Black Hole Paradox?

[modest]

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 very glad to hear it :)

 

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.

I'm not quite sure I follow. Do you mean that she could survive until it evaporates after which she could return to Bill (who was never in the hole)? Assuming I have that right, I would not agree.

 

I certainly agree that the average density inside a black hole can be quite small... comparable to densities that we deal with every day. But, the way I would think about it intuitively would be to imagine something with a low density (about the density of water let's say) that is very strong. It has the highest compressive strength imaginable.

 

If we collected a spherical arrangement of it then the Newtonian escape velocity on the surface would be

 

[math]V = \sqrt{ \frac{2GM}{r}} [/math]

 

we would have to calculate the mass from the density since that's what's constant so,

 

[math]V = \sqrt{ \frac{2G(\rho \frac{4}{3} \pi r^3)}{r}} [/math]

 

As r increases the escape velocity increases even with the same low density.

 

If we kept adding to it and making a larger spherical collection of the material eventually the escape velocity at the edge would reach c. That is the point at which you get the Schwarzschild radius,

 

[math]c = \sqrt{ \frac{2GM}{R_S}} [/math]

 

That is the Schwarzschild radius equation solved for c. A low density can get us there if r is large enough.

 

So it doesn't matter how high the compressive strength of the material is. Nothing can withstand the acceleration of gravity at that point regardless of what arrangement the matter takes because nothing can be given enough acceleration to reach that velocity. Nothing could stay stationary at the edge.

 

If it were a spherical shell then the outside of the shell would collapse in and things would keep collapsing from there.

 

A large collection of low density material, in other words, can provide a huge gravitational force... enough that no amount of acceleration (and no force providing that acceleration) could overcome.

 

 

"Everyone out of the universe! Quick!" :phones:

 

EDIT: upon sober reflection, I'm not convinced my intuitive reasoning makes any sense. I'm curious what you guys think.

[Qfwfq]

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?
  

It is essential to recal that the "radial" direction becomes timelike. The [imath]r=0[/imath] line is the inexorable future.

 

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.

Even if she met no harm falling in, she would never survive the evaporation. Even if a virtual Alice-Antialice pair got unvirtualized just outside, would the Alice be the continuation of the one that had fallen in? Would there even be a univocal mapping of their proper times?

[modest]

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]?

wiki's schwarzschild metric article has something on that too:

 

A particle orbiting in the Schwarzschild metric can have a stable circular orbit with r > 3rs. Circular orbits with r between 3rs / 2 and 3rs are unstable, and no circular orbits exist for r < 3rs / 2. The circular orbit of minimum radius 3rs / 2 corresponds to an orbital velocity approaching the speed of light. It is possible for a particle to have a constant value of r between rs and 3rs / 2, but only if some force acts to keep it there.

[Slinkey]

Sorry for the delay in replying. Been busy the last week or so with my course and family commitments. Lot of messages on this thread now so I will pick my way through them and address those that seem relevant to the discussion.

 

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.

 

No need to apologise. I could probably have framed my replies with more clarity. Glad that we have found some agreement though. Always a good thing. :)

 

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?

 

I'm not sure although I would be inclined to think the equation relates to an observer at infinity if the equation has a modification to take into account your gravitational potential. ie. where you are observing from.

 

First to recap (as I have been absent for several days and to save you reading back): Alice and Bill are at different gravitational potentials around the BH such that from Bill's perspective, Alice's clock is ticking at half the speed of his clock, and from Alice's perspective Bill's clock is ticking twice as fast as her clock.

 

If the equation for the lifetime of the BH is dependent on their location in the gravitational field we need to clarify what we mean here first:

 

Let's assume that the lifetime of a BH is calculated for the observer at infinity. Let's say Bill is at this (theoretical) point (or as near as damnit) and thus can use the equation for the lifetime of the BH without needing to alter it. He calculates that it will exist for N years.

 

As Alice is at a point where her clock ticks half as fast as Bill's clock (Bill's clock ticks twice as fast as Alice's) she has a modified equation that takes this difference into account and calculates the BH will exist for 0.5N years. ie. she multiplies the time by a factor to take their different clock rates into account.

 

Let's now move Alice closer to the hole so that her clock now ticks 4 times as slow as Bill's clock (Bill's clock is ticking 4 times faster than Alice's). She does the same equation and this time calculates the BH will exist for 0.25N years.

 

We can continue doing this, and we move Alice again so her clock now ticks 8 times as slow as Bill's (Bill's clock is ticking 8 times faster than Alice's) and she calculates 0.125N years for the existence of the BH, and so on.

 

I think you may be able to see where I'm going with this. As the clocks relate to each other Bill will see Alice's clock slow down more and more as she gets closer and closer to the EH. Conversely, Alice will see Bill's clock get faster and faster as she gets closer and closer to the EH.

 

At some point Bill will see Alice's clock approach infinity ie. it will tick infinitely slowly compared to his clock (Bill's clock is ticking infinitely faster than Alice's). This means that for Alice to calculate the existence of the BH she would have to multiply N by an infinitely small quantity which to all intents and purposes would reduce the time the BH exists to virtually zero from her location.

 

ie. the closer you get to the BH the less time it exists from your reference frame and if you reach the EH the BH will cease to exist (or at best exist for an infinitely short time).

 

Now let's assume that the calculation is the proper time of the BH. Again, let's clarify what this means as I'm not sure how this would apply to a BH. I'm assuming by this you mean that if you were at the EH you would calculate the existence of the BH to be as measured by a clock at the EH. Is this correct?

 

If this is so, then we still have a big problem. If you calculate the existence of the BH using a clock at the EH to be N years, and clocks at the EH move infinitely slowly to all observers above the EH, then the BH will exist for an infinite time for all observers above the EH.

 

What are your thoughts?

[Slinkey]

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?

 

Fortune favours the brave, as they say. :)

 

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.

 

Indeed. ie. they would favour one reference frame over another which, as I understand it, is not permitted in relativity. No reference frame has a prefered observation point. However, usually in relativity we assume an event and then work out how different reference frames would observe the event. In this case we are delineating the viewpoints first and then trying to ascertain what the event would be.

 

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.

 

OK. Please see my reply to JMJones above which addresses this.

 

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.

 

Well, now we're getting to the EH of my mathematical understanding of GR! I haven't worked through general covariant tensors and do not know how this would change what I have stated, but I would be interested to hear how this situation is addressed in that form, if you are of a mind to give me some ideas of how it would resolve the paradox.

[Slinkey]

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.

 

Sorry for taking just this part of your reply in isolation, but I wanted to ask a question about this.

 

Are you saying that we would not see any dilation of Alice's clock as she falls towards the EH from Bill's location?

[Slinkey]

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

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.

 

I don't like that last sentence which I have put in bold. Not that I see it as disrespectful, or that I am offended or believe there was intent to offend, but because it doesn't actually tell me where or how I have gone wrong. If I am lacking knowledge at some point then I would sincerely like to know what I am lacking. The idea of the thread was to address my idea and why my complaint (if complaint is the right word) is either correct or unfounded due to something I am not understanding.

 

For the record, I'm not expecting something unphysical to happen, rather I am taking two physical pictures and asking how they can be reconciled with each other satisfactorily (if they can indeed be reconciled), and if they cannot be reconciled then what does that say about the theory behind it.

 

Please take my reply here in the spirit of proper discourse with no slight intended or implied.

[Qfwfq]

Indeed. ie. they would favour one reference frame over another

Actually, that isn't what I said. Inobservability is not the same thing as just a different coordinate choice. When I posted that, I hadn't yet gathered exactly what your question was.

 

which, as I understand it, is not permitted in relativity.

Well, it isn't quite a felony, it's a very mild offence, and only when done without appropriate caution. :hihi:

 

For instance, deriving the Schwarzschild solution could hardly be accomplished in any other choice than Schwarzschild coordinates. Actually, it is even necessary to make an appropriate change of coordinates during the process. The principle of relativity, and of general covariance, is what makes it sane to make such choices without the essence having a restricted validity; it only requires one to know which coordinates are being used (how they are defined) and how to transform the results for other choices (which isn't always a simple task).

 

Slinkey, you asked a very good question but keep in mind that GR isn't such a simple topic, especially if you haven't yet done any differential geometry. I graduated in Physics and I did a GR course. It was not a first year course. It wasn't one of the fundamental, compulsory courses, I chose it as one of those that students could pick. It was even one of the Ph. D. courses that had been made available for the final year of the ordinary degree. Some of the concepts can be understood by carefully following explanations of them, but drawing conclusions requires a lot of caution.

[modest]

Sorry for taking just this part of your reply in isolation, but I wanted to ask a question about this.

 

Are you saying that we would not see any dilation of Alice's clock as she falls towards the EH from Bill's location?

No problem.

 

No, I'm certainly not saying that there is no time dilation. I'm saying that any conclusions drawn from the Schwarzschild metric are bound to be wrong once you reach the horizon. The infinities you reference come directly from that metric.

 

I think this is more of a problem than you might realize because in Hawking's derivation of Hawking radiation the radiation originates from outside the horizon. This means that the thought experiment you just outlined has a very specific and clear solution.

 

Assuming Alice is hovering very near the event horizon, she will look down and see radiation coming from it. That radiation, from her perspective, will be extremely high frequency gamma rays. From that observation she could calculate how long the black hole should live according to her clock. It won't be that long because very high frequency gamma rays have a lot of energy. The black hole is bleeding a lot of energy very fast. She won't expect the hole to survive that long.

 

From Bill's perspective, those gamma rays have redshifted a great deal. While she is observing gamma rays he might be observing radio waves. Because the hole, from his perspective, is leaking quite a bit less energy he would expect it to last a lot longer.

 

When Hawking derived the expected lifetime of a black hole he made the calculation with respect to a distant observer. It would be expected that anyone close to the horizon, and hovering above it, would calculate a shorter lifetime.

 

How would you feel about this version of the paradox?

 

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. Bill sees the black hole evaporate into nothing one million years after she jumped.
   

 

1 contradicts 2

[Qfwfq]

I don't like that last sentence which I have put in bold. Not that I see it as disrespectful, or that I am offended or believe there was intent to offend, but because it doesn't actually tell me where or how I have gone wrong.

Sorry, but I was arguing with Modest there.

 

I am taking two physical pictures and asking how they can be reconciled with each other satisfactorily (if they can indeed be reconciled), and if they cannot be reconciled then what does that say about the theory behind it.

Modest and I showed, later, how they can be reconciled in the exact scenario of your question, where Alice doesn't actually fall through the EH. In the scenario where she does (which I was talking about at first), there is no need to reconcile them because What happens to Alice after it will never be observable by Bill.

[modest]

I hadn't yet gathered exactly what your question was.

Ditto

 

Please ignore my first few posts on the topic.

[Slinkey]

Thanks for the replies, guys. I haven't got time to reply right at the moment but hopefully should be able to at the weekend and give the time to your replies that they deserve. In the meantime, have you seen this paper by Lawrence Krauss, Dejan Stojkovic and Tamnay Vachaspati?

 

http://arxiv.org/PS_cache/gr-qc/pdf/0609/0609024v3.pdf

 

The math is much too dense for me I have to admit, but I thought some of you guys might have a better handle on what they're saying which essentially seems to be that BHs cannot form! :blink:

[Qfwfq]

In the meantime, have you seen this paper by Lawrence Krauss, Dejan Stojkovic and Tamnay Vachaspati?

No, I hadn't seen it and I find it very interesting. I have recently thought about the basic idea they are tackling, put simply, of whether a BH will even form before it evaporates, but these guys seem to be doing a good analysis although not yet complete.

 

I thought some of you guys might have a better handle on what they're saying which essentially seems to be that BHs cannot form!

They are using some quite advanced math and I don't have great familiarity with all of it, but I took a look at their main line of argument and they make a good case.

[G Anthony Kent]

I wonder what people's thoughts are on this:

 

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. (I'm ignoring redshifting here so please no arguments about not being able to see it due to the light coming from the person being redshifted towards infinity).

 

For the person falling towards the BH they will cross the EH and be destroyed.

 

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?

 

 

This would also mean that a black hole cannot grow. But, certainly they do.

 

I think the answer lies in the idea that one should compare apples to apples. One should evaluate equivalent referenced times in each segment of each scenario. Crossing over to different reference frames AND scenarios is like mixing metaphor. It's a NO NO.

 

In other words, one must stay within a given scenario until one reaches an experimentally testable result, then one may compare results. While one may actually do what Einstein suggested and ride a photon or at least travel at the speed of light (if one had an infinite source of energy - which may be the whole universe itself) it may be "pulling one's pud" to imagine scenarios that are even theoretically impossible. Such would necessitate making false assumptions. False assumptions can be expected to achieve false results.

Edited December 4, 2011 by Qfwfq
shouting avoided

[Guest MacPhee]

This would also mean that a black hole cannot grow. But, certainly they do.

 

Surely that's the point. Made by CraigD, in his post #46. If matter doesn't really fall into black holes, they couldn't grow to become supermassive. But - do we know supermassive "Black Holes" really exist?

 

Personally, I don't believe in them. They sound like the 21st Century equivalent of epicycles, phlogiston, protoplasm, and caloric fluid.

All those concepts, served a useful purpose, in their time. They encouraged us to look for a rational explanation of the Universe. That's good.

But they were discarded, because they led to paradoxes. Just as "Black Holes" lead to paradoxes - as previous posts have shown.

 

And something which leads to a paradox, can't be logically true. "Black Holes" lead to paradoxes. So they can't be true, can they?

[modest]

Surely that's the point. Made by CraigD, in his post #46. If matter doesn't really fall into black holes, they couldn't grow to become supermassive.

If you have a small black hole with a radius [math]r[/math] and you throw a bunch of matter at it — from an outside perspective the matter will smear into a thin shell around (and outside of) the event horizon. Does this mean the black hole didn't grow?

 

Calculate the mass inside [math]r + 1[/math]. Both the new mass and the old mass will be inside that new, larger, radius. The mass can easily be sufficient for it to be the new (larger) radius of the black hole even though nothing crossed the old event horizon.

[Guest MacPhee]

If you have a small black hole with a radius [math]r[/math] and you throw a bunch of matter at it — from an outside perspective the matter will smear into a thin shell around (and outside of) the event horizon. Does this mean the black hole didn't grow?

 

Calculate the mass inside [math]r + 1[/math]. Both the new mass and the old mass will be inside that new, larger, radius. The mass can easily be sufficient for it to be the new (larger) radius of the black hole even though nothing crossed the old event horizon.

 

Thanks modest. I read your post through carefully several times. And now I think I see what you mean.

 

The black hole, does not get bigger, by new matter falling right down into the hole's centre. Rather, the new matter builds up around the hole's outer rim . So the hole expands, not from the centre, but by adding more "layers" to its outside.

 

Hope I've understood it correctly! But honestly, "black holes" still remind me of "caloric fluid"! :)