Tuesday, October 23, 2007

So I really haven't had much time lately to do much of anything which is not all that surprising considering I am taking 21 credit hours this semester. I thought I could spare a quick minute to make a post though mostly because I am having trouble with this calculation for number theory and I am taking a break.

So far I have been trying to make it clear how I have been ariving at results and giving a bit of background explanation to most things that I have been talking about. I am going to stop doing that for a second and just give you some straight results without talking about how they were derived.

If Jupiter were turned into a black hole space time immediately around the event horizon would be moving circularly at about 168 times the speed of light.

I estimate that the core densities and temperatures necessary to create a black hole are around 1028 g/cm3 and 1019 K which would need to be sustained for only 10-20 sec or so...

If a system of atomic bombs could be used to send pressure waves into Jupiters core the minimum necessary pressure could be provided by about 107 one megaton atomic bombs.

The method of creation using equally distributed h-bombs is not very economical but is the least invasive as it would cause a minimum of damage to the moons and surrounding systems.

I'll fill you in a bit on how I got these estimates later now its back to hw.

Friday, October 19, 2007

finitaphobia

The Greeks abhorred the infinite thinking that anything that was infinite was illogical. Zeno's famous paradox is a good example of how introducing infinity can cause logical trouble. Without allowing ourselves to use the infinite though we cannot arrive at calculus which uses the continuity of numbers as a necessary part of its operation. In some sense of course the Greeks allowed countable infinities grudgingly since they didn't think that there was a "last number" so to speak but allowing for unboundedness is a long way off from allowing the entity of infinity to exist as a philosophically analyzable entity. Allowing for unboundedness allows the Greeks at least partial access to the first order of infinity the countable infinities. For those of you who aren't terribly familiar with the concept of transfinite numbers there are different sizes of infinity the infinity that is the number of the counting numbers is called countable infinity because anything that is countably infinite can be put in a one to one correspondence with the counting numbers. That is countable infinities can be numbered while uncountable infinities are too large to be counted. If that doesn't make sense to you go find some material on cantor set theory.

The Greeks had a pretty good understanding of the rational numbers which are countably infinite. Saying that the Greeks were really only afraid of the real numbers (which are uncountable) is being too nice. They abhorred the existence of numbers like the square root of two since it could not be expressed as a rational number. The Greeks were quite happy to deal with the rational numbers as discrete objects but they would have frowned on talking about all of them as a whole. There is nothing disturbingly infinite about 2/3 but you cannot then haul off and start talking about all the numbers a/b that are between 0 and 1. The later the Greeks would have viewed as beyond the realm of logic because of its association with the infinite.

This inability to talk about questions which touched on the infinite or to use any sort of methods which invoked the infinite limited Greek mathematics. The reason of course for this abhorrence of the infinite goes deeper than the possibility of logical inconsistency, not everyone agreed with Zeno. The real reason is that in mathematics, logic, and geometry the Greeks did not just see disciplines of abstract thought. The profession of logician and geometer was to deal with the world as it really was. The Greeks took philosophy and mathematics as a way of uncovering deep truths about their own physical and metaphysical reality. The abhorrence of the infinite was rooted in the belief that the physical world cannot support the infinite and so any infinite argument or object does not have any reality and so does not deserve to be thought about.

I am afraid that in the intervening time physicists may have become afraid not of the infinite but of the finite. Mathematicians also though to a lesser extent tend to distance themselves from problems that are inherently discrete. We have learned how to deal with the infinite in a logical and rigorous way and we have found the tools that this dealing with the infinite has given us so useful that we have become at least partially unable to work without them. The assumption of the continuity of space is so basic to the world of physics that is essentially completely unchallenged in all of physics. Even in quantum mechanics where things become discrete instead of continuous the underlying space is still thought to be continuous. String theory is even worse since it assumes a completely flat subspace in which the strings are to interact. If you don't see the connection between continuity and the finite then I will just take a moment to point out that continuity is a concept that cannot be achieved without the use of something similar to the real numbers and that means the introduction of uncoutable infinities. Countable infinities are what you get when you allow discrete systems to be unbounded. So intrinsically countable systems are discrete. Since countable systems are necessarily discrete (even though any two rational numbers can approach each other as close as you like every set of rational numbers is disconnected)then that implies that countable systems are necessarily made up from finite objects.

What I really am talking about when I say that physicists have a fear of the finite is that they have a fear of methods that are based in discrete objects. Calculus with its dependence on uncountable infinity and the continuity of things that are being examined is totally indispensable to modern physics. I don't necessarily think this is a bad thing in the sense that calculus is a wonderfully robust and interesting tool and we should be not afraid to use it. However physicists should have a way around calculus no matter how painful and difficult there ought to be a way to do physics that is fundamentally discrete but there is no such thing. Mathematicians and computer scientists can play with the discrete but physics remains fearful of it.

Tuesday, October 16, 2007

Why a local black hole is a good idea

Despite my misgivings about the wanton destruction of an entire planet (and possibly several moons) a local black hole has a number of distinct advantages. By far the most certain and powerful reason to have a black hole is for its value to scientific study. Black holes are precisely the region of the physical world where we don't understand what is going on. Einstein's relativity and quantum mechanics don't get along well together, we know that on some level one theory or the other has to turn out to not be correct. String theory is grown mostly from the realm of quantum mechanics and so naturally assumes that it is special relativity that is the large scale limit of string theory. String theory however is built up of conjectures about objects that exist on distance scales of 10-33 m. Thanks to the Heinsenberg uncertainty principle in order to probe such small distances we need very very high energies. Say you wanted to be able to probe an object on the order of 10-33 m with light. You would need light particles with wavelengths at least as small and hopefully much smaller than the object you were looking at. plank's relation for a photon (a light particle) is E = hf where h is plank's constant `6.62 x 10-34 J*S, E is the energy of the light particle and f is its frequency. For a light particle c = L*f where L is the wavelength and again c is the speed of light. so to get a light particle of wavelenght 10-33 m the energy of the photon would have to be E = h*c/10-33 = 1.98 x 108 J When you consider that this is the energy of a single particle it becomes readily apparent that making particles with such energies is a difficult proposition at best. (if you want to do the same sort of thing for matter with mass then de broglie's relation is L = h/p , where p is the momentum of the particle)

The point is that probing validity of string theory based on reachable predictions of it is very difficult. In order to come up with a workable theory of quantum gravity we need to have access to very very high energies. Objects falling into a black hole get accelerated to incredible energies just before they pass the event horizon. A quick calculation concerning the energy gained by falling from 71000 km from the singularity to 1.3 m of the singularity (or in other words from the current outer radius of Jupiter to the event horizon) of M*3.846 * 1026 J so for the lowly electron with a mass of about 10-31 kg we get an increase in energy of 10-5 J which is about 1014 eV. Putting that number in perspective a little the lhc is going to be able to achieve collision energies for heavy ions of (hopefully) about 10^15 eV. The great thing about gravitational acceleration is that there is no loss due to radiation caused by the acceleration. Falling into a black hole is a free fall motion which means that the particle doesn't feel like it is experiencing acceleration. Achieving such high energies as 1014 eV on earth is a daunting prospect possibly not even feasible. For heavier ions just in free fall the particles can attain energies of about 1019 eV. I need to go take a physics test so I will stop there for now but expect a part 2.

Monday, October 15, 2007

Eccological Concerns

I don't know if it is today or if I missed it but there was supposed to be a day where as many people as possible said something about the environment in their blog. Now I don't know how friendly turning a planet into a black hole is exactly. First of all there is very likely no ecology on Jupiter at all though if there is one all plans to destroy the planet become rather ahem a-moral. It is unlikely though that Jupiter has any life at all on it at least it has no earth like life. However one of it's 63 moons just might have something crawling or swimming around on it. Europa has liquid oceans on it and I believe there is a very good chance that it currently supports some kind of primitive life. If Europa did in fact have life in it's oceans then turning Jupiter into a black hole would be a black act indeed. Let me take a moment though and say that turning Jupiter into a black hole wouldn't have any dire consequences for Europa necessarily since it would be perfectly capable of sustaining its orbit around a Jovian massed black hole in exactly the way it currently maintains its orbit around Jupiter. However since Jupiter cannot be transformed entirely into a black hole all at once but rather a black hole would first form somewhere near it's center and then suck the rest of the gas giant into it. This process would give off a great deal of radiation and great streams of high energy particles. I really don't know whether this would be enough to cause any theoretical life under the ice of Europa any trouble but it would very likely be enough to at least melt the ice and would no doubt cause something of an ecological uproar if Europa did in fact have something that could fairly be called an ecology.

I would here like to say that I am not in favor of destroying Jupiter, I am in favor of creating a black hole. My desires for a local black hole outweigh the desire I have to preserve Jupiter. Should there be life that would be destroyed by this procedure of creating a black hole then I would say that the costs would now outweigh the gains. If the process of creating a black hole out of Jupiter's mass was such a violent process as to cause harm to the earth then I would say again that such a method was not advisable.

Lastly I would like to say that I find it very likely that the destruction of Jupiter is an action that will never find itself under serious consideration by an entity with the power to carry it out. Of course I also think that there is a distinctly non-zero chance that in the future humanity will discover some means of faster than light travel which will require the use of a region of highly curved time space such as that found inside a black hole. Now the question arises that if the destruction of Jupiter would allow us a gateway into the rest of the cosmos then would its destruction be worth the costs? The answer is very probably no. The solar system is a large enough place to play host to a great civilization. In the aeon's that it would take to fully utilize the resources of our humble home I think humanity could live a life much more grand than its due. However that does not mean I do not favor a more grand scheme in which humanity spreads through the stars. We must be wary of growth that comes with such destructive costs. If humanity sweeps through the galaxy destroying worlds in order that it can continue to sweep through the galaxy then perhaps we should consider that growth should not always be sustained but one must learn to live with balance.

All that is just to say that I don't see the creation of a black hole by the destruction of Jupiter as necessarily a good thing. What I do see is a fascinating possibility and a great technical challenge. This blog is not so much devoted to the idea that blowing up Jupiter is good as it is to the question of how might it be done or is it even possible?

Friday, October 12, 2007

Since I don't really know a lot of this stuff myself I think I had better put up some statistics of our prey. I selected some relevant bits of information selected statistics from the all powerful wikipedia, if you need some more information here is the link.

Jupiter stats.

Equatorial radius: 71,492±4 km (11.209 Earths)
Polar radius: 66,854±10 km (10.517 Earths)
Surface area: 6.21796×1010 km² (121.9 Earths)
Volume: 1.43128×1015 km³ (1321.3 Earths)
Mass: 1.8986×1027 kg (317.8 Earths)
Mean density: 1.326 g/cm³
Equatorial surface gravity: 24.79 m/s² (2.358 g)
Escape velocity: 59.5 km/s
Sidereal rotation period: 9.925 h
Rotation velocity at equator: 12.6 km/s 45,300 km/h
Atmosphere
Surface pressure: 20–200 kPa (cloud layer)
Composition:
89.8±2.0% Hydrogen (H2)
10.2±2.0% Helium
~0.3% Methane
~0.026% Ammonia
~0.003% Hydrogen deuteride (HD)
0.0006% Ethane
0.0004% water

quoting the internal structure section of the wiki entry

"Jupiter may possess a dense, rocky core with a mass of up to twelve times the Earth's total mass; roughly 3% of the total mass. The core region is surrounded by dense metallic hydrogen, which extends outward to about 78% of the radius of the planet. Rain-like droplets of helium and neon precipitate downward through this layer, depleting the abundance of these elements in the upper atmosphere.

Above the layer of metallic hydrogen lies a transparent interior atmosphere of liquid hydrogen and gaseous hydrogen, with the gaseous portion extending downward from the cloud layer to a depth of about 1,000 km. Instead of a clear boundary or surface between these different phases of hydrogen, there may be a smooth gradation from gas to liquid as one descends.

The temperature and pressure inside Jupiter increase steadily toward the core. At the phase transition region where liquid hydrogen (heated beyond its critical point) becomes metallic, it is believed the temperature is 10,000 K and the pressure is 200 GPa. The temperature at the core boundary is estimated to be 36,000 K and the interior pressure is roughly 3,000–4,500 GPa."

Pay close attention to the part about most of the diameter of jupiter being metalic hydrogen that turns out to be important.

Thursday, October 11, 2007

I am really quite interested in what the result of the poll at the top of the page is going to be. I will let the same poll run every couple of days (if I can be trusted to be that consistent) Now what I am really interested in is not so much the result from any particular poll but how the numbers will change with time. Obviously the first few people to find my blog will consist almost entirely if not entirely of people that I know personally. Now of course the collection of people who know me personally is much much smaller than the people who do not. I forsee two possible outcomes for the popularity of this blog. The first and most likely outcome is that this blog will have a celebrity that extends only so far as a few friends of mine. The second case of course is that this blog grows in popularity to a much wider audience before it's growth again stabilizes. Of course in both cases since my group of acquaintances is more or less static then the number of yes responses should grow rapidly at first and then stabilize into a more or less linear pattern of growth. The initial rapid growth is due to the fact that those people I know (and have sent a link to) will rapidly find the site but since the circle of these friends is finite the rapid growth will come to a quick stop. the linear growth would be due to my gaining new acquaintances whom I would direct to the site and due to people I already knew who are just finding the site. What is really interesting though is the number of no responses this represents people who have found my site all on their own or who have been linked to it by people who know me. If my blog is good then I can expect the no column to at first experience little to no growth followed by a period comparatively large growth followed by again little to no growth. Of course if my blog is not so appealing then the number of no's should experience basically null growth indefinitely. The reason for the null growth is more or less obvious, the underlying premise for the s-curve shaped growth is that when people who don't know me find the site they are much more likely to share it with other people in their social sphere who are also likely to not know me. Thus after an initial period of anonymity the site could then break into social groups other than my own thus causing rapid growth of the number of nos. This sharing would also tend to saturate whatever groups it is introduced to and would again stabilize.

Black hole basics part 1

To fill in those of you who may not know a black hole is an object with such a great density of mass that the gravitational acceleration near enough to the center of mass is so great that light is unable to escape. So more or less a black hole is an object whose escape velocity is the speed of light c. Since a massive body would have to have infinite energy to achieve a speed of c there exists a region around a black hole such that once an object has entered that region it cannot again escape it. The surface of this region of no return is called the event horizon of the black hole. When one talks of the "radius" of a black hole one means the radius of the event horizon.

The event horizon is actually always perfectly spherical and so the radius of the event horizon is actually a perfect description of its geometry. The reason the surface of the region of no return is called the event horizon is that events that happen on the outside of it may have an effect on other events that happen farther away but events that happen inside of the event horizon cannot affect anything outside of it. This is because information can only travel at the speed of light and so information can't get transmitted from inside to outside a black hole.

The event horizon of course has no physical substance and is not the surface of a black hole in the sense that a star has a surface. In fact we do not know if there is a material surface of a black hole or not. It is generally considered that because the gravitational acceleration inside the event horizon exceeds the speed of light no other force could be sufficient to overcome it and therefore all matter inside a black hole must collapse to a single point. This point is known as the singularity. However it is possible that because of quantum effects or effects due to string theory there may be a point beyond which it is not possible to compress matter in which case there would be a tiny nugget of unimaginably dense material at the core of each black hole. As a point of reference it is interesting to note that the density of a neutron star is somewhere in the neighborhood of 1017 Kg/m3 and the primary agent against further collapse of neutron stars is in fact the pauli exclusion principle. So quantum effects are already needed to keep neutron stars from collapsing to a singularity.

The model of a black hole that I have been describing up to this point is what is called the non-rotating or Schwarzschild black hole. It is a solution to Einstein's equations for the case where there is no rotation and no charge. both rotating and charged black holes are fascinating critters but for the moment I am just going to keep on ignoring them maybe I will include a post about them later.

You may have heard the phrase that "black holes have no hair" this simply means that black holes have only three independent properties, mass, angular momentum, and charge. They have no hair in the sense that unlike any other macroscopic object they are totally indistinguishable but for those three properties. Since we are looking at non-rotating and non-charged black holes the only item of interest is the radius of the event horizon and its relation to mass. Interestingly enough the radius that classical mechanics suggests for such an object turns out to be the correct radius. This radius of the black hole for the non-rotating case is called the Schwarzschild radius.

If you compress a mass down to close to its Schwarzschild radius it will collapse into a singularity under its own gravity. The equation for the Schwarzschild radius is R = 2GM/c2 , where G is the gravitational constant and c is of course the speed of light. Now for the grand finale of the post we apply this equation to find the approximate Schwarzschild radius for Jupiter as being 2*6*10-11*1027/(9*1018). Which is about 1.3 meters. Now compressing the entirety of jupiter's mass into a volume of only 1.3 meters may seem just a tad on the side of implausible and I would tend to agree. However keep in mind that this is the radius that one would need to put matter in in order to make it collapse under its own gravitational field alone. At any rate though we have our first rough calculations of what it would take to make jupiter a black hole.

Sunday, October 7, 2007

Why Jupiter?

Thanks to what I decided to title the blog I feel I have a certain duty to make the first blog entry at least explain why on earth I would want to blow up Jupiter. To be precise I want to implode Jupiter not explode it. So my title is a bit misleading, but I don't feel that I am being too misleading since I do in fact want to bring an extraordinarily large explosive force to bear on that planet. The idea is to collapse the planet Jupiter into our very own local black hole. At about ~1027 kg Jupiter is about two million times less massive than it would need to be in order to collapse into a black hole by itself. Since Jupiter is the second largest mass in the solar system it is also the second easiest mass to create an artificial black hole from. Although the sun has a much larger mass than Jupiter (~1033 kg) it is somewhat less desirable to convert into a black hole for at least one very good, very obvious reason. Aside from the fact that it is desirable to keep the sun shining down on the earth there is the matter of the sun's internal fusion. The fusion in the sun would resist any collapse and it would be all but impossible to make that fusion cease. Jupiter then is very likely actually the easiest mass to convert into a black hole in the solar system despite its diminutive size. Since the core of Jupiter is believed to be made of heavy elements a resistive fusion force would not be a problem.

I still haven't said though why humanity needs a local black hole. Eventually mankind will need to have a local black hole in order to be able to really test the limits of physical reality. We spend hundreds of billions of dollars on accelerator projects that allow us to reach energies which are still tiny in comparison with the sort of experiments we could do at the surface of a black hole. I propose this as the ultimate accelerator project and as the only way we will ever be able to really experiment with extreme curvature of space-time.