Transformation of the Alpha Factor Between Inertial Frames

Suppose we have two inertial frames O and O' with O' moving along the x axis of O with a constant speedA particle moving with a velocityin the inertial frame O has a momentumwhereThe same particle in the inertial frame O' has a momentumwhere

In the inertial frame O'(1)

The velocity transformations from O to O' are

Substitution of these expressions into (1) gives

In the inertial frame Oso

Hence

and

Multiplying out the brackets in the numerator and simplifying gives

Now take the square root to give

hence