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. :)

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.

Limits of applicability

A question: In physics, the world is considered to be continues. We can see this for example in the fact that we use integrals instead of finite sums in mechanics. However, we know that the objects in our world are made of atoms, which are in turn made of more elementary particles so they cannot be considered continues in a mathematical meaning. So can we apply mathematical result that are based on the assumption that the world is continues, to solve physical problems?

Since as I already said we use integrals in mechanics, it should be obvious that we can do this. But this is not true in all cases. For example lets consider the following theorem:

For any compact and convex set K in R^n and any continues function F from K to K, there is a point x in K such that F(x)=x.

This theorem is known as the general version of the Brouwer Fixed Point Theorem.

It should be obvious that a glass of water can be viewed as a compact and convex set in R^3. Also mixing the water using a spoon can be considered a continues function. Therefore, according to the theorem there is a point that didn't move. But this is obviously wrong. This is because the theorem gives us a point, but that point doesn't have to correspond to a particle and therefore saying that it didn't move is meaningless.

Why can we consider the world to be continues in one case and not the other? The reason for this is size. When we want to solve a problem in our scale (that is, we are dealing with objects that we can see and with results that can be observed directly), we can safely assume that the world is continues and use the corresponding tools. But this is only because the atoms are so small that for us there is no practical difference if we assume that the objects are continues.
But when like in the example with the water, the result we want to calculate (in that case a point) is not something we can see with our own eyes (we cannot see if an atom moved or not) assuming that the world is continues is wrong.

The above classification is very basic and inaccurate but it gives a general understating of the situation. In the next post, I plan to continue with this topic by discussing applicability of theories to reality.

Sudoku solver and some other problems

I got a comment today on my post Algorithm for solving Sudoku puzzles with a link to a program for solving sudoku online. So if you have problems with a particular hard puzzle, give it a try. The program is called Sudoku solver.

I also stumbled on an excellent collection of simple physic problems. You can find the collection here. Unfortunately there are no solution to the problems there, but they seems all to be of the type that requires understanding of physics and not knowledge of math and formulas. Since I am trying to write a math blog, it would be pointless for me to write solutions to these problems. However, I want to solve the first problem:

You are given two identical steel balls of radius 5 cm. One ball is resting on a table, the other ball is hanging from a thin string. Both balls are heated (e.g., with a blow torch) until their radii have increased to the same value of 5.01 cm. Which ball absorbed more heat and why?

The answer - the ball that lays on the table will absorb more heat. The reason to this is that the heat can escape better from the ball when the ball touches another object, in this case the table. When the ball is suspended in the air, the heat also escapes but the rate of the escape is lower. This is due to the fact that air is much less dense that a table.
Since more heat escapes, more hear will be required to heat the ball on the table.

Running in the rain

I was sent a letter yesterday which asked me to look and comment on the last claim on this page. Since that page might not remain forever, I am copying the paragraph in question:

You get less wet by running in the rain. Actual mathematical equations devoted to this popular question have suggested it is true, though not for the simple reasons you might think. Complexities include factoring in the number of rain drops hitting the walker’s head versus smacking the runner’s chest.

Well, lets do the calculations. While this sound like a complex problem it is actually a simple case of Galilean relativity, so it is easy to calculate.
The first step in dealing with a problem like this is to formulate it correctly. As I understand this problem, we need to prove or disprove the statement that if you stand still under the rain for some time T, you will be more wet than if you were running for the same time T.
The second step is to simplify the problem. In this case the simplification is to assume that the rain is uniform. That is, the amount of rain drops per square cm is the same in all the are we are dealing with, and there are no sudden changes in the wind.
Now, what we want to calculate is how wet will a person become. Clearly this depends on many factors - the size of the drops, the body area of the person, the speed of the rain drops etc. Because of the uniformity assumption we made, we can say that all the factors except for the speed of the rain are fixed - they don't change with time, and they don't change if the person is moving. We can now define the "wetness" (B) as a simple multiplication between the matrix A and a vector v. "A" will be the constant which we get from the problem condition. Since we are in 3d space A is a diagonal matrix with A1,A2,A3 (the corresponding constant for each one of the three directions) on the diagonal. "v" is the speed of the rain ( it is a vector v=(v1,v2,v3)). The multiplication result, B, is also a vector. The size of B multiplied by the time T is then the answer.

So if the person stands still under the rain we will get:

B=Av

T|B|=T

Now lets look what happens if a person starts to run. Since we can choose the coordinate system in any way we want we can assume the the person runs on the x axis with some speed w (for simplicity, we will assume that the speed is constant). Now it is time to use the Galilean transformation. From the coordinate system of the person the speed of the rain is no longer v=(v1,v2,v3). Instead it now becomes v'=(v1-w,v2,v3). Since nothing else changed, we will get:

B'=Av'

T|B'|=T

It is perfectly possible that this is less than the previous result, but it can also be large. It all depends on the direction of w. If for example w=v1, you will get less wet by running. But if w=-v1 you will get more wet if you will run.

This solution was done under the assumption that the rain is uniform. However, while this assumption is not realistic it is very close to reality and on a short time interval it should be extremely close to reality. It is possible to solve the problem without this assumption, but it will only introduce extra steps without changing the final result.

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.

The genius of Newton

This is just a short joke I found today:

Archimedes, Pascal, and Newton are playing hide-and-seek.
Archimedes covers his eyes and starts counting.
Pascal looks around and hides behind a bush.
Newton grabs a stick and scrapes a one meter by one meter square in the dirt and stands in it. Otherwise he does not hide at all.
Archimedes opens his eyes and looks around. Of course, he immediately sees Newton and calls "I see Newton" Newton calmly says "But hang on, one Newton in a square meter is a Pascal!"

It is handy when you have physical units named after you......

Quantum cat

A quantum cat. I am sure Schroedinger didn't think that his little experiments will become the reason for so many jokes..

Science is fun sometimes, isn't it?

Since in this case we see the cat, we are sure that he is in the box (and we know he is alive). And therefore this is not a paradox. However, it is always good to think about puzzles of quantum mechanics...

Quantum physics in a grafiti form

For some reason, I feel that this simple pictures talks well about such properties of quantum physics as probability and our inability to observe the quantum processes.




What do you think about it?

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...

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.

Small scale and slow motion

A lot of research is done with the purpose to make computers faster and to generally improve them. One of the goals is to develop a fast optical computer. The advantages such a computer would have are clear - electricity heats the circuit it moves in, but not light. The problems needed to be solved in order for such a computer to be built are enormous, but surprisingly Nature managed to overcome at least some of them a long time ago - photonic crystals needed for such a computer were recently found in the shimmering, iridescent green scales of a beetle from Brazil. Read this article for the full story.

The reason I am writing about this is that it is an excellent example of an important principle at work. This principle is that all the inventions we made are already present in Nature in some form. It probably sounds a bit strange. But if we will look, we will see that the problems we use technology and science to solve, have a "more" natural solution already. For a simple example, lets look on nuclear fusion. We don't yet have control over it (probably it will happen soon, but not yet) but the problem we are trying to solve with it is a need of heat energy. The heat will be used to produce electricity, but we need to produce heat firstly.The nature has the some problem - in order for the universe to develop somehow, heat is needed. Heat is also needed for production of new, heavier elements. How did nature solve it? Nuclear fusion is exactly what heats stars. This example is cosmological, and it speaks more about properties of matter. Yet there are many more examples. The computer can be viewed as our attempt to recreate the human brain using technology, for example.

It can be argued that seeing such connection is a bit too much, and yet the problems we (as a race) face are in no way unique. And if so, there is good chance to see solutions of them already present in the Nature. But to find such solutions, we will need to look on the small scales.

The reason for this is that when viewed from another angle our world doesn't look the same. In the video below you will see a series of simple actions first in normal speed and then in slow motion. The actions stay the same, but there is a difference. It is well visible in the experiment with balloon and fire. In normal speed it looks like it was burn by the fire, but it looks differently in slow motion.

Inside LHC - Atlas detector

LHC is a particall colider. But to get data we also need a detector which will corectly detect what happened in the collusion. This detector is named ATLAS. In the video below you can see a simple animation that shows how it looks and how it is constructed.

Episode 1 - A new Hope



Episode 2 - The Particles Strike Back (Part 1)



Episode 3 - The Particles Strike Back (Part 2)




The next video is very short - but is shows very well the scale of LHC. Usually when we hear about such large constructions we cannot visualize them well. This video should make it easier.


From Space to Atlas

A bit of random staff

This post is a collection of links/reviews of some random (but interesting) staff I found on the web. I never did such a post on Math Pages before so we will see how it goes.

Firstly, a bit of humor. This ad was apparently seen and photographed somewhere in the USA:

It is hard to say what was the reason behind this ad - perhaps they simply didn't want anyone to come? I wonder if to actually get the help you need to past a reading test (being able to read 200 words per minute for example).

Also apparently somebody recently did an upgrade to Mathematics - we are now on version 2.12. I never heard about version 2, but from the release notes (if link doesn't work - download .txt here) it seems a good joke (written in bold because some people failed to understand it).
Examples of the fixes in the new version:
1. Pi now equals exactly 3.
2. The term "negative number" has been deemed offensive. The term "non-positive non-zero number" is now in use.
3. Fixed problem where 1 = .999...
4. Removed the Proof By Contradiction exploit.
5. Users may now enter the paradise Cantor created for us for a nominal monthly fee.


Now to some serious staff. For some reason I stumbled on a lot of surprisingly interesting articles this week - mostly about physics. What is also good about them is that they are very well written and explained. Enjoy:

Maxwell: Thermodynamics meets the demon - a very detailed explanation of what the Maxwell demon is. Also includes a short overview of the development of Thermodynamics.

Fusion 2.0 - "Fusion could one day generate limitless cheap energy from little more than water, while emitting no greenhouse gases. We look at its promise as the ultimate power panacea for a warming world." There are currently plans to build a large fusion reactor in France. It is expected to generate 500MW and the construction cost is 10 billion euros. It is easy to see from these numbers that this is a scientific experiment and not an economically feasible solution to the energy crisis - but it may prove to be a very important step.

The Mechanical Battery - the main idea is very simple. Instead of storing energy in a regular battery in chemical form why not store energy in a rotating wheel? This may sound very strange, but the idea is based on the fact that rotating objects store energy in direct proportion to their mass and RPM. The end result is more much more friendlier to the environment that batteries, and such a wheel can be both charged and recharged in a very short time without damaging itself. Perhaps, this technology will finally provide the necessary power for creating electrical cars...

Firefox God - over 300 excellent extensions in different categories. If you are using Firefox, it is a good idea to visit this page.

Million dollar problems - A list of the million dollar problems in math. They are all explained is a simple enough way.

Unsolved problems in physics - a very large and interesting list.


It would be a good idea to finish this post with something serious, but I didn't post anything funny for a long time so instead I will write about this page - 101 More Great Computer Quotes. Examples:

"I do not fear computers. I fear lack of them."
– Isaac Asimov

"A computer once beat me at chess, but it was no match for me at kick boxing."
– Emo Philips

"Getting information off the Internet is like taking a drink from a fire hydrant."
– Mitchell Kapor

"Programming can be fun, so can cryptography; however they should not be combined."
– Kreitzberg and Shneiderman

"Don't document the problem, fix it."
– Atli Björgvin Oddsson

Micro black holes

While this subject sound like something not very real (and for a good reason) it became somewhat important to the general public in the last years. The reason for this is the Large Hardon Collider - LHC. While it is still only a theory, many people think that it might produce micro black holes that would destroy our world. I already wrote what I think about this danger in the post The horror of LHC. There is simply no reason what so ever to expect that these black holes even if they will be created will posses any danger. You can also read more about it this nice article: Help! A black hole ate my lab.

On a more general note, I never managed to understand why people who clearly don't have even basic knowledge of physics tend to speak their opinions about it so loudly. It doesn't happen only in this field of science of course, but why does it happen at all? It is how the world works I guess...

I am now on winter vacation - this means that I have more free time, but I also have a lot of stuff to do. Hopefully I will manage to keep this blog updated with more math related content, for some reason I didn't wrote about math for a long time...

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.