“Relativity”. At first, relativity was denified by Einstein in 1905 with the term “Special”. This word represented a rectilineal uniform motion. Then in 1916 Einstein published a new theory: "The General Relativity”. It is about every kind of reference frame and it uses high mathematics. But, what is a reference frame? It was called by Einstein “Body of reference” because it was not something abstract. It is an intersection of three asses (x,y,z) and each point is identified by three numbers. It is very important because it represents an event which can be every kind of situation and it is measured by a meter stick (space) and a stop watch (time). It is described by the scene of events  that indicates where and when the event happens. There are particular reference frames like the Inertial reference frame where the law of inertia acts. When a body rests or is moving without stopping and there isn’t a force that is acting on it, we can say that it is under the law of inertia. How can we understand when we are or not in a inertial reference frame? We have no way to detect it and we can’t know if we are moving or not because it is not an accelerating system of reference.

 

Light and sound
The study of light phenomenon began in 1861 when Maxwell published his work “On physical lines of force”.  The scientist describes some of his questions about electromagnetic phenomenons. In particular it focuses on the importance of light that is an electromagnetic wave. By his important study, Maxwell discovered a formula that can describe the velocity of electromagnetic waves, and so of light. In the formula we can find two constants ε_o andμ. The first one is the dielectric constant and the second one is magnetism constant. The velocity of light is a constant of nature and it’s about 300000 km/s. Another important wave is the sound wave, which is a mechanical wave. Its velocity through air is different from the velocity of light: in fact it is about 340 m/s. But this fact can change in base of the situation. If a man moves closer to a sound wave, its velocity gets higher. Meanwhile, if the observer goes away from the sound wave, the velocity gets lower. The same experiments are valid to light. But the light propagates in a different space from the sound one. This space is called Ether, a place where light goes through and without acceleration.

Galilean Principle of Relativity
The starting point of Einstein’s theory is the “Galilean Principle of Relativity”. Galilei says that all the natural phenomenons run their course in a inertial reference frame, because laws of physics are equal in all of them. This theory goes against Maxwell’s formula, because this one is valid only when ether is still, and so velocity is constant. This system is called: “System of Stationary Ether”.

In our second lesson, we continued talking about phenomenons in a inertial reference frame. Their description changes in base of the different reference frame. In fact we saw a Kinematic phenomenon and the application of "Galilean law of addition of velocities". By taking an example, we can prove the differences of velocities in a inertial reference frame. There are two observers: O and O' . The first one is still and on the ground, meanwhile the second one is on a car, which is moving to O with a velocity Vo. If O' throws a ball with velocity u' to O, O will take the ball not with velocity u', but with velocity Vo + u'. This addition corresponds to the composition of Galilean transformations. 

 

Let’s imagine a sound wave by an organ and an observer. The velocity of the wave ( c ) never changes in base of the velocity of the source. It depends on different factors, for example on the position of the observer. If he’s nearer to the source, velocity will get higher. Meanwhile if the observer goes away from the organ, the velocity c gets lower.

Sound velocity also depends on the direction of the wind. In fact when sound and wind waves have the same direction, sound velocity corresponds to the addition of the two velocity ( c’= c+w ). If they haven’t the same direction, sound velocity is c’= c-w. 

 

After Maxwell’s equations, we discovered that light propagates with a medium, called Ether. But, if our earth is moving, we are supposed to perceive a wind, called the Ether wind. In 1887 two scientists, Michelson and Morley, made an experiment, which is considered one of the most important in physics history. They wanted to detect the Ether wind, and so Ether. They used the combination of two waves in two different position of earth and the calculated an interference. But the result of this experiment was NULL!! So the scientists started asking.. Does the ether exist? A lot of strange ipotesis were made to prove its existence, but they were vain. In 1905, Einstein gave the answer to this question: Ether DOES NOT EXSIT ALL!

In fact, light propagates in vacuum! It was also proved that the speed of light in vacuum has the same value in all inertial reference frame, so it’s a constant.

Time as Simultaneity
As consequence of the two postulates of the theory of special relativity, scientists came up with another important question: what is time? It can seem a question without answer, but we are not speaking about it philosophically, we want to estabilish it as a physical quantity (something that can be measured), so we can say that time is simultaneity of event and position of the clock, taking a simple example of a fire-cracker that blow up at 5 o’clock, we can see that the event is simultaneous to the position of the clock hands. It can be demonstrated with a taught experiment: if we take in example two points on the ground called A and B and we imagine that two lighting bolts will strike at the same time from these points, considering a point μ (the middle point), if the raises meet at μ it means that they are simultaneous.

But is it always valid? What happens if we take two observers? The first called 0, that is rest, claims that the two events are simultaneous to him.
The second 0’ is moving to B, so he reach first B than A, to him the events are not simultaneous.

What is the conclusion?
TIME IS RELATIVE!

If we take two observers, one on the ground called Cesare and one on the carriage called Luchetta, we can detect that time changes from the different points of view. To Luchetta the clock is rest and the proper time interval is:

To Cesare the clock is moving and the proper time interval is:

Because he is more distant from the light then Luchetta and also because he has an oblique visual of the clock, for this reason we can use Pitagora’s theory to explain his proper time arriving to this formula:

γ is called the Lorentz Factor and it represents the entity of time. Factor γ and velocity are connected, because, if the velocity gets higher, γ's value gets larger and larger. Another important simbol is β: it represents the rapport between v and c. In fact:

We can also say that beta and gamma are connected, because if β=0, so γ=1. Meanwhile if β=1, γ can be calculeted by a limit and it results that it is ∞. In fact if we have two objects, A at rest and B moving, time is different in base of the point of view. So if to A time is moving slowly, the same thing happens to B. That's why this defect is symmetircal.

There are many experimental evidences about this and Cosmic radiatons  are one of them. They demonstrate that SPACE is not empty but is filled by radiations (γ race) emitted by stars. By the collision of these cosmic radiations at the level of the atmosphere (6km), we see the creation of some particles, called muons. Muons have a very short life ( 2,2 μs ), because they decay with a velocity 0.99 c, similar to the light's one, in a distance of 700 m. Muons wouldn't have to arrive on earth, but, from a lot of experiments, the scientists noticed that we received a lot of them. This phenomenon is possible because muons' life is extended thanks to the high velocity, which can give them the possibility to ride a very long distance.
We can find another example of time dilation in satellites’ velocity, which is 4 km/s, and at this velocity they produce a delay of 7 μs a day. If we don’t count the delay, there will be an error of 2 km on a GPS. 

Now, bearing in mind what we have learned in the other lessons, we can talk about the Twin Paradox: one of the most famous apparent paradox of Special Relativity. There are two twins, Pipino and Merry, aged 25.  The first one, after an argument, decides to go for a travel in the space to reach a star.

 

The spaceship travels 10000 billion km with a velocity of 0,98 c for 24 light years. After getting the star, Pipino  decides to come back. When he gets on earth, notices that his brother Merry is not young anymore. But how is this possible? How can we explain this phenomenon? We know that the total of light years of the voyage is 48. By dividing this datum with the velocity of the spaceship (0,98 c), we find that time passed on earth is of 49 years! So, now the question is: how much time is passed for Pipino? Using a formula that describes a division between the years passed on earth ( 49 ) and the factor γ, which is 5, we can notice that time on the spaceship is of 10 years. Here there is a link that describes the voyage : 

http://www.andreaminini.org/data/andreamininiorg/paradosso-dei-gemelli-3.gif#.Ww7SIUagBYA.link. 

But who is right, Pipino or Merry? Of course... Merry! But WHY??  Because, during the time of the journey, he stays  on the same inertial reference frame, rather Pipino changes it two times!

But what we see, if an object is moving faster and faster? Well, its image is contracted, but why this phenomenon happens? Because, if we measure the length of anything moving relative to our frame ( the ground ), we find its length L to be smaller than the proper length L_0, that would be measured if the object were stationary. In fact, if we take the example of a ball, first at rest and then moving with high velocity, we see how that ball gets thiner and thiner.
By the study of all these phenomenons, the great Albert Einstein arrived to the formulation of one of the most important formula in physics' history : E = mc². It describes the energy of mass M at rest; meanwhile, if we want to calculate the energy of a mass M while is moving, the formula will be: E = γmc².