Could a Particle Accelerator Create a Black Hole That Destroys Earth?

[DaveC426913]

No, it wouldn't, because the matter starts out at rest and it will take time for it to cover the distance to the center. Roughly speaking, if we assume that the matter has zero viscosity for this purpose (since it's all going down the hole at the center so matter just crossing the horizon won't "push back" against matter behind it), the time for the matter at the Earth's surface to reach the center and get swallowed by the hole should be about 20 minutes--one fourth of the free-fall orbit time.

Mayhap, but that only applies once the BH is macro-scale. For most of the BH' life, the distance to atoms is on the atomic scale. i.e. the fall time would be in picoseconds.

 

[jbriggs444]

Mayhap, but that only applies once the BH is macro-scale. For most of the BH' life, the distance to atoms is on the atomic scale. i.e. the fall time would be in picoseconds.

When the black hole is micro-scale, it consumes even less.

 

[PeterDonis]

For most of the BH' life, the distance to atoms is on the atomic scale. i.e. the fall time would be in picoseconds.

I don't think this is true for a hole that is actually going to grow (i.e., it won't evaporate before it accretes enough mass to push the evaporation time far out into the future), at least not within any kind of ordinary matter like that of the Earth. I can't resist analyzing this in more detail, so here goes.

First, let's write down a general formula for the fall time to a hole of mass $M$ from a distance $R$. We can use the Newtonian equations for a point source of gravity for this because, as we'll see, the Schwarzschild radius of the holes we'll be discussing is so much smaller than the distances we'll be calculating that we can ignore relativistic corrections, since they will only be significant for a tiny fraction of the fall time. The Newtonian free fall time to a point mass $M$ from a starting radius $R$ is

$$
t = \frac{\pi}{2} \frac{R^{\frac{3}{2}}}{\sqrt{2GM}} = 1.36 \times 10^5 \sqrt{\frac{R^3}{M}}
$$

where I have plugged in all of the numerical factors, in SI units (so $R$ is in meters and $M$ is in kilograms, and $t$ comes out in seconds).

Next, the formula for the Hawking evaporation time of a black hole of mass $M$ (as has already been posted, I'm posting it again for convenience) is

$$
T = 5120 \frac{\pi G^2 M^3}{\hbar c^4} = 8.4 \times 10^{-17} M^3
$$

where again I have plugged in all of the numerical factors in SI units (so again $M$ is in kilograms, and $T$ comes out in seconds).

Evaluating the inequality $t < T$ gives us a first cut at a relationship between $M$ and $R$, i.e., within what radius $R$ surrounding matter would have to be at to be able to fall into a hole of mass $M$ before it evaporates. This relationship is

$$
R < 7.25 \times 10^{-15} M^{\frac{7}{3}}
$$

So, for example, a 1 kilogram hole would have to have surrounding matter within $7.25 \times 10^{-15}$ meters to have any of it fall in before the hole evaporates. Since that distance is roughly the size of an atomic nucleus, such a hole would evaporate well before any adjacent atoms had time to fall in; only if the hole happened to be created inside an atomic nucleus (or inside an object with density comparable to that of an atomic nucleus, such as a neutron star) would anything have a chance to fall in prior to evaporation.

To get $R$ to roughly the size of an atom, i.e., roughly $10^{-10}$ meters, requires a hole of about $59.4$ kg. So a hole with mass less than that would not have a chance of pulling in adjacent atoms before evaporating.

For a hole to have a significant chance of accreting enough matter to keep itself from evaporating, it probably needs to have a mass within the radius $R$ that is comparable to its own mass. If we assume Euclidean spatial geometry (which is a good approximation for a planet like the Earth or an ordinary star like the Sun, but might not be for something like a neutron star), then the mass inside radius $R$ is $\frac{4}{3} \pi R^3 \rho$; so if we want that mass to be the same as the mass $M$ of the hole, we plug the formula for $R$ in terms of $M$ above (using equality instead of less than) into the Euclidean formula, which, after some algebra, gives

$$
M = 9.25 \times 10^6 \frac{1}{\rho^{\frac{1}{6}}}
$$

So for a density of $1$ (the density of air, roughly), $M$ would be $9.25 \times 10^6$ kg, or 9250 metric tons. The $R$ corresponding to this is about 130 meters. For a density equal to the average density of Earth, about $5500$, $M$ reduces to about $2.2 \times 10^6$ kg, or 2200 metric tons. The $R$ corresponding to this is about $4.6$ meters. And the fall time $t$ for this $R$ and $M$ is about $5.9$ seconds. (Note that this hole has a Schwarzschild radius of about $3.3 \times 10^{-21}$ meters, still much smaller than an atomic nucleus.)

So if we imagine a black hole of mass 2200 metric tons that somehow got created inside a ball of Earthlike-density matter with the same mass, and radius 4.6 meters, it would take almost 6 seconds for the hole to swallow all of the mass; and for most of that time, the distances that matter was falling to reach the hole would not be microscopic. Even more so, if we imagine such a hole created at the center of the Earth, it would take 6 seconds for it to swallow all of the matter in a sphere 4.6 meters in radius around it; but it would take about 20 minutes (per my previous post) to swallow the entire Earth, and for almost all of that time, the distances would not be microscopic. (Note that we can't just plug the Earth's radius and 2200 metric tons for $M$ into the formula for $t$ above; we would need to figure out a formula for $t$ if $M$ is changing with time as the hole accretes matter. That's more than I have the time or patience for right now.)

 

[DaveC426913]

I don't think this is true for a hole that is actually going to grow (i.e., it won't evaporate before it accretes enough mass to push the evaporation time far out into the future)

Yes, I concede that evaporation puts the nail in the coffin of a subatomic runaway black hole.

I started this thread because I was thinking that evaporation doesn't seem to provide sufficient warm & fuzzies that it can't happen. I'll just have to take Hawking's word for it that it really happens at the rate predicted. (Has this been experimentally verified?)

I was looking for an answer more concrete than "it could grow - IF it weren't stopped by this other mysterious phenomenon". I was looking for a "it can't happen at all" kind of answer to mollify friends who read these stories and ask.

So for a density of $1$ (the density of air, roughly), $M$ would be $9.25 \times 10^6$ kg, or 9250 metric tons. The $R$ corresponding to this is about 130 meters. For a density equal to the average density of Earth, about $5500$, $M$ reduces to about $2.2 \times 10^6$ kg, or 2200 metric tons.

Is that a reasonable estimate of the pressure at the centre of the Earth?

But I guess, when we're talking about BH's, a few orders of magnitude in density is insignificant.

 

[DaveC426913]

When the black hole is micro-scale, it consumes even less.

Which might just balance out against the fact that the fall-time is virtually zero.
But I'm not certain which way it would go.

Even if it's only consuming a few atoms at a time, if that occurs every pico-second, how fast might it grow?

 

[PeterDonis]

I'll just have to take Hawking's word for it that it really happens at the rate predicted. (Has this been experimentally verified?)

No, nor is it expected to be any time soon. The evaporation rate from any hole we can observe (stellar mass or larger) is many, many orders of magnitude too small to measure. And we aren't likely to be able to manufacture a hole small enough that we could measure its evaporation rate any time soon.

Is that a reasonable estimate of the pressure at the centre of the Earth?

I assume you mean the density at the center--no, it's not, I was just using the Earth's average density for a rough estimate. Since the threshold mass goes as the inverse sixth root of the density, the difference between the Earth's average density and the density at the center of the Earth doesn't make much difference in this calculation.

Even if it's only consuming a few atoms at a time, if that occurs every pico-second, how fast might it grow?

Consider the hole for which $M$ is just large enough that an adjacent atom can fall in before it evaporates; this is the hole with mass $59.4$ kg as calculated in post #38. The number of atoms that can be adjacent, if they are packed as a solid, is of order unity; let's suppose it's 10, as an upper limit. Let's suppose each atom has mass of about $10^{-25}$ kg (this is an atomic weight somewhere around that of iron). Then the hole sucks in $10^{-24} / 59.4$ or a fraction of about $10^{-26}$ its own mass; this will not increase its evaporation time by enough to matter.

This is the reason why I said in post #38 that, in order to grow, the hole has to have mass large enough that a mass on the order of its own mass is within the radius $R$ calculated using the formula in that post; that will ensure that enough mass falls into keep the hole from evaporating. That mass is, as I calculated, around 2200 metric tons, or about 37,000 times the mass of the 59.4 kg hole discussed just above. Since the fall time goes like the inverse square root of $M$, this means the fall time for adjacent atoms into this hole will be about 200 times shorter than the fall time for adjacent atoms into the 59.4 kg hole. So something like an atom per 1/100 of a picosecond is how quickly a hole needs to suck in adjacent atoms in order to grow fast enough to keep itself from evaporating.

 

[DaveC426913]

I assume you mean the density at the center

No, I meant pressure. The atoms are not merely falling, they have a huge weight on top of them. They're being crammed into the void left by atoms that are consumed by the BH.

 

[LURCH]

Or to add another fantasy: It happened so often before until the right setup has been left.

That’s what I’m saying; we’ve found a new theory for dark matter. The universe is mostly filled with teensey-weensey black holes that used to be planets. (Lol)

Come to think of it, we may have explained the Fermi Paradox at the same time. That’s it, I’m rushing over to the sci-fi Forum to inspire some aspiring author!

 

[PeterDonis]

I meant pressure. The atoms are not merely falling, they have a huge weight on top of them.

Ah, ok. I didn't take any pressure effects into account since I was only looking at very rough orders of magnitude. I also didn't take into account that as the hole evaporates, its Hawking radiation exerts pressure outward.

 

[PeterDonis]

The atoms are not merely falling, they have a huge weight on top of them. They're being crammed into the void left by atoms that are consumed by the BH.

I'm not sure this is right. The force between the atoms in the normal case is there because the situation is static: none of the atoms can free-fall. If a void is created by the BH at the center, and the atoms start falling into it, the force between the atoms should vanish, at least to a first approximation (assuming a perfectly symmetrical fall process). The atoms on top could only "push" the atoms below if for some reason the atoms below were falling slower than the atoms on top; but that shouldn't be the case, at least to a first approximation.

 

[anorlunda]

If we go back to the original premise of this thread, that a BH could form in the LHC at CERN. In that scenario, the BH must be built one particle at a time. But as I showed in #20, the 2 proton BH event is explosive, not implosive. @PeterDonis analysis in #38, considered only free fall, and not the repulsive forces of radiation pressure. Even if the BH is embedded in ordinary matter, the net forces are initially repulsive rather than attractive. So I see no credible evolution to build a massive BH, adding one particle at a time.

Must we trust that Hawking was correct? If micro black holes did not evaporate, then we would find it difficult to explain why we don't find micro primordial black holes everywhere, or to explain the genesis of all the elementary particles at the origin of the universe instead of genesis of primordial black holes. If there was a non-zero probability of two particles forming a BH in the genesis, and if that was irreversible, then why not all particles becoming BH? I view the absence of primordial BH as an argument that the two-particle BH event (if it exists at all) must be time reversible.

 

[fresh_42]

I was looking for a "it can't happen at all" kind of answer to mollify friends who read these stories and ask.

I think you can say:
"Our best experts in the field have calculated various scenarios and none led to a result which was even close to a disaster."

 

[jbriggs444]

I'm not sure this is right. The force between the atoms in the normal case is there because the situation is static: none of the atoms can free-fall. If a void is created by the BH at the center, and the atoms start falling into it, the force between the atoms should vanish, at least to a first approximation (assuming a perfectly symmetrical fall process). The atoms on top could only "push" the atoms below if for some reason the atoms below were falling slower than the atoms on top; but that shouldn't be the case, at least to a first approximation.

Why would one not look at this in the Newtonian approximation as an example of a choked flow? You have a volumetric flow rate which can be calculated in terms of velocity multiplied by cross-sectional area.

Velocity scales as the square root of gravitational potential. [Newtonian] potential scales with inverse radius. So we are talking $v \propto \frac{1}{\sqrt{r}}$. Cross section scales as inverse radius squared. So we are talking $a \propto r^2$

In the absence of a pressure gradient, the volumetric flow rate at radius r is then given by $VFR(r) \propto r^{1.5}$. But since the equilibrium mass flow rate must be equal everywhere, we either need a pressure gradient to slow down the flow up top to match the choked flow rate at the bottom or a density gradient to accomplish the same thing. A density gradient implies a pressure gradient anyway.

Or am I missing something? [Obviously I'm missing that it's not a perfect equilibrium and that the Newtonian approximation breaks down near the bottom]

Edit: Boneheaded math mistake rectified.

 

[PeterDonis]

Why would one not look at this in the Newtonian approximation as an example of a choked flow?

Because the black hole at the center acts as an infinite sink, from a Newtonian standpoint. Matter goes in but never comes out; it gets destroyed in the singularity inside the hole. So there is no "choke point" in the flow. A more technical way of putting it is that the presence of the black hole means that the continuity equation is violated (flow lines all go into the hole, none come out).

 

[PeterDonis]

Even if the BH is embedded in ordinary matter, the net forces are initially repulsive rather than attractive.

They are for a hole with mass on the order of the energies available in the LHC, yes; you calculated that in post #20.

For the much larger holes I have been considering, let's see. The Hawking radiation power is

$$
P = \frac{\hbar c^6}{15360 \pi G^2 M^2} = 3.562 \times 10^{32} \frac{1}{M^2}
$$

We can convert this to a flux (in Watts per square meter) at a one-atom distance (which I am taking to be $10^{-10}$ meters) by dividing by the area of a 2-sphere with that radius, $4 \pi \times 10^{-20}$:

$$
F = 2.835 \times 10^{51} \frac{1}{M^2}
$$

We can then convert this to an effective energy density, and hence (since we're dealing with light) an effective pressure, by dividing by $c$ to obtain

$$
p = 9.455 \times 10^{42} \frac{1}{M^2}
$$

For $M$ equal to 2200 metric tons (or $2.2 \times 10^{6}$ kg), this works out to $p = 1.95 \times 10^{30}$ Pascals. This is indeed still huge, certainly large enough to keep matter from falling into the hole while it evaporates, despite my previous calculations (which assumed free falling matter).

From the above formula, if we wanted a pressure no greater than, say, atmospheric pressure at the Earth's surface, about $10^5$ Pascals, we would need a hole with mass $9.66 \times 10^{18}$ kg. This is about the mass of a moderately sized asteroid. Note that this hole has a Schwarzschild radius of $1.43 \times 10^{-8}$ meters, so at this point we would have to change the analysis anyway since the hole is larger than the one-atom distance I assumed above. But at any rate it seems clear that for the hole's Hawking radiation pressure not to easily prevent adjacent matter from falling in, the hole has to be of at least "small astronomical body" mass.

 

[DaveC426913]

Must we trust that Hawking was correct? If micro black holes did not evaporate, then we would find it difficult to explain why we don't find micro primordial black holes everywhere, or to explain the genesis of all the elementary particles at the origin of the universe instead of genesis of primordial black holes. If there was a non-zero probability of two particles forming a BH in the genesis, and if that was irreversible, then why not all particles becoming BH? I view the absence of primordial BH as an argument that the two-particle BH event (if it exists at all) must be time reversible.

That's a good point. Hawking has to have got it right.
We get more energetic collisions right here on Earth (occasionally) and they don't form runaway black holes.

 

[mfb]

Uh... let's clarify some things.

The newsweek article is terrible, and probably just written as book advertisement. One example here:

However, Rees also says we must be mindful of all eventualities: “Physicists should be circumspect about carrying out experiments that generate conditions with no precedent, even in the cosmos.
[...]
“Many of us are inclined to dismiss these risks as science fiction, but give the stakes they could not be ignored, even if deemed highly improbable.”

It is correct that we should be careful if we would create conditions with no precedent. But we don't do that. Cosmic rays lead to collisions of higher energy every day. In the article this is not mentioned at all - it sounds like we would do something beyond natural processes.

The relevant formula is:

$T(M) = 5120 \frac{\pi G^2 M^3}{\hbar c^4}$

So for two protons the evaporation time will be about 3 x 10^-96 seconds.

For one gram of matter, about 8 x 10^-26 seconds. I wonder how that compares to the reaction rate of a typical nuclear explosion?

That formula works for macroscopic black holes where Hawking radiation is dominated by massless particles for most of its lifetime but it fails for microscopic black holes where more particles contribute. They evaporate faster. At 1 gram Hawking radiation will include every particle we know (and potentially several we do not).

We would need QED and perhaps GR to calculate the behavior of nearby particles in that time window with those conditions.

We don't, there are no nearby particles. The next proton is at least 10^-15 m away, it cannot reach the black hole in less than 10^-24 seconds - ages compared to the lifetime of the black hole.

Why would it take billions/trillions of years? (Presumably, the vast majority of that time would be spent at subatomic size, with the last macro-scale gobbling happening in just moments.)
With matter crushing down on it at millions of atmospheres, why would it take so long to grow?

If Hawking radiation wouldn't exist a microscopic black hole would still need millions to billions of years to accumulate matter - it simply doesn't have any mechanism to attract other matter strongly. Gravity is the only way, and the gravitational attraction of something with the LHC collision energy is tiny no matter where the black hole is. Pressure doesn't matter here. The black hole will be in the wave functions of particles and the absorption probability is still tiny.

No, it wouldn't, because the matter starts out at rest and it will take time for it to cover the distance to the center. Roughly speaking, if we assume that the matter has zero viscosity for this purpose (since it's all going down the hole at the center so matter just crossing the horizon won't "push back" against matter behind it), the time for the matter at the Earth's surface to reach the center and get swallowed by the hole should be about 20 minutes--one fourth of the free-fall orbit time.

That requires more than zero viscosity. You would need infinite compressibility as well. That is not a good assumption. Tangential forces will slow the matter which then leads to radial forces on the matter behind it.

But at any rate it seems clear that for the hole's Hawking radiation pressure not to easily prevent adjacent matter from falling in, the hole has to be of at least "small astronomical body" mass.

There is another mechanism to consider which works even for stellar mass black holes: Radiation from infalling matter will slow down the collapse. This is the concept of (yet unobserved) Quasi-stars, stars with a central black hole that can last for millions of years.

If we assume a collision of equal energy particles moving in opposite directions (which AFAIK is the normal setup in an experiment like the LHC), a black hole that was produced could have zero momentum. That doesn't happen with normal collision products because the energy of the products is so much larger than their rest energy that they have to be moving very fast. But a collision that produced a black hole could have all of the collision energy converted to rest energy of the hole.

Black holes would be produced from parton collisions, the partons have a random fraction of the proton's energy, and in general collisions are asymmetric. They can be roughly symmetric, however, unlike cosmic rays.

Another wrinkle to consider is that a black hole of mass that small--well under the Planck mass--might not even be possible, depending on how quantum gravity turns out.

It is expected that black holes cannot form below the Planck mass. It would need extra dimensions, and while some people expect them to exist that is certainly not the mainstream view. Black holes at the LHC would need even more: Extra dimensions with just the right number and size to make black holes there possible but not at previous colliders.

I'll just have to take Hawking's word for it that it really happens at the rate predicted. (Has this been experimentally verified?)

Well, not without black holes in the lab... there is an equivalent phenomenon for sound, however, and there it has been observed.
If there are primordial black holes with just the right mass range we might see their evaporation today. Nothing found so far.
If collisions could form stable black holes they would consume neutron stars quickly. We can see neutron stars.

 

[PeterDonis]

That requires more than zero viscosity. You would need infinite compressibility as well.

I'm not sure I understand this. The matter falling into the hole is not going to be compressed. A given "shell" of matter at a particular radius from the hole is going to fall into the hole slightly faster than the shell just above it, and slightly slower than the shell just below it. So compressibility should not be an issue.

(Note that here I'm talking about an idealized collapse that is perfectly spherically symmetric. See below.)

(Note also that I'm assuming that the shells of matter do not all start falling at the same time: each shell starts falling a little bit before the shell above it, and a little bit after the shell below it. This is because it takes time for each shell to "know" that the shell below it is no longer pushing it up. This is different from, for example, the Oppenheimer-Snyder model of stellar collapse, where it is assumed that the entire object starts collapsing inward at exactly the same time--more precisely, "at the same time" in the frame in which the object is initially at rest.)

Tangential forces will slow the matter

In the idealized case I was considering, everything is spherically symmetric, so there are no tangential forces. But I agree that in any real object there will not be perfect spherical symmetry (and also not perfectly zero angular momentum), so tangential forces will exist and will slow things down, yes.

There is another mechanism to consider which works even for stellar mass black holes: Radiation from infalling matter will slow down the collapse.

This happens because the infalling matter does not have zero angular momentum, so it doesn't fall radially into the hole, but swirls around it, giving time for more matter to fall in on top of it and create shock waves, etc.

In a real object, I agree these effects would contribute. That just means the numbers I was calculating are only rough lower bounds on the mass that a hole inside an object would have to have to not evaporate before it could accrete matter: in actual cases the required hole mass would be larger.

 

[Charles Link]

I'm joining this discussion rather late, but just a comment to the OP @DaveC426913 : There really is no guarantee when we start playing around with things of higher energies such as accelerators, that we won't tap into some kind of unforeseen source of energy. For example, in the fission process, the larger atoms are basically like a large spring with a latch on them that keeps them in an energy potential well. Once that latch is released by splitting the atom, an enormous amount of potential energy is released. $\\$ There is no guarantee when we start giving particles enormous amounts of energy that it won't encounter some kind of system of stored energy that is transparent to particles of lower energies. That's my 2 cents. If just black hole type phenomena are considered, it might be impossible there, but, IMO, it is not out of the question that some other source of energy won't be found by providing particles with the kind of energies that we give them with these accelerators.

 

[PeterDonis]

There is no guarantee when we start giving particles enormous amounts of energy that it won't encounter some kind of system of stored energy that is transparent to particles of lower energies.

Where is this magic system of stored energy going to come from? There is no mystery about where, for example, the energy released in nuclear reactions comes from: it comes from the rest mass of the nuclei. That was known and understood even before people knew the actual mechanisms of particular nuclear reactions, and the possibility of turning rest mass into energy was well known and appreciated based on Einstein's discovery of special relativity some time before. There was never a time where people discovered new nuclear reactions that tapped some "system of stored energy" that nobody knew about before and that came out of nowhere.

Also, to be clear, a hypothetical process that creates a tiny black hole by colliding high energy particles is not "unlocking" any new source of energy. The energy of the hole (its mass) comes from the energy in the colliding particles. If the hole radiates Hawking radiation, the energy in the radiation comes from the mass of the hole. There is no mystery about any of this either; it's not some new source of energy that comes from nowhere.

 

[DaveC426913]

There is no guarantee when we start giving particles enormous amounts of energy that it won't encounter some kind of system of stored energy that is transparent to particles of lower energies.

Well, that is exactly the kind of fear-culture I'm trying to forestall. People who get their physics from the newspaper headlines often have unfounded fears.

The fissioning of the atomic bomb did not catch us by surprise.

The universe has been ticking along for 13.7 billion years before humans came along. It is implausible in the extreme that such a spring has been coiled for all that time, waiting for us to come along and release it in a particle collider.

 

[mfb]

I'm not sure I understand this. The matter falling into the hole is not going to be compressed. A given "shell" of matter at a particular radius from the hole is going to fall into the hole slightly faster than the shell just above it, and slightly slower than the shell just below it. So compressibility should not be an issue.

Consider a homogeneous Earth. Gravitational acceleration increases with radius. The outer parts have to fall in faster than the inner parts. Not only do the shells get a smaller radius, they also get closer together. Free fall needs immense compression. Earth is not homogeneous - the gravitational attraction is nearly constant until you reach the core. That still means the shells get smaller without increasing in radial distance.

In the idealized case I was considering, everything is spherically symmetric, so there are no tangential forces.

Spherical symmetry doesn't rule out spherically symmetric tangential forces (pressure).

I'm joining this discussion rather late, but just a comment to the OP @DaveC426913 : There really is no guarantee when we start playing around with things of higher energies such as accelerators, that we won't tap into some kind of unforeseen source of energy. For example, in the fission process, the larger atoms are basically like a large spring with a latch on them that keeps them in an energy potential well. Once that latch is released by splitting the atom, an enormous amount of potential energy is released. $\\$ There is no guarantee when we start giving particles enormous amounts of energy that it won't encounter some kind of system of stored energy that is transparent to particles of lower energies. That's my 2 cents. If just black hole type phenomena are considered, it might be impossible there, but, IMO, it is not out of the question that some other source of energy won't be found by providing particles with the kind of energies that we give them with these accelerators.

We know the total energy stored in the protons. 6.5 TeV per proton. The release of more is only possible if our vacuum is not the lowest energy state. If 13 TeV collisions would be sufficient to change the vacuum then cosmic rays would have done so long ago.

 

[epenguin]

A lot of needless panic that these experiments may create a black hole that would swallow the Earth or the universe has been created.

I've re-done the calculations and I found that at most it would only swallow Switzerland, so there is not much to worry about.

 

[.Scott]

I'm not sure this is right. The force between the atoms in the normal case is there because the situation is static: none of the atoms can free-fall. If a void is created by the BH at the center, and the atoms start falling into it, the force between the atoms should vanish, at least to a first approximation (assuming a perfectly symmetrical fall process). The atoms on top could only "push" the atoms below if for some reason the atoms below were falling slower than the atoms on top; but that shouldn't be the case, at least to a first approximation.

I doubt this analysis - at least in terms of its completeness. If small black holes are anything like their big brothers, there will be a huge amount of energy released. And if the big brother analogy doesn't hold, then there is the issue of the Hawking radiation. In either case, a significant portion of the matter heading for the black hole will be converted to energy before or after reaching the event horizon - energy that will vaporize neighboring rock. So the BH would almost immediately become surrounded in a high pressure gas. I expect this will result in pressures that will soon exceed those at the center of the Earth - resulting ultimately in an exploding Earth. I expect that some portion of the Earth's mass would reach escape velocity and never be consumed.

 

[Nugatory]

If small black holes are anything like their big brothers, there will be a huge amount of energy released.

How much? Don't guess, calculate!

 

[DaveC426913]

I expect that some portion of the Earth's mass would reach escape velocity and never be consumed.

So, we'd be safe after all!

 

[.Scott]

How much? Don't guess, calculate!

I'd be happy to. What's the formula?
The calculation for the Hawking radiation is easy. Since we are talking about a borderline case where roughly the same amount of mass is being consumed as being radiated - it's e=mc^2. Plenty enough to disturb the notion of matter simply dropping into it unobstructed.

But when stars drop into a BH, an accretion disc can formed. Sine the Earth is spinning, something like that should form around our smaller BH. In other cases, jets form along the spin axis. I don't understand the mechanism and I don't know the formulae. Educate me.

 

[.Scott]

Here some math:
Earth velocity at the surface at the equator due to rotation: Ve=460m/s
Earth radius: Re=6371 Km
BH with Earth mass radius: Rb=8.87mm
Estimated velocity due to rotation at the surface of the equator of the black hole: Vb = Ve Re/Rb

Vb = 460m/s * 6371Km / 8.87mm = 460*6371/8.87 10^6 m/s
Vb = 327.6 * 10^6 Km/s (well over c)

So there will be an accretion disk.

 

[.Scott]

So, we'd be safe after all!

Better than that, we have an up close look at some real Physics !

 

[PeterDonis]

BH with Earth mass radius: Rb=8.87mm

What does this have to do with the discussion in this thread? We're not hypothesizing a hole with the mass of the Earth. We're hypothesizing a much, much smaller hole that forms somewhere inside the Earth.

Estimated velocity due to rotation at the surface of the equator of the black hole

What does this have to do with the discussion in this thread? Nobody has hypothesized a spinning hole, much less a hole with both the mass of the Earth (see above) and the same angular velocity of rotation as the Earth (which, as you show, is impossible).

 

[PeterDonis]

Consider a homogeneous Earth. Gravitational acceleration increases with radius.

Ah, yes, you're right, I forgot to take into account the Earth's gravity as well as the hole's.

Spherical symmetry doesn't rule out spherically symmetric tangential forces (pressure).

But it does rule out tangential motion. Tangential pressure can exist, but it has to be static.

 

[.Scott]

What does this have to do with the discussion in this thread? We're not hypothesizing a hole with the mass of the Earth. We're hypothesizing a much, much smaller hole that forms somewhere inside the Earth.

What does this have to do with the discussion in this thread? Nobody has hypothesized a spinning hole, much less a hole with both the mass of the Earth (see above) and the same angular velocity of rotation as the Earth (which, as you show, is impossible).

You participated in the discussion about how atoms would feed into the black hole. In particular, I responded to your post #45. Then Nugatory asked me for some math. So I obliged.

So we were talking about what would happen when material started falling into the BH. And there I was right to assume an accretion disk. So I did as much math as I could - determining whether there was enough angular momentum for an accretion disk to form. There is - and therefore, at some point well before the BH diameter reaches 9mm, atoms are not going to simply fall into the BH.

I was also assuming that the BH started out without a spin. But it will pick up a spin - and more importantly, the material falling into it will be spinning at relativistic speeds. In any case, if you want to calculate how long we would have before dying from the tiny BH (which some posters were trying to do), these are factors that need to be considered.

 

[PeterDonis]

I responded to your post #45. Then Nugatory asked me for some math. So I obliged.

You claimed that a small BH would release a lot of energy. @Nugatory asked you to calculate how much. You haven't done that.

The point @Nugatory was trying to get you to realize is that a BH can't radiate more energy than its mass, and a tiny black hole hypothetically formed by a collision in the LHC has a tiny mass--a few TeV. So that's all the energy it can radiate, and if we're talking about possible effects on a macroscopic amount of material, that amount of energy is negligible.

we were talking about what would happen when material started falling into the BH. And there I was right to assume an accretion disk

No, you weren't, because the general conclusion of the discussion before you entered it was that, unless the BH is at least the size of a small asteroid, the radiation pressure from the hole's Hawking radiation will prevent matter from falling towards the hole at all before the hole evaporates. And since we're talking about a hypothetical hole formed somewhere inside the Earth by LHC-type collisions (or even cosmic rays), we're talking about a hole massing a few TeV, not a macroscopic hole at all.

 

[mfb]

But it does rule out tangential motion. Tangential pressure can exist, but it has to be static.

No one was talking about tangential motion (well, now .Scott started it, but that is a separate discussion). Pressure is just another name for the forces discussed.

You claimed that a small BH would release a lot of energy.

I'm quite sure the claim was that infalling matter would release a lot of energy in the process of falling in. For a rotating Earth that is certainly the case - most of the matter has too much angular momentum to fall in directly. We get an accretion disk, we get radiation from this disk, the radiation pressure will slow further collapse. The accretion disk has to get rid of angular momentum for more matter to fall into the black hole. The calculation to show that has been done.

A black hole with nearly maximum spin is highly efficient in converting infalling matter to radiation - something like 20-30% if I remember correctly. Earth's gravitational binding energy is just 2*10³² J, or 4*10^-10 times its total energy. Something like a billionth of the mass falling into the black hole could release sufficient radiation to evaporate the rest of Earth.

 

[PeterDonis]

I'm quite sure the claim was that infalling matter would release a lot of energy in the process of falling in.

The claim was that infalling matter would form an accretion disk, thereby releasing a lot of energy. But in order to form such a disk, first, the hole would have to be spinning; and second, the matter would have to get close enough to the hole's horizon to form such a disk and convert an appreciable fraction of its rest mass to energy.

For a hole formed by particle collisions, I would expect its spin to be small. But that's not really the major issue.

The major issue is that, for holes that could possibly be formed by any kind of particle collision process, their Hawking radiation pressure will, by many orders of magnitude, keep any matter from getting anywhere near close enough to their horizon to form an accretion disk and start radiating energy, before the hole itself evaporates. That's what the discussion that's already been had in this thread shows.

For a hole with the mass of a small asteroid or larger, yes, it might be possible for matter to get close enough, without Hawking radiation pressure blowing it away, to form an accretion disk (and for the hole to acquire enough spin to make the energy conversion rate non-negligible) and start radiating energy. But such a hole is not going to form inside another body like the Earth by any conceivable process.