Black holes

[sol invictus]

As always, i appreciate your corrections. The spacetime curvature is radical at the central singularity, and the cumulative effect of the curvature within the event horizon has a radical effect on what outside observers see as their probes approach the event horizon, but I should have said what you said about curvature at the event horizon itself.

Exactly right.

[size=-1](A couple of days ago, I hand-calculated the Christoffel symbols for Lemaître coordinates, but I haven't yet calculated the components for the Riemann or Ricci tensors. The purpose of this tedious calculation is to motivate myself to learn the identities and more sophisticated methods.)[/size]​

If you've got the Christoffel symbols, you're more than halfway there. You can easily use them to calculate the acceleration necessary to stay out of the hole, and you'll see it goes to infinity at the horizon.

The next step is to calculate the Ricci scalar - but sadly, it is exactly zero and won't tell you anything (except that you've correctly solved Einstein's equations). So to get a sense of how big the curvature is, you'll need to calculate something like the Riemann tensor squared. Depending on your technique the calculation may be a bit hairy, but the answer is extremely simple and anyway can be guessed (up to numerical coefficient) from dimensional analysis.

 

[sol invictus]

This has always puzzled me; if this is true, then wouldn't it apply to the mass of the collapsing star itself? If so, then there should be no black holes?

Think of a intrepid investigator fish shouting over its shoulder as it approaches and then crosses the sound horizon. To its friend upstream (swimming against the current so that it holds level with the banks), the shouts get fainter and fainter and spaced farther and farther apart as the investigator fish approaches the horizon.

Its last shout right as it crosses will take a very long time to reach the friend, because the current was so strong the shout was barely moving (relative to the banks) when it was emitted, so it took a long time to get upstream. Measured in units of sound time, therefore, nothing ever falls into the sound horizon - time slows down and stops there, in fact.

But did the investigator fish "really" never cross the horizon? Is it "really" frozen forever just above the horizon? That seems like an absurd conclusion, and that goes equally for black holes. The only real difference is that we're pretty sure that it's fundamentally impossible for anything to ever escape from a black hole, and that turns out to make that view logically self-consistent - but it doesn't make it necessarily correct.

 

[Tubbythin]

Black holes are also interesting because you can play pong with them.

 

[Brian-M]

No, it can't. The event horizon is by definition a surface that nothing can cross out of. One way to visualize that is to think of it as moving out at the speed of light. Anything emitted from inside the hole can't catch up with the horizon. You might wonder how that can make sense, given that the black hole isn't expanding - but it's because the gravitational field is so strong, moving out at the speed of light is just enough to stay in place at the horizon.

A useful analogy is a river with a current that grows stronger and stronger as it approaches a waterfall. Let's say the speed of the current exceeds the speed of sound in water at some fixed distance from the waterfall. Mini-submarine probes swimming around in the river will never be able to send any sound pulses back across that line - it's a sound horizon.

In a physics-for-poets metaphor: The black hole's gravity is so strong that it's pulling space itself into the central singularity faster than outwardly directed light can travel through that space.

So the message I'm getting from this is that space itself is expanding at the speed of light at the edge of the event horizon?

Let me explain how I was thinking of this, so you guys can tell me where I'm going wrong.

Imagine the classic rubber-sheet analogy for relativity. A black-hole would be a very deep depression in the sheet.

I was thinking of the event horizon as being the equivalent to the depth at which a ball would have to be thrown at faster than the speed of light in order to leave the depression completely. So someone below that point couldn't write a message on a ball and throw it to someone outside the hole, the ball would just fall back down before reaching the top. But someone else above that point could read the message and throw their own message-ball to someone outside the hole.

So translating this analogy to the real-world, photons could pass out from the event horizon, but either their path would be so curved that they'd pass back into the event horizon, or they'd end up redshifted down to nothing within a finite distance of the horizon. Either way, someone very close to the event horizon would be able to intercept and retransmit a message before the information effectively disappeared.

So this way of looking at it is completely wrong?

 

[vladi]

I was thinking of the event horizon as being the equivalent to the depth at which a ball would have to be thrown at faster than the speed of light in order to leave the depression completely. So someone below that point couldn't write a message on a ball and throw it to someone outside the hole, the ball would just fall back down before reaching the top. But someone else above that point could read the message and throw their own message-ball to someone outside the hole.

The analogy with the rubber sheet is right but everyone that is above the point, i.e. above the event horizon, even a millimeter is still outside the hole and can't receive messages from inside the hole. So someone above the point can never read the message on the ball because the ball can never reach him. And he can't just "see" the message on the ball through the event horizon, because no light can escape.
You can't cheat because the very definition of event horizon is the distance from the center of an object such that, if all the mass of the object were compressed within that sphere, the escape speed from the surface would equal the speed of light. And nothing can travel faster than light so nothing can escape. Events inside cannot affect an outside observer. So if you can receive messages from someone it means that he is still outside the event horizon.

 

[Vorpal]

Depressions on rubber sheets aren't a very good analogy to begin with, but they're even less useful for the interior of black holes, where spacetime is not stationary (but rubber sheet are). You won't get a signal out no matter how many relays there are.

On the other hand, sol invictus's analogy of a sound horizon in water that travels supersonically with respect to the riverbank is pretty exact; in the frames of observers freefalling from rest at infinity, it does look like space itself is falling towards the black hole at the local escape velocity. Throwing balls against against the current won't enable you to to make progress with respect to the riverbank unless you can throw them faster than the current to begin with.

 

[dlorde]

So this way of looking at it is completely wrong?

Not completely, but I don't think it represents the situation as well as the waterfall or whirlpool analogy where the medium (water/spacetime) itself is accelerating into the 'hole'.

 

[Brian-M]

You can't cheat because the very definition of event horizon is the distance from the center of an object such that, if all the mass of the object were compressed within that sphere, the escape speed from the surface would equal the speed of light. And nothing can travel faster than light so nothing can escape. Events inside cannot affect an outside observer. So if you can receive messages from someone it means that he is still outside the event horizon.

It's my understanding that escape velocity is the minimum speed that an object at a given point needs to be traveling so that it will continue to travel away from the mass indefinitely, instead of falling back in.

For example, the escape velocity from the surface of the moon is about 8568 km/h (5324 MPH). If you were sitting in a small crater so that you were just a few meters below the surface of the moon, and threw a ball directly upwards at 20km/h the ball would fall back down onto the surface... eventually^*. In the meantime, it'll have passed above the surface, and can be seen by observers above the surface.

So just because an object is traveling at less than escape velocity of the surface does not mean that it can't temporarily rise above surface height. As I understood it, in the case of a black hole, the event horizon is simply a name for an imaginary surface defined as the point where escape velocity is equal to C.

If space were Newtonian, there's no reason why objects below the event horizon couldn't temporarily rise above the event horizon, just like the ball temporarily rising above the surface of the moon.

Your explanation is unhelpful because it gives no reason why this isn't the case.

But what the others are saying is that space itself is endlessly stretching out, making moving at a velocity of C away from the center of mass more like an aeroplane capable of flying at a maximum 500km/h trying to fly into a headwind of 500km/h or greater.

Which gives me a good idea as to why it isn't possible.


^* But if you built a catapult to fling the ball upward at 9000 km/h, the ball would just keep on going, and never fall back down to the moon.

 

[phunk]

The event horizon isn't the point where escape velocity is light speed. If that were the case, you could escape still. The event horizon is the point where even moving at light speed, you can't move any farther away from the center at all.

 

[Brian-M]

The event horizon isn't the point where escape velocity is light speed. If that were the case, you could escape still. The event horizon is the point where even moving at light speed, you can't move any farther away from the center at all.

Thanks for explaining that. I was just coming back to ask that very question. (I'd added it to my post as an ETA, and then removed it so I could think about it a bit first.)

ETA: I'd been under the false impression that the event horizon was the point where escape velocity became greater than C. My idea for the probes would work to relay information back from that point.

 

[Halfcentaur]

Something I've wondered recently, how do black holes emit things? I hear of black holes emitting energy, but how does this escape their gravity?

 

[vladi]

Black holes are thought to evaporate through Hawking radiation
And also the matter accelerated towards the black hole emits x-rays http://en.wikipedia.org/wiki/Black_hole#Accretion_of_matter and that's one way to detect black holes.

 

[Brian-M]

Something I've wondered recently, how do black holes emit things? I hear of black holes emitting energy, but how does this escape their gravity?

I can answer that one. Virtual particles. Pairs of particles that spontaneously pop into existence and immediately cancel each-other out. Except when one of them falls into a black hole, leaving the other one as a real particle that appears to have been emitted from the event horizon.

Or something like that.

ETA: Looks like Vladi beat me to it. From the article he links to...

A slightly more precise, but still much simplified, view of the process is that vacuum fluctuations cause a particle-antiparticle pair to appear close to the event horizon of a black hole. One of the pair falls into the black hole whilst the other escapes. In order to preserve total energy, the particle that fell into the black hole must have had a negative energy (with respect to an observer far away from the black hole). By this process, the black hole loses mass, and, to an outside observer, it would appear that the black hole has just emitted a particle.

 

[Roboramma]

Something I've wondered recently, how do black holes emit things? I hear of black holes emitting energy, but how does this escape their gravity?

A similar question that I once asked and had answered, but can't really recall the answer to (or maybe just never understood): how do black holes manage to have electromagnetic fields? How can the world outside "see" their fields, if nothing can escape from the black hole?

 

[Vorpal]

Nothing can escape the black hole, and that includes all information about what the charge does once inside the horizon. Thus, the external field must stay the same as it was when the charge approaches the horizon. So the no-escape property of the black hole, rather than something to be overcome, is actually exactly what enables black holes to have them in the first place.

In somewhat more specific terms, the electric field of a charge near the horizon is so strongly distorted by the black hole that it appears radial to a far observer, and the horizon ends up acting as if the charge is smeared across its entirety.

 

[Libra]

Just as ignorant and on this, as the OP poster, but just as interested in the topic too.

The rubber anology, but with the BH as a well in the rubber, still assists visualisation of it for me. If the gravity inside the BH appraches infinity, would this increase the density of the matter in the BH to infinity too ? If so, if the density in the BH approaches infinity, would that not mean that the dimensions of what ever matter there is in it would then approach NIL ? Is it still matter if this is the case ?

A very interesting hypothetical layman scenario that I read some time ago, still fascinates me. The author proposed that if the BH is big enough, like the size of our solar system, the tidal forces at the event horizon of such a BH would not be that strong, to such an extent that the hypethetical observer entering such a BH might even then stay in-tact and not go through the elongation as one would in a small BH with much larger tidal forces at the EH.

Once in, the observer would then accelerate until C and time will slow down drastically. This is where the "suspended-animation object, a "frozen star" " description from the top helps me too.

The drastic time-dilation would then mean that if the observer, beyond the EH, looks forward, he would then see all the objects that has fallen into the BH in the past, and if he looks back the way he came in, he would see all the objects that are still to fall into the BH in the future.

 

[Corsair 115]

Black holes... dark matter... dark energy... the universe is populated with some pretty weird things. Which I guess is what makes it so interesting.

 

[Vorpal]

If the gravity inside the BH appraches infinity, would this increase the density of the matter in the BH to infinity too ? If so, if the density in the BH approaches infinity, would that not mean that the dimensions of what ever matter there is in it would then approach NIL ?

Yes, it would, but whether that actually happens in reality is anyone's guess at this point.

Once in, the observer would then accelerate until C and time will slow down drastically. This is where the "suspended-animation object, a "frozen star" " description from the top helps me too.

But that's not really helpful. See post #22 here.

The drastic time-dilation would then mean that if the observer, beyond the EH, looks forward, he would then see all the objects that has fallen into the BH in the past, and if he looks back the way he came in, he would see all the objects that are still to fall into the BH in the future.

The first part's true in a very idealized sense, but the second part is just plain wrong.

 

[sol invictus]

The event horizon isn't the point where escape velocity is light speed. If that were the case, you could escape still. The event horizon is the point where even moving at light speed, you can't move any farther away from the center at all.

Thanks for explaining that. I was just coming back to ask that very question. (I'd added it to my post as an ETA, and then removed it so I could think about it a bit first.)

ETA: I'd been under the false impression that the event horizon was the point where escape velocity became greater than C. My idea for the probes would work to relay information back from that point.

It's not a false impression, and your idea does not work at that point. The event horizon is the point where the escape velocity is light speed, in the sense that light (or a very highly relativistic particle) emitted at any larger radius can escape to infinity.

What's going wrong with your intuition is that you're forgetting that c is a special speed, and relativistic corrections are extremely important here.

Instead of characterizing escape of some projectile by a velocity, use kinetic energy. Then the condition for escape is simple - it's that the kinetic energy must be equal to or greater than the projectile's gravitational potential energy at the surface. The smaller a fraction of the potential energy the kinetic energy is, the less far up away from the surface the projectile will make it before falling back.

So what happens on the surface of a black hole? The gravitational potential goes to minus infinity. That means the requisite escape kinetic energy is infinite. So any object with finite energy will not travel up at all.

All of the above is slightly wrong - you can't really talk about gravitational potential at a horizon, since potential is a Newtonian concept - but hopefully it shows you where your intuition may be breaking down.

 

[sol invictus]

A very interesting hypothetical layman scenario that I read some time ago, still fascinates me. The author proposed that if the BH is big enough, like the size of our solar system, the tidal forces at the event horizon of such a BH would not be that strong, to such an extent that the hypethetical observer entering such a BH might even then stay in-tact and not go through the elongation as one would in a small BH with much larger tidal forces at the EH.

Once in, the observer would then accelerate until C and time will slow down drastically. This is where the "suspended-animation object, a "frozen star" " description from the top helps me too.

The drastic time-dilation would then mean that if the observer, beyond the EH, looks forward, he would then see all the objects that has fallen into the BH in the past, and if he looks back the way he came in, he would see all the objects that are still to fall into the BH in the future.

As Vorpal says, the last part is wrong (the whole "frozen star" thing can be very misleading).

Here's what I wrote in another thread on this topic:

Alice and Bob step out of the hatch of their spaceship near a black hole. The ship is on autopilot, and continually blasting its rockets just enough to stay a fixed distance from the horizon. Alice steps out first, followed a few seconds later by Bob. Neither has any means of maneuvering in space.

What they will see is the rocket accelerate away from them, picking up speed as it moves away. They themselves will stay almost exactly a fixed distance apart, at relative rest. This continues as they approach and then cross the horizon. They will continue to see the rocket the entire time (although it moves further and further away), and they will remain almost at rest with respect to each other.

In other words, what they will see is almost identical to what they would see if they stepped out of an accelerating rocket in flat, empty space - unsurprisingly, since that's more or less a consequence of the equivalence principle.

The difference is that some time after crossing the horizon they will start to feel some unpleasant stretchings as their bodies are acted on by tidal forces. Shortly thereafter, they will be spaghettified and ripped apart as they approach the singularity (ouch).

This ignores any other objects falling in, stars behind the hole getting lensed, etc., and it also assumes the hole is fairly big.