Infinite processes in the real world

A long time ago, in ancient Greece, one of the philosophers asked a simple yet very important question - is matter infinitely divisible? He of course formulated the question in a much more intuitive way: what will happen if you take a stick and break it in half, than take one of the halves and break it in half again and so on. Thinking about this problem, he concluded that at some point we will not be able to continue breaking the stick. According to him, after a finite amount of time we will reach an indivisible component of matter. He named this indivisible component "atom".
As with any new idea, there were those who believed in it and those who concluded that this idea is wrong. Likely for both sides, there were no means to actually check it so they could argue as much as they wanted.

Even though we are much more advanced today we still don't know the answer to this problem. Ironically we have discovered particles which we named atoms only to find out that they can be split apart as well only a few years later. Although, to be really precise, we need to remember that the problem can be formulated as the "atom" being the basic component of a specific type of mater. In other words, one possible understanding of the problem is that it asks to find a "part" that if divided further looses the recognizable properties of the object we started with. If we formulate the problem in this way, then there are indeed such "atoms" - molecules.

At this point you are probably wondering what is this about and how is it connected to infinity. To understand this lets look on a somewhat famous paradox - the Thomson lamp. Consider a lamp with a toggle switch. Flicking the switch once turns the lamp on. Another flick will turn the lamp off. Now suppose a being able to perform the following task: starting a timer, he turns the lamp on. At the end of one minute, he turns it off. At the end of another half minute, he turns it on again. At the end of another quarter of a minute, he turns it off. At the next eighth of a minute, he turns it on again, and he continues thus, flicking the switch each time after waiting exactly one-half the time he waited before flicking it previously. The sum of all these progressively smaller times is exactly two minutes.
So, in the end, is the lamp on or off?

It turns out that there is no clear answer to this problem. While we know the state of the lamp at any time during the process, we cannot tell what is the state at the end. Now lets return to our original problem. Lets suppose for a second that "atoms" don't exist. With this in mind we can take the being from the lamp paradox and instead of it toggling the switch we will make it break sticks in half. Since there are no atoms, the process doesn't end before two minutes pass. But what do we have after two minutes?

In this case it is rather simple to look on the problem mathematically. Lets substitute the stick for the line [0,1]. The whole process can be described then as just a limit of [0,2^(-n)] when n goes to infinity. The limit is a single point, so that would mean that we will get a "particle" with size and mass equaling zero. However, that would suggest that the matter is build from particles with zero mass, and this is a rather bizarre conclusion.
The only possible result we can get from this line of thought is that if such a being actually exists then there are "atoms". However, if there is no such being then we cannot say anything.

While I would like to finish this post with at least a partial solution to the problems I presented, there is no solution as far as I know. There is, however, a funny "solution" to the Thomson lamp paradox. Lets assign numbers to the states of the lamp - 1 and 0. If we do this then the state of the lamp after n steps is: 1-1+1-1+...+(-1)^n.
Therefore, if we take the limit when n goes to infinity, we will get the state of the lamp after two minutes. So lets see what the limit is.

A=1-1+1-1+1-....

1-A=1-1+1-1+1-....=A
2A=1
A=0.5

As you can see, after two minutes the lamp is half on. :)

End of the Semester

Today I went to the last lecture of this semester. As it is somewhat typical with last lectures, the professor talked about interesting problems that are somewhat above the scope of the course. If only those problems didn't tend to be more complex that what can be explained in a 45 minute lecture... Luckily, this is of little importance. While the problems discussed were interesting, I rather spend my time working on staff that is more relevant to me now. With the semester finally over, I now have tests to worry about, but I should also have plenty of time to write new posts. Actually, I have a few posts already in the making, I just need some time to actually finish writing them. With the semester over, I finally have time to do so.

To be honest, this year was for some reason really difficult for me. I never was good with making timetables for myself, so I ended studying till I was too mentally tired to do anything else. While I am pretty sure that I managed to do well in all of my courses, I barely kept up with my activity on the net. Both this blog and my stumble upon blog were not active most of the year. Hopefully next year will go in a more normal fashion.

In other news, I am considering to close my Windows Live account. It is not very useful to me, and I noticed that I am getting a lot of spam from it. Initially I opened it in order to have access to free online storage for my files. However, I cannot say that I am satisfied with the service, and therefore I will likely close this account. To be honest, I sometime think about closing my Facebook account as well, but it is slightly better than Windows live. And what is more important is that I can login to other sites using my Facebook account.

I have also started a little project. About two weeks ago, I got an invite to Dropbox. Basically, it is a file sharing site, but it has two features that make it nearly perfect for my uses. Firstly, Dropbox integrates into the desktop. That is instead of having to upload your files to the site manually, all you need to do is to put the files in a specific folder on your computer and Dropbox will upload them to the web and then sync them with your other computers.
Secondly, and much more importantly for me, Dropbox officially supports Linux and works well on it. I tried to find other similar services, but they all either don't support linux or they worked horrible. This even includes Ubuntu One (at least the version I tried about half a year ago).
The only downside it has is that they only give 2GB to free users. However, it is possible to get more space by inviting others - go to the site if you are interested in details, I am pretty sure that this policy will not last for a long time so there is no point to write much about it.

Right now I am using Dropbox to backup and share some video files (documentaries about dinosaurs and other scientific topics) that I have collected. I never cared much about video quality, so I encode the video files in low quality (all the important details are still there) add subtitles and then upload them. In one case, I managed to compress 3 hours into 300MB. As long as I watch them on the computer display, it is perfectly fine.

Correct math wrong results

In the previous post I mentioned that in some situations even the math we used is correct, the result we arrive at might not be correct or it might not apply in the real world. In this post I intend to discuss three such examples.

The first example is known as the Tompson lamp problem. Imagine the following situation: You switch a lamp on, than after one minute you switch it off. After 30 seconds you switch it on again, and then after 15 seconds you turn it off. We continue like this for two minutes. Now, is the lamp on or off? There is no real mathematical solution to this problem. One proposed solution is to say that the state of the lamp after two minutes is independent of its state before. So for all we know, after two minutes the lamp could have mutated into a pumpkin. Seriously.
Another solution originates from noticing that the behavior of the lamp can be though of as the infinite sum: 1-1+1-1+1-1+.... So if we find the sum we will get the solution. Consider the following:

S=1-1+1-1+1-1+...
1-S=1-(1-1+1-1+1-1+...)=1-1+1-1+1-1+...=S
1-S=S
1=2S
S=0.5

And thus we found the sum. The result is usually interrupted as the lamp being half on. I don't know about you but I never saw a lamp being in that state.
What would happen if we are to do this switching in reality? Thats simple - the switch will break.
As you can see in this case modeling the situation mathematically fails completely.

The second example is an implementation of a theorem to a situation it cannot be applied in. The theorem I am talking about is the Brouwer's fixed point theorem. It is one of many fixed point theorems, which state that for any continuous function f with certain properties there is a point x0 such that f(x0) = x0. Sometime ago I read a statement that according to this theorem, if you take a glass of water and then mix it in anyway, there always will be a small part of the water that didn't change its position. If you are not careful this may seem reasonable. After all, mixing the water is a continues process. Unfortunately, this is simply wrong. The theorem itself is of course correct, but it cannot be used to discribe water in a glass. For this theorem to be used the body it is used on must be continues, but the water is discrete - it is composed from atoms. Because of this the theorem cannot be applied to such a situation, and the result we get by applying it forcefully is wrong.

The third example is known as the Banach–Tarski paradox. It is a theorem in set theoretic geometry which states that a solid ball in 3-dimensional space can be split into a finite number of non-overlapping pieces, which can then be put back together in a different way to yield two identical copies of the original ball. The reassembly process involves only moving the pieces around and rotating them, without changing their shape. Obviously this is not something that is possible to do in reality.
Again, the theorem is perfectly fine. The problem is that it cannot be applied to actual balls. During the proof of the theorem (at least the proof I am familiar with) we get a countable infinity of finite degree polynomials of the form p(sinx)=q. We need to choose x in such a way that for all the polynomials q is not 0. Since any polynomial have a finite number of roots, there is at most a countable infinity of values for x that doesn't give us what we want. Therefore we can always choose a value that will work.
Unfortunately x is the angel of rotation of the ball. If we want to choose a specific x we need firstly to make sure that we can rotate the ball by such an angel. Surprisingly this is not always possible. The reason for this is physical and not mathematical, so I am not going to explain it in detail, but the main idea is that we cannot make "moves too small" in the real world.

Mathematics is often said to be describing the real world. Personally I don't think so. Those and other similar examples have one common trait - correct math that cannot be applied in our world (at least not the way we want it to). But it can be used to describe a world of math.

Windows XP on Ubuntu with qemu

If you were reading this blog for a long time you already know that I ditched Windows and moved to linux years ago. I have been using Linux for about 3 years now and I never regretted moving to it. However, annoyingly enough, there is one tiny thing that I cannot do on linux but I can do on Windows. In the Hebrew University, in order to get the exercises you need to download them from the university site. Most of the site works just fine in Firefox, but for some reason it is impossible to download the exercise with it. It is only possible to do so in IE.

The students have been asking the university to fix the site for the last two years, but without much success. It is of course possible to go to an university computer and to download everything there, but it is inconvenient. I tried installing IE using wine, but it didn't work. So in the end the only option I could think about was to install Windows using qemu. This post is a short how-to that shows how to install and then configure windows under qemu for it to work without doing problems.

The first step is obviously to obtain a windows cd or an iso file. The next is to install all the required packages:

sudo aptitude install qemu kqemu-common kqemu-source samba smbfs

This will install qemu and also samba for sharing files between Windows and Ubuntu. Next is to configure kqemu - it is an accelerator used by qemu:

sudo module-assistant prepare
sudo module-assistant auto-install kqemu
sudo addgroup --system kqemu
sudo adduser $USER kqemu
sudo modprobe kqemu

Now you need to log out for the changes to take effect. After logging in, the next step is to create a disk image. The minimum size is 4GB (because of the SP3 and other updates), but I used 10GB because I want to be able to install programs latter on without worrying about free space. The image will change it size dynamically, so don't worry about throwing away too much space. To create the image type:

qemu-img create -f qcow windows.img 10G

Now we can start the install process. If you have a CD insert it and type:

qemu -localtime -cdrom /dev/cdrom -m 512 -boot d windows.img

I set memory to 512MB but you can enter something else, preferably at least 384. If you want to install from an iso, put the iso in your home directory and type:

qemu -localtime -cdrom cdimagefile.iso -m 512 -boot d windows.img

After doing this, the install process will start. It worth to note that it will go on for longer than a regular install, so you will have to be patient. Once the install is done, you can start using Windows. To run windows you type:

qemu windows.img -localtime -m 512

Optionally you can create a launcher on the panel or the desktop. For this you will need an appropriate icon. I used this one:



As we all know, windows is much less safe than Linux. Luckily qemu has some options that allow us to protect the installation. The first such option is to create an overlay. This will allow you to use windows while saving all the changes in the overlay file and not in the original img file. If you do this and at some point at time Windows becomes corrupt you can just delete the overlay and create a new one without installing Windows from the beginning. To create an overlay type:

qemu-img create -b windows.img -f qcow windows.ovl

To use the overlay you will need to type:

qemu windows.ovl -localtime -m 512

For increased safety you can also use the snapshot mode. In this mode all the changes you make are written to a temporally file which is removed when you close qemu. To use this just add "-snapshot" to the end of the command.
The final step is to make it possible to share files between Windows and Ubuntu. To do this we can use samba. The first step is to create a shared folder. In my case I created a folder named AnatolySharedFiles in my home directory. Now we can setup samba to share this folder:

sudo addgroup samba
sudo adduser $USER samba
sudo aptitude install system-config-samba

Now you can go to System->Administration->Samba, and use it to add the folder you want to share and to add a user password for yourself. Now all that remains is to configure Windows. The first step is to go to My Computer->Properties->Computer name and enter the correct data. Then go to Network places to create the network. The next step is to mount the shared folder as a network drive (this seems to be the best option to me,buy you can access it from network places as well), this is done by clicking on My Computer->Map network drive.

After doing all this you hopefully have a working Windows and I have an easy way to get my math homework...

Tautology and theories

What is a tautology? Simply put it is a statement that is always true. More specifically, it is a statement that is true because of its structure. Usually such a statement is not informative. The easiest way to explain this is by example, so lets look on the following statements:

1. All apples are round.
2. All apples are either round or not round.

I have never seen an apple that wasn't round, so the first statement is a correct one. However it is not a tautology. It is perfectly possible for a square shaped apple to exist, you just need to make it grow inside a box. Therefore this statement is not always true.
The second statement is always true, and therefore a tautology, but it doesn't say anything. That is, if all we know about apples is that an apple is either round or not round we don't know anything about apples.
It doesn't mean that tautologies are useless, they have both use and importance in certain cases.

Lets look on tautologies in a more formalistic way. In the example above I assumed we have an object "apple" and a property "round". However, this is not necessary. All we need to write this example is two symbols: P, Q. If we rewrite the example using this symbols we get:
1. P->Q
2. P->(Q(or)(notQ))

I don't want to explain the notation, if you don't know it you are welcome to use wikipedia.
This notation is far better that the previous one. With this notation we are free to chose the meaning of the symbols. For example we can suppose that the meaning of P is x=7, and the meaning of Q is x+4=0. In this case 1 is usually false (but not always) but 2 is true.
From this we see that the second statement is completely independent from reality. It doesn't matter if we use it to tell something about apples or about mathematics, it will always remain true. This is the true meaning of being a tautology.

Sometime ago I read a post that claimed that all theories are tautologies. From the above it should be obvious that this is wrong. But there is a certain moment that causes this confusion. A theory is basically a collection of statements (theorems) that can be logically concluded from a certain set of prepositions. Those prepositions are divided in two groups - tautologies and axioms. (We don't really need to include tautologies, but because they are always true they get included automatically). Axioms are not tautologies. They are just "rules" that we choose by ourselves. In physics, those rules are based on reality and experiment, but this doesn't have to be the case. For example - the parallel postulate of Euclid. It seems natural to us to think that it is correct. But it is possible to built a geometry without it. Naturally in such a geometry all the statement that were derived from the parallel postulate are no longer correct (they might be correct, but not necessarily).

To sum up this discussion, a theory is correct as long as the axioms from which it was built are correct. However, if we try to apply it to a "world" in which the axioms are not true, the theory will not work. By the way, this is a major problem for physics and economics. They can build wonderful theory only to find out that the world we live in is different from their initial assumptions. In mathematics this is not a problem because there is no desire to describe our world, but to built a mathematical structure.
This also makes easy to explain the statement that a theorem once proved is true forever - a theorem is dependent only on its own conditions, so as long as they are satisfied the theorem will always be true.

Amazingly funny

I just read a post on Astroengine about a recent development concerning LHC. I rare write about post written on other blogs, but this time I felt is was just unfair not to share this. The main idea is that some person named Tia Aumiller decieded to open a group called: "People for the Ethical Treatment of Hadrons" (PETH).

It is not 1 July today, but I really hope that this is not a prank... Anyway, this organization has already protested in front of CERN. Their claims are:

“You’ve got these subatomic particles accelerated at great speeds for the sole purpose of being destroyed. No one thinks of the ethical implications of this. There’s a limited supply of hadrons in the universe. Do we just want to go around destroying them? What if we run out? What if the hadrons can feel pain? Will we look back at this hundreds of years from now and regret it? Kinda like we do with the killing of bacteria with antibiotics now.”

It is just unbelievable and extremely hilarious. I really have no idea how crazy somebody has to be to really believe in this.

Update: After checking this a bit, this story turned out to be fake. However it is still very funny so I am not removing this post from my blog. I guess I should learn a lesson from this - no posting of things that look fake, even if they are funny...

Warp Drive Engine

I didn't intend to write any more about faster than light travel, but I was sent a link which is too good not to write about. I am not going to comment on the science involved, I am not qualified to talk about anything that concerns 11 dimensions. However, this article raises an interesting point which I didn't mention in my posts about faster than light travel. If you read those post, you know that I started this topic with a simple proof of the impossibility of faster than light travel. I also showed that there are exceptions to this rule because quantum mechanics and relativity doesn't play well together. But what about large objects? Is there a way for them to somehow escape relativity?

In the article I linked, you can read about a simple (but nearly impossible to do) idea - expanding the space behind the object and shrinking it in front of the object. Thus you will create a bubble that slides in space. Inside the bubble the object will move slower than light, but the bubble itself can move faster, because it is not even matter but space itself. This is called a wrap drive.
This sounds possible, but it is easy to see that the method I used to show to show the impossibility of time travel, still works for this example.

However, in this particular case my prove doesn't apply. The reason for this is very simple - while it is bot obvious the proof is build on a simplistic assumption that the universe is the same on large scales. Unfortunately, this is not true. This is only a simplification used for ease of calculations, although it is very close to being correct. If you allow universe to be extended and contracted (and this is exactly what ruins this assumption), there is nothing that forbids faster than light travel.

To finish this post - a little joke:
An experimental physicist finished running a very complex experiment. After plotting the data on a graph, he got to the conclusion that he doesn't understand why the graph looks the way it does. So he went to a theoretical physicist, showed him graph and asked to explain the very high peak in one of the points. The theoretical physicist looked on the graph for a second and said "Oh, there is a perfectly good reason for this peak". And started talking. During the explanation the experimental physicist suddenly looked on the graph and said" Wait a a second it is with the wrong side up". The theoretical physicist looked on it, and said "Oh, there is a perfectly good reason why there is this very law value in this point" And started talking.....

Einstein paradox

This post is a little follow up for the series of posts I wrote about faster than light travel. In one of the posts on this subject I brought a simple example of how in Quantum mechanics faster than light travel is possible, despite it being impossible according to relativity. The example I talked about in that post was very simple, and it was easy to explain why this is indeed what happens. But this example talked only about faster than light travel on a very small scale. In this post I want to talk about another example, which is far more complex but it shows that faster than light travel is possible also on large, even cosmic distances.

Lets consider the following situation. Suppose you have two balls, one is pink and the other is green. However, the color property of the balls is quantum - both of the colors are in superposition, so both of the balls are pink and green in the same time. But if you will measure one of them, the superposition will collapse to one of the options. What is interesting is that if you will measure one, you will cause the other one also to collapse, because you now know its color as well so it is no longer in superposition.
There are no such balls in the real world, however it is possible to create particles with all the required properties. I don't want to talk about a specific example in this post so we will agree that the balls stand for some object that have a quantum property which we will call color.

Now, lets suppose that you create two such ball in the laboratory and give one of these balls to your friend. Lets label this ball A. You friend happens to be an astronaut and he flies to the moon with this ball. When he gets there, you measure your ball (B) and discovery that it is pink. This measurement causes your ball to collapse - it is no longer in superposition of ping and green, but it also causes the second ball to collapse, in the exact same instance.But even light travel to the moon in over a second. So, something changed in the second ball, A, without a reason to this being in its "Cone of light". This again means that the information of the measurement traveled faster than the speed of light.

This is known as the Einstein paradox. He originally presented it in an attempt to prove that quantum mechanics is incomplete. He claimed that it is not correct that quantum processes are probabilistic - "God doesn't play dice". In this mind experiment there is nothing impossible from the position of quantum mechanics, but allowing faster than light travel we allow time travel, and give place to a lot of other paradoxes. He offered a solution, to add a unknown property lambda which we cannot yet measure but that decides the outcome of the measurement. This works because quantum mechanics says that from all the properties of the object we now about we cannot deduct its state (in our example the ball color) so until we measure it, the object is in superposition. But if we allow for such lambda to exist we get that there is no superposition the balls are always the same color - there is no longer probability involved, all is determent from the beginning.

Interestingly, Einstein was wrong. It took some time but eventually a test that checks if such lambda exist was performed. The test was an experiment that returned a value, we will cal it S. If S is less or equal to two, then there is lambda. If S is bigger there might be lambda, but faster than light travel (in the case of the Einshtein paradox) is possible. To be even more specific, if quantum mechanics is correct . However, in physics to show that something is equal exactly is nearly impossible, so the main point here is to check if S is less than two or not. This experiment was performed a lot of times. In the beginning the equipment wasn't sensitive enough, but after a few decades the result was that . Since this is large than two, the case was closed, faster than light travel is possible. It is still unknown if there is lambda. It turned out that we can design a theory with lambda and without it, and they both will work always. They both manage to explain all the results of all the experiments conducted until now.

Relativity and quantum mechanics

In my previous post on this topic I have shown that while faster that light travel is impossible, it is possible for the electrons to move from one energy level to another in zero time. The distance it travels is like the distance between planets on our scale, however the time is zero.

The reason for such a result is very simple - relativity and quantum mechanics cannot be used together. It is not possible to apply relativity where one should use quantum mechanics. When we discussed individual electrons the relativity theory simply stopped working. The results that were correct for large scale become wrong on this scale.

But why is it so? There is after all a general agreement that a theory that works only under specific conditions should transform gradually to a different theory when the conditions it requires are changed. This part probably sounds a bit confusing, so here is a simple example:

Photo by wili_hybrid

A long time before Einstein, people noticed that for two system that are moving with a constant speed compared to each other, the system of coordinates has to be transformed when you move from one system to another. If, for example, you are on a train that moves with 50 kph east relative to the earth and you see someone who is sitting on the field outside, than from his system of coordinates you are moving with speed 50 kph east, but from your point of view he is moving with the same speed to the west. The transformation used to move from one system to another is the Galileo transformation. If your coordinates in one system are (x,y,z,t) than your coordinates in a system moving away from you with a constant speed v are (x',y',z',t'). If at the moment t=o the both observers where in the same place and the movement is only on the x axis we get that:

x'=x-vt
y'=y
z'=z
t'=t

However, according to relativity this is not correct when v is big enough. In relativity we use Laplace transformation instead of Galileo's. Under the same condition we will get:

y'=y
z'=z

The formulas look very different. Partially this is because the of units used. However, if we will go to the limit were c is significantly large than v (that is c is regarded as infinite), they will turn into the Galileo transformation. It is very easy to see - the only impostarnt part is to notice that the units need to be balanced, after this it is trivial. We can say therefore that relativity turns into classical mechanics when the speeds are low in comparison to the speed of light.

However, this doesn't happen with quantum mechanics. It is divided from relativity by a scale barrier, and when this barrier is approached the two theories start to contradict each other. A lot of work have been done to solve this problem. The main approach is to try to unify all the basic forces. Those forces are - Electricity, Magnetism, Gravitation, Strong and Weak. The first two are already unified for a lot of time. The weak force also can be unified with them. I also heard that the strong force was unified with the weak force, but I don't know the details. Gravitation is a problem however. For the other forces particle carriers where found - but not for gravitation. In fact, the question what gravitation really is, is still without answer. It is a mystery waiting to be solved...

A bit of trivia - It is a surprising fact that Einstein contributed a lot to both of these theories, but while he helped quantum mechanics to take roots he wasn't happy with the result. He was the one who proposed the duality of the photon, and he was the one who helped to promote the understanding that all particles have this duality.

I ended my previous post with a question - Was Einstein wrong? The answer should be clear, but I will say it anyway. He wasn't wrong. It is simply that by going to this very small scale I left the domain of relativity and there the rules are different.

You probably noticed that this post raises a very interesting question. Since faster than light travel is equivalent to time travel as I have shown, does the fact that the electron can move such great distance (on his scale) in zero time means time travel is possible? Nope. There is no time travel in this case. The reason for this is simple, but it requires getting used to. The electron belongs to the "quantum world". We can think about this world as being separated from our world by a "shield". This shield is called The Heisenberg Uncertainty principal. What it says is very simple - the uncertainty in the location and energy are always bigger than some constant number. It means that we simply cannot see too well what is going on in this "quantum world".
In fact the way I used to show that faster than light travel is equivalent to time travel depends on accurate measuring of distance and time. Since we cannot do this with the electrons, even this general result just doesn't apply.

By the way, if you want to be remembered forever in the history of science, finding a way to unify electricity and gravitation will surely achieve this goal...

Faster than light

I finished my post about time travel paradoxes with a promise to write about one very specific example in which faster than light travel (which is equivalent to time travel) is possible. I am going to write it in a slightly weird way - firstly I intend to explain why this particular example is not possible, and then to show that it happens nevertheless.


We all know how an atom looks like - it is just a little ball (nucleus) surrounded by a cloud of electrons. Most of the atom is empty space. The atom has a very interesting property: the electrons that orbit the nucleus can be only on a finite number of orbits = energy levels. I am not going to talk about why it is so, but what is important to note is that the distance between any two energy levels is very large. In fact, the atom can be viewed as a solar system in miniature - the distances between the orbits are comparable to the distance between planets. The electrons decide on which energy level to be according to the energy they have. The more energy an electron has the higher (further from the nucleus) it will be..

Thee is a simple way to cause an electron to jump from orbit to orbit - we just need to hit it with a photon. If the energy the photon has is the energy needed for the electron to move to a higher orbit, it will absorb the photon and move to this higher orbit. After discussing this we can go to the main question I will deal with in this post:

How much time does it take for the electron to move from one energy level to another?

Answer: Zero.

To explain this, lets firstly look on why this time must be more than zero. In the following paragraph I will develop a mathematical formula - if you don't understand what I am writing just skip it and go to the next paragraph.

Lets look on a regular piece of wire. If we will connect it to a battery (if you want this battery to be of any use lately don't do it) a current will flow in it. How much current? The usual way to define current is to look on the area of a cut of the wire = A. The current than is the amount of charge that passes in infinitesimal time=dt. To define it precisely we need first to define the density of charge flow. The density of charge flow (J) is the number of free electrons in the wire multiplied by their speed and by their charge: J=nvq. Note - J is a vector. The current is than defined as:

I=

dA is an area vector of an infinitesimal part of the cut area A. If, as it is with regular wire J=const we get that I is simply: I=JA. The formula with dA is good for any object in which a current can flow. Because of this we can use it in a slightly different way. Lets think about a general bounded 3d object. This formula is good for it - but instead of saying what the current is, it tells what is the rate of change in the charge of this object.
What the formula says exactly? It says that the rate of change of the charge is an integral on the charge flow density. Thus, the charge (electrons) that left this object is exactly the charge that passed the surface of this object - the integral is basically a flow throw the surface area of our object. This means that electrons cannot just jump outside - they must cross the surface somewhere. This conclusion seems obvious, but it isn't. Without this formula, having only the law that the total charge is conserved, we can say that if the electron jumped outside the total charge in the space is the same - so it is possible. But this simple formula says no. However, it doesn't say that it is impossible for the electrons to jump inside our object. To show that this is also impossible we need another formula, so lets develop it.

What is Q? It is the total charge of our 3d object. Therefore it can be expressed as:

In this integral rho is the charge density (it is not J). If we will put this expression into the previous formula, and also use the Gauss theorem to change the right integral in our formula from JdA to divJdV we will get:

The both integrals are dV, so we get that:

It is usually written in this form:

The plus sign is a convention - it doesn't really matter, because the positive and negative flow direction is defined according to what is easier to work with.
What does this new formula says? It says one important thing - the change in the density of charge in an infinitesimal is equal to minus the divergence of J. This is exactly what we were looking for - now even the electrons inside our 3d object cannot jump. They all must flow passing all the midpoints.
Therefore it is impossible for the electron to jump from one energy level to another in zero time.

But is this correct? No. The math I used is correct, but there is one single thing that is left out - quantum mechanics. The formulas I developed don't take it into account. They are still correct, but they cannot be used on very small scales.

It is now time to explain what happens with out electron. When the electron is being hit by a photon, it consumes the photon and gains its energy. This energy is exactly the energy needed for it to jump to the next energy level (otherwise the photon will not be consumed). The collusion between the photon and the electrons happens in one instance - it is not a process. What this means is that the electron gets energy that corresponds to the next energy level. Therefore it must be in a higher energy level. It cannot just stay. If there would be no photons, than the electron would consume the light energy gradually and would "rise" to the next level. But the energy is transfered in a collusion.
Mathematically, the potential energy is space must be continues. Because of this it is not possible for the electron to have a higher energy while being in a place that corresponds to a lower energy. The reverse is also true - when the electron descends to a lower energy level it releases a photon and jumps down - in zero time.

So, was Einstein wrong in saying that faster than light travel is impossible? Were I wrong when I said in the previous posts that faster than light communication (or travel, it doesn't matter) is equivalent to time travel? This I will answer in the next post.

Time travel paradoxes

I recently wrote a post about why faster than light communication is impossible. I finished this post showing that such communication would be equivalent to time travel, and it is logical to assume that time travel is impossible.
However, is this indeed true? Time travel is often talked about in science fiction, but is there a reason to think that it is possible?

Photo by jonrawlinson

Firstly we need to distinguish between two types of time travel - to the past and to the future. We all know that it is possible to go to the future. We all do it, it is called living. It is also possible to travel to the future with a "faster speed" than normal life. All you need for this is to accelerate yourself to a high enough speed, and according to relativity you will travel to the future - when you will return to the place you started your travel you will see that your watch is late. How late depends on your speed, it might be a second or 1000 years. This effect is very real, we even have to take it into consideration when we communicate with satellites.

Traveling back to time is a different story. Unlike traveling to the future it was never done. So the only thing we can do is to discuss different theories. There are three main theories about traveling to the past:
1. Time travel is not possible - there are two versions: either there is no way or it will destroy the universe in the process.
2. Time travel is possible but the past cannot be changed.
3. Time travel is possible, but it will destroy part of the universe.

Lets look on all of the three theories:

1. Time travel is not possible:
There are two versions of this theory. The first one comes from a literal understanding of time travel. According to it in order to travel back one second in time, you need somehow to return the whole universe to the exact some state it was one second ago. It means that you need to place every single particle exactly at the same place. However this is clearly an impossible task.
The second version of this theory is based on time travel paradoxes. The paradox I will talk about here is called the grandfather paradox: Lets suppose that time travel is possible. Lets also suppose that someone (Mr X) traveled back in time. While he was in the past he 9accidently) killed his grandfather. In doing so he preventing himself from being born in the first place. But if he wasn't born how could he travel back in time?
There is also a more general version of this paradox - by traveling back in time you change the world past, so in the very moment you will get to the past the world you come from (the future) will no longer exist. And therefore, you never traveled back in time.
The conclusion from this is that if time travel is possible, you will destroy the whole universe by traveling back.

2. The past cannot be changed:
This theory is an answer to the question arising from the previous one. In the grandfather paradox, we assumed that it was possible for Mr X to kill his grandfather and to prevent his own birth. But what if this is not true? What if there is a law that not allows people to influence the past? In this particular example, we can suppose that Mr X will be stopped by police just in the right moment, or it would turn out that the men he killed wasn't his grandfather at all. For the more general version of this paradox, we can assume that either the results of the activities of time travelers just slowly vanish so they don't affect the future in any way, or that there is a fixed time line in which time travel appears together will all other things and nothing can be changed.
This theory allows for time travel and solves the paradoxes I presented, but there is a problem with it. Lets do the following thought experiment: Suppose you have a time machine, and a laser that shoots a bit of light into the time machine. The machine send the light back in time, so it goes out of it on the opposite side and two minutes before the laser was fired. On the wall after the time machine there is a detector that when hit by the laser been will send a signal to put a barrier between the laser and the time machine. It looks like this:Now if the time machine works, the laser will prevent itself from firing, but this would mean that the detector didn't close the barrier so the laser worked - and this is a contradiction. Note that it doesn't matter how the time machine works, and to how long ago the light pulse is sent.
If, as the theory says, we will assume that somehow "it all worked" it follows that something is broken - if for example the detector is broken, no paradox will be created. However, this is a very simple system. The only thing that is likely to always malfunction is the time machine itself. This means that the time machine doesn't work, and therefore time travel is impossible.

3. Local destruction:
This one is the attempt to unite the previous two theories. Basically it says that time travel is possible but because of the paradoxes described above, it will destroy the universe. However, the universe is a very large thing. So only a small part of it will be effected. Time travel will create a "wave of destruction" which will move over some finite distance, destroying everything. As it moves it will slow down and become less distractive, so after some finite distance it will just stop. Beyond this distance (this is, beyond a sphere with the time machine in the center), the universe will remain as it was. Inside this sphere however nothing will exist - a singularity will be formed.
From this is should be obvious that this theory also doesn't allow for time travel - because it can be used only as a weapon, and it is not possible to return or to do anything.

Conclusion:
Time travel is not possible, and therefore faster than light communication is also impossible.
But - not always. In the next post about this topic, I will write about some very specific situations in which faster than light travel and time travel are possible.

Why faster than light communication is impossible

Lets consider the following situation - You are 4 light years away from the Earth, on a distant planet X. While you there aliens land on that planet and capture you. They make you tell them from where you came, and tell you that they will go and destroy Earth. You manage to escape, and get to your spaceship. You cannot stop them, and they don't want to look for you - they just go to Earth.
Lets suppose that:
1. Their spaceship can fly at a speed lower than light, but faster than yours spaceship.
2. If the people on Earth are informed in advance about these aliens coming, they will likely be able to defend themselves.
3. You have a device that allows you to send message to Earth that will travel there faster than light.
4. You use the device and send the message in the exact same moment that the aliens leave that planet.

Now, after we have all the information, lets see what will happen.
From the view point of out story hero, he send the message and now goes home.
From the view point of Earth the message is received, and they prepare to fight the aliens.
But what the aliens see?

Lets denote the starting point (the planet X) and the starting time (sending of the signal) by (0,0) respectively, and the end point (Earth) and end time (signal received) by (x,t).
The speed of the alien spaceship is v<1 c="1)" style="text-align: center;">x'=

t'=

For point (0,0) we get (0,0), which is not surprising. For point (x,t) we get ( ,).
Firstly lets suppose that t=0. This means that the message went to Earth in zero time. We will get: (). The minus sign means that this event happened before t=o. Thus the signal was received before it was sent. This is clearly impossible - unless a time travel is involved.

Now, what happens if the message is not instant but is still faster than light? In this case t=x/(1+h) - time is distance divided by speed - where h is a positive number (remember I am working with c=1).
Lets look what is the condition for t'>0, for any v (we must find condition for any v because we don't know what is the speed of the alien spaceship):

>0

We can divide:

>0

And finally we get:

1>v+vh

Rearrange:

1-v>vh

If we will choose v=1-<1>

>(1-)h

If we will now take the limit when epsilon approaches zero, we will get that h=o. This means that it is not possible to send messages faster than light, because otherwise there are always be an observer for whom the order of the events changes - which means time travel.
Note that the way the signal is send is unimportant. The only thing I used is a spaceship which is totally unrelated to the way the device works.

Strictly speaking, this doesn't prove that it is impossible to comminicate faster than light - but the only way to do this involves time travel. Therefore it is logical to assume that it is simply impossible for anything to travel faster than light, including information.

Update: Read part two of this post - Time travel paradoxes.

Two amazing videos

I just stumbled on two excellent videos. The first one is about the Briggs-Rauscher oscillating reaction:
From Wikipedia: "The Briggs-Rauscher oscillating reaction is one of a small number of known oscillating chemical reactions. It is especially well suited for demonstration purposes because of its visually striking color changes: the freshly prepared colorless solution slowly turns an amber color, suddenly changing to a very dark blue. This slowly fades to colorless and the process repeats, about ten times in the most popular formulation, before ending as a dark blue liquid smelling strongly of iodine."

You can watch the video on GoogleVideo or on StumbleUpon Video.

The second video is a complete movie - Life after people. If you ever wondered how the world will look like if the human race will just disappear in a second, don't miss this movie. You can watch it on GoogleVideo. It also includes a lot of scenes of collapsing buildings...

The Eiffel tower - in what was the center of Paris...

I wanted to watch this one for a long time,. and I glad I finally did. It documents the effect of time and the ability of nature to adapt extremely well, in my opinion.

A summer place

Our world is a unique place... even if there are millions of other planets with intelligent lifeforms on them, the planet we live on is still unique - because any other place will be different. We have a lot of conflicts and difficulties but there is also rest and beauty of nature. The video below was made by my friend CRISSANCA67. For me the important part in it is the images - I never liked music and I don't listen to it. The images however manage to tell an interesting story - they manage to capture a lot of unique moments.

In other news I clearly managed to find myself a lot of problems. I decided to study History of Mathematics this semester, hoping that it will be an easy and simple course. Unfortunately, it turned out that I am supposed to do a lot of extra reading for the course. I have to read the whole History of Mathematics by Boyer. Normally this wouldn't be a problem, but it is a 700 pages book and I have to read it in the library. The book is well written, and it is interesting to read - if you happen to take a course in this subject it is a good choice. It begins with what is known about mathematics in ancient Egypt, then jumps to Mesopotamia and Greece. I thought about buying it, but it would mean ordering it from Amazon or eBay, which even with its relatively cheap price I don't want to do. I managed to get a copy of History of Mathematics by David M. Burton, but since the exam will be based on Boyer there is not much reason to read Burton. Hopefully I will manage to read Boyer enough each week. I read it for only about an hour last week because I had to go to a restaurant with some friends of mine, whom I didn't see for a long time... It seems that I will have more time this week. I should probably try to read half of it by the end of next week.

A bit of news

After experiencing a rather unpleasant plan crash, I resolved to study more and to procrastinate less. Gladly I didn't stop on this. I decided to sit in University from 8:00 to 4:00 everyday. The extra hours should be enough for me to do a significant part of my homework during the week. No more doing-it-at-the-weekend mentality.
Overall it went well, although not without surprises. On Monday I came to the library at about 8:20 only to be told, with a rather surprised face, that the library opens at 9:00. That was a pity.. I went to an empty classroom and studied there instead, so not too much time was lost.

In other news, I am getting a lot of traffic to my posts about Linux. I am happy to get any visitors at this point, but it is a bit disappointing to see that while people come to this blog there is no reason for them to return. I will blog about Linux in the future, but it will never be the central theme of this blog.

I have also installed Firefox RC1 today. Almost all of my extensions are incompatible with it, but after using Nightly tester tools they all agreed to work. I failed to notice anything different, so there is not much to talk about now.

The horror of LHC

At this point almost everybody knows what LHC is. There is more than enough media hype around it, and especially around its "danger to the whole planet". There is even a lawsuit about it already... While it is nice that people who are far from science are interested in this topic, it is a bit annoying when people try to give advice without even understanding the subject they are speaking about. Unfortunately, this is exactly the case with the LHC.

According to the media, there are two ways in which LHC can destroy our planet: by creating micro black holes (that would tear the planet apart) or by creating a so called "strange matter" capable of converting our planet into a "strange star". I am not going to attack the people who are behind this ideas, but I do think that this ideas don't have a sufficient logical basis.

Lets examine both of the ideas.

Micro black holes: It is not yet even an accepted theory that they may be created in the conditions provided by LHC. It would be wrong to disregard it because of this, but even if such holes will indeed form there is no reason to be afraid from them.
Black holes are dangerous because of their gravitational pull. This pull is so strong that even light cannot escape the event horizon of a black hole. But why black holes have such a strong gravitational pull? In order to have gravity you need mass, the bigger the mass the more force you will get. The cosmic black holes we often hear about have the mass of at least 3 suns (this is the theoretical minimum, usually they are much large). LHC is not capable of creating such a cosmic black hole - it would require a totally insane amount of energy.
Micro black holes are very different from the cosmic black holes. Their mass is very small, and they also don't light escape their event horizon. How they manage to do this if their mass is very small? The answer is very simple. A black hole must satisfy the following equation (more precisely it should be less than and not equality) :

0.5c^2=GM/R

R - radious of the black hole. M - mass of the black hole. G - gravitational constant. c - speed of light.
Any object that satisfies this equation is a black hole. G and c are constants, M is the mass of the micro black hole created by LHC and is therefore also unchangeable. So the only way to create a block hole is by making the radios smaller. The micro black holes are nothing more than a very, very well compressed matter.
LHC is capable of very high energies but they are very small when compared to mass energy.. The energy needed to produce even 1 kg of matter is much more that what is LHC capable of.
Even if a micro black hole would be produce this way it would have no impact on the world around. More precisely, the effect of such a block hole on the objects around it would be exactly the same effect its mass was causing before it was compressed. Noticeable difference would be visible only a very tiny scales - radius of an atom for example. Creation of such a black hole, if it is possible to to create a all, would in no way harm the world - unless it will start growing.
It is very easy to prove that this will not happen, but it is more difficult to understand. The reason is the same as with the "strange matter" .

Strange matter : First of all it is a totally theoretical object. It was never seen, not even in a lab. This is not enough to say that it doesn't exist, but it is enough to say that LHC will not produce any. This is so because LHC energy is much lower than Cosmic rays energy. If it is possible for LHC to destroy the earth by producing micro black holes or strange matter, so it is for these rays. Since we are still not destroyed, there is no reason to worry.

Visualizing large numbers

This is a very hard thing to do. While it is easy to think about small numbers it is very difficult to imagine large numbers. It is largely because they seem to be totally unrealistic and not related to reality. For me it is sometimes difficult to visualize even not very large numbers - 100 for example. And visualizing say 10^50 is nearly impossible for everyone.

A bit of trivia: what is the largest named number?

Surprisingly it is a very well known one - googolplex. It is .
Why I said it is well known? Well the Google main office complex is named after it. So it is a very well known number (for such a huge and totally not connected to reality number) .
At this point you are probably wondering why I said that it is totally not connected to reality, that is, there is nothing in the physical world that it describes. After all the universe is so huge. So why wouldn't it contain a googolplex of something? However it turns out that there is only about atoms in the whole universe. This is almost zero compared to the googolplex!!

When I first learned that there is only atoms I actually said: "and this is all? Such a small number... " I said so because I couldn't visualize it. It had no meaning for me, and I new that there were numbers much bigger.

Lets see another example. Suppose we take a sheet of paper 0.1 mm thick and we will cut it in half, stack it and then cut in half again. We will repeat this 100 times.
In the end we will have a stack of paper sheets. Since the stack height is representing the number, it should help us visualize it, right? Well lets see how high will the stack be.

=1,267,650,600,228,229,401,496,703,205,376

This is in 0.1 mm. In light years it is about 13.4 billion light years. This is close to the radius of the whole universe...

The problem with visualization of such numbers should be rather obvious now. Nothing we have in our world can help us visualize them. But what about smaller numbers - 10^6 for example?
Such numbers are possible to visualize using small objects. But it is still difficult.
For example you can visit this site to see how a 10^18 pennies looks like.

For most of us this is not an issue. We don't use such numbers were often, and we are not asked to visualize them. However, there is an interesting underlaying fact in all this. Those numbers can be written, we can use them in equations (I see no reason why but it is possible) but we have no use for them. They don't describe anything in our world. Then why do they exist? And do they exist at all, if they are useless? As I already said, I believe that math is discovered and that it has a "life of its own". While the existence of such numbers is not a proof to this view, it is still something to think about - if most of the numbers that we have are of no use to us then why we invented them, if they were invented by us?

Strange theories

Recently I stumbled on some strange and surprising theories. Personally I like a lot to read about theories or facts that defines normal logic. However, while I do agree that the world is a strange place I think that it follows some logic. Perhaps we cannot see this logic right now with the tools we have but, it is still there. And if it is there it can be found. Eventually.

The first theory is a very straight forward one. It simply asks the question - is there an opposite to absolute zero? Now we all know that absolute zero is -273 Celsius. At this temperature there is no movement at all. But is there are a maximum temperature?
The question may seem a bit strange, but well, the answer is even strange.
While nobody knows exactly what it is, there are 4 contenders. The first one is plank temperature.
This is the temperature believed to be at the beginning of our universe. It is about 10^32 kelvin. the other two contenders are 10^30 and 10^17.
However the last contender is a real surprise - it is 0 kelvin, that is -273 C. And no this is not a typo. There are actually people who believe that absolute zero can also be the maximum possible temperature. While it sounds shocking the underlaying idea as I see it, is very simple: we don't know what happens at absolute zero. The laws of physics we know stop working there. The some think happens when we reach plank temperature. Thus going both ways we get to the same situation.
You can read about it more here.

The second theory is .. well it is barely a theory actually. It is more like a mind game. Before saying what it is about we will need some background:
While we never met with an alien civilization, many speculations about this subject exists. One of such speculations is an attempt to rank them according to their development level. This whole attempt is based mostly on our current understanding of physics so it may be right but it also may be completely wrong.
According to the current rank system there are four ranks of civilizations.
A rank one civilization uses all the energy of its planet. A rank two uses all the energy of its star. And rank three - of the whole galaxy.
As you can see we are not even rank one civilization. We only recently started to use wind and solar energy and such use is hardly widespread. In fact electricity is not widespread enough yet. There are areas on earth that are almost totally untouched by civilization.
Now when we now this we can discuss the theory I mentioned. We as a civilization are in constant threat of a catastrophe capable of destroying all of us. An ice age or an asteroid, and we will disappear. A rank one civilization is mush more stable. A change in weather cannot destroy it. With such energy at its disposal it should be capable of controlling the weather. But still even such civilization can be destroyed by an asteroid or another sufficiently great catastrophe. A rank two civilization can be destroyed only by a supernova. Rank three has colonized all the galaxy so even this is not a threat for it. But what if the universe has an end? This is, what if the universe itself will be destroyed?
The theory I mentioned stays that a type three civilization should have enough energy and knowledge to create a new universe for itself. And thus, such a civilization is truly immortal. Perhaps you don't see anything weird in this line of thought. But to me the idea of creating a new universe seems strange. It is one thing when such claims are made from an religious perspective. But here this is not the case.

The third one is probably the most bizarre. It says that time is slowing down and will stop at some point. It is offered as an alternative to dark energy. At lest the theory claims that our planet will be long gone by then...
You can read about it here.