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This question appeared quite a time ago and was inspired, of course, by all the fuss around "LHC will destroy the Earth".

Consider a small black hole, that is somehow got inside the Earth. Under "small" I mean small enough to not to destroy Earth instantaneously, but large enough to not to evaporate due to the Hawking radiation. I need this because I want the black hole to "consume" the Earth. I think reasonable values for the mass would be $10^{15} - 10^{20}$ kilograms.

Also let us suppose that the black hole is at rest relative to the Earth.

The question is:
How can one estimate the speed at which the matter would be consumed by the black hole in this circumstances?

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    $\begingroup$ This is a nice but also hard question, I suppose :) $\endgroup$ – Robert Filter Jan 12 '11 at 17:11
  • $\begingroup$ Well, actually, I have some thoughts about it. But I'll wait for some suggestions first... $\endgroup$ – Kostya Jan 12 '11 at 17:14
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    $\begingroup$ I think I tried to make a blog post about this once but I couldn't figure out how to do a realistic calculation (at least, not without spending an excessive amount of time on it). So I'd be very interested to see what people come up with. $\endgroup$ – David Z Jan 12 '11 at 18:43
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    $\begingroup$ The location of the hole on the earth's surface should be completely irrelevant, actually. The Earth's gravitational field should cause the hole to fall into the planet, constantly accelerating until it reached the core. At that point, the inertia that the hole has picked up should carry it back out towards the other end to about the same altitude that it was created at (using your LHC theory) and would effectively pass back and forth through the planet ad nauseum. The only way the black hole would be "at rest" relative to Earth is if it were created in the Earth's core. $\endgroup$ – GWLlosa Jan 12 '11 at 20:04
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    $\begingroup$ This question has attracted a lot of incorrect answers, including the accepted answer. Related: physics.stackexchange.com/q/424929 $\endgroup$ – Ben Crowell Dec 9 '18 at 20:38
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In the LHC, we are talking about mini black holes of mass around $10^{-24}kg$, so when you talk about $10^{15}-10^{20}kg$ you talk about something in the range from the mass of Deimos (the smallest moon of Mars) up to $1/100$ the mass of the Moon. So we are talking about something really big.

The Schwarzschild radius of such a black hole (using the $10^{20}$ value) would be

$$R_s=\frac{2GM}{c^2}=1.46\times 10^{-7}m=0.146\mu m$$

We can consider that radius to be a measure of the cross section that we can use to calculate the rate that the BH accretes mass. So, the accretion would be a type of Bondi accretion (spherical accretion) that would give an accretion rate

$$\dot{M}=\sigma\rho u=(4\pi R_s^2)\rho_{earth} u,$$

where $u$ is a typical velocity, which in our case would be the speed of sound and $\rho_{earth}$ is the average density of the earth interior. The speed of sound in the interior of the earth can be evaluated to be on average something like

$$c_s^2=\frac{GM_e}{3R_e}.$$

So, the accretion rate is

$$\dot{M}=\frac{4\pi}{\sqrt{3}}\frac{G^2M_{BH}^2}{c^4}\sqrt{\frac{GM_e}{R_e}}.$$

That is an order of magnitude estimation that gives something like $\dot{M}=1.7\times10^{-6}kg/s$. If we take that at face value, it would take something like $10^{23}$ years for the BH to accrete $10^{24}kg$. If we factor in the change in radius of the BH, that time is probably much smaller, but even then it would be something much larger than the age of the universe.

But that is not the whole picture. One should take also in to account the possibility of having a smaller accretion rate due to the Eddington limit. As the matter accretes to the BH it gets hotter since the gravitational potential energy is transformed to thermal energy (virial theorem). The matter then radiates with some characteristic luminosity. The radiation excerpts some back-force on the matter that is accreting lowering the accretion rate. In this case I don't thing that this particular effect plays any part in the evolution of the BH.

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    $\begingroup$ Really nice work. I have to re-check your calculations, though :) $\endgroup$ – Robert Filter Jan 13 '11 at 15:42
  • $\begingroup$ :) Be my guest. As I’ve said, it is more of a qualitative calculation, but I would like to add that one could easily integrate the differential equation of the accretion rate (which I didn’t originally do for my time estimation). The result would be something like 5 orders of magnitude smaller, if I haven't missed any zeros... :P, which is still much bigger than the age of the universe. $\endgroup$ – Vagelford Jan 13 '11 at 15:55
  • $\begingroup$ woudn't gravity overwhelm the thermal velocities. A particle falling from say 2x the Schwarzchild radius to the event horizon would be traveling at near the speed of light. The real issue, is you would form some sort of accretion disk, with relativistic jets at the poles. Also the tremendous release of energy, might blow some of the material away. Shouldn't this whole thing end up resembling a miniature version of a gamma ray burst. $\endgroup$ – Omega Centauri Jul 29 '11 at 15:45
  • $\begingroup$ See above where I commented on the question. This is not relevant to the LHC because it assumes 3+1 dimensions. $\endgroup$ – Ben Crowell Jul 30 '11 at 16:06
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    $\begingroup$ I think there is also a problem with this answer's estimate of the accretion rate. If you look at Giddings, arxiv.org/abs/0806.3381 , eq. 4.31 and appendix A, you'll see that they use an effective radius that they refer to as the Bondi radius where you use the Schwarzschild radius. The Bondi radius is greater by a factor $(c/c_s)^2$, and that causes their accretion rate to be greater than your estimate by a factor of $(c/c_s)^4$, or about $10^{18}$. $\endgroup$ – Ben Crowell Dec 9 '18 at 20:10
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Since I have much better answer from Vagelford -- I'll write my own version.

When matter falls on the black hole it gets fractioned and radiates. As far as I know (correct me if I'm wrong) one can estimate the radiated energy as $\simeq 0.05mc^2$. Where $m$ is the mass of the falling matter.

The Earth's matter is pulled by the black hole gravitation and pushed away by the radiation. Moreover, for the matter flow $J$ we have "negative feedback" system:

  • bigger $J$ -> more radiation -> more matter is "pushed away"
  • smaller $J$ -> less radiation -> more matter is "pulled in"

The equilibrium between those forces corresponds to already mentioned Eddington luminosity:
$L (J/s) = 1.3\cdot 10^{21} \frac{M}{M_{sun}}$

Equating $L=0.05Jc^2$ and going to $r_{sh}(m) = 3000 \frac{M}{M_{sun}}$, I obtain:

$J (kg/s) = 100 r_{sh}(m)$

It is remarkable, that the "consumption speed" for the $10^{20} kg$ black hole ($r_{sh} = 148.5\mu m$, look here) will give you $1.48\cdot10^{-5}$ kg/s. Which is just order of magnitude larger than the estimate by Vagelford.

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    $\begingroup$ The accretion luminosity in a black hole is given as $L_{acc}\approx\dot{M}c^2$. The Eddington luminosity on the other hand is as you say the luminosity at the equilibrium between the radiation and the matter, which is $L_{Edd}=1.3\times10^{31}(M/M_{sun})J/s$. The luminosity that I calculate with my accretion rate is $\approx10^{10}J/s$ while the $L_{Edd}$ is $\approx10^{12}J/s$. So, what you are doing is imposing the Eddington luminosity to your accretion, while my accretion rate seems to be smaller than the one required for that luminosity. $\endgroup$ – Vagelford Jan 13 '11 at 21:50
  • $\begingroup$ If you calculate the fraction of $L_{acc}/L_{Edd}$ in this case you will see that it is something like $10^{-22}M_{BH}kg^{-1}$, which means that the luminosity gets to the Eddington limit when the BH will get to the $10^{22}kg$. Then the accration rate will be the one that the Eddington luminosity dictates. $\endgroup$ – Vagelford Jan 13 '11 at 22:01
  • $\begingroup$ Well, I wouldn't take an order of magnitude discrepancy too seriously. $\endgroup$ – Kostya Jan 14 '11 at 8:21
  • $\begingroup$ The possibility of an Eddington limit is discussed in Giddings and Mangano, arxiv.org/abs/0806.3381 . See section 4.5 and Appendix B, section B.2.2. They find that there is no Eddington limit here. The estimate here is about $10^{19}$ times bigger than what I get in my answer using the results from the Giddings paper, so I don't think this answer is right. $\endgroup$ – Ben Crowell Dec 9 '18 at 20:51
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If the black hole simply swalled matter, and didn't lose any energy, it probably isn't too hard a calculation, just assume the earth is unsupported mass that falls into the BH, which grows in mass as it adds more stuff. The problem, is we know this isn't how it would happen, and some significant fraction of swalled mass will be released as energy, maybe one to a few percent of mC**2. So the energy liberated from swallowing mass, is orders of magnitude greater per unit mass than an H bomb. Clearly most of the planets mass would be blown away, and only a small amount would end up incorporated into the BH. I'd bet this would happen extremely rapidly, and the shock wave that rips the planet apart would probably only take a few seconds. Note freefall time to the center of the earth is probably more like a half hour (order of magnitude), so most of the planet wouldn't even begin to fall before the released energy blasted it apart.

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This question is addressed in Giddings and Mangano, http://arxiv.org/abs/0806.3381 . See eq. 4.31 and appendix A. For a $10^{20}$ kg black hole, the rate of accretion comes out to be about $10^{13}$ kg/s.

This is greater than the estimate in Vagelford's answer by a factor of $10^{18}$. The reason for this factor is that Giddings essentially uses the Bernoulli equation to model the mass flow, and in this model the mass does not just flow in at the speed of sound $c_s$ all the way until it hits the event horizon. If I'm getting the gist of the calculation right, mass flows in at the speed of sound up until it reaches a certain radius, which is greater than the Schwarzschild radius by a factor of $(c/c_s)^2$. Even without digging in to the details of the calculation, this sort of makes sense. Infalling matter is going to move at relativistic velocities, $\sim c$, as it approaches the horizon, not at $c_s$. They refer to this effective radius as the Bondi radius, and the difference between their estimate and Vagelford's is basically that they use this radius where Vagelford uses the Schwarzschild radius. This causes their accretion rate to be greater than Vagelford's estimate by a factor of $(c/c_s)^4$, or about $10^{18}$.

Using the Giddings result, it takes on the order of years for the black hole to double its mass. I haven't integrated the relevant differential equation, but since the rate goes like the square of the black hole's mass, it looks like it would only be a matter of decades before the black hole consumed a significant fraction of the earth's mass. (It may not consume all of it because of conservation of angular momentum, the ejection of some mass, and other astrophysical processes that occur near the end where the earth's structure is severely disrupted.)

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It would take a long time if we do a back of the envelope calculation.

  • the black hole would exert a force of 1g at around 20 km (assuming 10^20 kg of mass).
  • if we can reasonably assume that the mass inside this sphere is going to be absorbed quickly, that would mean the black hole mass increases correspondingly.
  • on the other hand this extra mass can be calculated to be around 10^20 kg too. So we can expect the 1g radius not to increase significantly.
  • I believe that mass with less than 1g of pull will take a long tine to spiral inside the black hole, as its size (Shwartzchild radius) would be in the micrometer scale and the sizes involved in the kilometre scale.

Hope this helps!

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    $\begingroup$ The pressure in the core is several megabars, and the density is circa ten g/cm**3, so I think if the material below a given point were removed, the layer above would exceedingly rapidly be pushed down. The real kicker is the messy eating habits of the BH, which means enough energy is liberated to overwhelm the other dynamics. $\endgroup$ – Omega Centauri Jan 12 '11 at 22:10
  • $\begingroup$ Agreed that everything would.blow up as you said, I was just trying to answer the question, not to model black hole dynamics ;-) $\endgroup$ – Sklivvz Jan 13 '11 at 2:00
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The black hole of mass $10^{20}$ kilograms is not as dangerous as it may look. It is $10^{28}$ Planck masses, so the radius is $10^{28}$ Planck lengths or $10^{-7}$ meters. The gravitational acceleration near its event horizon is $10^{-10}\times 10^{20} / 10^{-14}$ which is $10^{24}$ meters per second per second. Even meters from the black hole horizon, the acceleration reaches the Earth's acceleration on the surface. Centimeters from the black hole horizon, the acceleration would be enough to break the matter.

The black hole would obviously try to find the minimum of the gravitational potential produced by the Earth, so it would sit and eventually stabilized at (and oscillated around) the Earth's center. If the Earth's center were imagined to be solid, I find it plausible that the black hole would break and eat some matter that is meters away from the micrometer-sized black hole. And the rest of the solid Earth could just sit around.

However, this is clearly not what would happen because the center of the Earth is liquid - because of the immense pressure. The liquid metals from meters around the black hole would simply flow to the black hole by reasonable speeds. The black hole would drink a lot of this fluid and its density would decrease in much of the Earth's core. It's questionable whether the solid layers of the planet could survive this reduced pressure. I can imagine that a black hole could actually be sitting inside the Earth and slowly drinking the liquid iron.

On the other hand, we know for sure that this is not what generically happens inside the celestial bodies because it would also take place inside gas planets, and those could be fully swallowed in a relatively shorter time, producing lots of rotation along the way.

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  • $\begingroup$ How can the density of the core decrease? This is ridiculous. The core would simply contract. There are no forces to counteract the core's contraction, and the pressure from the rest of the Earth is immense. $\endgroup$ – Peter Shor May 8 at 11:58
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I just want to add that oscillations about the center of the Earth are dampened due to the momentum of the entering mass.

Figures for volume of the mass continually eaten by the black hole differ by orders of magnitude going by previous posters. But the consumed material as it falls will depend on the cross section of this volume times the radius of the Earth, or cross section times density for linear mass density of path of destruction.

That mater has zero kinetic energy and a potential energy as a function of height. Gravitational field is directly proportional to radius (due to continuous spherical distribution) so potential energy is $r^2$ function. I write gravitational potential of BH as $C R^2 m$ where R is radius of Earth and m is mass (kg) of BH and C is some constant I'm not going to address. Denote linear density of path of destruction as $l$ (kg/m), and integrate $C r^2$ potential to find $1/3 C R^3 l$ to center of Earth, or $2/3 C R^3 l$ to other side of Earth.

Assume it eats material perfectly and there are no other interactions. It begins with $C R^2 m$ energy (Newtonian!) and m mass. It acquires $2 l R$ mass in one trip (assuming acquired mass is small relative to total and thus nearly touches surface again). We find the deficit in specific potential energy at the end of its trip: (P.E._end/end_mass) / (P.E._start/start_mass)-1.

$$\frac{ \frac{ C R^2 m+\frac{2}{3} C R^3 l}{m+2 R l} }{ \frac{C R^2 m}{m} } - 1$$

$$=\frac{m}{m+2 R l} \frac{R^2 m+\frac{2}{3} R^3 l}{R^2 m} - 1$$

$$=\frac{1+\frac{2}{3} \frac{R l}{m} }{1+2 \frac{R l}{m}} - 1 = \frac{1+2 \alpha }{ 1+\frac{2}{3} \alpha} - 1$$

where $\alpha = R l/m$ dimensionless parameter representing fraction of initial mass added by trip.

We assume $\alpha \ll 1$ and Taylor expand at $\alpha=0$ to find

Specific energy deficit after one trip $= -4/3 \alpha$

Looking more closely at alpha, write $\alpha = R A \rho/m$, where A is the cross sectional area I referred to and rho is the density of Earth.

$$R = 6.4 \times 10^6 m$$ $$A = 1 cm^2$$ $$\rho = 4.0 g/cm^3$$ $$m = 10^{20} kg$$

$\alpha = 2 R l/m = 2.56 \times 10^{-13}$ (fraction of BHs mass accumulated in half-trip, sounds good)

For change in height due to trip, use mgh approx and find

$$( - \frac{4}{3} \alpha ) 6,400,000 m = 2.18 x 10^{-6} m$$ Lower

It falls 2.18 micro meters lower at the end of the trip. Now, this scales directly with the area eaten, and thus with the square of the radius at which material is captured. To get a factor of 1e6, that radius would need 10 meters versus 1 cm.

Thus, dampening really IS SMALL, and the fate of the Earth would be dictated by how it eats matter while traveling at high speeds through the core. I'm going to go tell people now that the reason the LHC is underground is so that a BH won't poke out the surface if an accident happens. I love spreading disinformation.

Edit: This was my first answer given on physics SE, so I've gone back and put the equations in the right format, although the organization of the answer probably reflects its bizarre history.

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  • $\begingroup$ I think your estimate of $A$ is too small by about a factor of $10^7$. This is based on the fact that the Bondi radius calculated by Giddings and Mangano, arxiv.org/abs/0806.3381 , is ~100 m, not 1 cm. So actually I think the damping is rather large, and the black hole will only take a matter of days to settle down to the earth's core. $\endgroup$ – Ben Crowell Dec 9 '18 at 20:57

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