Theory of black hole composition

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sevenperforce
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Theory of black hole composition

Postby sevenperforce » Fri Jan 08, 2016 9:58 pm UTC

A while back we were kicking around some ideas in the RELATIVITY QUESTIONS thread and it got me to thinking about black hole formation and a lot of the unsolved problems surrounding black holes. Specifically, the question arose whether black hole formation could be in any way represented from an "inside-out" approach, or if the "collapsing spherical shell" approach was the only way to look at it.

When a black hole forms from gravitational collapse of a dense, heavy object, there are a few different possibilities for how the density goes up at the center of the mass distribution. If the mass distribution stays at some maximum density and builds up into a sphere until it reaches its own Schwarzschild radius (case 1 below), then the "collapsing spherical shell" model would seem to be correct (though that might have additional repercussions as I'll discuss later). However, if all fermions break down and degeneracy pressure disappears, then density would increase without bound at the center of the mass distribution until a singularity state is reached (case 2 below). That's the possibility that interests me.

two black hole cases.png

The critical density for the formation of a black hole as a function of radius is ρ(R) = 3c2/8πGR2. So as long as the density during collapse goes up at a rate faster than inverse-squared, a black hole will form from the inside-out rather than the outside-in.

What are some of the properties of such a black hole? Well, if you're starting from what is essentially the minimum possible size of a black hole, then there are a few things you might run into. For example, black hole evaporation. Consider a black hole so small that when it evaporates, it releases a shell of Hawking radiation with the same mass-energy density that the black hole itself had:

equal density evaporation.png
equal density evaporation.png (1.5 KiB) Viewed 5521 times

Obviously, this takes place when the volume of the spherical shell and the volume of the inner sphere are equal, which takes place when the outer radius of the spherical shell is 21/3 times the radius of the inner sphere. The evaporation time Tev of a black hole with mass M can be found as Tev = 5120πG2M3/ħc4, so the thickness of the evaporation shell can be found as the product of Tev and c.

Setting the thickness of the shell equal to (21/3 - 1)*R where R = 2GM/c2 allows us to solve easily for this "critical density" black hole:
Spoiler:
Tevc = (21/3 - 1)R

5120πG2M3/ħc3 = (21/3 - 1)*2GM/c2

2560πGM2/ħc = (21/3 - 1)

M2 = ħc(21/3 - 1)/2560πG

M2 = hc(21/3 - 1)/5120π2G

M = sqrt(hc(21/3 - 1))/32π*sqrt(5G)
Work out the math, and it comes to M = 1.237e-10 kg, roughly the mass of a small grain of sand.

A black hole with a mass of 1.237e-10 kg would have an event horizon at 1.8e-37 m, a temperature of 9.9e32 K, and a lifetime of 1.6e-46 seconds. While these values are all outside of Planck limitations, they are only outside of it by a relatively small margin, so they may still be physically meaningful. However, the blackbody spectrum peaks at a wavelength of 2.9e-36 m, meaning that a single photon emitted at the peak of the radiation curve would have an energy 6,000 times greater than the total mass-energy of the black hole, which throws a wrench into things.

At what point does a photon at the peak of a black hole's radiation curve actually have an energy equal to the mass-energy of the black hole? That's not too hard to figure out; setting hc/λ equal to mc2 and using Wien's displacement constant to find the wavelength as a function of temperature gives M = hc/4π*sqrt(bGkB) = 9.67e-9 kg, just under half a Planck mass. Such a black hole would have an evaporation time of 7.6e-41 seconds, about 1400 Planck times.

What if all this is moot because black holes form from the "outside-in" instead? Well, that prompts the question of what happens inside such a black hole as it is formed. This is not a meaningless question; due to the shell theorem, anything outside a particle's radial position is gravitationally invisible to that particle, and so the event horizon does not exist from the perspective of a particle inside the distribution of matter that would become a black hole. Thus, it will continue to obey the same physical laws it would have obeyed if the event horizon had not yet formed. The closer you get to the center of the black hole, the more of the black hole's mass becomes gravitationally invisible, and thus the density at the center of the collapsing object will continue to rise without bound as discussed above:

nested regression.png
nested regression.png (26.13 KiB) Viewed 5521 times

In case it's not clear: as an event horizon forms around the uniform-density core, the inside of that core fails to notice and continues to collapse, repeating the process by iteration until it forms an arbitrarily-low-mass black hole.

So here's the hypothesis.

Stable black holes form when macroscopic distributions of matter collapse to a density between 7.8e95 and 4.8e99 kg/m3. At this density, a black hole quanta forms. Because of its extremely small size, the quanta's behavior is dominated by quantum effects and it immediately tunnels out of its own event horizon, producing a wavefunction with a mass-energy distribution having an average density on the order of the density of the black hole quanta.

Subsequent collision between the mass-energy distribution of the Hawking radiation and infalling outside matter, however, immediately causes the wavefunction to collapse into a black hole quanta again, due to the high density of the surrounding medium. This process repeats over and over and over again, rapidly transforming the mass distribution into a cloud of constantly-evaporating-and-collapsing black hole particles.

The propagation of this cloud is at first chaotic, but although the quanta themselves are not affected by radiation pressure, the infalling matter is, and so the spherical symmetry of the original mass distribution causes the formation of a spherical shell of black whole quanta. The black hole quanta themselves are just outside their collective Schwarzschild radius.

Why don't the quanta fall in? Well, their evaporation time is so miniscule that they don't have time to be pulled inside. Hawking radiation produced by each of the evaporating quanta falls inside, but since there is nothing inside, the shell theorem means it feels no gravitational force and may escape through to the other side:

black quanta cloud.png
black quanta cloud.png (9.79 KiB) Viewed 5521 times

It is impossible for a singularity to form at the center of this distribution because the quanta must necessarily evaporate through quantum tunnelling more rapidly than it would be able to interact with anything at the center, and Hawking radiation released from evaporation by quanta inside the Schwarzschild radius escapes freely. Collapse is only possible by interaction with other quanta or with infalling matter outside the would-be event horizon, so the center of the sphere is merely a diffuse photon gas.

Evaporation happens so rapidly that the outward progress of the quanta cloud is necessarily many orders of magnitude faster than infalling matter. Therefore this distribution of black hole quanta expands outward, swallowing up anything it encounters and converting it to more black hole quanta, rather than just sitting still and allowing matter to fall into it as with a conventional black hole. It will continue to grow as it swallows up new matter, but no collective event horizon ever forms. This growth continues until it runs out of matter to swallow up.

At this point, the quanta forming the surface of the shell are still evaporating...but the radiation they produce is affected by the collective gravity of all the remaining black hole quanta. This redshifts the Hawking radiation, reducing the collective power output of the cloud and decreasing the temperature of the Hawking radiation to match the blackbody spectrum of a classical black hole. As with classical objects, however, this is a statistical match, not an absolute match as would be the case with a true classical black hole.

What about when the cloud runs out of matter to swallow up? Won't the black hole quanta evaporate away? Well, yes...but recall that the energy density of the Hawking radiation is on the order of the black hole quanta themselves. Therefore, as the radius of the collective cloud becomes many many orders of magnitude greater than the radius of a black hole quanta, the density of the emitted radiation becomes more uniform, while also becoming cooler:

sizes increase.png
sizes increase.png (2.86 KiB) Viewed 5521 times

Because the density at the surface increases while the temperature of the Hawking radiation decreases, the infalling radiation from the CMBR is sufficient to maintain the re-collapse of the black hole quanta. The black hole never really stops growing, it just grows more and more slowly.

This handily solves the black hole information paradox, because the no-hair theorem no longer applies. There is no black hole and no macroscopic event horizon; there is only a spherical shell of black hole quanta which are continually evaporating and re-collapsing. To an outside observer, the cloud will appear to act in many ways like a classical event horizon black hole, but there is no singularity. A Kerr black hole forms from the collective orbit of the many black hole quanta. While this isn't a complete theory of quantum gravity, it's a step in that direction.

So, that's the hypothesis.

Predictions? Well, the math needs to all work out. I'd have to go in and figure out whether the redshift would affect the spectrum properly, narrow down the "rest mass" of each black hole quanta, and so forth. I have to figure out how Hawking emission can be represented as quantum tunneling and then solve the square-well problem to figure out whether that side of things works. There are a lot of qualitative explanations that will need to be rigorously tested to determine whether they are quantitatively sound as well. But I'm optimistic.

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Re: Theory of black hole composition

Postby SuicideJunkie » Mon Jan 11, 2016 10:02 pm UTC

Two sets of questions that jumped out at me:
At what point does a photon at the peak of a black hole's radiation curve actually have an energy equal to the mass-energy of the black hole? That's not too hard to figure out; setting hc/λ equal to mc2 and using Wien's displacement constant to find the wavelength as a function of temperature gives M = hc/4π*sqrt(bGkB) = 9.67e-9 kg, just under half a Planck mass.

Would you not require the emission of two or more particles to maintain conservation of momentum?
And would that system of 2+ particles then add up to 1 Plank mass perhaps?

Why don't the quanta fall in? Well, their evaporation time is so miniscule that they don't have time to be pulled inside. Hawking radiation produced by each of the evaporating quanta falls inside, but since there is nothing inside, the shell theorem means it feels no gravitational force and may escape through to the other side:
Even if there isn't matter inside, there is still a lot of energy from those photons. Does the concentrated energy of those photons not provide a gravitational field?
If you presume photons don't have mass in that way, wouldn't you be able to make a gravity wave transmitter by cyclically producing and annihilating antimatter pairs?

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Re: Theory of black hole composition

Postby Copper Bezel » Tue Jan 12, 2016 6:21 am UTC

SuicideJunkie wrote:Even if there isn't matter inside, there is still a lot of energy from those photons. Does the concentrated energy of those photons not provide a gravitational field?

All of the other questions in this thread are harder, but that one's relatively straightforward - the gravitational field depends on energy density, irrespective of the particles' being massive or not.
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Re: Theory of black hole composition

Postby sevenperforce » Tue Jan 12, 2016 2:31 pm UTC

Copper Bezel wrote:
SuicideJunkie wrote:Even if there isn't matter inside, there is still a lot of energy from those photons. Does the concentrated energy of those photons not provide a gravitational field?

All of the other questions in this thread are harder, but that one's relatively straightforward - the gravitational field depends on energy density, irrespective of the particles' being massive or not.

Correct. The energy density of the photon gas inside the would-be black hole definitely produces a gravitational field; I am just fairly certain that it won't ever be dense enough to produce anything close to an event horizon. All emission trajectories should have equal probability:

evaporation trajectory.png
evaporation trajectory.png (3.67 KiB) Viewed 5271 times

So only some percentage of photons will ever cross the would-be event horizon, and those that do will not necessarily be pointed at the center, so you're not going to have a large enough buildup of photons to have a smaller event horizon form inside.

SuicideJunkie wrote:
At what point does a photon at the peak of a black hole's radiation curve actually have an energy equal to the mass-energy of the black hole? That's not too hard to figure out; setting hc/λ equal to mc2 and using Wien's displacement constant to find the wavelength as a function of temperature gives M = hc/4π*sqrt(bGkB) = 9.67e-9 kg, just under half a Planck mass.

Would you not require the emission of two or more particles to maintain conservation of momentum?
And would that system of 2+ particles then add up to 1 Plank mass perhaps?

Good question.

There isn't necessarily a concrete physical significance to the peak of the black hole's radiation curve. The blackbody radiation curve is, in classical quantized thermodynamics, a statistically emergent phenomenon: the photons emitted from a thermally-radiating object appear to be random, but end up "collecting" in a blackbody curve. Hawking radiation, on the other hand, is expected to be perfect blackbody radiation. But when you're dealing with micro black holes, the quantization of photons starts to make that a bit difficult. Conservation of energy prevents the black hole from evaporating into a photon with an energy greater than its own mass-energy, but if a cap on photon energy exists, how can it follow a blackbody curve at all?

I'm sure there's been some prior research on the last moments of the evaporation of a black hole...gonna dig around a bit.

(digs around)

Hmm...less than I expected. One note: as the temperature of the black hole increases, it is speculated that it could begin to produce neutrinos and other non-massless particles. But since emission still has to follow an electromagnetic blackbody curve, I'm not sure how that's supposed to work. It's possible that existing theories simply break down at these energies and so any attempt to resolve them is like trying to model a nuclear bomb as a chemical reaction...but it's worth looking into.

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Re: Theory of black hole composition

Postby PM 2Ring » Wed Jan 13, 2016 10:34 am UTC

FWIW, the Kugelblitz blog of David Horgan has lots of info about black holes made from photons. Note that you need a very high energy density to get photons to collapse gravitationally: the photon gas needs to have a temperature greater than the Planck temperature.

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Re: Theory of black hole composition

Postby sevenperforce » Wed Jan 13, 2016 5:39 pm UTC

SuicideJunkie wrote:Would you not require the emission of two or more particles to maintain conservation of momentum?
And would that system of 2+ particles then add up to 1 Plank mass perhaps?

Let's use this question of conservation of momentum to solve for the most energetic particle(s) an evaporating black hole can create. Since we're necessarily dealing with relativistic particles, we can go straight to relativistic momentum and relativistic kinetic energy since rest masses will be negligible. The blackbody radiation curve is the curve of power output to wavelength, so let's do this in terms of λ whenever possible. The two Hawking particles must have equal momentum PH, and their energy must sum to the total mass-energy of the black hole, so we can say that EBH = 2PH*c.

Setting EBH = M*c2 and PH = h/λ gives:

M*c2 = 2hc/λ

Solving for wavelength...

λ = 2h/M*c

Now, the only way that two identical particles can satisfy a blackbody curve is to have their wavelength at the peak. The peak wavelength λmax is found as λmax = b/TH, where TH is the Hawking radiation temperature. However, this is the wavelength perceived by an observer at infinity due to redshifting, which throws a wrench into our equations. As far as I've been able to tell, classical Hawking radiation is emitted from the event horizon (which should result in infinite redshift), but it has enough quantum uncertainty to allow emission from a "fuzzy" range outside the event horizon, resulting in finite redshift, a smooth emission spectrum, and a nonzero power flux. How, then, can we model micro black hole evaporation by particle-pair quantum tunneling?

Black Hole Quanta Evaporation Reference Frames.png
Black Hole Quanta Evaporation Reference Frames.png (10.63 KiB) Viewed 5120 times

In the above diagram, we see the initial black hole quanta (QB) followed by the evaporation of the QB into two particles of equal and opposite momentum. However, as shown at the bottom, each emitted particle "sees" itself leaving behind a black hole half the size of the QB. Thus, we can characterize its redshift at infinity according to the Schwarzschild metric, λf = λ0/sqrt(1 - RS/R) where λf is the final wavelength and λ0 is the initial wavelength. In the above diagram, RS/R is simply 1/2, so we end up with λf = λ0*sqrt(2). Thus the Hawking radiation wavelength as observed from infinity must be divided by the square root of 2 to find the actual wavelength at emission:

λ0 = λf/sqrt(2)

Thankfully, conservation of energy is not violated; the lost energy is spent climbing out of the gravitational potential.

Thus:
Spoiler:
λ0 = λWein max/sqrt(2)

And since λ0 = 2h/M*c and λWein max = b/TH with TH = ħc3/8πGMkB, we combine equations:

2h/M*c = 8πbGMkB/sqrt(2)*ħc3

and solve for M.

M = ħc*21/4/sqrt(2*GbkB)
This evaluates to a black hole quanta mass of 1.627e-8 kg, three quarters of the Planck mass.

Such a black hole would have a Schwarzschild radius of 1.5 lP and an evaporation lifetime of around 6700 tP. It would evaporate into two particles, each having an initial mass-energy of 7.311e8 J with an initial wavelength of 2.717e-34 m (about 17 lP) redshifting to 3.842e-34 m (about 24 lP) at infinity.

Having these integer multiples/factors of lP and mP is tentatively encouraging, especially considering that they don't arise from accidents of dimensional analysis (derivations of the lP and mP depend on the square root of ħ). I'm particularly interested in the Schwarzschild radius of 1.5 Planck lengths, as the photon sphere for a metastable circular orbit around a black hole is exactly 1.5 times the Schwarzschild radius. I wonder whether a metastable photonium molecule might satisfy these parameters...

PM 2Ring wrote:FWIW, the Kugelblitz blog of David Horgan has lots of info about black holes made from photons. Note that you need a very high energy density to get photons to collapse gravitationally: the photon gas needs to have a temperature greater than the Planck temperature.

A Kugelblitz black hole is typically envisioned as a more stable arrangement than what I'm suggesting here.

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Re: Theory of black hole composition

Postby sevenperforce » Wed Jan 13, 2016 10:07 pm UTC

In the last graphic, I depicted the evaporative photons leaving behind a reduced event horizon which, from their perspective, was located at the center of the former black hole quanta. However, since we are dealing with evaporation lifetimes well in excess of the Planck time, it may be that the reduced event horizon will be located at the opposite edge of the former black hole quanta. If that was the case, RS/R would end up being 1/4, which would rework the math slightly. The mass of a single black hole quanta would evaluate to 2.736e-8 kg, about 30% greater than the Planck mass, with a Schwarzschild radius of 4.063e-35 m, about two and a half Planck lengths. Emitted particles would have a mass-energy of 1.229e9 J and an initial wavelength of just 10 Planck lengths, redshifting to 20 Planck lengths at infinity. Not sure which is more likely.

While the possibility of the black hole quanta matching the parameters of a metastable binary photon pair is tantalizing, I don't have quite the QM-fu necessary to probe it at the moment. Instead, I'll take a look at whether the spherical-quanta-shell model could accurately predict the Hawking blackbody spectrum of a large, stable black hole.

Consider a lunar-mass (7.3459e22 kg) black hole, with its event horizon of 1.1e-4 m, its surface area of 1.52e-7 m2, its Hawking radiation temperature of 1.668 K, its blackbody peak at 1.73e-3 m, and its power output of 6.6e-14 W. We'll use the earlier numbers for this.

First of all, might the blackbody peak match? An evaporating black hole quanta with its center of mass resting on the surface of the would-be event horizon releases two particles, one of which escapes and one of which is captured. The emission takes place, then, at 1.5 Planck lengths above the collective Schwarzschild radius. The Schwarzschild metric says the emitted radiation will be redshifted by a factor of 2.13e15, meaning that the peak emission of 2.717e-34 m will be redshifted to 5.787e-19 m at infinity.

Now, that's still sixteen orders of magnitude less than the peak wavelength of the classical...but that's not immediately damning, since the peaks of the two different spectra may not necessarily line up. But what about the power output? A black hole quanta emits a particle which is redshifted down to 5.787e-19 m with an energy of 3.433e-7 Joules, and it does so in just 3.612e-40 seconds. Not looking great so far. However, time dilation comes into play here. Time passes very slowly at 1.5 Planck lengths above the event horizon of a lunar mass black hole -- it's the same Schwarzschild factor again. So those 3.612e-40 seconds stretches into 7.694e-25 seconds. Unfortunately we're still dealing with a power output thirty-nine orders of magnitude too high. Moreover, there are roughly 4.5e30 quanta in a lunar-mass black hole, so that power output jumps to 69 orders of magnitude more than we want.

Saving grace, though: the probability of an evaporating quanta having an emission axis aligned along an escape trajectory is going to be vanishingly small. The surface area of the black hole quanta "poking out" of the surface of the putative event horizon is 2πR2 or 3.7e-69 m2 with a circumference of 7.62e-35 m. However, if you are nearing the event horizon of a black hole, the warping of spacetime essentially wraps the entire event horizon around you from behind. This reduces that 7.62e-35 m by the Schwarzschild factor of 2.13e15 to a tiny disc with a radius of just 1.79e-50 m, reducing the available escape area from 3.7e-69 m2 to 1e-99 m2. Thus, emission will be oriented onto an "escape" trajectory in just 1/3.7e30 instances. Power output is now down to 39 orders of magnitude too high.

What about the particle itself? With an initial wavelength of 2.717e-34 m, will it be able to fit through an "aperture" that is 3.58e-50 m wide? It will have to be diffracted through. I don't recall off the top of my head how to calculate the probability of diffraction for a particle with a wavelength larger than the opening it is attempting to pass through, I think it's proportional to the square of the ratio. In this case, the square of the ratio is 1.7e-32, bringing the power output within a few orders of magnitude of our target. Moreover, diffraction would "break up" the high energy particle into many lower-energy particles, which will hopefully allow us to hit that thermal blackbody curve.

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Re: Theory of black hole composition

Postby Hypnosifl » Wed Jan 13, 2016 10:46 pm UTC

One reason I would be cautious about some of this reasoning is that you seem to assume that the formula for Hawking radiation as a function of radius, which predicts some very intense radiation would be emitted by sufficiently small black holes, still holds for the initial formation of a black hole at the center of a collapsing star (or other collapsing matter/radiation sphere)--to see if this is valid one would have to look into the specifics of the derivation, it's possible it's derived based on assuming a Schwarzschild metric with fixed radius for example. From what I understand the simplest GR metric used to describe an expanding or contracting black hole is the Vaidya metric (it's a purely classical model, so contraction can only be modeled using the assumption of negative mass flowing into the black hole), and with some googling I found this paper on Hawking radiation in the Vaidya metric which seems to say the results are somewhat different (for example, they say the temperature is assigned not to the absolute horizon but to an 'apparent' horizon). There's also an analysis of Hawking radiation in the formation of a black hole starting on p. 81 of "Modeling Black Hole Evaporation" which can be read on google books, but the section available on google books cuts off before they reach their conclusions. I've found there's a lot of knowledgeable people at physicsforums.com, you might consider posting to ask about this question about how Hawking radiation works when the event horizon first forms at the center of collapsing matter there (though they tend to crack down on personal speculations that haven't been published in any peer-reviewed source, so if you do post there be sure to just ask an open-ended question rather than advancing a particular hypothesis).

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Re: Theory of black hole composition

Postby sevenperforce » Thu Jan 14, 2016 5:36 pm UTC

The complexity of these different approaches certainly makes me feel like I'm trying to model a thermonuclear weapon with a 2nd-grade chemistry set.

But hey, why not give it a shot?

I think that even before evaluating the validity of my application of Hawking radiation, I should probably try to get a better idea of whether a minimum-mass-black-hole formation event even makes sense in the context of a collapsing stellar-mass object.

So far, I've only ever seen stellar-mass-BH formation modeled/depicted as an outside-in event, where you have a sphere of reasonably uniform density which grows until its Schwarzschild radius meets its outer radius. For example, a 3-solar-mass neutron star has a radius of 13 km and a Schwarzschild radius of 8.9 km, so that its photon sphere (13.35 km) is actually just outside its actual surface. If mass was slowly added to the neutron star and automatically crushed to the neutron star's average density, then at approximately 10.73 solar masses, the Schwarzschild radius would equal the surface radius and the neutron star would collapse into a black hole from the outside in.

But we don't see neutron stars larger than about 2 solar masses, and it's expected that the pressure at just 3 or 4 solar masses would exceed degeneracy pressure at the core and lead to core-collapse. So it's an open question. Clearly, black holes must have a minimum size, but could that minimum size be realized during core collapse, or is the black hole already macroscopic/stable when its formation conditions are met? If the latter is the case, then either there must either be some degeneracy pressures stronger than those in a neutron star to limit the central density and allow it to build up to its own Schwarzschild radius, or the collapse happens with a specific velocity gradient so that the central density never has time to skyrocket. But a specific velocity gradient would presumably require some sort of mediating degeneracy pressure between the collapsing elements (thanks, shell theorem) which makes this seem less likely.

If something collapses with NO interaction between the collapsing particles then every single particle will be in a parabolic orbit through the center of mass of the object.

I had seen that book on Google Books and where the preview cut off...really quite disappointing.

I'll ask around on physicsforums.com about collapsar-formed black holes.

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Re: Theory of black hole composition

Postby Neil_Boekend » Fri Jan 15, 2016 8:51 am UTC

sevenperforce wrote:If something collapses with NO interaction between the collapsing particles then every single particle will be in a parabolic orbit through the center of mass of the object.

I am far from an expert but that doesn't seem to hold up in black hole conditions, or even near black hole conditions. During near black hole conditions the particle slows down (from an outside perspective) due to time dilation as it gets to it's perhelion. This means that a particle will spend a longer time (again, outside perspective) close to the barycenter than far away from the barycenter. This means that the particle density is higher close to the barycenter, more particles means higher gravity, that means higher time dilation and thus an increased effect. Thus the effect kinda stacks and increases over time.
Not to mention the weird orbits the time dilation itself may cause (as this is way beyond my armchair physicist's knowledge).
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Re: Theory of black hole composition

Postby doogly » Fri Jan 15, 2016 1:30 pm UTC

Yeah, parabolas are solutions in Newtonian gravity. That is so not what's going on here.
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Re: Theory of black hole composition

Postby sevenperforce » Fri Jan 15, 2016 2:10 pm UTC

Neil_Boekend wrote:
sevenperforce wrote:If something collapses with NO interaction between the collapsing particles then every single particle will be in a parabolic orbit through the center of mass of the object.

I am far from an expert but that doesn't seem to hold up in black hole conditions, or even near black hole conditions.

doogly wrote:Yeah, parabolas are solutions in Newtonian gravity. That is so not what's going on here.

Yeah, I misspoke there -- it's not going to be anything close to parabolic. The dominating factor is actually going to be radiation pressure; incoming radiation will be hella blueshifted and outgoing radiation will be hella redshifted, so it's going to squeeze everything together at the center pretty damn tightly. I can't see how the collapsar can avoid its central density increasing without bound.

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Re: Theory of black hole composition

Postby doogly » Fri Jan 15, 2016 2:16 pm UTC

sevenperforce wrote:I can't see how the collapsar can avoid its central density increasing without bound.

Why should it avoid that? We're talking black hole formation here.
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Re: Theory of black hole composition

Postby sevenperforce » Fri Jan 15, 2016 4:54 pm UTC

doogly wrote:
sevenperforce wrote:I can't see how the collapsar can avoid its central density increasing without bound.

Why should it avoid that? We're talking black hole formation here.

Well, perhaps to make the point more clear: I can't see how the collapsar can avoid its central density increasing without bound before meeting the necessary conditions for the formation of a black hole. The combination of radiation pressure and gravitational force would seem to dictate that the density gradient be greater than the inverse-squared relationship between density and mass of a black hole.

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Re: Theory of black hole composition

Postby doogly » Fri Jan 15, 2016 5:52 pm UTC

But there is no such relationship.
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Re: Theory of black hole composition

Postby sevenperforce » Fri Jan 15, 2016 6:22 pm UTC

doogly wrote:But there is no such relationship.

Pretty sure there is.

The density of any object is its mass divided by its volume. For a sphere, this is the mass divided by 4πr3/3.

At the moment a spherical object becomes a black hole, its radius is equal to 2*G*M/c2.

So, the critical density ρc of an object at the moment it becomes a black hole, as a function of mass, is:

ρc(M) = 3c6/32πG3M2

Plotting this function, we get:

black hole region.png
black hole region.png (8.25 KiB) Viewed 4842 times

An object becomes a black hole whenever it crosses from the lower region to the upper region. Note that it can cross in a few different ways: by increasing mass while keeping density constant, by increasing density while keeping mass constant, or by increasing both mass and density together.

The kicker is that in a core-collapse situation, density will not be constant with respect to radius, nor will the increase in density necessarily be linear with respect to radius. So you can have a situation where the core crosses this line before the object does as a whole, leading to a black hole forming at the center of the collapse. Depending on the nature of the density gradient, this region of the core could be arbitrarily small...perhaps even at the minimum mass of a black hole.

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Re: Theory of black hole composition

Postby doogly » Fri Jan 15, 2016 6:27 pm UTC

That's not the volume of a black hole. This formula for volume in terms of radius works for balls in Euclidean space, but a black hole is not like this. You have to specify who is doing the measurement of volume and what they mean by it, and do integrals.

Even if there were some other simple formula that you could use instead, you could still only say something like average_density = M / V. This wouldn't give you anything meaningful as a function of position though, it's just an average.
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Re: Theory of black hole composition

Postby sevenperforce » Fri Jan 15, 2016 6:33 pm UTC

doogly wrote:That's not the volume of a black hole. This formula for volume in terms of radius works for balls in Euclidean space, but a black hole is not like this. You have to specify who is doing the measurement of volume and what they mean by it, and do integrals.

Even if there were some other simple formula that you could use instead, you could still only say something like average_density = M / V. This wouldn't give you anything meaningful as a function of position though, it's just an average.

Then let me specify.

For a spherical distribution of constant mass M, collapse into a black hole will take place when the average density as measured from that mass distribution's reference frame exceeds 3c6/32πG3M2.

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Re: Theory of black hole composition

Postby doogly » Fri Jan 15, 2016 7:05 pm UTC

You're using the Euclidean volume formula again.
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Re: Theory of black hole composition

Postby sevenperforce » Fri Jan 15, 2016 7:17 pm UTC

Are you suggesting that there is no reference frame in which it is meaningful to calculate the volume of space within the Schwarzschild radius corresponding to a given amount of mass?

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Re: Theory of black hole composition

Postby doogly » Fri Jan 15, 2016 7:30 pm UTC

Well, you can do something that has meaning in one reference frame. Things that aren't invariants are often pretty boring, but you can do them.

But even when you do choose to live this way, you have to do the integrals, not jsut take the Euclidean formula. Take some coordinates that cover the space you're interested in (so painleve maybe, not schwarzschild) and integrate,
V = int sqrt(g) dx ^ dy ^ dz

Yeah, 3-volumes aren't really well behaved.
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Re: Theory of black hole composition

Postby sevenperforce » Fri Jan 15, 2016 7:38 pm UTC

I don't think it much matters for these purposes, as long as the integral of the volume function from 0 to r1 is necessarily less than the integral of the volume function from 0 to r2 for all r1 < r2.

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Re: Theory of black hole composition

Postby sevenperforce » Sat Jan 16, 2016 1:32 am UTC

In case I was in some way unclear...

I'm talking about density and volume prior to the formation of a black hole, not afterwards. The blue region in the graphic above is undefined, and I understand that. It's the behavior below the line that I'm interested in. Are you suggesting that it is meaningless to talk about the density or volume of various layers of a neutron star?

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Re: Theory of black hole composition

Postby doogly » Sat Jan 16, 2016 2:23 am UTC

It's wrong to use Euclidean geometry, yes.
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Re: Theory of black hole composition

Postby sevenperforce » Sat Jan 16, 2016 2:50 am UTC

doogly wrote:It's wrong to use Euclidean geometry, yes.

Odd, then, that the density gradient inside a neutron star receives so much attention.

If you balk at using Euclidean geometry to define the sufficient conditions for the formation of a black hole, how would you define these conditions? In every source I've seen, a distribution of mass is considered to have become a black hole when it falls within its own Schwarschild radius. Is such a definition lacking some vital caveat? And, more to the point, would such a caveat somehow prevent those conditions from being met at the infinitesimal center of a collapsing star?

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Re: Theory of black hole composition

Postby Link » Sat Jan 16, 2016 12:16 pm UTC

Don't confuse the r coordinate in e.g. the Schwarzschild metric with the radius of a sphere. Although there's spherical symmetry, a black hole or relativistic star isn't a sphere in the usual sense of the word because space is curved. Calculating the density or mass of a strongly self-gravitating object at any point is not something you can do with pen and paper unless it has an unphysically trivial equation of state (e.g. constant density).

The relation of the energy density ρ on the "radial" coordinate r is important, but r isn't identical to a sphere's radius, nor does the integral of 4πr²ρ(r) from 0 to some constant R give you the mass enclosed in the sphere of radius R. That integral can give you a star's mass measured from infinity, if you integrate it up the r where it stops changing (which corresponds to the surface of the star), but apart from that, it doesn't correspond 1-to-1 to what we generally call mass.

Source: "Gravity: An Introduction to Einstein's General Relativity", James B. Hartle, Ch. 24. (This chapter also has some info about estimating the upper bound on the mass of neutron stars; it's worth a read.)

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Re: Theory of black hole composition

Postby sevenperforce » Mon Jan 18, 2016 4:05 pm UTC

Link wrote:Don't confuse the r coordinate in e.g. the Schwarzschild metric with the radius of a sphere. Although there's spherical symmetry, a black hole or relativistic star isn't a sphere in the usual sense of the word because space is curved. Calculating the density or mass of a strongly self-gravitating object at any point is not something you can do with pen and paper unless it has an unphysically trivial equation of state (e.g. constant density).

The relation of the energy density ρ on the "radial" coordinate r is important, but r isn't identical to a sphere's radius, nor does the integral of 4πr²ρ(r) from 0 to some constant R give you the mass enclosed in the sphere of radius R. That integral can give you a star's mass measured from infinity, if you integrate it up the r where it stops changing (which corresponds to the surface of the star), but apart from that, it doesn't correspond 1-to-1 to what we generally call mass.

Ah, I see.

Well, suppose that we have some function M(r) representing the definite integral from 0 to r, with r properly defined, such that M(r) represents the mass enclosed within radius r. Not too terribly worried about finding M(r); just need to make sure we agree on what it is. Then, since RS(M) is presumably still going to be equal to 2GM/c2, so we can define RS(r) = 2GM(r)/c2. Or, if that's somehow no longer applicable, then we can merely define RS(r) = RS(M(r)). Again, not too worried about needing to actually evaluate this.

The question is, given r1 and r2 such that r1 < r2, properly defined for strong-self-gravitating curved space, is there any aspect of the curvature of space which would prevent RS(r1)/r1 > RS(r2)/r2? Or, if it makes the math easier, r1 - RS(r1) < r2 - RS(r2)? Either way, it's the same qualitative condition: that there is some r1 wherein the mass contained within r1 is "closer" to being a black hole than the mass contained within r2 where r1 < r2. Hopefully that makes sense.

If it could be proven that this condition is impossible, that would be pretty interesting in its own right.

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Re: Theory of black hole composition

Postby Link » Mon Jan 18, 2016 5:01 pm UTC

I don't see why that shouldn't be possible, but then again I have only a basic working knowledge of GR so I could well be wrong. In any case, your reasoning appears to be similar to what Hartle uses for the aforementioned neutron star mass bound, though he explicitly assumes the core isn't a black hole, while you're interested in the case where the core does become a black hole. I don't really have the time to look into it at the moment, though -- you've already seriously nerd-sniped me at this point. :P

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Re: Theory of black hole composition

Postby sevenperforce » Mon Jan 18, 2016 5:07 pm UTC

Link wrote:I don't see why that shouldn't be possible, but then again I have only a basic working knowledge of GR so I could well be wrong. In any case, your reasoning appears to be similar to what Hartle uses for the aforementioned neutron star mass bound, though he explicitly assumes the core isn't a black hole, while you're interested in the case where the core does become a black hole. I don't really have the time to look into it at the moment, though -- you've already seriously nerd-sniped me at this point. :P

Mission accomplished!

Thanks for the help, though. :)

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Re: Theory of black hole composition

Postby KarenRei » Tue Jan 19, 2016 1:18 pm UTC

All of this assumes that there's a such thing as "inside a black hole".

Hopefully some of you with more knowledge than I can chime in, but I've noticed a lot of movement in recent years toward the concept that all matter falling into a black hole exists at (or more accurately, just beyond) the event horizon, that it never falls in - and its information is still correlated with our universe as it's released as the black hole boils off. This would include the material that formed the black hole in the first place. Basically, like a region of nearly frozen time. In such a situation then you really don't need to deal with any questions of what is "inside" a black hole, you just need a Lie algebra wherein gravity distorts the space metric such that an external-observer's perception of the location of matter near the event horizon shows it moving outward as the black hole grows and inwards as it shrinks.

To an infalling observer, then, there's "no drama", just a fall into a collapsing star. But since an outside observer perceives their relative time pass at a crawl, then the inside observer must perceive the boiloff of the black hole at a much faster rate for causality to work. This also requires a significant expansion of space. Anything concerning the apparent expansion of space directly maps to inflation. Because inflation calls for the significant expansion of space in the early (aka dense) stage of our universe. Our universe seems to be compatible with space metrics that alter in apparent size when density is high, so why not assume that it's also applicable near black holes?

If that would be correct, that the Big Bang was directly analogous to a black hole boiloff, then a curious side effect would be that there could be pre-universe information existing in our universe - although too distorted to ever make use of. It would also make black holes would have "black hole nucleosynthesis", unification of the forces, etc directly mapping to the Big Bang - just heavily distorted from our perspective by relative perceived time differences.

Regardless of whether one subscribes to the above, a point to consider: by the time matter currently near the event horizon of a supermassive black hole is released, the universe will long have since died a heat death and drifted apart to nothingness. From the perspective of whatever condenses out of that energy, it will be a new universe.
Last edited by KarenRei on Wed Jan 20, 2016 11:26 am UTC, edited 1 time in total.

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Re: Theory of black hole composition

Postby doogly » Tue Jan 19, 2016 1:30 pm UTC

This is a little jumbled. If you are out at asymptotic infinity and you drop a ball into the black hole, you won't see it cross. But the ball doesn't see just a collapsing star, it sees bona fide black hole stuff. If the ball were equipped with rocket thrusters, there is no amount of thrust they could fire to get back out of the horizon (this being the defining thing with black holes.) This particular misunderstanding does seem to be in vogue though.

The big bang stuff then gets further off, not sure what the basis even is for that stretch.
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Re: Theory of black hole composition

Postby sevenperforce » Tue Jan 19, 2016 2:22 pm UTC

For why the universe is not a black hole, we turn to Sean Carroll.

doogly, do you happen to know whether there is some aspect of the curvature of space in/around a strongly self-gravitating body which would prevent that previously-discussed condition from being met -- specifically, that given R1 < R2 < R, the mass contained within R1 has a Schwarzschild radius which is larger in proportion to R1 than the Schwarzschild radius of the mass contained within R2 is in proportion to R2?

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Re: Theory of black hole composition

Postby doogly » Tue Jan 19, 2016 2:35 pm UTC

What's the unsubscripted R?
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Re: Theory of black hole composition

Postby sevenperforce » Tue Jan 19, 2016 2:46 pm UTC

doogly wrote:What's the unsubscripted R?

Oh, sorry, I meant to make that Rmax, corresponding to the surface of the object, such that all the mass of the object is at a "height" lower than Rmax.

And obviously I recognize that these distances aren't necessarily linear compared to Euclidean space...but since we are talking about a very compact object rather than a black hole, these distances should still be continuous, right?

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Re: Theory of black hole composition

Postby doogly » Tue Jan 19, 2016 3:26 pm UTC

So anything which was less dense in the center than at higher radius would violate this condition, yeah? I can think of such things.
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Re: Theory of black hole composition

Postby KarenRei » Tue Jan 19, 2016 3:51 pm UTC

doogly wrote:This is a little jumbled. If you are out at asymptotic infinity and you drop a ball into the black hole, you won't see it cross. But the ball doesn't see just a collapsing star, it sees bona fide black hole stuff.


Says...?

You're simply taking your premise and asserting it as fact, rather than arguing it.

If the ball were equipped with rocket thrusters, there is no amount of thrust they could fire to get back out of the horizon (this being the defining thing with black holes.)


It can (according to Hawking) never pass or even get to the event horizon - that's the latest solution to the firewall paradox. Hence the concept of "thrusting out from beyond the event horizon" is ill defined.

The concept of black holes basically being an area of nearly-stopped time (from an outside observer's perspective) is perfectly compatible with observations, and it keeps it fully thermodynamically coupled with our universe without any complications.

The only thing that it requires is localized inflation, as the event horizon is perceived to change in size relative to the mass of the black hole; any matter that was on the event horizon while the black hole was smaller can't suddenly move into undefined space as the black hole grows; space itself has to inflate. But we know that space itself can inflate in particular circumstances.

The big bang stuff then gets further off, not sure what the basis even is for that stretch.


Oh please, don't act like there's not quite a number of physicists who have proposed a Big Bang / black hole equivalence and anyone who suggests the same thing is an idiot.

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Re: Theory of black hole composition

Postby doogly » Tue Jan 19, 2016 4:15 pm UTC

Oh, this is what my doctorate is in (not actually touching myself, as the title below the username may seem to indicate. that's just a bonus.)

Reports of firewall paradox are vastly overstated. Unless you believe in string theory microstate counting more than the equivalence principle, and even then, I don't know that this actually resolves everything. There are some weak claims being made in press releases as well, so not all of the confusion here is second hand.

Big Bang / black hole equivalence is... idiot is a very strong word, but yeah, the link posted above to Carrol's blog is pretty set.
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Re: Theory of black hole composition

Postby sevenperforce » Tue Jan 19, 2016 4:21 pm UTC

doogly wrote:So anything which was less dense in the center than at higher radius would violate this condition, yeah? I can think of such things.

Well, I'm not asking whether it's possible for this condition to ever conceivably fail; I'm asking whether it's possible for this condition to ever be met.

I hesitated to say "density" because of the whole issue with non-Euclidean geometry. Also, it needs to be more than merely "deeper is denser"; the depth/density relationship would need to be steep enough that RS(M1)/r1 is greater than RS(M2)/r2, not just where ρ1 > ρ2. I was trying to explain this earlier quantitatively, but that's where the non-Euclidean geometry started to mess things up. Thanks for the feedback; I'm in a little over my head here.

I do know that it can go either way in a Euclidean arrangement. For example, if ρ(r) ∝ 1/r, then even though the density is increasing as you approach the core, the mass at or below a given r-coordinate decreases more rapidly than r, so that the RS for the mass below r1 is a smaller proportion of r1 than the RS for the mass below r2 in comparison to r2. Alternatively, if ρ(r) ∝ 1/r3, then the amount of mass at or below a given r-coordinate will decrease more slowly than r, so that RS for the mass below a given r-coordinate becomes closer and closer to that r-coordinate.

I'm wondering whether the non-Euclidean relationship between radius and volume would somehow prevent the latter scenario from ever being realizable.

KarenRei wrote:
doogly wrote:This is a little jumbled. If you are out at asymptotic infinity and you drop a ball into the black hole, you won't see it cross. But the ball doesn't see just a collapsing star, it sees bona fide black hole stuff.


Says...?

You're simply taking your premise and asserting it as fact, rather than arguing it.

He's not asserting a premise as fact. That's required by the equivalence principle.

If the ball were equipped with rocket thrusters, there is no amount of thrust they could fire to get back out of the horizon (this being the defining thing with black holes.)

It can (according to Hawking) never pass or even get to the event horizon - that's the latest solution to the firewall paradox. Hence the concept of "thrusting out from beyond the event horizon" is ill defined.

Given a black hole with event horizon at RS as observed from infinity, an object dropped into the black hole from infinity will not appear to cross RS within finite time to an observer from infinity, but the object itself will most definitely cross RS in its own reference frame. Now, it's not going to notice anything special at RS because when it gets to RS the event horizon of the black hole below it will appear to have shrunk per the equivalence principle, but that doesn't mean it hasn't crossed RS, and once it has reached that point (even though it didn't notice anything locally special), no amount of thrusting will get them to a higher altitude.

The firewall paradox is that the release of Hawking radiation from the event horizon would cause the event horizon to produce a great deal of energy and be quite noticeable to anything crossing it, violating the equivalence principle. Hawking's qualitative solution is that quantum fluctuation can cause the apparent horizon to behave chaotically, so that an infalling observer will not be able to see a discrete point at which Hawking radiation reaches a maximum and thus equivalence is preserved. This in no way prevents one from crossing RS.

The big bang stuff then gets further off, not sure what the basis even is for that stretch.

Oh please, don't act like there's not quite a number of physicists who have proposed a Big Bang / black hole equivalence and anyone who suggests the same thing is an idiot.

Then where are the papers? Also, see the link I posted earlier.

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Re: Theory of black hole composition

Postby KarenRei » Tue Jan 19, 2016 4:30 pm UTC

doogly wrote:Reports of firewall paradox are vastly overstated.


The number of physicists actively publishing on the issue says otherwise.

Big Bang / black hole equivalence is... idiot is a very strong word, but yeah, the link posted above to Carrol's blog is pretty set.


Carroll's blog is based on the premise of A) a universe currently inside a black hole, B) based on the similarity of the Hubble radius and the Schwartzchild radius

Why you would think that an appropriate reply to someone who A) argued that there's no such thing as "the inside of a black hole" and B) made no mention of said similarity, is beyond me. An entire blog post about the inside of black holes, in reply to a person who argued that there's no such thing as "the inside of a black hole" - seriously, that's your only argument?

Care for something other than ipse dixit?

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Re: Theory of black hole composition

Postby KarenRei » Tue Jan 19, 2016 4:58 pm UTC

KarenRei wrote:You're simply taking your premise and asserting it as fact, rather than arguing it.


He's not asserting a premise as fact. That's required by the equivalence principle.


What a person experiences falling into a black hole depends on the spacetime metric. If the spacetime metric, as constructed from an outside perspective, equates to "space inflates and time slows to a near stop", then the equivalence principle for an infalling observer is "no drama". Apart from the drama of being killed by falling into a supermassive chunk of mass-energy.

If he wants to argue "drama", then he needs to argue that such a spacetime metric is incompatible with observations. He can't just simply assert it and call that done.

Given a black hole with event horizon at RS as observed from infinity, an object dropped into the black hole from infinity will not appear to cross RS within finite time to an observer from infinity, but the object itself will most definitely cross RS in its own reference frame.


Again, you're simply asserting a particular spacetime metric as fact without arguing why it must be considered as fact. If, from the perspective of an outside observer, space itself is inflated and time slowed to a near stop, then no, they really never do cross any sort of event horizon. They just see themselves falling into a collapsing star.

In said scenario, if we have an infalling craft and an immortal outside observer with flawless ability to reconstruct distorted information, said observer sees the entire journey of the infalling craft. The infalling observer sees it happen quickly and undistorted; the outside observer sees it spread out over incredible timescales and distorted to extremes. There is never a break, in said metric, between the black hole and the outside world, just tremendous temporal and spatial distortion. The infalling observer really is falling all the way in, and the outside observer really will see it, all the way - only spread out over tremendous timescales and with tremendous distortion.

Hawking's qualitative solution is that quantum fluctuation can cause the apparent horizon to behave chaotically, so that an infalling observer will not be able to see a discrete point at which Hawking radiation reaches a maximum and thus equivalence is preserved. This in no way prevents one from crossing RS.


http://arxiv.org/abs/1401.5761

“I take this as indicating that the topologically trivial periodically identified anti deSitter metric is the metric that interpolates between collapse to a black hole and evaporation. There would be no event horizons and no firewalls. The absence of event horizons means that there are no black holes — in the sense of regimes from which light can't escape to infinity. There are however apparent horizons which persist for a period of time. This suggests that black holes should be redefined as metastable bound states of the gravitational field.”


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