The Einstein in 1905 introduce the concept of Special Theory of Relativity. In our daily life we see the effects of the relative motion, for an example if suppose there are two cars one is BLACK and the other RED and both are moving into the same direction. If the speed of black car is 50 km/h and for the red is 70 km/h, then the person who are sitting in black will see that the red car is passing to him by the speed of only 10km/h in forward direction while the passenger of red car will see that black car is going in opposite direction by the same speed i.e. 10km/h.
One more example we can consider here, that is about the trees which seems to move in opposite direction when we travel by the train or car. This is also an observation of a person who is sitting inside the train coach, see that tree and plants are moving in opposite direction. But not true for the person who is standing in the field, he see only that train is moving and passengers are sitting inside to it.
These are observations which we see in our daily life because of the relative motion. One can be true for the first observer but not necessarily true for the others. All these or such kind of phenomenon can be explain very easily with the help of Einstein’s Special Theory of Relativity (STR). But what is the important point here when we play with the STR, it is applicable actually only when the relative speed is comparable to the speed of light.
In the Special Theory of Relativity (STR) we found the relationship between the two different observations, that observed by the observers in stationary and in moving frame of references for any particular event(s). Now to understand it we will required to know some basic terminology of the STR. In this, 1.) is the Frame of Reference, 2.) is the Observer and 3.) is the Event. A frame of reference can be a car, bus, train, an airplane, spacecraft, rocket, a house, a tree or anything from where an observer basically observe or see the event. An observer is the person who note the event or see the event in his frame of reference or somewhere at other place from his frame of reference. An event is basically can be an explosion, your clapping, a collision, anything which can be define in terms of spatial coordinates and time coordinates like (x,y,z,t). So if suppose an even occur in a frame of reference, you will define that by P(x,y,x,t), mathematically in this way.
Now, the idea of frame of reference, an observer and an event is clear to you. We will now transform the daily life observations in mathematically form by using the frame of reference, observer and an event. How ? let us see it by a simple example, suppose you are standing across the road and a person who is sitting on bike pass to you with 50 km/h speed, this speed of the bike is constant and relative to you. He throw a stone in forward direction, the speed of the stone by which he throws is 10km/h.
From this example:
A. There are two frame of references
a. The place where the first person, who is standing
b. The bike, is second frame of reference and it is moving relative to the person with constant speed, this is relative velocity (v=50km/h)
B. There are two observers
a. one who is standing on the surface of the earth
b. second who is sitting on the moving bike
C. there is one vent
a. when the biker throw a stone
Now, in this chapter the first topic is of Galilean Transformation, here through this we have to develop a relationship between the two observations, or you can say that we have to transform one coordinate system into the other. One is un prime frame (S) and second is the primed frame (S’). The primed frame is a moving frame and un prime is stationary frame i.e. is in rest relative to the moving one.Where O and O’ are observers in S and S’ frames respectively. S’ frame is moving along the positive X direction with the relative speed of v. P is any event that occur in a moving frame. The coordinates of the event according to the observer O in S frame will be (x, y, z, t) and for the observer O’ in S’ frame is (x’, y’, z’, t’).