Galilean Relativity Transformation
It was observed by Galileo, in multitude of experiments, that mechanical phenomena occur identically in all inertial frames. (Later, this was experimentally verified for electromagnetic phenomena as well.) These observation of Galileo were stated by Newton in the following words: “The motions of bodies included in a given space are the same among themselves, whether that space is at rest or moves uniformly forward in a straight line.”
The above statement, known as Galilean principle of special relativity, means that all phenomena in nature should appear same from all inertial frames. In modern terminology, it says that the laws of nature must remain form-invariant as we go from one inertial frame to another.
The measurements of the motion of a particle P (i.e. its position, velocity and acceleration at any instant) from two different inertial frames S and S’ leads to two sets of values. The relationships between these two sets of measurements are called Galilean transformation equations.
One of the reference frame S is (arbitrarily) called stationary while the other S’ is called moving. One frame must be moving with a constant velocity V relative to other. Otherwise, the two frames become same. Both the frames (or observers) S and S’ may use Cartesian co-ordinate systems for measurements.
Let us denote the co-ordinates of particle P in frame S by r = (x, y, z) and time t, and in frame S’ by r’ = (x’, y’, z’) and time t’; t and t’ denote the time read by clocks attached in frames S and S’, at the instant when position co-ordinates of P are respectively r and r’.
In particular, if we assume that origins O and O’ of both the co-ordinate systems coincide at t = t’ = 0, then at any later time t = t’, the origin O’ of co-ordinate system S’ will be at a distance R = Vt from the origin O of co-ordinate system S, where V is the constant velocity with which S’ moves relative to S. Hence, if r and r’ are position vectors of P as observed from S and S’ at that instant, then we have
r = R + r’ = V t + r’
Or, r ‘ = r – V t
The above relation, along with t = t’, are known as Galilean transformations.
As a special case, if the frame S” moves parallel relative to the frame S with X’-axis coinciding with X-axis, as shown in fig., the Galilean transformations are given in the components as following:
x’ = x – Vt
y’ = y
z’ = z
t’ = t
Galilean transformations are not the transformation equations between two co-ordinate systems within a single inertial frame, like e.g. which relate rectangular to spherical polar co-ordinates; these are relations between two different inertial frames.
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