable/survivable if the black hole were large enough (and thus the singularity were far enough away). From the standpoint of a bunch of cavemen living on such a black hole, it wouldn't look much different than living on earth. You throw stuff up, it falls down.

I think you're confusing the concept of a Newtonian 'dark star' with the concept of an Einsteinian 'black hole'. A black hole isn't JUST something so heavy that the escape velocity exceeds light a certain distance from it - it's also a curvature of spacetime. Once a particle reaches the event horizon, it's going nowhere except in. It isn't going away from the singularity, not even by a fraction of a millimetre.

 

[sol invictus]

Nothing keeps you from "standing" on the "surface" of a black hole and jumping, but you'll always fall back to the surface.

That isn't quite right. You'd have to jump at the speed of light to get up off the horizon at all.

which means that it takes an infinite amount of energy to get infinitely far away from the black hole.

That's correct - the horizon is the surface from which escape to infinity would cost infinite energy. But that actually implies that it's completely impossible to go from on or inside the horizon to any region outside, no matter how close.

The reason is that the gravitational potential energy goes to zero at infinite distance (it doesn't rise indefinitely because the spacetime far from the hole isn't affected by it, and because Newtonian gravity applies there and tells us the energy tails off like 1/r). So the only way the horizon can be infinitely far down a potential energy well like you say is if the gravitational potential goes to minus infinity at the horizon, meaning it would require infinite energy to move any distance off of.

 

[Fredrik]

Can you explain this to me? I understand that if your head is past the EH and you decide to wiggle your toes, you won't be able to because the original nerve signal is also past the EH and therefore can't escape.

However, I don't understand how that would change once your entire body was past the EH. Surely as long as your head stayed closer to the singularity than your feet, the nerve signal would never be able to reach your feet.

You already got some good answers, in particular from 69dodge, but I'll give you my answer as well. You're of course right that the nerve signal can't get from inside the horizon to outside, but we're talking about a person in free fall here. The nerve signal traveling from his head to his feet is at no point moving away from the singularity. It's just moving towards the singularity at a slower speed than the rest of the body, which is in free fall. That's why it can reach his feet. When it does, the feet will be closer to the singularity than the head was when the signal was emitted.

The same thing happens if he uses a gun and tries to shoot himself in the foot. The gun can't possibly increase the distance between the bullet and the singularity, but it can put the bullet on a path that reaches the singularity a bit later than the falling person.

 

[Fredrik]

Nothing keeps you from "standing" on the "surface" of a black hole and jumping, but you'll always fall back to the surface.

Forget about jumping, you won't even be able to stand. The event horizon isn't a solid surface of course, so you would actually have to stand on the floor of a spaceship, and its engines would have to deliver an infinite thrust, which gives the floor an infinite acceleration, which would crush you to infinite flatness.

 

[69dodge]

My understanding is that this is not correct. I may be misremembering my physics, but as I remember it, a black hole is defined as having an escape velocity exceed the speed of light, which means that it takes an infinite amount of energy to get infinitely far away from the black hole.

That does not mean that it takes an infinite amount of energy to get six inches away from the black hole.

Nothing keeps you from "standing" on the "surface" of a black hole and jumping, but you'll always fall back to the surface.

Well, how high can you jump, then?

(Rhetorical question. You really can't jump at all. Starting from six inches above the event horizon, it takes only a finite amount of energy to get infinitely far. So all the rest of the infinite required energy is compressed into those six inches.)

My understanding is that you could even have a black hole with the surface gravity (and tidal forces at the surface) being reasonable/survivable if the black hole were large enough (and thus the singularity were far enough away). From the standpoint of a bunch of cavemen living on such a black hole, it wouldn't look much different than living on earth. You throw stuff up, it falls down.

Surface gravity and tidal force are very different things. At a black hole's event horizon, tidal forces can be small but the surface gravity is always infinite. It's impossible to remain stationary there, unless you're a photon that's "moving" directly up. Otherwise, you're falling through the horizon at the speed of light.

If you're falling freely, you feel only the tidal forces. In that sense, things don't look much different at the event horizon than anywhere else. For example, as you fall through the event horizon, a photon that's "stationary" there passes you at the speed of light, just as photons always do.

 

[shadron]

It wasn't completely wrong - no need to apologize.

The "Sorry" was for stating it without having a reference, the only mortal sin allowed to atheistic skeptics.

 

[bellonax]

It doesn't have to travel to where your feet were when your brain sent it. All that's required is that, at some point in time, it and your feet be in the same place.



Yes, that's right.



Yes, nerve impulses are being dragged towards the black hole too, and they do fall in, but the rest of you falls in even faster. So the impulses and your feet still meet.

You already got some good answers, in particular from 69dodge, but I'll give you my answer as well. You're of course right that the nerve signal can't get from inside the horizon to outside, but we're talking about a person in free fall here. The nerve signal traveling from his head to his feet is at no point moving away from the singularity. It's just moving towards the singularity at a slower speed than the rest of the body, which is in free fall. That's why it can reach his feet. When it does, the feet will be closer to the singularity than the head was when the signal was emitted.

The same thing happens if he uses a gun and tries to shoot himself in the foot. The gun can't possibly increase the distance between the bullet and the singularity, but it can put the bullet on a path that reaches the singularity a bit later than the falling person.

Ah, I get it now. Many thanks for your patience

I was under the impression that everything would fall at the same speed after an Event Horizon. I don't know why.

ETA: Just to clarify - something can use it's speed or strength/power etc to slow it's descent towards a Black Hole, but not to escape it, once it has passed the Event Horizon?