How fast can a small black hole eat?

[halfeye]

A black hole of 2385 tons lasts one second, so there's presumably no saving that.

However, with increasing mass, the input to save the hole decreases, so somewhere in there, there's a point where the input can exceed the output.

If a one year lifetime black hole has a radius of 9.9 * 10^-17 is that big enough to feed in enough mass to keep it going if it's at the centre of a star? assuming degenerate matter how much can be pushed through a hole that small per second?

[Battleship789]

A black hole of 2385 tons lasts one second, so there's presumably no saving that.

However, with increasing mass, the input to save the hole decreases, so somewhere in there, there's a point where the input can exceed the output.

If a one year lifetime black hole has a radius of 9.9 * 10^-17 is that big enough to feed in enough mass to keep it going if it's at the centre of a star? assuming degenerate matter how much can be pushed through a hole that small per second?

For a black hole with that lifetime placed at the center of the sun (this Hawking radiation calculator places a black hole with a lifetime of 1 year at ~90000 metric tons and a Schwarzschild radius of 1.31 x 10^-19 meters), the black hole would evaporate in approximately one year (if not exactly one year) and no material would enter the black hole. Even with a year-long lifetime, the black hole's Hawking radiation easily overwhelms even the intense pressures at the center of the sun. The equilibrium mass is approximately 10¹⁵ kg; black holes above this mass will absorb material from the sun (to a very rough approximation).

Spoiler: Calculations

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Radiation pressure, p, is given by the formula p = I/c = P/(cA), where I is the intensity/irradiance of the source, P is the power of the source, A is the area of the surface to be calculated (in this case, the black hole's event horizon), and c is the speed of light. For our black hole (using the calculator above), P = 4.61 x 10¹⁶ W and A = 2.14 x 10^-37 m². Hence, the radiation pressure is a staggering 7.18 x 10⁴⁴ pascals (N/m^2). For comparison, the pressure at the center of the sun is estimated to be around 2.65 x 10¹⁶ pascals (see, for example, the Solar core Wikipedia article) and so the Hawking radiation of the black hole pushes the sun's core away from the event horizon.

We may calculate the equilibrium radius r_eq at which the solar core and Hawking radiation pressures cancel out by choosing an appropriate surface with area A = 4 π r_eq². This yields a radius of approximately 21.5 μm (micrometers) and no material from the sun gets close to the black hole (in a relative sense; the equilibrium point is roughly 100 trillion Schwarzschild radii away from the even horizon).

Using the formulas from the Hawking radiation calculator, we see that a black hole's radiation pressure as a function of mass M is p = ℏ c⁹/(245760 π⁴ G⁴ M⁴) ~ 4 x 10⁷⁶/M⁴ (where M is in kg and the numerical coefficient has units such that the result is in pascals). Setting this value equal to the sun's core pressure, the equilibrium mass is ~1.13 x 10¹⁵ kg, corresponding to a blackhole with a lifetime roughly 8 orders of magnitude longer than the current age of the universe. Hence, any black hole more massive than this threshold will grow (at the very most basic of levels, anyway; this is an extremely simplified calculation).

[Trafalgar]

My son eats like a black hole. I will time him at dinner tonight.

[Chronos]

As a first approximation, you could set the Eddington luminosity equal to the Hawking luminosity.

[halfeye]

For a black hole with that lifetime placed at the center of the sun (this Hawking radiation calculator places a black hole with a lifetime of 1 year at ~90000 metric tons and a Schwarzschild radius of 1.31 x 10^-19 meters), the black hole would evaporate in approximately one year (if not exactly one year) and no material would enter the black hole. Even with a year-long lifetime, the black hole's Hawking radiation easily overwhelms even the intense pressures at the center of the sun. The equilibrium mass is approximately 10¹⁵ kg; black holes above this mass will absorb material from the sun (to a very rough approximation).

That's useful. I was interested as a digression from the Earth if the sun became a black hole thread, because a small black hole falling into the sun and growing seems like the only way that could happen.

It seems small black holes aren't black, they're really very bright.

[Manga Shoggoth]

It seems small black holes aren't black, they're really very bright.

Black holes themselves are indeed black. Since light cannot escape from them they can't be any other colour (or lack thereof).

The "bright" part is from the accretion disk, that gets very hot, and as a result starts emitting visible (and higher) radiation. Calling the accretion disk the black hole is a little like calling the Rings of Saturn, Saturn.

I find it rather ironic that a black hole is literally a black body, while the accretion disk acts as a black body, but is very bright.

A black body is a perfect absorber and emitter of radiation - absorbing all radiation that falls on it, and emitting radiation depending on its temperature. A black hole only absorbs radiation (Yes, I'm aware of (theoretical) Hawking Radiation, but that isn't directly emitted by the black hole itself while black body radiation is directly emitted by the body in question; Hawking Radiation also appears to be dependent on mass rather than temperature).

[Chronos]

We're not talking about the luminosity from accretion, here. We're talking about the Hawking radiation. One way to think of Hawking radiation is that a black hole is, in fact, perfectly black... and therefore it radiates in a perfect blackbody spectrum. For a large black hole, the temperature is so low (about a millionth of a kelvin, for a stellar-mass hole, and much less than that for a larger one) that this blackbody radiation is completely undetectable, but for a sufficiently-small black hole, it could be much more significant. Where does a black hole smaller than a star come from? Nobody knows (or even if they exist at all), but they might have been formed in the very early Universe.

[Lord Torath]

My son eats like a black hole. I will time him at dinner tonight.

He throws most of his food around the room? Black Holes are messy eaters.

[Bohandas]

https://www.youtube.com/watch?v=Q6kJaMf3Lgo

[halfeye]

Black holes themselves are indeed black. Since light cannot escape from them they can't be any other colour (or lack thereof).

The "bright" part is from the accretion disk, that gets very hot, and as a result starts emitting visible (and higher) radiation. Calling the accretion disk the black hole is a little like calling the Rings of Saturn, Saturn.

We're not talking about the luminosity from accretion, here. We're talking about the Hawking radiation.

Wot 'e sed.

You are technically correct that the hole is black,.Hawking radiation comes from particle pairs spontaineously appearing one inside, one outside. When they do that at the rate which anihilates 2,000+ tons in a second, it's true that none of the particles which are emitted come from inside the event horizon, but that's so small you couldn't find it with an electron microscope, and all of that energy makes the area around the event horizon brighter than a supernova.

[Radar]

Wot 'e sed.

You are technically correct that the hole is black,.Hawking radiation comes from particle pairs spontaineously appearing one inside, one outside. When they do that at the rate which anihilates 2,000+ tons in a second, it's true that none of the particles which are emitted come from inside the event horizon, but that's so small you couldn't find it with an electron microscope, and all of that energy makes the area around the event horizon brighter than a supernova.

Interestingly, Hawking radiation results in a spectrum exactly like that of a perfect black body. This is also why temperature can be assigned to a black hole.

One thing I am wondering (as I did not read nor could understand the relevant papers), how does the Hawking radiation works out for small (atomic size or smaller) black holes. General relativity model of a black hole probably does not work on such scales, but I wonder if there are some additional effects at small scales even within that model.

[Chronos]

The short answer is, we don't know.

The long answer is, we have no bleeping idea whatsoever.

The longer answer: General relativity, by itself, doesn't care about size at all, but general relativity, by itself, doesn't predict Hawking radiation at all, either. Usually, when discussing Hawking radiation, we use what's called semiclassical gravity, which means that we assume that the background spacetime is classical and static, and then do quantum field theory within the context of that static, classical background (this is actually only a hair more difficult than doing QFT in flat spacetime). So what happens when you try to do semiclassical gravity with a tiny (close to the Planck mass or smaller) black hole? You find that the black hole isn't static, and is in fact changing very rapidly (even on the timescales the quantum stuff is working at), and that therefore semiclassical gravity isn't a valid approximation any more.

What you would actually need to describe a black hole that small would be a theory of true quantum gravity, where both the spacetime and the fields within it are subject to quantum mechanics. Which leads to the problem that we don't have any such theory, yet. There are attempts at developing such a theory, but none of them have made any significant progress.

[halfeye]

The short answer is, we don't know.

The long answer is, we have no bleeping idea whatsoever.

The longer answer: General relativity, by itself, doesn't care about size at all, but general relativity, by itself, doesn't predict Hawking radiation at all, either. Usually, when discussing Hawking radiation, we use what's called semiclassical gravity, which means that we assume that the background spacetime is classical and static, and then do quantum field theory within the context of that static, classical background (this is actually only a hair more difficult than doing QFT in flat spacetime). So what happens when you try to do semiclassical gravity with a tiny (close to the Planck mass or smaller) black hole? You find that the black hole isn't static, and is in fact changing very rapidly (even on the timescales the quantum stuff is working at), and that therefore semiclassical gravity isn't a valid approximation any more.

What you would actually need to describe a black hole that small would be a theory of true quantum gravity, where both the spacetime and the fields within it are subject to quantum mechanics. Which leads to the problem that we don't have any such theory, yet. There are attempts at developing such a theory, but none of them have made any significant progress.

I personally am not that interested in planck mass black holes. A one kg black hole has a lifetime of approximately 1 x 10^-19 seconds and a event horizon of approaching the planck length, that's a bomb in no uncertain terms, and whether there's a planck mass residue that can't eat is more or less irrelevant to the universe at large.

[Bohandas]

Shouldn]t the putput of hawking radiation be limited by the density of vacuum energy? My understanding it that it's an effect of the black hole's interation with spontaneously produced virtual particle pairs arising from the vacuum energy. The energy actually comes from the vacuum but by some means that likely involves math that I can't do the energy deficit untimately winds up getting taken out of the black hole instead. But despite the debt untimately being transferred to the black hole shouldn't the density of energy available in the vacuum still limit the process?

[halfeye]

Shouldn]t the putput of hawking radiation be limited by the density of vacuum energy? My understanding it that it's an effect of the black hole's interation with spontaneously produced virtual particle pairs arising from the vacuum energy. The energy actually comes from the vacuum but by some means that likely involves math that I can't do the energy deficit untimately winds up getting taken out of the black hole instead. But despite the debt untimately being transferred to the black hole shouldn't the density of energy available in the vacuum still limit the process?

As I understand it, which is vaguely, this is correct, but the limit is just that high that the evaporation can proceed at an exponentially increasing rate.

There's a thing about vacuum energy as a power source because this energy is supposed to be so high, allfdgedly the are particla pairs emerging and anihilating all the time, but because the pairs meet up and die, there is no overall energy except when a black hole interferes.

[Bohandas]

AFAIK they really don't have any idea how much energy is in the vacuum. IIRC they tried to estimate it by different means and got resukts that differ by over 100 orders of magnitude, depending on how they were estimating it.

[Manga Shoggoth]

Trouble is, every source I have read has described Hawking Radiation as being so faint as to be undetectable by telescopes, even for near solar-mass black holes (the smallest we know about). If you are seeing light from the location of a black hole, it should be accretion disk or jets, not Hawking radiation.

Even the photographs we've seen (admittedly of supermassive black holes) show a dark disk (of course, there may be colourising of the image in play).

As far as I know Hawking Radiation hasn't been proven to exist except by analogy in an interesting sounding experiment using sonic black holes and Bose–Einstein condensates. It's like the Higgs Boson a few years ago - the theory is there, and looks good, but we can't get the measurements to prove it.

If there's proof to the contrary I'd like to hear about it - this was one of the more interesting parts of my degree in the days of my mis-spent youth.

[halfeye]

Trouble is, every source I have read has described Hawking Radiation as being so faint as to be undetectable by telescopes, even for near solar-mass black holes (the smallest we know about). If you are seeing light from the location of a black hole, it should be accretion disk or jets, not Hawking radiation.

Even the photographs we've seen (admittedly of supermassive black holes) show a dark disk (of course, there may be colourising of the image in play).

As far as I know Hawking Radiation hasn't been proven to exist except by analogy in an interesting sounding experiment using sonic black holes and Bose–Einstein condensates. It's like the Higgs Boson a few years ago - the theory is there, and looks good, but we can't get the measurements to prove it.

If there's proof to the contrary I'd like to hear about it - this was one of the more interesting parts of my degree in the days of my mis-spent youth.

Black holes haven't been proven ti exist either, they're just a best quess given what we see, and the theorists like them, and it's pretty much the same with Hawkinf radiatiog, we can't say we've seen it, but it seems pretty likely. I first heard of Hawking radiation a long time ago. The copyright date of the book I read about it in is 1978, I probably read it in 1980 or later, I''be been familiar with the table of black hole durations in it for a long time now.

[Lord Torath]

Black holes haven't been proven ti exist either, they're just a best quess given what we see, and the theorists like them, and it's pretty much the same with Hawkinf radiatiog, we can't say we've seen it, but it seems pretty likely.

We've taken photos of them, and we've now seen them 'eating'. How much more evidence do you need?

[Bohandas]

I think they're using a strict definition that demands the interior solution to be as commonly understood, not just the accretion disk and event horizon

[halfeye]

We've taken photos of them, and we've now seen them 'eating'. How much more evidence do you need?

I was comparing their provedness to that of Hawking radiation. As I understand it, Hawking radiation is not controversial or generally disputed in scientific circles.

As for what proof is, I was thinking of Lewis Carroll's "What the Tortoise Said to Achilles" in which proof is alledged to be an infinite regression.

https://en.wikipedia.org/wiki/What_t...id_to_Achilles

Spoiler

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Achilles had overtaken the Tortoise, and had seated himself comfortably on its back.

"So you've got to the end of our race-course?" said the Tortoise. "Even though it does consist of an infinite series of distances? I thought some wiseacre or another had proved that the thing couldn't be done?"

"It can be done," said Achilles; "It has been done! Solvitur ambulando. You see, the distances were constantly diminishing; and so—"

"But if they had been constantly increasing?" the Tortoise interrupted. "How then?"

"Then I shouldn't be here," Achilles modestly replied; "and you would have got several times round the world, by this time!"

"You flatter me—flatten, I mean," said the Tortoise; "for you are a heavy weight, and no mistake! Well now, would you like to hear of a race-course, that most people fancy they can get to the end of in two or three steps, while it really consists of an infinite number of distances, each one longer than the previous one?"

"Very much indeed!" said the Grecian warrior, as he drew from his helmet (few Grecian warriors prossessed pockets in those days) an enormous note-book and a pencil. "Proceed! And speak slowly, please. Short-hand isn't invented yet!"

"That beautiful First Proposition of Euclid!" the Tortoise murmured dreamily. "You admire Euclid?"

"Passionately! So far, at least, as one can admire a treatise that wo'n't be published for some centuries to come!"

"Well, now, let's take a little bit of the argument in that First Proposition—just two steps, and the conclusion drawn from them. Kindly enter them in your note-book. And in order to refer to them conveniently, let's call them A, B, and Z:—
(A) Things that are equal to the same are equal to each other.
(B) The two sides of this Triangle are things that are equal to the same.
(Z) The two sides of this Triangle are equal to each other.
Readers of Euclid will grant, I suppose, that Z follows logically from A and B, so that any one who accepts A and B as true, must accept Z as true?"

"Undoubtedly! The youngest child in High School—as soon as High Schools are invented, which wlil not be till some two thousand years later—will grant that."

"And if some reader had not yet accepted A and B as true, he might still accept the sequence as a valid one, I suppose?"

"No doubt such a reader might exist. He might say, "I accept as true the Hypothetical Proposition that, if A and B be true, Z must be true; but, I don't accept A and B as true." Such a reader would do wisely in abandoning Euclid, and taking to football."

"And might there not also be some reader who would say, "I accept A and B as true, but I don't accept the Hypothetical"?"

"Certainly there might. He, also, had better take to football."

"And neither of these readers," the Tortoise continued, "is as yet under any logical necessity to accept Z as true?"

"Quite so," Achilles assented.

"Well, now, I want you to consider me as a reader of the second kind, and to force me, logically, to accept Z as true."

"A tortoise playing football would be—" Achilles was beginning

"—an anomaly, of course," the Tortoise hastily interrupted. "Don't wander from the point. Let's have Z first, and football afterwards!"

"I'm to force you to accept Z, am I?" Achilles said musingly. "And your present position is that you accept A and B, but you don't accept the Hypothetical—"

"Let's call it C," said the Tortoise.

"—but you don't accept
(C) If A and B are true, Z must be true."

"That is my present position," said the Tortoise.

"Then I must ask you to accept C."

"I'll do so," said the Tortoise, "as soon as you've entered it in that note-book of yours. What else have you got in it?"

"Only a few memoranda," said Achilles, nervously fluttering the leaves: "a few memoranda of—of the battles in which I have distinguished myself!"

"Plenty of blank leaves, I see!" the Tortoise cheerily remarked. "We shall need them all!" (Achilles shuddered.) "Now write as I dictate:—
(A) Things that are equal to the same are equal to each other.
(B) The two sides of this Triangle are things that are equal to the same.
(C) If A and B are true, Z must be true.
(Z) The two sides of this Triangle are equal to each other."

"You should call it D, not Z," said Achilles. "It comes next to the other three. If you accept A and B and C, you must accept Z."

"And why must I?"

"Because it follows logically from them. If A and B and C are true, Z must be true. You don't dispute that, I imagine?"

"If A and B and C are true, Z must be true," the Tortoise thoughtfully repeated. "That's another Hypothetical, isn't it? And, if I failed to see its truth, I might accept A and B and C, and still not accept Z, mightn't I?"

"You might," the candid hero admitted; "though such obtuseness would certainly be phenomenal. Still, the event is possible. So I must ask you to grant one more Hypothetical."

"Very good. I'm quite willing to grant it, as soon as you've written it down. We will call it (D) If A and B and C are true, Z must be true.
Have you entered that in your notebook?"

"I have!" Achilles joyfully exclaimed, as he ran the pencil into its sheath. "And at last we've got to the end of this ideal race-course! Now that you accept A and B and C and D, of course you accept Z."

"Do I?" said the Tortoise innocently. "Let's make that quite clear. I accept A and B and C and D. Suppose I still refused to accept Z?"

"Then Logic would take you by the throat, and force you to do it!" Achilles triumphantly replied. "Logic would tell you, "You ca'n't help yourself. Now that you've accepted A and B and C and D, you must accept Z!" So you've no choice, you see. "

"Whatever Logic is good enough to tell me is worth writing down," said the Tortoise. "So enter it in your note-book, please. We will call it
(E) If A and B and C and D are true, Z must be true. Until I've granted that, of course I needn't grant Z. So it's quite a necessary step, you see?"

"I see," said Achilles; and there was a touch of sadness in his tone.

Here the narrator, having pressing business at the Bank, was obliged to leave the happy pair, and did not again pass the spot until some months afterwards. When he did so, Achilles was still seated on the back of the much-enduring Tortoise, and was writing in his note-book, which appeared to be nearly full. The Tortoise was saying, "Have you got that last step written down? Unless I've lost count, that makes a thousand and one. There are several millions more to come. And would you mind, as a personal favour, considering what a lot of instruction this colloquy of ours will provide for the Logicians of the Nineteenth Century—would you mind adopting a pun that my cousin the Mock-Turtle will then make, and allowing yourself to be re-named Taught-Us?"

"As you please!" replied the weary warrior, in the hollow tones of despair, as he buried his face in his hands. "Provided that you, for your part, will adopt a pun the Mock-Turtle never made, and allow yourself to be re-named A Kill-Ease!"

[Manga Shoggoth]

I was comparing their provedness to that of Hawking radiation. As I understand it, Hawking radiation is not controversial or generally disputed in scientific circles.

And I said as much. The point is that we have plenty of proof that we have observed black holes, and none for Hawking Radiation. The theory is good, but we haven't seen it yet.

(And yes, for very small (primordial?) black holes the Hawking Radiation would be much bigger. Trouble is, we 'ain't seem them yet either...)

[Chronos]

In a way, Hawking radiation not existing would be a bigger disruption to what we know of physics than black holes themselves not existing. As soon as you have black holes, Hawking radiation falls out naturally from the laws of thermodynamics. And in this house, we obey the laws of thermodynamics.