LORENTZ TRANSFORMATION, LORENTZ TRANSFORMATION EQUATION, WHAT IS LORENTZ TRANSFORMATION?
Lorentz Transformation is a set of equations which correlates space-time coordinate of an event occurring in space as observed by two observers in different frame of reference moving with constant velocity relative to each other.
To derive equation let us consider two inertial frame s and s’. s’ is moving with velocity v, relative to s along the direction of x-axis. Let us consider that at an instant the origin o and o’ of two frames. Suppose a pulse of light is generated at t=0 at the origin. It will spread into the space and consider the situation when pulse reaches at P. Observers at o and o’ measure coordinate of P as (x,y,z,t) and (x’,y’,z’,t’). When pulse is observed by an observer at s then
As (z,z’) and (y,y’) axes are perpendicular to the direction of motion coordinate will remain unaffected.
Now y = y’ and z = z’
So that from Eq. (1) and (2), we get
Where k is independent of x and t.
If we suppose that the system s is moving relative to s’ with velocity v along +x direction. Then
Putting Eq. (4) into Eq. (5) we get
Putting x’ from Eq. (4) and t’ from Eq. (6) in Eq. (3) we get
This is an identity and hence the coefficient of various power of x and t must vanish separately. Equating the coefficient of t*t to zero, we get
Putting this value in Eq. (5) we get
Also putting in Eq. (6), we get the equation
Therefore Lorentz Transformation Equations are:
Inverse Transformation Equations are: