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Thursday, 29 December 2011

The Irregularity of Time [2/2]

A satellite orbits the earth. Due to it's speed, special relativity says time should be going slower than if it was on earth for it, but in fact, it goes quicker. Why?

This is the third of three posts on time. We have already discussed The Special Theory of Relativity in the first post, and the concept of space-time in the second one. I would recommend reading them first, for this most may not make much sense if you have not. In true "TheCompBlog" style (a great tech blogger/friend-of-mine), this post is being written whilst on a airplane flight, and just how topical it is! We learnt in the aforementioned posts, that the faster an object move, the slower time runs for it. So up here, in this plane, time for me is moving quicker than for someone standing below me right? Well, there is more to it than that. You see, another of Einstein's discoveries was that not only is time affected by speed, it is also affected by gravity. How does this come about? well really, it is just applying Special Relativity:

The best way to explain Gravitational Time Dilation, this process of time being slowed by gravity, is through imagining a perfect vacuum in the middle of space. You are in a rocket in this vacuum. No external forces, including gravity, are acting on the rocket. At the front end of rocket is a lamp, emitting regular pulses of light every second. At the other end is a target.
The pulses arrive at the target at the same rate as when they left the lamp, namely 1 per second. The time at the front of the rocket would be moving at exactly the same rate as the time at the back. However, what would happen if the rocket started accelerating? By the time the pulse reached the other end of rocket, the rocket would have aquired a speed. We know that velocity equals acceleration times time taken, or v=at, and the time it takes for the light pulse to travel the distance is distance divided by the speed of light, or t=xc. Therefore the speed gained whilst the pulse is traveling is v=a xc. Well firstly we have to look at something called the Doppler shift. We have all experienced the effect when an ambulance passes you. The siren sound seems to drastically change as it is approaching, passing and moving away. This is because the frequency changes. This change can be measured with this formula:

f'=f(1±  v  )
c
The bottom of that fraction is similar to that from our other equations from Special Relativity, though this is only because, just like in time in special relativity, the frequency is affected by velocity. For very large velocities, time dilation has to be taken into account, but in this situation, the speeds are not large enough for it to matter. The equation can be rearranged to be
(f'-f) ≈ -fvc
and from our previous equation for v, we get
(f'-f) ≈ -fahc2
The pulses of light reach the target at a different frequency as when they started:

Now imagine that the rocket is not in space but instead on a earth. The lamp is on the floor of the rocket now, and the target is on the top:
Gravity is, in essence, just an acceleration. If a feather and hammer are dropped in a near vacuum, they will fall at exactly the same rate. On earth, that rate is 9.81 m/s2. So the rocket, in this situation, is in fact feeling an acceleration downwards, just as we feel pushed towards the ground. The person standing at the top would conclude that the pulses are being released at longer intervals a person standing next to the lamp would measure the frequency to be, this is called the gravitational redshift. Now imagine that the pulses are some sort of clock, each one emitted a second after the last. Seeing that the frequency of this would be less for the person at the target, he would, correctly, conclude that the time closer in the gravitational field was moving slower.

Many experiments have been made to prove this theory. In 1960, Robert Pound and Glen Rebka verified it by firing gamma radiation up and down a 22.5m tower. The shift would only have been about -2.5 x 10-15 but it was verified with a margin for error of only 1%. When the first GPS satellites were launched they did not take into account time dilation from gravity or from speed and were therefore very far off when calculating someones position after only a day or two.

I really hope you like this post and this series of posts. I may revisit it one more time, just to run over the main points, sort of as a conclusion, but I hope that this has interested you in the mean time. Please email me at theaftermatter@gmail.com, comment or get in touch on Twitter if you have any suggestions,  complaints or just want to have a chat about physics/maths/engineering, I love hearing from you.

Check out my last two posts: 
What is Energy? - The word energy is used so loosely these days, so what does it actually mean?
"We don't serve neutrinos in here!"...A neutrino walks into a bar - What are neutrinos and what is all the fuss about?

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, 7 December 2011

What is Energy?

Renewable Energy, Wasting Energy, 'He's Got Too Much Energy' - None Of These Are True. So What Is Energy?

We have all heard of energy. A word used in all sorts of situation. Energy in food, people and objects are all very common things to hear, but what exactly is energy? Well the easiest way to describe energy is through the statement: Energy is the ability to do work. Ok...That doesn't bring us any closer to knowing what energy actually is, so let me first start of by explaining Work:

Before we all get confused, physical Work is not work in the sense that we use it everyday. The incredibly clever person in your class, who does extra homework does not have more energy than you. Work, shown with the letter W, can be expressed easily with this simple formula:
W = Fx
In words, this means that the work done is the Force applied to an object multiplied by distance.  For example, if I am pushing an object with a force of 10 Newtons, and I push it 5 meters, the work done would be 50 Joules, 10 x 5, because F (the force) is 10 and x (the distance) is 5. 
As you can tell Work is very simple. The only thing you need to remember when thinking about Work is that the distance is not just any distance, it is the distance in the direction of the force. This may not seem very significant, but let me give you this example: If I have a steel ball suspended by a string. I start spinning it around, in a circle of circumference 5 meters, with the same force as before, 10 newtons, and it does one rotation. Surely the work done would be the same as in the first example?

 
Firstly, when looking at the work done, you have to analyse the forces. As we can see, the only two forces acting on the ball is that of me, which is F1 and that of the string, F2. The string is pulling the ball towards me, whilst, to keep the ball spinning, my force is at a right angle to F2. The resultant force is in a different direction than either F1 or F2, therefore, there is no work done.

So now that we know what work is, we can move on to energy. If I asked you to list all types of energy you knew, what would you say? If you said Nuclear, chemical, electric, thermal etc. you would be right, but they can be put more simply into two groups, potential energy, Ep, and kinetic energy, Ek. An object has kinetic energy when it is doing work, this can be in the case of electrical energy, in an exothermic reaction or simply when something is moving. An object has potential energy when it has the capacity to do work due to it's position or configuration. This can be in the case of a compressed spring, something being held any distance above the ground or, in fact, any matter, as famously described in Einstein's E=mc2, anything with mass also has energy.

Both of these types of energy have formulas to work out their values. These will change for certain situations, for example, with electrical energy, but in simple mechanics the formula for kinetic energy is Ek=12mv2 and the one for gravitational (potential energy possessed by an object due to it being pulled by gravity) potential energy, in it's simplest kind, is Ep=mgh. Both of these equations can be obtained using the equation for work, as previously mentioned, and Newton's second law: F=ma.
The potential energy equation is very simple. If you substitute ma, from F=ma, into the definition for work , Fx , for the F, you get: Ep=max. When using this to describe gravitational potential energy, you can substitute h, the height the object is lifted too, in for x, which just means distance. As we know, gravity is just an acceleration. The earth's gravitationally constant is 9.81m/s2 so we can put g in for a. That is how we get to the equation Ep=mgh.
The kinetic energy equation is also very simple. One of the main equations of motion is that final velocity squared equals initial velocity squared plus 2 times the acceleration times the time taken, or v2=u2 + 2ax. I do not feel it is relevant to explain how we got this equation, although, if the idea is popular, I may do a post on the basics of mechanics. Mainly Newton's laws of motion and the equations of motion, but anyway, kinetic energy: In a system with kinetic energy, the object must have started at a stop. Therefore the initial velocity is 0 and can be removed from the equation. Now we have v2=2ax. The equation F=ma can also be expressed as a = Fm and therefore we can change our equation to v2=2Fxm. Look what we have found! Above the m is Fx, or work done, and energy is the just the ability to do or be doing work, so, by rearranging the equation we get: Ek=12mv2.
So now we understand energy, right? Well first there are a few things to go over. Energy does not get destroyed or created. When I drop a pen, no energy is being lost, or created. Instead the potential energy of the pen is being converted into kinetic energy. In fact, if we forget about friction, all of the potential energy is lost over the fall. Let me give you an example: I lift a pen, mass 1kg (It is a heavy pen) 3 meters and then drop it. What is the velocity when it hits the ground?
 
Since we are ignoring friction, we know that:
   Loss in Ep= Gain in Ek
    mgh =12mv2
1Kg x 10m/s2 (We are rounding this up, gravity is actually more like 9.81 m/s2) x 10m= mgh
         12mv2 = 12 x 1 x v2  
0.5v2 (Kinetic Energy) = 30J (Potential Energy)
   v2= 60J
     v ≈ 7.45 m/s

The amount of energy in a isolated system, even in the air or other apparently inactive objects, must be the same, no matter what form they are in. This is the law of conservation of energy!

Now we understand energy. It isn't a physical particle or even a force, but instead a capability to do something. For the next section, let me ask you a question. If you take a large piece of coal and burn it, and then take a spatula of gunpowder and light it. Which reaction would you say is more violent? Probably the gunpowder, but the coal released more heat, and therefore, we will assume, it had more energy. Why does this happen? Well it is all down to something called Power:
Power is the rate at which work is done. It is given by the formula: P=Wt . P is power, W is work done and t is time taken to do the work. Therefore, even though the coal does more work, the gunpowder does it's work in a shorter time and is therefore more powerful.

So those are the basics of energy. Of course, there is further you can go, looking into the different forms of potential energy, for example. In fact, string theory, something I looked at in my last post, says that ever bit of matter, when broken down as far as you can, past atoms, protons, electrons and quarks are just 1 dimensional strings of energy. So next time you hear someone talk about an energy crisis, renewable energy or wasting energy, you can correct them, our energy is not running out, instead, we are just running out of easy sources of it.

Anyway, I hope this cleared up any misconceptions you had about energy, and helped you understand one of the most important physical aspects of the universe. Thank you for reading.
Check out my last two posts: 
"We don't serve neutrinos in here!"...A neutrino walks into a bar - What are neutrinos and what is all the fuss about?
The Irregularity of Time <1.5/2> (You might want to check out the first post on this topic before this) - Why time isn't as constant as you think.

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  )
c2
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  )
c2

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  )
c2
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  )
c2
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

2=1?

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:
2=1

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
0/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!

Theo




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'
c2
t=t'√(1-  v2  )
c2
and then:
t'=t√(1-  v2  )
c2
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.