Black Hole Paradox?

[CraigD]

Call for consensus, and some clarification

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.

:QuestionM Can we get consensus on this? Not on the validity or resolution of the paradox, just on its basic description?

 

Scientists say the laws of physics break down at the event horizon. In truth the laws probably start breaking down someplace just outside of it.

I’ve never read a scientific suggestion about classical physics, extended by relativity, breaking down at the event horizon of a black hole. I think you’re confusing this claim with the one that these laws break down near the singularity hypothesized to be at the center of a black hole. Can you check to see if you’re recalling a reference to them breaking down at the event horizon, and if you find them, post a link or reference?

 

We both know the paradox that we speak of is not possible just as the grandfather paradox is not possible in our Universe.

I think bringing up the grandfather paradox is off topic, and, though a discussion-worthy topic, distracts from the one at hand.

 

The black hole paradox we’re discussing in this thread is, I hope, as I described above.

 

The grandfather paradox involves a person traveling back into their own (or their ancestor’s) past (following a closed timelike curve worldline) and interacting with events in that past that contradict their history in a manner that makes their act of traveling into the past impossible – for example, by killing their past self, or their parent of grandparent.

 

The black hole paradox requires a black hole. The grandfather paradox requires a way of traveling backward in time. These are two very different requirements.

 

My guess as to what's going on is a proton or an electron approaching an event horizon will get spaghettified into a wave whose wavelength is equal to the energy of the particle at that time.

To understand this discussion, it’s important to understand precisely what “spaghettification” means, and that it is not required by physical law to occur near the event horizon of a black hole.

 

Spaghettification occurs when the gravitational force on one end of a body differs so greatly from that on the other that the body is stretched beyond its breaking point. All macroscopic bodies experience this difference in force – standing on or orbiting around Earth, the gravitational force on your feet is slightly greater than that on your head, but so slightly compared to the strength of your body, it’s unnoticed. In orbits, the force, though small, can be noticed, such as when the long axis of an orbiting object becomes tidally locked in a vertical attitude.

 

This tidal force can be calculated simply, with different results depending on how the body being stretched is represented, but all result give about [imath]F = \frac{GMmL}{4r^3}[/imath]. Applying this to masses M large enough to be black holes, we find that F for a couple of 1 kg masses separated by a thin 1 m cable at the event horizon is on the order of 10¹⁰ N for a star-mass black hole, but on the order of only 0.1 N for a super-massive black hole like the Milky Way’s.

 

So, for a sufficiently large black hole, tidal force is small enough that Spaghettification doesn’t occur. Your guess can only be applicable to small black holes.

[Slinkey]

Call for consensus, and some clarification

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.

:QuestionM Can we get consensus on this? Not on the validity or resolution of the paradox, just on its basic description?

 

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.

 

Does that help?

[Qfwfq]

When the BH has evaporated Alice collocates with Bill and compares clocks.

If this happens, it means that Alice did not pass the event horizon. It can't be that she did according to her own view, if she did not, according to Bill's view; no change of coordinates could provide for it.

[Slinkey]

If this happens, it means that Alice did not pass the event horizon. It can't be that she did according to her own view, if she did not, according to Bill's view; no change of coordinates could provide for it.

 

Indeed. Hence the paradox. Every book I have read about BHs says that Alice will never be seen to cross the EH by Bill, yet they also say that Alice, from her reference frame, will cross the EH and meet the singularity.

 

Thus Alice must always be above the EH. From her view something else is happening.

[Qfwfq]

Hence the paradox.

No. There's no paradox at all. If she does cross the border, they won't ever meet again. If they do, then she can't have crossed the border. From whence do you deduce that both things ought to happen?

[Slinkey]

No. There's no paradox at all. If she does cross the border, they won't ever meet again. If they do, then she can't have crossed the border. From whence do you deduce that both things ought to happen?

 

I don't deduce it myself. Scientists tell us this happens. Or more precisely, science tells us that we will see Alice above the EH for the entire remaining existence of the BH, AND that she will cross the EH and meet the singularity.

 

What I'm saying is that if she is not seen to cross the EH then she can't have crossed it and that something else must be happening from her point of view other than crossing the EH.

[JMJones0424]

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.

 

Does that help?

 

The bold portion of the text is the question that is being addressed by Craig, and while interesting, I still think your conceptual problem is more fundamental because of some of the other replies you have made in this thread, such as, "What I'm saying is that if she is not seen to cross the EH then she can't have crossed it and that something else must be happening from her point of view other than crossing the EH."

 

 

1) Alice falls into the blackhole and becomes a part of the singularity, whatever that might be. It is called a singularity because there are infinities involved and the math breaks down.

 

2) Bob observes Alice to slow down while approaching the event horizon. The closer Alice gets to the event horizon, Bob observes more and more time between the ticks of Alice's clock.

 

There is no paradox here, consider for a moment why the event horizon is called "event" "horizon". It is the line at which all events on the side of the singularity will never be able to be observed on the other side. Just as a ship disappears over the horizon when sailing away from port. Bob is not in an appropriate place to observe Alice's transit across the event horizon or what happens to her afterward. This does not mean, however, that Alice never crosses the event horizon.

 

If your next question is why, then the answer requires a more broad explanation of relativity, a feat that I have tried several times and failed more often than succeeded. I highly suggest you read a layman's level book that deals with relativity, such as Stephen Hawking's A Brief History of Time. Alternatively, try enrolling in an entry level physics course dealing with relativity or browse youtube for the many available uploaded courses that introduce relativity to non-physics majors.

 

_______

 

 

Do we here at Hypography have a well-written thread that goes over the basics of relativity, from Galilean relativity and the hidden assumption of universal time to the consequences of the invariant speed of light and special relativity, and finally to the equivalence principle and general relativity? I'd like to bookmark that thread to refer to in the future. If it does not exist, I am tempted to try to write such a thread, though of course I will likely need help.

[Slinkey]

1) Alice falls into the blackhole and becomes a part of the singularity, whatever that might be. It is called a singularity because there are infinities involved and the math breaks down.

 

Agreed.

 

2) Bob observes Alice to slow down while approaching the event horizon. The closer Alice gets to the event horizon, Bob observes more and more time between the ticks of Alice's clock.

 

Agreed. Do you also agree that Bill will never see Alice cross the EH, as this is what every book I have read says?

 

There is no paradox here, consider for a moment why the event horizon is called "event" "horizon". It is the line at which all events on the side of the singularity will never be able to be observed on the other side.

 

I agree. It also marks a spot where a clock being observed by Bill will cease to tick. ie. it will take an infinite amount of time for it to move.

 

Just as a ship disappears over the horizon when sailing away from port. Bob is not in an appropriate place to observe Alice's transit across the event horizon or what happens to her afterward. This does not mean, however, that Alice never crosses the event horizon.

 

If she crosses the EH then I agree. However, can Bill see her cross the EH? According to everything I have read he cannot see her cross the EH. Thus the event of Alice crossing the EH does not occur from Bill's viewpoint. I would argue that Bill cannot even see Alice reach the EH. Her clock will continue to slow down the closer she gets to the EH for an infinite amount of time from Bill's viewpoint.

 

Let's try another thought experiment:

 

Let's assume that Alice and Bill are both above the EH but Alice is closer to it than Bill such that when Bill looks at Alice's clock it is running at half the speed of Bill's clock, and conversely when Alice looks at Bill's clock it is running twice as fast as her clock.

 

Bill does a measurement to see how massive the BH is and finds it is 10 billion solar masses. From this he can infer that the BH will have a lifetime of (approx) 2.1 * 10^97 years.

 

Similarly, Alice does a measurement to see how massive the BH is and also finds it is 10 billion solar masses. She also infers that the BH will have a lifetime of (approx) 2.1 * 10^97 years.

 

Let's imagine Alice and Bill are immortal (ignoring the technicalities of fueling their respective spaceships and food requirements for this thought experiment) and remain at the same locations such that their clocks run at the same rates as described above.

 

Bill watches the years tick by and eventually the BH evaporates. He sends a message to Alice to confirm this is the case. Alice, however, has only seen half as much time pass on her clock as Bill's clock, and for her the BH has only reached the half way point and still exists. She sends a message back to Bill telling him that the BH still has (approx) 1.05 * 10^97 years to go. However, they both measured the mass of the BH and both inferred it would exist for 2.1 * 10^97 years.

 

Where has my logic gone wrong?

 

If your next question is why, then the answer requires a more broad explanation of relativity, a feat that I have tried several times and failed more often than succeeded. I highly suggest you read a layman's level book that deals with relativity, such as Stephen Hawking's A Brief History of Time. Alternatively, try enrolling in an entry level physics course dealing with relativity or browse youtube for the many available uploaded courses that introduce relativity to non-physics majors.

 

Thanks for the pointers. I've read several of Stephen Hawking's books. I've also read Kip Thorne's "Black Holes and Time Warps", and Leonard Susskind's "The Black Hole War" amongst many others, and I am currently a first year Natural Science (Physics) degree student with the Open University. I wouldn't dare call myself an expert on this but I wouldn't call myself a novice either.

[JMJones0424]

I apologize if I misunderstood your previous statements. With minor qualifications, such as an increasingly long period of time is not equal to an infinite period of time, I agree with your agreements.

 

In your new thought experiment, where both observers remain forever outside of the event horizon, I see no fundamental difference between that and the instance of an observer orbiting the Earth and on the surface of the Earth. Replace the amount of time required for the evaporation of a black hole with n oscillations of an atomic clock.

 

I suspect, though there is certainly a great possibility that I am incorrect, that the equation wikipedia lists for the amount of time necessary for a blackhole of mass M_o to dissipate is in proper time, and because Alice is at a significantly lower gravitational potential, she should have to adjust the amount of time she expects to pass for the blackhole to dissipate.

 

Is this correct?

[Qfwfq]

What I'm saying is that if she is not seen to cross the EH then she can't have crossed it

This is a very bold step in your logic. If no man heard the sound of the tree falling in the jungle, did it make a sound?

 

Some epistemologists would argue that it makes no sense for Bill to even ask whether Alice goes through the EH and this does not contradict the fact that it does make sense for her to say she did (assuming she survives it, of course), because there will never be a possibility for these two observers to compare their observations.

 

Do we here at Hypography have a well-written thread that goes over the basics of relativity, from Galilean relativity and the hidden assumption of universal time to the consequences of the invariant speed of light and special relativity, and finally to the equivalence principle and general relativity? I'd like to bookmark that thread to refer to in the future. If it does not exist, I am tempted to try to write such a thread, though of course I will likely need help.

Once upon a time, Erasmus00 made an attempt at this but it floundered in disruption, right from after his first post. In a sense, things like this would require a slightly different setup than the usual rules for the usual kind of thread. No doubt it would take a competent person to do it, who could also afford the time and effort.

 

Bill does a measurement to see how massive the BH is and finds it is 10 billion solar masses. From this he can infer that the BH will have a lifetime of (approx) 2.1 * 10^97 years.

 

Similarly, Alice does a measurement to see how massive the BH is and also finds it is 10 billion solar masses. She also infers that the BH will have a lifetime of (approx) 2.1 * 10^97 years.

You raise an interesting point which shows how it isn't simple to work things out. No doubt that the different coordinates also mean that the duration of the same thing is differently observed, so it would be wrong for the two of them to calculate the same duration according to their own coordinates. Indeed those equations are argued out specifically in Schwartzschild coordinates, so they must be valid in these. They are not in general covariant tensor form.

[modest]

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 think the key is that it would require Alice an infinite amount of acceleration to get out of the hole (or even to stay stationary inside the hole as it evaporated). If you could accelerate an infinite amount then it shouldn't be a surprise that clocks further in the direction of acceleration run infinitely fast, like two people in a rocket.

 

Both people expect the same thing then. If Alice gets out, bill will be infinitely old.

 

Bill knows this which is why he moved on and married the girl with the stutter from Yorkshire. She has her faults, but at least she's not jumping in the first black hole she finds.

 

EDIT -->

 

I realize that was one of those worst kinds of explanations that would only make sense to those for whom it is unnecessary.

 

Imagine a ladder 1 light-year long. It has a rocket at the bottom capable of providing infinite thrust. Alice is at the bottom and Bill at the top. The bottom end is lowered into the black hole. For Alice to stay stationary or to get back out she'll need to provide infinite thrust and infinite acceleration.

 

The time dilation equation for the bottom and top of a latter undergoing acceleration is [math]e^{ah/c^2}[/math] where a is the acceleration and h the height. The clock at the top runs faster than the bottom by a factor of [math]e^{ah/c^2}[/math]. As acceleration goes infinite in the equation, the clock at the top (Bill's clock) runs infinitely fast relative to the bottom.

 

Therefore, if she could get back out she would expect bill to have aged infinitely. Bill would expect the same.

 

But, infinite acceleration is asking quite a lot.

[Qfwfq]

I think the key is that it would require Alice an infinite amount of acceleration to get out of the hole (or even to stay stationary inside the hole as it evaporated).

Given that you are considering Craig's scenario, there are two questions to answer in this case:

1. How much acceleration would it take for her to go backwards in time (or even just halt her own proper time)?
   
2. How would she survive the black hole evaporating?
   

I think Alice, once past the EH, could only resign herself to the idea of that other ***** swiping her guy! :lol:

 

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...

Edited November 22, 2011 by Qfwfq
reply to modest's edit

[modest]

How much acceleration would it take for her to go backwards in time (or even just halt her own proper time)?

infinite +1.

 

:)

 

No, I'm not sure relative to who you mean.

 

How would she survive the black hole evaporating?

you mean the radiation hurting her?

 

I think Alice, once past the EH, could only resign herself to the idea of that other ***** swiping her guy! :lol:

And she could watch it happen too Like the rich man watching Lazarus with the great gulf fixed between them

[Qfwfq]

I'm not sure relative to who you mean.

Oh it doesn't matter who. Recall what happens to the [imath]rr[/imath] and [imath]tt[/imath] Schwartzschild metric coefficients for [imath]r<R_{\rm S}[/imath].

 

you mean the radiation hurting her?

Actually, no. I mean her being part of it.

[modest]

Oh it doesn't matter who. Recall what happens to the [imath]rr[/imath] and [imath]tt[/imath] Schwartzschild metric coefficients for [imath]r<R_{\rm S}[/imath].

Right. That proves the impossibility of the whole thing. The schwarzschild metric is static. It assumes that the observer is stationary relative to the mass. That's why it breaks down at the horizon and why time dilation goes infinite there. It takes infinite acceleration to stay stationary at the horizon. It would, by extension, take 'more than infinite' anywhere under the horizon which is impossible and why she can't get out of the hole.

 

You would get the same paradox if you said "a rocket has infinite acceleration for 5 seconds according to the person in the rear of the rocket". That would leave the person in the nose of the rocket aging an eternity. It's the same thing.

 

Actually, no. I mean her being part of it.

I think the origin of Hawking radiation involves antiparticles getting stuck in the hole. If she could get out (of course, impossible) I can't think of anything about evaporation that would physically hurt her.

[Qfwfq]

You would get the same paradox if you said "a rocket has infinite acceleration for 5 seconds according to the person in the rear of the rocket".

What distance would the rocket travel, in coordinates where it was initially at rest?

 

Let's say it is a similar thing but not the same; Alice and Bill don't have the same acceleration and a constant distance, Alice doesn't undergo an infinite acceleration for a finite duration. It's full of subtleties anyway. If she is freefalling, her [imath]\frac{dp}{d\tau}[/imath] approaches infinity when reaching the EH and this corresponds to the fact that she must reach it at [imath]c[/imath]. This is not the same as saying [imath]\frac{dv}{dt}[/imath] approaches infinity. If instead she is held at a constant [imath]r=R_{\rm S}+\epsilon>R_{\rm S}[/imath] it takes a force which approaches infinity in the [imath]\epsilon\rightarrow 0^+[/imath] limit. It scarcely makes sense to say what happens if she is held at [imath]\epsilon=0[/imath] (even if she is pointlike).

 

If she could get out (of course, impossible) I can't think of anything about evaporation that would physically hurt her.

It is impossible without her being part of the evaporation and it is somewhat unlikely she would recondense exactly as she was before.

[modest]

What distance would the rocket travel, in coordinates where it was initially at rest?

many times infinite +1 :)

 

depends on how long it takes to run into something :D

 

No, if you're serious, I don't know. It's an unphysical premise. I'd recon the answer would be c times however long it takes an infinitely slow clock to reach 5 seconds.

 

Let's say it is a similar thing but not the same; Alice and Bill don't have the same acceleration and a constant distance...

oh, but if they have constant distance we wouldn't expect them to have the same acceleration in the rocket. That is to say, a rocket accelerating in deep space. She, in the rear of the rocket, would be at the rindler horizon with infinite acceleration. He would be distant from it with much lower acceleration. It is, by the equivalence principle, exactly the same as the black hole—to the point even where hawking radiation is analogous to unruh radiation. Maybe that's not what you meant... I should keep reading...

 

, Alice doesn't undergo an infinite acceleration for a finite duration. It's full of subtleties anyway. If she is freefalling, her [imath]\frac{dp}{d\tau}[/imath] approaches infinity when reaching the EH and this corresponds to the fact that she must reach it at [imath]c[/imath]. This is not the same as saying [imath]\frac{dv}{dt}[/imath] approaches infinity.

I'd agree that dp/d-tau -> infinite doesn't imply the same for velocity and coordinate time, but you've lost me as far as what context you've put that. I don't understand "freefall... reach it at c"

 

If instead she is held at a constant [imath]r=R_{\rm S}+\epsilon>R_{\rm S}[/imath] it takes a force which approaches infinity in the [imath]\epsilon\rightarrow 0^+[/imath] limit. It scarcely makes sense to say what happens if she is held at [imath]\epsilon=0[/imath] (even if she is pointlike).

 

It is impossible without her being part of the evaporation...

Right. Like I said, that's why the event horizon is a singularity in the Schwarzschild metric.

 

When I say "it takes an infinite amount of acceleration to stand on an event horizon" I meant that it is impossible. It is not a sensible thing to say. 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.