Black Hole Paradox?

[Qfwfq]

Gosh, I think we're getting a bit lost now and it's getting late over here.

 

You have to ignore that impossibility to get on with Slinkey's paradox. Both Alice and Bill, in other words, are expecting Bill to age infinitely more than Alice (despite the fact that the black hole is short lived) because they are expecting something unphysical to happen.

Er, I'm not sure what Alice and Bob are expecting, they never told me, but maybe it's Slinkey that's expecting something unphysical to happen. There is no paradox because they can't compare elapsed times (as with the twin paradox in SR). If instead Alice activates her rocket before it's too late, they can meet again; she will have aged less than him but the ratio will be finite. There is no problem for those who properly understand the principles.

[modest]

Gosh, I think we're getting a bit lost now and it's getting late over here.

but, that's the best time to be lost.

 

Er, I'm not sure what Alice and Bob are expecting, they never told me, but maybe it's Slinkey that's expecting something unphysical to happen.

Right.

 

There is no paradox because they can't compare elapsed times (as with the twin paradox in SR). If instead Alice activates her rocket before it's too late, they can meet again; she will have aged less than him but the ratio will be finite. There is no problem for those who properly understand the principles.

I certainly hope I didn't imply otherwise.

 

The point is that both (Alice and Bill) agree that Alice cannot get out in a finite amount of Bill's proper time. Solving from either perspective gives that result. If both Alice and Bill make the same unreasonable assumption (the assumption of the paradox) they both come to the same unreasonable conclusion.

 

Let me put this another way... the OP is lost in the idea that Alice is spending an infinite amount of Bill's proper time at the event horizon. But, Bill would never come to that conclusion unless he first assumed that she was using an infinite amount of acceleration to sit at the horizon. If she is freefalling then he could use freefall coordinates in which time dilation doesn't go infinite.

 

Alice would likewise not assume that Bill's clock is running infinitely fast unless she were sitting on the horizon. It is the infinite acceleration that makes it impossible and it is the infinite acceleration that makes time dilation go infinite. That is the whole source of the problem.

[Qfwfq]

Nah Modest, that was the best time to go to bed and get a few winks! :hihi:

 

Anyway, I didn't think Slinkey was seeing it that way so I went to check the OP and discovered that it wasn't Craig that brought in Hawking radiation and I hadn't taken due notice of:

According Hawking-Bekenstein Radiation, a BH does not exist for an infinite amount of time because it will eventually evaporate and we can calculate a finite time for its existence.

 

As it takes an infinite amount of time to watch someone reach and cross the EH for all reference frames outside the EH, and the BH will evaporate in a finite time, can we ever see anyone reach and cross the EH if we are outside the EH?

So now I see what the problem is, and let me take a stab at it.

 

If Bill does hang around long enough to see the BH evaporate, he can't apply the Schwartschild coordinates for a constant M and this must be allowed for in the computations of coordinates. It means that [imath]R_{\rm S}[/imath] is decreasing too. This can complicate things greatly, according to the actual quantities in question and I'm unsure if anyone has ever undertaken a full analysis of how it is described in the freefalling coordinates. One thing is sure: the time ordering of events can't change according to physically meaningful coordinate choice. If the BH evaporates before she falls through the EH, it must be so in her coordinates too because, in them, the duration of the BH must be accordingly less than in Bill's coordinates.

[JMJones0424]

Yes, but I don't think the original question was what happens if the black hole evaporates before Alice crosses the event horizon.

 

Imagine we are watching someone fall towards a black hole (BH) from a given distance above the event horizon (EH).

 

According to every book I have read, for observers outside the EH it will take an infinite amount of time for the person to reach the EH due to gravitational time dilation. ie. we will never see them cross the EH.

 

...

 

According Hawking-Bekenstein Radiation, a BH does not exist for an infinite amount of time because it will eventually evaporate and we can calculate a finite time for its existence.

 

As it takes an infinite amount of time to watch someone reach and cross the EH for all reference frames outside the EH, and the BH will evaporate in a finite time, can we ever see anyone reach and cross the EH if we are outside the EH?

 

If we cannot see someone reach and cross the event horizon from one reference frame, then how can they fall in and be destroyed from their reference frame?

 

Would there not be two distinct histories?

 

For my edification-

 

1) The bold section of the original post, that there are two distinct histories... There are obviously two distinct histories of observations, but does this mean that there are two distinct histories? Alice only has one history, despite the fact that some of it lies outside of Bob's possible observation.

 

2) In the underlined section, Bob will never see Alice cross the event horizon, even given an infinite amount of time, and even if the blackhole doesn't evaporate.

 

3) Because black holes do not have an infinite lifespan, the event horizon will recede as the mass is reduced. As this happens, does Bob see increasingly more of Alice's fall towards the singularity? If Bob watches the black hole until it entirely disappears, would he be able to see the entire history of Alice's doom? Or is this question akin to asking how many angels can dance on the head of a pin, since redshift makes observation past a certain point impossible? If so, is this not a loss of information, which was the whole problem with Hawking radiation in the first place? Or is information of events outside of your future light cone irrelevant, even if the event's future lightcone is warped back into your future lightcone as the black hole dissipates?

Edited November 24, 2011 by JMJones0424

[modest]

If Bill does hang around long enough to see the BH evaporate, he can't apply the Schwartschild coordinates for a constant M and this must be allowed for in the computations of coordinates. It means that [imath]R_{\rm S}[/imath] is decreasing too.

That's a very good point. A good way of looking at it might be to compare the usual type of diagram to one where the horizon is shrinking. Ordinarily...

 

[source - near the end of the page]

 

before she reaches the horizon she crosses all of the lines of simultaneity in Bob's future. She could get close to the event horizon then turn about and meet up with him anywhere in his future. The only conceptual limit to how far in his future would be how close she got to the horizon.

 

More to Slinky's scenario, if she crosses the horizon while sending out a message, the message would be diluted into eternity.

 

But, if the black hole is evaporating it might be more like...

 

 

where she wouldn't cross S9 or any subsequent instant of Bill's proper time. She would never be simultaneous with events that happen after the hole evaporates, and if she were sending Bill a signal while crossing the Horizon it would only be diluted up to S6. The last photon Bill would get from the signal would be (R-R_S)/c after S6. After that time he would see nothing of her.

 

Would you agree?

 

EDIT:

 

The ladder would need to have infinite tensile strength when it reaches the EH and puh-leeze let's not discuss about when it were to pass through it...

I forgot to mention the ladder was made of adamantium :doh:

[modest]

Alice only has one history, despite the fact that some of it lies outside of Bob's possible observation.

What is Bob doing during that history? How would she describe him?

[JMJones0424]

What is Bob doing during that history? Observing the black hole from outside of the event horizon and settling in for a long wait for it to evaporate.

 

How would she describe him? He'd be increasingly blue shifted until she either gets ripped apart by gravitational forces or gets fried by infalling, and now increasingly energetic, radiation and particles.

 

I should have said-

 

1) The bold section of the original post, that there are two distinct histories... There are obviously two distinct histories of observations, but does this mean that there are two distinct histories for Alice? Alice only has one history, despite the fact that some of it lies outside of Bob's possible observation.

 

Right?

[modest]

1) The bold section of the original post, that there are two distinct histories... There are obviously two distinct histories of observations, but does this mean that there are two distinct histories for Alice? Alice only has one history, despite the fact that some of it lies outside of Bob's possible observation.

 

Right?

I think Slinky would say that Alice never leaves Bill's observation. He would say... if the black hole lives 100 years according to Bill's clock, Alice will never cross the event horizon in that time so she survives. The black hole evaporates before she can reach the horizon. Alice would come to a different conclusion. She survives in one history and dies in the other, which would be truly paradoxical.

 

What is Bob doing during that history? I don't know, it doesn't matter.

 

How would she describe him? He'd be increasingly blue shifted until she either gets ripped apart by gravitational forces or gets fried by infalling, and now increasingly energetic, radiation and particles.

His clock speeds up right? By how much would you say?

[JMJones0424]

continuing in the line of thought that Alice and Bob are able to make observations that are actually impossible to make...

 

I think Slinky would say that Alice never leaves Bill's observation. He would say... if the black hole lives 100 years according to Bill's clock, Alice will never cross the event horizon in that time so she survives. The black hole evaporates before she can reach the horizon. Alice would come to a different conclusion. She survives in one history and dies in the other, which would be truly paradoxical.

 

OK, grasping at straws here, but as the black hole evaporates, the event horizon recedes, and Bob observes Alice's progression towards the receding event horizon as the black hole loses mass, until the moment at which the black hole finally evaporates. At the instant that Bob observes the final disappearance of the black hole, he will observe Alice reaching the event horizon.

 

His clock speeds up right? By how much would you say?

 

I couldn't begin to do the math to figure it out, but wouldn't it depend on the mass of the black hole and Alice's distance from the singularity? Alice observes a finite amount of time passing before she falls into the singularity, so she would never see Bob's clock increase to an infinite speed.

[modest]

continuing in the line of thought that Alice and Bob are able to make observations that are actually impossible to make...

 

OK, grasping at straws here, but as the black hole evaporates, the event horizon recedes, and Bob observes Alice's progression towards the receding event horizon as the black hole loses mass, until the moment at which the black hole finally evaporates. At the instant that Bob observes the final disappearance of the black hole, he will observe Alice reaching the event horizon.

That still sounds like two distinct histories. If she never crosses the horizon, I don't see how the singularity could hurt her.

 

I really don't think Slinky's problem is assuming "if you can't see her cross the horizon then she doesn't cross it". The issue seems more about what he sees rather than what he doesn't see. Slinky is assuming that Bill sees the hole evaporate while he sees her approaching the horizon which would be problematic if it didn't happen that way according to Alice.

 

I couldn't begin to do the math to figure it out, but wouldn't it depend on the mass of the black hole and Alice's distance from the singularity? Alice observes a finite amount of time passing before she falls into the singularity, so she would never see Bob's clock increase to an infinite speed.

According to the Schwarzschild metric, Bill's clock approaches infinitely fast from a position infinitely close to the event horizon. In other words, clocks that are distant from the black hole run almost infinitely fast from the perspective of an observer who is just outside the event horizon.

 

This would allow Slinky to turn the thought experiment around and completely avoid Bill's observational problems. Alice can see Bill. If (this is a really big "if") she sees Bill's clock speed up infinitely as she reaches the horizon then he could send her a message from any point in his future. He could send a message saying "I just saw the black hole evaporate". If she gets that message before the black hole kills her then that would be paradoxical.

 

The problem is that these assumptions come from a static metric that assumes the observers aren't moving and the mass isn't changing, but this thought experiment relies on both those assumptions being false.

[Qfwfq]

Would you agree?

Yes. :thumbs_up

 

That's a good page on the topic and it voices my epistemological take about whether we can assert the actual existence of black holes but, gee Modest, I've never read Marvel Comics and I had to look adamantium up. :lol:

 

As this happens, does Bob see increasingly more of Alice's fall towards the singularity? If Bob watches the black hole until it entirely disappears, would he be able to see the entire history of Alice's doom? Or is this question akin to asking how many angels can dance on the head of a pin, since redshift makes observation past a certain point impossible? If so, is this not a loss of information, which was the whole problem with Hawking radiation in the first place? Or is information of events outside of your future light cone irrelevant, even if the event's future lightcone is warped back into your future lightcone as the black hole dissipates?

Well it is a complicated matter. The evaporation of a black hole would not mean that the inner events become observable. It wouldn't even make sense to say that particles in Hawking radiation are the same particles that had been inside. The information issue is the reason why entropy computations have been done on black holes.

 

As Modest points out, in the case where she does fall in, the time according to Bob will be longer but finite.

 

I hope the posts by Modest and the site he linked have helped to clarify matters.

[CraigD]

Both Alice and Bill, in other words, are expecting Bill to age infinitely more than Alice (despite the fact that the black hole is short lived) because they are expecting something unphysical to happen.

I agree. This is consistent with my attempt at a description of the paradox,

I think we’d all benefit from restating in clear, succinct words, this thread’s title paradox. Here’s my take:

The black hole paradox we’re discussing in this thread is due to the theoretical prediction that an observer, Alice, collocated with another, Bill, then falling past the event horizon of a black hole, then somehow later again collocating with Bill far from the black hole (say, after the black hole evaporates) and comparing their clocks, would find their clocks differed by an

infinite

duration of time, a physical impossibility.

I can't think of a way of describing this any better than I have above, but I will try.

 

From Alice's viewpoint: Alice and Bill are collocated. Alice falls towards the BH, crosses the EH and encounters the singularity. Bill and Alice can never again collocate.

 

From Bill's viewpoint: Alice and Bill are collocated. He watches Alice fall towards the EH. He will never her see her cross the EH and the BH will eventually evaporate. Alice is seen by Bill to always be above the EH and has not crossed it. When the BH has evaporated Alice collocates with Bill and compares clocks. The difference in the their clocks will be finite.

Although Bill can never see Alice cross the EH because of gravitational redshift of the light from her by which we would, I don’t think it follows that, after the BH has evaporated, Bill will see that Alice was always it. The reasoning that leads to the conclusion that it does isn’t based in physics theory, but on a strange epistemological position resembling Heinlein’s fictional “fair witnesses’”, a child’s prior to their acquisition of the object permanence schema described in Piaget’s theory of cognitive development, or the Pondicherry interpretation of quantum physics. In relativistic classical physics, events that can be calculated can be considered known, even if they can’t be seen via light images or signals.

 

It seems to me that your take on this, Slinkey, is essentially that nothing falling toward a black hole can, as observed by a distant observer (let’s stick with calling him “Bill”), can every cross its event horizon. This doesn’t make sense to me, because if it were true, I don’t see how supermassive black holes, which are formed not from a single supermassive star collapse to have radius less than its Schwartzchild radius, but from many black holes and other bodies combining, could form.

 

In short, I think “Bill will never see Alice (or any body) cross the EH” is an incorrect, a-theoretical and nonphysical assumption. It’s also one that, until rethinking in the course of reading and posting to this thread, I believed. In backtracking to see why I believed it, I recalled that it is a key plot device in Fredrik Pohl’s ca. 1980 Gateway science fiction novel series. I believe I encountered it there, incorrectly accepted it as an accurate application of General Relativity, not questioning it until now.

 

:doh: Argh, the pitfall of imagining a good SF novel a sort of physics textbook!

 

Invalidating this assumption renders “my take on resolving the paradox of crossing the event horizon of a black hole, which an observer in free fall across the horizon measures as taking a fairly short time, while a distant observer measures it as taking an infinite time”, which I gave in post #2, unnecessary.

 

The paradox, as I’ve described it, seems to still have life. Key to it being a “real paradox”, rather than one based on assuming that something nonphysical is actually physical (I suspect this assumption is “[Alice] somehow later again collocating with Bill”) is, I think, what occurs, as observed by Alice, within the EH of a BH, and whether any conditions exist in which she could survive (by survive, I mean some information describing her clock survive) until the BH evaporates as observed by Bill.

 

I’m aware of, but don’t understand in enough detail, calculations asserting that no sort of information-bearing configuration of matter-energy can long exist within a BH. Intuitively, I’ve a hard time believing this, mostly because the observed universe itself is only a few times less dense than necessary for it to be the interior of a single very large black hole (having an SR of about 10,000,000,000 ly vs. a radius of about 46,000,000,000 ly. I find it hard to imagine that, were the average density of the universe about [imath]4 \times 10^{-25} [/imath] rather than [imath]8 \times 10^{-26} \,\text{kg/m}^2[/imath], it would be so dramatically different than it is. (sources for “universe as a black hole” data: wikipedia articles Schwarzschild radius and universe)

[modest]

Would you agree?

Yes. :thumbs_up

Fantastic :)

 

but, gee Modest, I've never read Marvel Comics and I had to look adamantium up. :lol:

I actually wrote "it's dolomite baby" first, but that's an even more obscure reference.

 

 

 

It seems to me that your take on this, Slinkey, is essentially that nothing falling toward a black hole can, as observed by a distant observer (let’s stick with calling him “Bill”), can ever cross its event horizon. This doesn’t make sense to me, because if it were true, I don’t see how supermassive black holes, which are formed not from a single supermassive star collapse to have radius less than its Schwartzchild radius, but from many black holes and other bodies combining, could form.

 

In short, I think “Bill will never see Alice (or any body) cross the EH” is an incorrect, a-theoretical and nonphysical assumption.

Good point about a growing black hole. Just like a shrinking black hole, it would seem that Slinky's premise (that Alice approaches the horizon for an eternity of Bill's proper time) has to be violated. If I were to restate my understanding of Slinky's paradox it would go something like:

 

1. 
    
   
2. Alice jumps in a hole (or Bill pushes her—I'll leave that to the jury). As she approaches the horizon she slows down from Bill's perspective. If she had a laser beacon that was programmed to flash once per second, Bill would have to wait longer and longer to see each flash. First he waits a million years between flashes. Then he waits a billion years for the next flash, and a trillion years for the next one. After several hundred trillion years of doing this he gets bored and moves on to do something else.
    
   
3. Also, he sees the black hole evaporate into nothing one million years after she jumped.
   

 

Since the first paragraph is a common description from popular science sources and the second is the correct result of Hawking radiation, but the two clearly contradict each other, I think Slinky's paradox is an understandable concern.

 

The problem is that the first paragraph always seems to be given without any caveats. The paragraph is 100% correct (Bill would end up waiting trillions of years between the beacon flashes that Alice sent from outside the horizon) but it applies only to a static black hole so there would be no paradox as this paradox decidedly hinges on the black hole shrinking.

 

 

Here is my take on all three scenarios from both perspectives:

 

Static black hole:

 

Alice

:

 

An instant before crossing the event horizon she sends a laser flash directed at Bill. The photon moves away from her at c, but like a fish swimming against the current of a river, it makes infinitely slow progress relative to Bill.

 

Bill

:

 

Time is dilated near the event horizon to such an extent that he essentially has to wait an eternity before receiving the photon.

 

 

Growing Black Hole

 

Alice

:

 

An instant before crossing the event horizon she sends a laser flash directed at Bill. The photon moves away from her at c, but like a fish swimming against the current of a river, it makes infinitely slow progress relative to Bill. She falls into the singularity and dies.

 

As the photon struggles to get away from the horizon, new mass falls past it into the horizon. This increases the mass inside the horizon and creates a new, larger, horizon above the photon. It loses its struggle and starts going backwards into the hole.

 

Bill

:

 

Time is dilated near the event horizon to such an extent that he essentially has to wait an eternity before receiving the photon. While he is waiting, new mass approaches the horizon. Enough of it gathers around the old horizon to make a new horizon at a larger value of r. The new horizon is further from the center of the black hole than Alice was when she sent the photon so he correctly deduces that he will never see it.

 

Alice's story is finished for him. He will see nothing more of her.

 

 

Shrinking Black Hole (the resolution of Slinky's paradox)

 

Alice

:

 

An instant before crossing the event horizon she sends a laser flash directed at Bill. The photon moves away from her at c, but like a fish swimming against the current of a river, it makes infinitely slow progress relative to Bill. She falls into the singularity and dies.

 

As the photon struggles to get away from the horizon, negative energy (whatever that is—I don't understand Hawking radiation) falls past it into the horizon. This lowers the mass inside the horizon and creates a new, smaller, horizon now further below the photon. It wins the struggle and breaks free. At no point was the photon inside a horizon.

 

Bill

:

 

Time is dilated near the event horizon to such an extent that he essentially has to wait an eternity before receiving the photon. But, as he waits, he sees that Hawking radiation is coming from the black hole. He subtracts its energy from the hole and deduces that the horizon must be shrinking to accommodate the lowering mass.

 

The spot from which Alice sent the last photon is no longer time dilated to an near-infinite extent. The last photon makes it out and he sees it before the hole evaporates.

 

Alice's story is now finished for him. He will see nothing more of her.

 

That is my understanding.

 

I’m aware of, but don’t understand in enough detail, calculations asserting that no sort of information-bearing configuration of matter-energy can long exist within a BH. Intuitively, I’ve a hard time believing this, mostly because the observed universe itself is only a few times less dense than necessary for it to be the interior of a single very large black hole (having an SR of about 10,000,000,000 ly vs. a radius of about 46,000,000,000 ly. I find it hard to imagine that, were the average density of the universe about [imath]4 \times 10^{-25} [/imath] rather than [imath]8 \times 10^{-26} \,\text{kg/m}^2[/imath], it would be so dramatically different than it is. (sources for “universe as a black hole” data: wikipedia articles Schwarzschild radius and universe)

 

But, if the universe had a slightly higher density it would be above the critical density. It would eventually collapse into a singularity (big crunch). I don't think it would be dramatically different at the moment, but its future would be.

[Qfwfq]

I actually wrote "it's dolomite baby" first, but that's an even more obscure reference.

Obscure? Nah, I live a rock's throw from the Dolomites.

 

(the real ones) :hihi:

[modest]

Obscure? Nah, I live a rock's throw from the Dolomites.

 

(the real ones) :hihi:

"rock's throw" :hihi:

 

I think you out-referenced my reference ;) I didn't know it was a mountain range. I was thinking of a Futurama episode where Farnsworth says "it may yet survive... it's dolomite, baby!". I'm pretty sure they were making a pop culture reference of their own—a 1970's movie called 'Dolomite'... It was a stereotypically black culture movie... so I don't know, is the mountain range or the rocks there dark in color by chance?

 

~modest

[Qfwfq]

I don't know, is the mountain range or the rocks there dark in color by chance?

We're getting off topic aren't we? :lol: You need to ask Monsieur Dolomieu...

 

If you want to see the best of them, go to places around the South Tyrol, Cortina d'Ampezzo, Belluno and Feltre (the one nearest to me) but the area is a bit wider than that. There are two massifs of "lesser dolomites" on the fringes of them, around 50 km or so from me as the rock throws. These are visible on a good day from my parts, providing obstacles are avoided, with a nick of the Vette Feltrine peeping through the vally between them.

Vette Feltrine

Other Dolomites

http://www.dolomiti.org/dengl/welcome.html

[CraigD]

I actually wrote "it's dolomite baby" first, but that's an even more obscure reference.

I fear no Futurama reference is too obscure for much of the audience for this thread. :) I didn’t even need to check its wiki to recall this one ;)

 

I’m aware of, but don’t understand in enough detail, calculations asserting that no sort of information-bearing configuration of matter-energy can long exist within a BH. Intuitively, I’ve a hard time believing this, mostly because the observed universe itself is only a few times less dense than necessary for it to be the interior of a single very large black hole (having an SR of about 10,000,000,000 ly vs. a radius of about 46,000,000,000 ly. I find it hard to imagine that, were the average density of the universe about [imath]4 \times 10^{-25} [/imath] rather than [imath]8 \times 10^{-26} \,\text{kg/m}^2[/imath], it would be so dramatically different than it is.

But, if the universe had a slightly higher density it would be above the critical density. It would eventually collapse into a singularity (big crunch). I don't think it would be dramatically different at the moment, but its future would be.

What I’m looking for here is a proof that Alice, as observed by herself, can survive after falling through a black hole’s event horizon.

 

Questions for thought:

• If most of the BH’s mass is concentrated in a singularity or singularity-like non-zero volume at its center, can she follow a fairly circular orbit around that with [imath]r_s > r > \frac12 r_s[/imath]?
  
• Can most of the BH’s mass orbit its barycenter such that the shell theorem applies, and most bodies in the BH’s interior, such as Alice, experience little net gravitational force? in which case the BH’s interior might contain ordinary formations such as star and planetary systems?
  

The typical popularized science description of falling through a black hole reads something like this:

Alice falls thought toward the BH’s EH. Tidal forces stretch her into “spaghetti” then rip her apart. Even if she survives spaghettification, it’s a short-lived reprieve, as she falls into the singularity at heart of the BH, where the very laws of physics as we know them cease to exist.

Many article and forums, including ours, have shown that tidal forces can be very small for very massive BHs. What I’d like to show is that the “short-lived reprieve” of typical popular descriptions of BH physics is also not necessarily inevitable, and that supermassive BHs may contain small universes with astronomical structures much like our greater one.

 

If this is the case, this thread’s paradox is poignant, as nothing in principle prohibits Alice surviving within her BH, which from Bill’s perspective eventually (it’s important, I think, to visualize how very slowly large BH’s evaporate: a supermassive one’s like our galaxy’s has a evaporation time of about 10⁷⁴ years, a mind-bogglingly vast period) evaporates due to Hawking radiation. Nonetheless, Alice and Bill should eventually be able to collocate and compare clocks, and the paradox of Bill’s clock being predicted by General Relativity to show a infinite amount of time, rather than a mere 10⁷⁴-ish years, to Alice’s small reading, is glaring.