Thursday, 29 November 2012

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:

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