Monday, 28 November 2011

"We don't serve neutrinos in here!"...A neutrino walks into a bar

Neutrinos going faster than the speed of light? Is it possible and, if it is true, what are the implications?

Hey guys. We have all been listening to the claims that neutrinos can go faster than the speed of light but many of the media explanations have been confusing or just plain wrong so I will explain:
Let me start by explaining what a neutrino is. A neutrino (which means "small neutral one" in Italian- thank you wikipedia) is a subatomic particle. They are very similar to electrons except they are not charged, because of this they are not affected by electromagnetic forces and can therefore move through mass almost unaffected. It was disputed, but mostly accepted now, that neutrinos do have mass, although it is extremely small, essentially making it massless. Neutrinos are expressed with the Greek letter nu, ν, and there are three types, each of which is like an electron, or the larger versions of electrons, the muon and the tau. The three types are the electron neutrino, νe, the muon neutrino, νμ, and the tau neutrino,  ντ. For most of the experiments we are looking at it is mainly the electron neutrino that is being talked about. 
Now lets talk about the speed of light. As you will know if you read my first The Irregularity of Time post, the speed of light is the same no matter what speed the source of the light or the observer is moving at. This leads to some strange equations that are the backbone of the special theory of relativity. Like when, in the aforementioned post, we came to the formula:
t'=t√(1-  v2  )
Which meant that the closer to the speed of light an object moves the slower time moves for it, and once it reaches the speed of light, time comes to a halt. The
√(1-  v2  )

in the equation turns up many times in special relativity, even appearing in the famous equation E=mc2. This equation is not as simple as it seems. The m does not simply represent mass, it represents this formula:
m=mo √(1-  v2  )
mo is the rest mass of the object. This shows that as an object increases in velocity, its mass increases too. The closer the object moves to the speed of light the larger its mass gets, and once it reaches the speed of light its mass becomes infinite. We also know, from my Is Infinity Possible post, that ∞ x anything is always ∞ so an object moving at the speed of light has infinite energy.

Now for the experiments: We first heard about this experiment in September, it was plastered all over the newspapers, with headlines boasting a disproving of everything that Einstein ever stood for and that the world was going to collapse etc etc. You see, a lot of what Einstein did, mainly his theory of relativity, relies heavily on the fact that the speed of light is the speed limit of the universe. As I mentioned before, in his equation E=mc2, the m stands for:
m=mo √(1-  v2  )
So if something was travelling faster than the speed of light, you would have to have the square root of a minus number on the bottom, an impossible idea, at least in the physical universe (I am not an advanced enough mathematician to go further into this). Therefore if something can move faster than the speed of light, this theory cannot be true. However there are situations, though unlikely, in which these equations do not apply. If relativity is true then these results will instead drastically change our view of neutrinos or possibly prove string theory, but more on that later. The neutrinos were created at CERN and shot through 454 miles of mountain, because, as I said before, neutrinos can pass through matter, before hitting a detector at the INFN-Gran Sasso laboratory in Italy. The time measured for the neutrinos to cover this distance was 60 nanoseconds quicker than the time it would have taken light to travel it. After the initial excitement people became sceptical, despite the 16000 times the test was done. The objection that held the most ground was that the neutrinos, which decay, that were used were particularly long lasting ones which could produce errors in the results. To correct any problem this could have caused, they changed the way the neutrinos were created so that they lasted for a short time. Strangely this second experiment showed the same result as the first. The researchers are now asking for other teams to do the same experiment. All of these results are particularly strange because when tracking supernovas in the past, neutrinos have been shown to be moving slower than the speed of light.

One of the biggest revolutions that could come from this experiment is that it could provide at least a small amount of proof towards a new scientific theory called String theory. String theory states that subatomic particles are not zero dimensional but are in fact one dimensional vibrating strings. The calculations in string theory work, but only in a space that has not only our four dimensions, which is called a membrane, but an additional six. The idea of ten dimensions can be very strange, especially so because we can not even see the forth one. Luckily, a man called Rob Bryanton has started a project to explain the ten dimensions. He has written a book about it. On his YouTube channel he has two videos covering the ten dimensions, here is the first:

And the second is here.
He is also doing individual videos going into depth on each dimension. I would really recommend checking out his channel.
Anyway, I got a little off topic, some string theorist are taking the results of the neutrino experiment as evidence of this. The idea is that the neutrinos may not be actually traveling at, or faster than, the speed of light, but, because of the remarkable circumstances, are moving a certain distance, and then taking a shortcut through another dimension, and then popping back into our dimension at a different location. This does not have any proper proof, accept for the strange times, but it is an idea which could hold water. String theory is not yet proven and not all physicists agree on it so we can only take this with a pinch of salt.

I hope you liked this post, I will keep looking for other things to do with this and other physics or mathematical news. You can hear about all my new posts or any interesting things I found, on my Twitter. Thank you for reading and I hope I shed some light on this confusing subject.

Friday, 18 November 2011

Irregularity of Time <1.5/2>

I was born in the past, I will die in the future and I'm living in the present...So my being born is not longer happening and I haven't died yet...right?

Firstly I want to apologize for this post being late. My laptop decided to die on me and I am having the write this mainly on my iPod.

If you read the first "The Irregularity of Time" post (which I recommend doing, otherwise some of this will not make sense to you), I said I would be talking about general relativity this week but I realized I had a lot more to talk about in the realms of Special Relativity.

We found out last time that time is not constant for two people moving at different speeds viewing the same event. But all this can get confusing, lets say I'm on the phone to a friend of mine in New York who has just walked one step up a flight of stairs, from the perspective of an astronaut in space my friend has walked up two steps and from the perspective of a different astronaut my friend hasn't even started walking up the stairs yet. These differing perceptions of time and space can be confusing. We have all heard someone, trying to get one up on someone else, most often, saying "Everything is relative", they would be very wrong. You can not summarize relativity that simply. Despite this, all measurements of space, time and speed are related through the equations we have worked out and others we can derive in other situations illustrated in special relativity.
We all have different perceptions of the space we are in or the object we are looking at. For example, if two people are looking at a pencil being held up in the middle of the room:
Someone looking from position P would see the pencil like it is shown at position PV where as someone looking from Q would see the pencil in position QV. PV and QV are projected views but we are used to seeing 3D objects in a 2D way, in fact, it is this affect that 3D movies, shows and games are aspiring to get. The new range of auto-stereoscopic TVs in development track head movements to apply this.

Ok less about games, more about relativity. This example with the pencil shows our different perceptions of space and time. Three years after Einstein published his theory of special relativity, one of his teachers, Hermann Minkowski, decided to approach the idea differently, proposing that what relativity was saying was that we did not just have the three dimensions of space, and then the one dimension of time, instead, we had four dimensions of space-time. The dimensions of time that we can see and the dimension of time that we can measure with a clock are only appearances. 

Instead of in the original three dimension, things are defined as events, having a place in all three dimensions and a certain time. Now let me give you something quite incredible to think about: If we can move, with enough energy, to any point within the three dimensions of space, then, if space and time dimensions are intertwined, should we not be able to move to any point in time? Yes...according to the space-time theory, time travel is totally possible...

I'm sorry that this is late and not very long but I did not have very much time this week. Check out our next post on the kerfuffle about neutrinos going faster than the speed of light. The problems with the idea, the theory behind the unusual results and how it relates back to what we have been talking about in my Irregularity of time posts. The irregularity of time [2/2] is here, check it out! I hope you enjoyed this post, bye.

Main source for this and the other articles based around relativity is the book about relativity here. It goes into more detail on all these subjects and the other books they do are brilliant too, I recommend them a lot.

Wednesday, 16 November 2011


Hey! Before I start this post, I better introduce myself, I'm Theo, and I'm a new writer on this blog along with Ned. Just a word of warning, this post does get quite technical later, but I do think you guys will like it, so please persevere, and enjoy!

I start with a problem for you guys...
Let a = b
a2 = ab
a2 +a 2 = a2 + ab
By subtracting 2ab from both sides we get:
2a2 - 2ab = a2 - ab
Then we can factorize to get:
2(a2 - ab) = a2 - ab
Which simplifies to:

Millenia of maths disproved? Not exactly...
Obviously there is a fallacy here, but can you see where and why?

If you wanna figure it out, don't read ahead yet, if you have done or want to cheat read on!

The problem here is the very last stage, from "2(a2 - ab) = a2 - ab" to "2=1." We have already stated that a2=ab, and therefore a2 - ab=0, meaning that simplifying that last stage is actually simplifying 2(0)=1(0) by dividing both sides by 0, which gets you into a bit of a mess...

And that leads me nicely on to 0 and its properties.

The furthest back we can trace the concept of '0' is in the Babylonian sexagesimal (base 60) number system, but rather than being considered a number in its own right, it was simply used as a place holder, a way of differentiating 101 from 11 for example.

0 was first considered as a number with which arithmetic could be done by Brahmagupta in 628 AD. Brahmagupta established some of the basic rules and definitions of 0...

0 + a positive number = that same positive number
0 + a negative number = that same negative number
x + -x = 0
However, he somewhat ducks the issue on  x÷0; he states simply that it equals x/0, and then contradicts the modern position further by saying that 0÷0=0.

If we treat division as the inverse of multiplication, then by simple substitution we can see that it's not as simple as that, as any number multiplied by 0 will equal it (3x0=0, then it seems logical that 0÷0=3 as well), so 0÷0 could be any number between −∞ and +∞, it is completely indeterminate. This is what is meant when it is said that 0÷0 is undefined: it has an uncountably infinite number of solutions and it could be any one of them.

Somebody called Dr James Anderson has recently said that he has come up with a solution to this problem: define the result as "nullity," or Φ, which "lies off the number line," essentially encompassing any number from negative to positive ∞. However, this doesn't do anything except attach a name to this; it is not a constant, and therefore nullity is essentially shorthand for any other number, not much better than the computer's "NaN" response.

He also works on the assumption that a positive number divided by 0 is ∞, and a negative number divided by 0 is -∞, both of which are untrue and both of which can be disproved.

The proof from this arises from the fact that 0 is neither positive nor negative, and again relies on division being the inverse of multiplication. Put simply, it uses the rule that if xy=z, then z/y=x. It is already clear that if y = 0, then x cannot be less than ∞, and 0 multiplied by a finite number is 0. However, as I am about to show, even if it is, we run into problems.

What we are essentially trying to do is, by repeated multiplication, make y closer to z and eventually equal it. For example, if we take y to be 3 and z to be 24, the by multiplying y by any number greater than 1,(let's use 2 for this example to start with), we make it closer to z (3<3×2<24), and then by multiplying by 2 again we make it closer still (3<3×2<3×2×2<24) until eventually we reach a value that is equal or greater than z, in this case, x=8.

However, if y = 0, then we have a problem. 0 × 2 = 0. No matter how many times we multiply 0 by another number, it will not get any closer to the number we are trying to achieve. Since it will never get closer, even multiplying it by 2 an infinite number of times won't help. Written algebraically, this says that y(n)≠z for non-zero z and n>1. Since n=∞, we can re-write that equation as × ∞ ≠ z, so therefore even if x=∞, it still isn't possible.

Thus, we have proved that division by zero gives neither a finite nor infinite result, that is to say, it has no result at all. And if you do dare divide by zero, then you find that 2=1, and maths as we know it collapses!

Thanks for reading guys, I know that was quite heavy at times but I hope you found it interesting, and Ned and I will post more soon!


Hey guys, it's Ned again. I hope you enjoyed this post. If you did, please take a moment to visit our last two articles:
The Irregularity of Time <1/2> - The first of two posts about why time isn't a constant as we think.
Is Infinity Possible? - What is there not to love about a never-ending number?

Friday, 11 November 2011

The Irregularity of Time <1/2>

Time is the same for anyone and everyone...Isn't it??

How weird would it be if, when someone ran past you, that time for them would be going slower than it was for you, on your watch it says 12:15, but for them its only 12:10? Its seems ridiculous right? Yeh...but it is totally and utterly possible...Ok, to be honest on that scale it won't happen, but as soon as you start launching satellites into space, the difference is huge.
You see, time isn't as constant as we like to think. Of course, if you're sitting on a park bench with your friend, they won't suddenly start turning into Gandalf in front of your eyes. It requires certain conditions and the change is measurable. The way we know this is through The Theory of Relativity, which I will be taking two blog posts to explain very briefly. I will warn you, this is a big and complicated subject, I'm only scratching the surface of what relativity is. This is Einstein's paper on only the first half of it...yes, I know...But please, read on, I think it will really interest you.

Oh good, you have chosen to continue, now we will get into the real stuff. Relativity is split into two types, Special Relativity and General Relativity. Today I'm going to be talking about the former, a theory put forward in 1905 by Einstein in his aforementioned paper, "On the Electrodynamics of Moving Bodies" and don't worry about the other yet. One of the things that Special Relativity proves is that for an object moving at a high speed, the time within that object is slower than that of the time within a object traveling at a different speed, or that is stationary.
Special Relativity relies on two main axioms, or assumptions. The first axiom is that:
The laws of physics are the same for all observers in uniform motion relative to one another (principle of relativity).
As Wikipedia so fantastically puts it, but to make it simpler, the same laws of physics apply to two things moving at a constant velocity relative to one another. It would be extremely weird if when your friend got up from the bench and walked about, a different force of gravity worked on them.
The second axiom is a lot more counter intuitive and this is where all the strange results we get from Special Relativity comes from. It is:
The speed of light in a vacuum is the same for all observers, regardless of their relative motion or of the motion of the source of the light.
Another remarkable description from Wikipedia but to put it simpler: A beam of light will always be calculated to be at the speed of light, no matter the direction or velocity of the observer or the source of the light. Now, this seems ridiculous, "If I'm traveling at a speed near the speed of light and a light-wave passes me, it must appear to be going slower, mustn't it?". And that is where a huge amount of complexity lies, its not logical, or at all observable to someone without state of art equipment, so it is very difficult to believe this axiom. But equations using the axiom have been proven true and it is widely accepted as true throughout the scientific community so you may just have to accept it and see the wondrous results.

Ok, now we get to the good part, this is why time runs slower in a fast moving object.

So, the easiest way to explain this is through an example, a train passes through a station, with one person sitting in the end of train (S) and the other on the platform (S'):
The train is moving incredibly quickly, otherwise the effect is not really noticeable. Inside the train,S is holding a torch and on the opposite wall to them is a mirror. He turns on the torch and times the time it takes for the light to return back to him.
In the diagram "d" is the distance from S to the mirror. To work out the time taken we apply the equation Time=DistanceSpeed and from that we can work out that the time taken for the light to travel to the mirror and back is t=2dc with c being the speed of light. Now we are going to look at the same event but from the perspective of S' sitting on the platform. For them, the torch is turned on in the same place as for S but when it reaches the mirror the train has moved:

This continues for the whole journey and the track of the light ends up looking like this:
As you can tell, the distance to the mirror is longer than for S, we will call this distance d'. Of course, there is nothing remarkable about that, its very logical really, but the next step is both the incredible and the illogical. We have to use Time=DistanceSpeed and equation we find is t=2d'c, because, due to the second axiom, the speed of light is a constant for any observers, irrelevant to the speed on the source of the light. This shows that the time it took for the light to travel from S to the mirror and back to S, is different from within the train and from outside of the train. Time for S if running slower than it is for S', although S won't notice, it isn't like he will suddenly be moving in slow-mo, S will just have elapsed less time than S'.
This is the measurements without the train:
You can use Pythagoras' theorem in order to find in what way these values connect. In the diagram, t is the time to light ray takes to get to the mirror, but not back, or dc:
Now using a2+b2=c2 to for an equation:

c2t2 = (c2-v2)t'2
t2= = (1-  v2  )t'
t=t'√(1-  v2  )
and then:
t'=t√(1-  v2  )
This means that the closer the velocity of the moving object is to the speed of light, the slower the time goes inside the object relative to a stationary object.

The thing is, all this seems ridiculous. In no way does this make sense but sadly, there is no to make it any other way. It is nonsensical and illogical but it works. Both the two axioms have been tested and confirmed, and experiments have been done to prove the overall concept. For example, muons are particles that breakdown into an electron and two neutrinos in two milliseconds. An experiment was done at CERN in Switzerland, they shot some muons around a fourteen meter diameter loop at a velocity of 0.9994c, the results were both remarkable and, based on our previous calculations, totally logical: the average lifetime of one these muons was measured to be 29.3 times longer than that of a stationary one.

I hope you enjoyed this post, it took me a while to complete but I'm very happy with it now. I'm only scratching the surface of an immensely huge subject, but I find it incredibly interesting, and I hope that i have at least shown to you just how much lies beneath the basic physics we know instinctively and how, sometimes, our instinct is wrong...
You can read the second and third posts now!

Main source for this and the other articles based around relativity is the book about relativity here. It goes into more detail on all these subjects and the other books they do are brilliant too, I recommend a lot.

Tuesday, 8 November 2011

Is Infinity Possible?

Is there such thing as something with no end?

If I'm honest, I've never really thought about this very much. I know what you are thinking: " What is this guy on about? He thinks he can write about maths but he hasn't even considered what is one of the biggest, literally, subjects in maths?" and for you, I have no answer, It just never seemed me to be that interesting, nothing to prove wrong and calculations to make, infinity is infinity, a never ending number, and as brilliant that is, it has never had a real meaning to me...until yesterday:

As I sat down in maths yesterday, I expected the usual (if you need to know more about my maths class, look at my last post). Little was I to know that it would be my inspiration to start this post and, in fact, this blog as a whole. It was my friend who started the process, telling a maths trick. Here it is:

"I'm going to prove that x in this equation, x={12+14+18+...}, actually equals 1.
Ok so first what I want to do is divide both sides by 2, from that we get 12x={14+18+116+...}
Now if we add 12 to both sides you get 12+12x={12+14+18+...}, and look, the right side in that last equation is the same as the right side in the first one, therefore 12+12x=x and x=1."

This is the beauty of ∞, that shouldn't work. Every logical part of you says that that couldn't work but no matter how many times you work over it you always receive the answer that x=1. See the problem is that when we add 12 to the set, we expect there to be one more number in it, like if we had a set of {1,2,3} and then you added 4 to it, there would be four numbers in the set. But here is the magic, there are infinity numbers in that set and infinity is not a number, it is an idea and, as it is the highest number, theoretically, ∞ + 1 = ∞, this is how it works, the number of numbers in the set cannot increase because it is already the highest it can be. The maths is sound, we just don't have the brainpower to think naturally about infinity.

Good links to read up more on infinity:
Link 1 (Quite Complex)
Link 2 (Simpler)

Thanks for reading this, please leave your opinions in the comments.