Exploring the Safety of Artificial Black Holes

[yuiop]

Hi,

In the safety case for the new Large Hadron Collider (LHC), the possibility of creating artificial black holes is considered. The main argument for the case that there is nothing to worry about, is that cosmic rays collide regularly with atoms in the upper atmosphere with much higher collision energies than can be produced in the LHC... and yet the Earth is still here...

That seams a reasonable argument. We can also note that generally speaking stars do not disappear randomly on a regular basis with no apparent cause.

However, it is reasonable to ask why the Earth is still here if black holes are occasionally formed in the upper atmosphere. Some LHC critics argue that any possible micro black holes formed in the upper atmosphere have high velocities and would pass through the Earth with little fuss because they are moving too fast to significantly gravitationally interact with the Earth. There concern is that the collisions in the LHC are due to two particle beams moving in opposite directions would have a net momentum of zero and any micro black hole formed would be more dangerous than a natural cosmic black hole because it not moving fast and has more time to interact gravitationally with its surroundings. Of course that counter argument is based on the assumption that natural cosmic black holes are basically harmless because they pass rapidly through the Earth with greater than the required escape velocity. I do not find that a convincing argument. In the history of the Earth at least one natural micro black hole would in all probability have had a succession of chance collisions with particles in the air and in the body of the Earth itself, sufficient to slow the black hole to below the escape velocity. The high velocity of any natural black holes does not seem sufficient enough argument to explain the observed survival of the Earth.

The other argument in the defence of the LHC safety case is Hawking radiation. The half life of a black hole is inversely related to its mass and for a micro black hole the lifetime is very brief before its evaporates in a burst of radiation. Critics counter that Hawking's radiation is only theoretical and has never actually been observed so in their view the safety case is based on something that not been proved empirically. I would add that even with Hawking radiation a rapid series of chance collisions after the formation of a natural black hole would allow it gain mass faster than it loses it through radiation...and yet the Earth is still here...

I would like to add a third argument for consideration here. Imagine a micro black hole has managed to survive evaporating away at birth and finds itself at the centre of the Earth. The interior Schwarzschild solution tells us that the gravitational gamma factor at any location with a a massive body is determined by the enclosed mass AND the mass outside the enclosed volume. A quick calculation shows that when the mass of the Earth+micro black hole is taken into account there is no longer an event horizon anywhere inside the Earth and the micro black hole is in fact destroyed (in that it no longer has the properties of being a black hole) by the mass of the Earth rather than the other way round.

It turns out that micro black hole holes are quite delicate creatures that have to be carefully nurtured to survive. At the baby stage they have to be fed frequently at the correct intervals to avoid evaporating between meals until they get to a juvenile stage where their mass is self sustaining because they gain energy from the CMB faster than they evaporate. Even when they survive to that stage, care has to be taken not to overfeed the black hole, as it seems too big a meal (where the mass of the meal greatly exceeds the mass of the black hole) at anyone sitting can cause them to choke and die.

 

[JesseM]

I would like to add a third argument for consideration here. Imagine a micro black hole has managed to survive evaporating away at birth and finds itself at the centre of the Earth. The interior Schwarzschild solution tells us that the gravitational gamma factor at any location with a a massive body is determined by the enclosed mass AND the mass outside the enclosed volume. A quick calculation shows that when the mass of the Earth+micro black hole is taken into account there is no longer an event horizon anywhere inside the Earth and the micro black hole is in fact destroyed (in that it no longer has the properties of being a black hole) by the mass of the Earth rather than the other way round.

Can you really "destroy" a black hole (revealing its interior) by putting a shell of matter with sufficient mass around it? I would have thought this would have come up in discussions of the BH information loss paradox...do you have a reference for this?

 

[yuiop]

Can you really "destroy" a black hole (revealing its interior) by putting a shell of matter with sufficient mass around it? I would have thought this would have come up in discussions of the BH information loss paradox...do you have a reference for this?

Nope, it is just my conclusion of applying the interior Schwarzschild solution and coupling that with the observation that we do not see many stars or planets mysteriously disappearing.

It does not really reveal a naked singularity as such. I will post the equations if you are interested.

 

[CFDFEAGURU]

Could you post your equations?

 

[yuiop]

Could you post your equations?

Sure. This is the interior Schwarzschild solution stripped of the rotational elements so that only radial location is considered:

$$dtau/dt = 3/2\sqrt{1-R_s/R_m}-1/2\sqrt{1-R_sR_o^2/R_m^3}$$

where
Rs is the Schwarzschild radius,
Rm is the radius that includes all the mass,
Ro is the radius where the gamma factor is being measured

and it is assumed the observer is at infinity (as in the exterior Schwarzschild solution).


The location of the event horizon (where dtau/dt=0) is given by:

$$R_o = \sqrt{9R_m^2-8R_m^3/R_s}$$

and it can be seen if total mass has a surface radius Rm that is greater than Rs*9/8 that the event horizon is imaginary and so does not exist. In short the gravitational gamma factor at a given radius is not only determined by the mass below that radius but also by the mass above the radius and the presence of the spherical shell beyond the black hole effectively destroys the property of the black hole having an event horizon so technically it is no longer a black hole.
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Sorry for being lazy. The above is basically a copy and paste from an earlier post. More background information can be found in this thread https://www.physicsforums.com/showthread.php?t=223730&page=2
which contains links to some papers.

The interior Schwarzschild solution can also be expressed as :

$$\frac{dtau}{dt} = \frac{3}{2}\left(1-\frac{2GM}{R_mc^2}\right)^{0.5}-\frac{1}{2}\left(1-\frac{2GM}{R_oc^2}\frac{(Enclosed Mass)}{(Total Mass)}\right)^{0.5}$$

The derivation for this is shown here: https://www.physicsforums.com/showpost.php?p=1769139&postcount=23

A related paper: http://odarragh.astro.utoronto.ca/Schwarzschild.pdf

 

[JesseM]

The Schwarzschild solution is a stationary spacetime describing an object that is infinitely old...I think that historically, physicists were not really convinced of the existence of black holes until it was shown that the event horizon and singularity would form in the dynamic case of a collapsing star. Similarly, even assuming your solution is correct, I don't think it necessarily proves that if you have an existing black hole you can erase its event horizon by surrounding it with a shell of matter.

 

[yuiop]

The Schwarzschild solution is a stationary spacetime describing an object that is infinitely old...I think that historically, physicists were not really convinced of the existence of black holes until it was shown that the event horizon and singularity would form in the dynamic case of a collapsing star. Similarly, even assuming your solution is correct, I don't think it necessarily proves that if you have an existing black hole you can erase its event horizon by surrounding it with a shell of matter.

All I can say is that the interior Schwarzschild solution (not my solution) does not care about the distribution of the matter within the enclosed mass below Ro, just as the Newtonian equation for gravity does not care whether all the mass of the Earth is contained in a point at the centre or distributed evenly when calculating the gravitational force at the surface of the Earth. Of course the equation given is the view point of a distant observer not significantly affected by the gravity of the Earth, but if he does not see the Earth destroyed by the micro black hole then the people on Earth will not see the Earth destroyed by the micro black hole either. Normally when analysing black holes we prefer to analyse the point of view of the observer free falling into the black hole and ignore the viewpoint of an external observer making coordinate measurements. In this case I think the coordinate measurements are perfectly valid, and the facts that it concludes that an atom sized black hole can not destroy the Earth shouldn't be so hard to swallow (oops).

P.S. Yes, it does have something to say about the information loss paradox. It says information is not lost in black holes. I think even Hawking's has come to agree with that.

 

[JesseM]

All I can say is that the interior Schwarzschild solution (not my solution) does not care about the distribution of the matter within the enclosed mass below Ro, just as the Newtonian equation for gravity does not care whether all the mass of the Earth is contained in a point at the centre or distributed evenly when calculating the gravitational force at the surface of the Earth.

What does this have to do with whether an event horizon is still present when you surround the black hole with matter?

Of course the equation given is the view point of a distant observer not significantly affected by the gravity of the Earth, but if he does not see the Earth destroyed by the micro black hole then the people on Earth will not see the Earth destroyed by the micro black hole either. Normally when analysing black holes we prefer to analyse the point of view of the observer free falling into the black hole and ignore the viewpoint of an external observer making coordinate measurements. In this case I think the coordinate measurements are perfectly valid, and the facts that it concludes that an atom sized black hole can not destroy the Earth shouldn't be so hard to swallow (oops).

Huh? I don't follow you at all, I thought you were saying that surrounding the BH with mass would destroy the event horizon, but here you seem to be talking about what happens in different coordinate systems (the event horizon is a physical entity that does not appear or disappear depending on your choice of coordinate systems).

And what's wrong with the conventional notion that the reason the Earth is not destroyed by mini black holes is that any that formed would lose mass to Hawking radiation faster than they'd gain it by outside matter falling into them?

P.S. Yes, it does have something to say about the information loss paradox. It says information is not lost in black holes. I think even Hawking's has come to agree with that.

Hawking believes information is not lost, but his reasons for thinking that are quite different from your argument, I believe. If you think you have an original argument for why information isn't lost in black holes, then it should really be presented in the Independent Research forum.

 

[yuiop]

What does this have to do with whether an event horizon is still present when you surround the black hole with matter?

Because an event horizon is defined by where dtau/dt=0 and the equations I have given show that dtau/dt is no longer zero at the original Schwarzschild radius of the black hole when the mass outside greatly exceeds the mass inside and nor is dtau/dt zero or less anywhere inside the black hole when you take mass outside the black hole into account. The interesting thing about the Schwarzschild solution is that mass in the shell outside the enclosed mass does have an effect unlike the Newtonian model. If dtau/dt is greater than 0 everywhere inside the black hole, it is no longer a black hole.

Huh? I don't follow you at all, I thought you were saying that surrounding the BH with mass would destroy the event horizon, but here you seem to be talking about what happens in different coordinate systems (the event horizon is a physical entity that does not appear or disappear depending on your choice of coordinate systems).

That is exactly what I was saying. If the distant observer using coordinate measurements (The interior Schwarzschild metric) does sees the event horizon as being destroyed then all observers should agree with trhat or we have different histories. However some people would disagree with you and say the event horizon is not a physical entity and is a false singularity whereby an infalling observer would not notice anything special about the event horizon. Coordinate measurements say the event horizon is special, while proper measurements of a free falling observer say there nothing unusual about the event horiozn. Personally I prefer the coordinate measurements.

And what's wrong with the conventional notion that the reason the Earth is not destroyed by mini black holes is that any that formed would lose mass to Hawking radiation faster than they'd gain it by outside matter falling into them?

Perhaps there is nothing wrong with that explanation beyond the fact Hawking radiation has never been observed. That in itself does not prove that what I am talking about is not true. However, I will go far as to concede that hawking radiation is very likely the dominant reason any black holes created by collisions in the upper atmosphere by cosmic rays are not normally a hazard as they do not last more than a very small fraction of a second. I was just extending the discussion to what would happen in the unlikely event a black hole made it to the surface of the Earth. For example it may have its lifetime extended by relativistic time dilation in the same way muons created high up survive the journey to the surface despite their very short lifetimes. If such a black hole made it to the centre of the Earth, the radiation from the magma at the centre of the Earth would be very much greater than the CMB radiation and the high pressure and density at the centre of the Earth provides a ready source of mass.

So basically I am raising this question. Would a small black hole that was of sufficient size not to evaporate away instantly yet still of significantly less mass than the Earth be able to completely consume the Earth? The question could be further extended to include the case of microscopic black holes, if for some reason they did not rapidly evaporate as predicted by Hawking.

For what it is worth, I think Hawking radiation does occur and any black holes produced in the LHC will evaporate almost instaneously. I just think it would be more reasuring to critics, if the LHC safety case did not depend entirely on an effect that has never been observed in nature or in a laboratory.

Hawking believes information is not lost, but his reasons for thinking that are quite different from your argument, I believe. If you think you have an original argument for why information isn't lost in black holes, then it should really be presented in the Independent Research forum.

I am just saying if my argument suggested information was lost in black holes then there would be a problem, but that is not the case.

 

[JesseM]

Because an event horizon is defined by where dtau/dt=0 and the equations I have given show that dtau/dt is no longer zero at the original Schwarzschild radius of the black hole when the mass outside greatly exceeds the mass inside and nor is dtau/dt zero or less anywhere inside the black hole when you take mass outside the black hole into account.

In Schwarzschild coordinates this may be the definition (although are you sure this is always the correct definition even when there is matter outside the Schwarzschild radius of the singularity? Aren't you suggesting there's a naked singularity here?), but the physical definition is that for all events inside the horizon, everything in their future light cone must also lie inside the horizon. This definition doesn't rely on any particular coordinate system.

That is exactly what I was saying. If the distant observer using coordinate measurements (The interior Schwarzschild metric) does sees the event horizon as being destroyed then all observers should agree with trhat or we have different histories. However some people would disagree with you and say the event horizon is not a physical entity and is a false singularity whereby an infalling observer would not notice anything special about the event horizon. Coordinate measurements say the event horizon is special, while proper measurements of a free falling observer say there nothing unusual about the event horiozn. Personally I prefer the coordinate measurements.

But there is nothing incompatible between the idea that infalling observers don't see anything unusual at the horizon, and the idea that once they have crossed the horizon nothing in their future light cones can escape the horizon. As I said above, I'm not sure if your coordinate definition is sufficiently general. And even if it is, there's still the point I made before: the Schwarzschild metric always describes an unchanging arrangement of matter which has existed forever, while physicists use different metrics to describe dynamical black holes which form out of collapsing stars and whose event horizons may change position over time. Even if you can find an unchanging Schwarzschild metric where the interior looks like a black hole but there's no event horizon, that doesn't prove that you can destroy the event horizon of an existing black hole in a dynamical way by putting a shell of matter around it.

I was just extending the discussion to what would happen in the unlikely event a black hole made it to the surface of the Earth. For example it may have its lifetime extended by relativistic time dilation in the same way muons created high up survive the journey to the surface despite their very short lifetimes. If such a black hole made it to the centre of the Earth, the radiation from the magma at the centre of the Earth would be very much greater than the CMB radiation and the high pressure and density at the centre of the Earth provides a ready source of mass.

But if you haven't actually done any calculations, you have no good reason to think that even at the pressure that exists at the center of the Earth, matter would be falling in faster than the BH was losing mass due to Hawking radiation. When a BH gets sufficiently small, it radiates very intensely and shrinks very quickly! This page says the evaporation time is inversely proportional to the cube of the mass, and the wikipedia article on Hawking radiation mentions that a BH with mass 2.28 * 10^5 kg would radiate away in just 1 second, and according to this calculator that would be a Schwarzschild radius of just 3.4 * 10^-22 meters. This is much smaller than the radius of a proton (around 8 * 10^-16 meters), but much larger than the Planck length (about 1.6 * 10^-35 meters). I have no idea whether the "micro" black holes that some theories postulate could be produced by cosmic rays or particle accelerators would be bigger or smaller than this.

So basically I am raising this question. Would a small black hole that was of sufficient size not to evaporate away instantly yet still of significantly less mass than the Earth be able to completely consume the Earth?

But are there any natural processes that would be postulated to produce such black holes intermediate in size between those produced by collapsing stars and those which might be produced by high-energy particle collisions?

 

[Dale]

All I can say is that the interior Schwarzschild solution (not my solution) does not care about the distribution of the matter within the enclosed mass below Ro

Except that it be spherically symmetric.

 

[yuiop]

Except that it be spherically symmetric.

Good point. I agree.

In Schwarzschild coordinates this may be the definition (although are you sure this is always the correct definition even when there is matter outside the Schwarzschild radius of the singularity? Aren't you suggesting there's a naked singularity here?), but the physical definition is that for all events inside the horizon, everything in their future light cone must also lie inside the horizon. This definition doesn't rely on any particular coordinate system.

My interpretation of the Schwarzschild metric is that there is not a singularity of infinite density in the middle of a black hole. The Case Western Team reached a similar conclusion that there is nothing inside a black hole and nothing ever crosses from outside to inside the event horizon in a paper that will be published in Physical Review. See http://arstechnica.com/news.ars/pos...-the-black-hole-information-loss-paradox.html and http://arxiv.org/abs/gr-qc/0609024
There viewpoint is basically that a black hole will evaporate before an infalling observer gets to the event horizon.

But there is nothing incompatible between the idea that infalling observers don't see anything unusual at the horizon, and the idea that once they have crossed the horizon nothing in their future light cones can escape the horizon. As I said above, I'm not sure if your coordinate definition is sufficiently general. And even if it is, there's still the point I made before: the Schwarzschild metric always describes an unchanging arrangement of matter which has existed forever, while physicists use different metrics to describe dynamical black holes which form out of collapsing stars and whose event horizons may change position over time. Even if you can find an unchanging Schwarzschild metric where the interior looks like a black hole but there's no event horizon, that doesn't prove that you can destroy the event horizon of an existing black hole in a dynamical way by putting a shell of matter around it.

Sure, the full calculations have to take motion into account, but it is circular reasoning to assume what that motion is before you do the calculations. The static solution gives a good indication of which direction the motion takes place before you apply the full dynamic solutions. So start with the unbiased assumption that nothing is moving initially, and calculate the accelerations and gravitational potentials to work out what the motion should be.

But if you haven't actually done any calculations, you have no good reason to think that even at the pressure that exists at the center of the Earth, matter would be falling in faster than the BH was losing mass due to Hawking radiation. When a BH gets sufficiently small, it radiates very intensely and shrinks very quickly! This page says the evaporation time is inversely proportional to the cube of the mass, and the wikipedia article on Hawking radiation mentions that a BH with mass 2.28 * 10^5 kg would radiate away in just 1 second, and according to this calculator that would be a Schwarzschild radius of just 3.4 * 10^-22 meters. This is much smaller than the radius of a proton (around 8 * 10^-16 meters), but much larger than the Planck length (about 1.6 * 10^-35 meters). I have no idea whether the "micro" black holes that some theories postulate could be produced by cosmic rays or particle accelerators would be bigger or smaller than this.

I concede that if Hawking radiation really occurs (and it seems reasonable that it does) then any black hole created in a collider would evaporate before it could do any serious damage. A rough estimate is that the one second black hole you refer to would have a msss of 228 tonnes and would have to consume about 76 tonnes of Earth material per second just to break even thermodynamically. All the fossil reserves of the world could not produce a black hole of the that mass so Hawking radiation alone is enough to assure the safety the LHC. A black hole that had enough mass to be be thermal equilibriul even at the 5000-7000K temperatures at the centre of the Earth would have to be many orders of magnitude larger again, with a mass of about 2.4*10^16 tonnes which equates to about 0.0004% of the mass of the Earth.

But are there any natural processes that would be postulated to produce such black holes intermediate in size between those produced by collapsing stars and those which might be produced by high-energy particle collisions?

Yes, there appears to be a large gap between the size of atmosperically or collider created black holes that last for billionths of a second and collapsed star black holes that seem to have a minimum mass of about one solar mass, but can we be sure that we know everything that is going on "out there"?

Anyway, the LHC case is almost a distraction of what I really wanted to talk about, because if you accept Hawking radiation as proven fact then then there are no safety concerns for the LHC. Basically I am asking the (interesting to me) question of would, say a 1 solar mass black hole consume a 10 solar mass active star instaneously? Hawking radiation is not such a big issue at the temperatures involved in this scenario. Assume there is a direct low speed collision and that the black hole does not orbit about the star gradually acreting mass from the star.