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# Smallest stable black-holes

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Like Jay said, antimatter has exactly the same mass-gravity property as normal matter. Besides, if anti-matter were gravitationally repulsive, they would never fall together to fall a black hole in the first place.

I'm not completely with you here. As far as I know, the only difference between matter and antimatter is that the charges are reversed. Electrons are positive in antimatter, and protons negative. If black holes only have mass, charge and entropy, then an antimatter black hole should be able to annihilate a matter black hole? Seeing as the only difference between the two is charge, and that's a property retained by a black hole?

Matter has the opposite charges to normal matter, charges are quantum numbers - the electric charge is the most physically accessible quantum number to our intellect. Other charges are also reversed when swapping from matter to antimatter.

Let me spell this out a little bit: Black holes will suck in all sorts of matter like electrons and protons, if it sucks in more electrons the black hole will have a net negative charge. An antimatter black hole may have sucked in positrons and negatrons, if it sucked in more positrons the black hole will have a net positive charge. These two black holes with opposing charges would attract each other - but this does not mean they annihilate when they collide, as a black hole made of electrons and one made of protons certainly wouldnt annihilate.

There is a slight hole in my argument. Black holes are said to destroy information, ie they only keep the bare minimum of information of what they have sucked in. This stops our conservation laws from been violated: the black hole must at least take on the basic quantum numbers of the particles it sucks in. But this is conjecture on my part as the no-hair theorem states only electrical charge is conserved.

Im not sure if I am getting any closer to answering peoples questions or not.. so I will stop here for the moment.

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Jay-qu, would this be consistent with GR?:

If you could enclose a hydrogen bomb in an impervious and infinitely strong box, the box would curve spacetime equally before and after the bomb in the box exploded. Before it exploded the box would contain mostly hydrogen and after the explosion it would have mostly photons, but the total energy of the box would be the same.

I'm thinking that it isn't so much mass that curves spacetime, but energy (stress & momentum not being a factor here). Since energy is conserved in an annihilation, whatever matter / antimatter annihilations happen behind a horizon would not affect the gravitational field. I'm curious if this makes sense to you.

~modest

EDIT: Sorry Hasanuddin, I didn't realize I should have posted this in the other thread. My bad.

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Yes Modest, that is correct. People often misinterpret Einstein's famous equation E=mc^2 ( or E^2=(mc^2)^2+(pc)^2 for those familiar with the full equation) as showing how mass and energy can be converted back and forth. Really it is showing a deeper equivalence, they are but the same thing only in different forms. So yes energy bends space just as mass does.

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As to the primary question of the smallest size that a black-hole can be and exist stably.
For example, a 2e5 kg black hole has a lifetime of about 1 second, a 7e5 kg one about 1 year, and a 1e11 kg one about 1.4e10 s, about the current age of the universe according to Big Bang model. By way of comparison, 1e11 kg is just a bit more than the mass of a large artificial structure like the Three Gorges Dam.

Okay, let me just clarify… you are talking about only the evaporating side of the equation… correct?

Yes. The most simple evaporation time equation is just an integral of the power – the rate of Hawking radiation of mass/energy, given by

$P = \frac{k_P}{M^2}$,

where $k_P=\frac{\hbar c^6}{15360 \pi G^2}$,

and $M$ is the black hole’s mass

– of a black hole as its mass, and consequently the radius and surface area of its event horizon, which determine that power, changes, decreased by the outflowing Hawking radiation. It doesn’t include inflowing radiation.

Note that it’s common and useful for power to be expressed as an equivalent temperature of a black body of given surface area, so it’s common to see the above expressed in, for example, degrees Kelvin (K), rather than watts (W) or other common units of mechanical power.

Note that the term “radiation” isn’t limited to photons, but describes anything that carries mass/energy from one body to another – and that “body” can refer to a well-defined volume such as within the event horizon of a specific black hole, or all of space not within that specific black hole.

What about the other side of the dynamic? Consumption. Does a black-hole consume more when the density of compactable material is high? Deductively the answer must be, “Yes.”

This answer has to be approached carefully, taking special care not to confuse mass and density (mass/volume).

As noted above, and in the linked wikipedia article, incoming and outgoing radiation must be considered for a full description of the mass of a black hole over time. Density, however, other that a critical value, isn’t a term in this description.

Assuming a small rate of rotation and nearly neutral net charge, the event horizon of a black hole is nearly spherical, its radius determined by its mass only, by the very strait-forward Schwarzschild radius,

$r_s = k M$,

where $k_s = \frac{2 G}{c^2}$

Density is important only in that it must be sufficiently large, on average, that the mass/energy responsible for the black hole’s gravity is within the volume defined by its event horizon. Because the Schwarzschild radius is proportional to M, and volume is proportional to $r_s^3$, the average density of a black hole can be arbitrarily small. For example, at about 130,000,000 solar masses, the average density of a non-rotating black hole is about equal to that of water, and rather curiously, if the visible universe were a single massive black hole, its required density would be very close to the very hard vacuum given by various predicted values of its total mass.

It’s unlikely, according to sensible physics, that the mass within a black hole is anything close to evenly distributed, so average density of its entire volume is almost certainly very different than the average density of various sub-volumes, the most accepted guess being that most of the volume is near vacuum, with a tremendously dense core - possibly an infinitely dense singularity, though, as the saying goes, nature – especially when viewed with the formalism of quantum physics - abhors infinities, so my guess is for “tremendous” of “infinitely”. However, the question of densities within the event horizon of a black hole isn’t important to the physics of anything outside of it, or, in theory, very knowable.

If a black-hole is in a pure vacuum then nothing could be consumed.

Here, we must be careful to agree on the meaning of “pure vacuum”.

The most common meaning is a volume that contains no fermionic matter (atoms or free nuclei or electrons, etc), but may contain bosonic matter (photons, etc).

However, because typical black holes Hawking radiate with such low power, a small influx of photons is sufficient to more than equal the outgoing radiation. This is what I mean by my statement (the numbers lifted directly from the wikipedia article)

In principle, a small black hole could be stable – that is, be at equilibrium, neither gaining nor losing mass – if the power of its infalling matter and radiation equals that of its Hawking radiation. For the cosmic background radiation – which all objects are more or less guaranteed to receive – a black hole of about $4 \times 10^{22}$ kg – about the mass of Earth’s moon has about this equilibrium.

So a black hole with the mass of the moon will, barring some bizarre “shadowing” phenomena, gain mass just via absorbing the CMBR. One smaller might still be stable or gain mass, if the influx of photons was greater – say from a nearby bright star – or if more than a small amount of matter – such as in typical near-stellar space – fell into it.

However, a black hole much smaller than this will, barring an extraordinary influx of bosons and/or fermions, Hawking radiate much more than it absorbs, so lose mass at an ever increasing rate. The main consequence of this is that, according the theory, small black holes, which can in principle be formed by such small-scale phenomena as man-made particle accelerators, will exist for only very short durations – a reassuring prediction, as it reduces our worries of swarms of tiny black holes from cosmic sources, or created by high-energy physics experiments, devouring our Sun or planets.

Perhaps I am misreading you, but it appears that you are saying that equilibrium for a black-hole only assimilating/accreting the energy from the CMB, but nothing else, will be stable at a mass of 4e22 kg. Am I reading you correctly? So, a smaller, yet still stable black-hole could be achieved under conditions where more energy/mass are being accreted. Is that correct?

Yes, as I hope the preceding explains more completely.

(infalling matter couldn’t be used to stabilize an arbitrarily small black hole, because the exclusion principle limits the amount of fermionic mater that can occupy a given volume of space.)

The problem is, following the link provided offers no evidence to support the notion that infalling energy would be interchangeable infalling mass to achieve black-hole stability. After all doesn’t the famous equation E=mc2 suggest an interchangeability between mass and energy? Nowhere on the link discussing the Pauli Exclusion Principle is it suggested that E=mc2 does not apply.

What I’m trying to explain with this reference is that, although under usual conditions, streams of matter (fermionic) are much higher power than streams of photons (bosons), fermionic matter streams have an upper limit to the power they can supply to a surface of given area, while photon streams don’t. This is because the Pauli Exclusion principle – AKA Fermi-Dirac statistics – limits how many fermions can occupy a given volume, while no such limit applies to bosons – or, to put it simply, the number of photons that can be contained in a given volume is unlimited.

What this all gets at concerning the minimum size of a black hole, is that while no know natural phenomena can prevent a black hole much smaller than $10^{22}$ kg from losing mass, fairly quickly “evaporating” completely, it’s conceivable that one might artificially sustain a very small black hole by, put simply, shining a very bright light on it.

Using the Hawking radiation power and Schwarzschild radius formulae above, we can calculate their constants for standard units,

$k_P = \frac{\hbar c^6}{15360 \pi G^2} \dot= 7.1 \times 10^{32} \,\mbox{W} \cdot \mbox{kg}^2$

$k_s = \frac{2 G}{c^2} \dot= 1.5 \times 10^{-27} \,\mbox{m/kg}$

, chart the various masses M, Schwarzschild radii r, Hawking radiation power P, and evaporation time t:

[font="Courier New"]M (kg)  r (m)    P (W)    t (s)   (y)      Comments
1.0e0   1.5e-27  7.1e32   8.4e-17 2.7e-24
1.4e3   2.1e-24  3.6e26   2.3e-7  7.3e-15  Power of the Sun
1.0e4   1.5e-23  7.1e24   8.4e-5  2.7e-12
1.0e5   1.5e-22  7.1e22   8.4e-2  2.7e-9
5.0e5   7.4e-22  2.9e21   1.0e1   3.2e-7   500 tons, 1 second
1.0e6   1.5e-21  7.1e20   8.4e1   2.7e-6
1.0e7   1.5e-20  7.1e18   8.4e4   2.7e-3
1.0e8   1.5e-19  7.1e16   8.4e7   2.7e0
8.0e8   1.2e-18  1.1e15   4.3e10  1.4e3    Most powerful laser
1.0e9   1.5e-18  7.1e14   8.4e10  2.7e3
1.0e10  1.5e-17  7.1e12   8.4e13  2.7e6
1.5e10  2.2e-17  3.2e12   2.8e14  8.9e6    Power of human civilization
1.0e11  1.5e-16  7.1e10   8.4e16  2.7e9    About 1/4th age of the universe
1.0e12  1.5e-15  7.1e8    8.4e19  2.7e12   Proton’s radius
1.0e13  1.5e-14  7.1e6    8.4e22  2.7e15   Uranium nucleus’s radius
3.6e16  5.3e-11  5.5e-1   3.9e33  1.2e26   Hydrogen atom’s radius
4.5e22  6.7e-5   3.5e-13  7.7e51  2.4e44   Moon’s mass, hair’s radius
3.0e24  4.5e-3   7.9e-17  2.3e57  7.3e49   Small ball bearing’s radius
6.0e24  8.9e-3   2.0e-17  1.8e58  5.7e50   Earth’s mass
2.0e30  3.0e3    1.8e-28  6.7e74  2.1e67   Sun’s mass[/font]

, and consider what would be required to artificially stabilize various small black holes.

From the first rows of the table, we can see that until we get an initial black hole with a mass of over about a million tons (1.0e9 kg), the power necessary to sustain it is prohibitively high for a civilization of our technological level. At these initial masses, the black hole is fairly long-lived by human standards - about 2700 (2.7e3) years - so for practical purposes, there’s not much point in bothering to sustain it.

Although black holes of these masses are tiny, with event horizons about 1/1000th the size of a proton (1.5e-18 m), their close-in gravitational fields are very strong – for a 1e9 kg black hole, the acceleration of gravity exceeds Earth’s surface’s at a distance of about 8 cm (0.08 m). This makes them potential doomsday objects, as, per $E=mc^2$, their daunting 7.1e14 W sustaining power requirements translate to only about 0.47 kg/minute (7.8e-3 kg/s) of matter.

Finally, there’s the engineering problem of how to make (or find) a sub-subatomic size, million ton black hole in the first place. If exploding stars are any indication, you’d need system masses several powers of 10 larger than the resulting black hole, and power many times greater than whole stars. We might hold out hope that in the many varieties of supernova’s in our galactic neighborhood, if we can manage to build spacecraft to visit some of these events, we might get lucky and find a small black hole, but quantum mechanics – among them those pesky Fermi-Dirac statistics again - predicts this is impossible, and observations showing the lack of strong x-ray emitters from sub-typical black hole size supernovae support these theoretical predictions.

What we’re left with, as best I can surmise, is the prospect that very small (less than 1000 kg) black holes with very short natural evaporation times might be possible to artificially produce and sustain by a technological civilization that could project focus radiation with the power of many sun’s on tiny targets. Otherwise, it appears the usual minimum mass prediction – about 1.4 solar masses – applies, resulting in black holes that won’t evaporate until the entire universe becomes nearly completely dark and on the order of $10^{100}$ years pass – the end of the “black hole era” predicted by some cosmological models.

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Thanks Craig for that informative post :)

So the Hawking radiation that leaves a black hole will be thermal? will the black hole act like a black body? :D

I am just wondering about the prospect of using a micro black hole as a power source, though likely the amount of energy that needs to be put in initially would outweigh its use as an output..

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Wow, honestly, thank you for the informative and well-thought out post. There was a part, however, that you misinterpreted the line of my inquiry and spent much time tangent.

What about the other side of the dynamic? Consumption. Does a black-hole consume more when the density of compactable material is high? Deductively the answer must be, “Yes.”

I still adhere to the assertion. I was not referring to the density of the black-hole, I was referring to the density of the matrix of the region of space surrounding the baby black-hole. Let me lay it out a little more syllogistically:

1: If there is no matter/photons/other in the vicinity of the bitty black-hole, then the MBH will receive nothing.

2: If there is matter/photons/other within the vicinity of the MBH, it will accrete/absorb.

2b: If it absorbs energy/mass, then it gains energy/mass

2c:All gains in energy/mass, offset possible losses through proposed evaporation.

2d: Therefore, a lower total initial mass is needed by the MBH to sustain stable equilibrium between evaporation and consumption.

2e: Therefore, shouldn’t areas of highest neighborhood density of accretable mass/energy also be the areas where a black-hole could exist in the smallest of sizes indefinitely?

So the question is, what size would that be? I mean chose the extreme: hot Pb (lead). Suppose we has a giant block of very hot lead under high pressure. Within those conditions, how small is the smallest size that a black-hole can be?

For the sake of this question, please put the question of photonic energy aside. The last post seemed to detail the ability of light to sustain black-holes for a very long time. Cool. However, I am left wondering the role of accreted matter in sustaining an MBH. So, the question of light aside, how small is the smallest black-hole to be under mass-dense and high heat conditions?

The closest we got in approaching the true question I am trying to ask is when:

although under usual conditions, streams of matter (fermionic) are much higher power than streams of photons (bosons), fermionic matter streams have an upper limit to the power they can supply to a surface of given area,

But you didn’t follow the natural thought progression. Please try. Under conditions of high fermionic surrounding matrix density (not the MBH’s density), under high pressure and high energy, what is the smallest size for an MBH to be able to stably persist?

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I am just wondering about the prospect of using a micro black hole as a power source

Brilliant! I've never heard or considered that. It seems like it would be the perfect power source. You put in matter and it's converted to pure thermal radiation. It'd be the perfect rocket engine :D

Of course, attaching it to the rocket might present a problem :hihi:

~modest

P.S. I've attached Craig's table with formulas in Excel if anybody wants to play with the #s

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Do smaller black holes, such as the minimum mass of 3.8 solar masses, have the same center point reference as huge black holes? The larger black hole will have more mass, but once blackholes reach a point center, is that reference the same for both?

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The closest we got in approaching the true question I am trying to ask is when:

But you didn’t follow the natural thought progression. Please try. Under conditions of high fermionic surrounding matrix density (not the MBH’s density), under high pressure and high energy, what is the smallest size for an MBH to be able to stably persist?

I think if you paid a bit more attention and read his post thoroughly you would have got your answer.

To keep a very small black hole stable you must put in as much energy/mass as it is spitting out - simply an equilibrium of input/outputs.

The smaller a black hole gets the greater the rate at which it spits out hawking radiation. So you have to cram more mass back into the black hole at an much higher rate. There is a limit where the black hole becomes so small that you cant cram in mass fast enough due to limits imposed by the Pauli exclusion principle. This limit only occurs for fermions, so you could conceivably continue cramming energy/mass in with bosons.

The logical conclusion of this is, you can make an arbitrarily small black hole, so long as you have to means to continually pump in a very high flux of bosons.

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There's a couple of important issues that have been left out of this.

One is that duration for micro black holes under large extradimensional string theory cannot be obtained by using the classical formulae for power and duration of black holes given by CraigD (this page). Secondly, even the type of large extra dimensional string theory can effect the result. This latter occurs where the extra dimension calculation takes into account the effect of gravity upon the radiated particles. When this is done it is called the 'microcanonical interpretation', where - for a given black hole mass - the energy, not temperature, is held fixed. I fail to see any reason why not taking this 'backreaction' effect into account can be regarded as valid.

The fact that within extra dimensions the duration of micro black holes is presented so much by CERN and elsewhere as extremely rapid, is a terrible disaster..

When the 'Randall-Sundrum' theory was considered in the paper of 2002 (also published in the 'International Journal of Modern Physics') http://arxiv.org/abs/hep-th/011025 by applying this alternative interpretation, the duration of a minimum possible 1TeV[/c^2] black hole in isolation is given at upto 30years. Though this theory is also the basis for the 2009 paperhttp://arxiv.org/abs/0901.2948, given in Hasanuddin's first thread post, parameters used in the earlier are not calculated in the new. Yet in the new, the other parameters are still described as 'Another possibility..'. This later neglect of calculation for risk evaluation demonstrates:

i) a terrible neglect. (Such undue neglect doesn't seem to have helped get the 2009 arXiv successfully peer reviewed either.)

ii) weariness of considering what sort of danger would be presented by taking both very slow evaporation/radiation along with accretion.

Why danger?

Alongside the evaporation, would be the Hawking radiation. One would need only an accretion rate - which would be gradually increasing - of .57kg/s to obtain a radiation level of 5x10^16W - extremely dangerous no doubt even if from within the earth's core. Such an accretion rate and luminosity has been obtained [0808.1415] On the potential catastrophic risk from metastable quantum-black holes produced at particle colliders by using accretion rate formulae given in a CERN risk paper.

I'm not seeing any way that analogous effects could be detectable on compact stars receiving similiar black hole creating collisions from cosmic rays. Also, the accretion rate would become limited so that whole star accretion would not have time to occur.

Note that the Fabi et al 2009 paper is not peer reviewed.

Which one would you (plural) rely on to address safety, concerning the Randall/Sundrum theory that claims to resolve a number of problems in physics;

Casadio/Harms 2002 or Fabi et al 2009?

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• 1 year later...

Sorry for the break from this dialogue, life called. Please understand.

I’d like to absolutely agree with Eric’s concern.

The fact that within extra dimensions the duration of micro black holes is presented so much by CERN and elsewhere as extremely rapid' date=' is a terrible disaster..

[/quote'] That is why I so hastily wrote the syllogism/book THE DOMINIUM Sequencing antimatter and gravity effects; Big Bang to black hole and implications for a manmade near-future doomsday: End-of-life on Earth back in the summer of 2007. LHC was just nearing completion. What a tense time that was.

Today LHC is running. That is true. Nothing much has happened. …literally. Yes, lot’s of data, but basically a lot of more-of-the-same, just at a larger scale. That is not something to tote.

Instead, CERN got in the media by a hoopla about containment of anti-Hydrogen.

http://public.web.cern.ch/press/pressreleases/Releases2010/PR22.10E.html

http://arstechnica.com/science/news/2010/11/researchers-trap-antihydrogen-atoms.ars

The problem w/ these stories is that they portray this as some kind of “new” achievement. It’s not. When I was at CERN in 2000 they were actively producing anti-Hydrogen in an ongoing and routine basis. A decade ago CERN rountinely created anti-hydrogen and even opened a full facility for this process. (notice the dates in these citations:)

Despite the lack of any significant NEWS originating from CERN, nothing addresses Eric’s concern regarding mini-black-holes (synthetic or naterual.) The fact that there is no knowledge of the generation of any black-hole material at LHC has occurred does not negate the questestion, whether black-hole material is dangerous. All it means is that LHC has been not yet more successful than previous machines. This certainty only refers to the track-record of a paricular, but unsuccessful, machine. This fact says nothing to the question of whether black-holes are stable or not. Hence the question of this thread, what is the smallest size, in kg, that a black-hole need be to exist longer that a second.

One side of the argument, is the idea that small black-holes will safely evaporate” away via “Hawking Radiation.” Cool, sound great. The problem has always been: the hypothesis of Hawking’s Assumptions has no basis in tangible evidence/experimentation. The irony has been the backers of the H-Assumptions have hoped that LHC itself, would prove the H-Assumptions by watching/recording a synthesized back-hole evaporating away. So far, they too have been disappointed, LHC has shown no signs of any black-holes. Again, this lack of data only conclusively shows the failure of LHC… not to any of the theories regarding black-holes once/if they are formed.

The reason why I kept going back to this idea of what is the smallest mass-size of a black-hole (either synthetic or natural)?? is because I know that until it is actually recorded then no-one can know for certain.

This, then leads to the ethical questions concerning future use/direction of giant scientific endevours, such are the one going on at CERN and countless labs across the world. How to proceed efficiently (i.e., maximize potential for major scientific leaps) ethically with regard to risks to humanity. The reason for this concern, for me, comes from the deductive conclusions of the Dominium Model, especially the assertion that black-hole material is the 5th, and final phase of matter. All the known trends (density, energy-content, and most importantly, stability) would continue effortlessly if this were true.

The beauty and simplicity of this ominous assertion demanded to be written down and disseminated using unconventional means, such as publishing it as a book and went via social-media (such as hypography.com) rather than by formal journal article. That way, the potential for danger could be known/assessed sooner rather than later.

Oh well, the failure of LHC to weigh in either way does not to abate the potential of mini black-holes to be dangerous. Nor does it answer the central question of this thread concerning the minimum "safe" size of black-hole material.

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