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The frame of reference S is considered as the static frame and S’ is considered as the moving frame of reference.S’ is moving with a speed v relative to the frame S. Suppose that an object has a speed ux’ measured in the S’ frame so that:
Using the expressions for x’ and t’ we get:
But  Thus, the expression for velocity in S’ frame will be,
Similarly, if an object has velocity components along the y and z axes, the components as measured by an observer in S’ are:
When ux < < c then we have,
And, when ux = c, then we get ux’ = c. Thus, an object moving with a speed c relative to an observer in S also has a speed c relative to an observer S’- independent of the relative motion of S and S’. Hence, the speed of light is the ultimate speed.
Consequences of the Lorentz Transformation: Following are the consequences of Lorentz transformation: -
(i) The speed of light in vacuum measured with respect to any observer is always c, irrespective of the motion of the source or the observer.
(ii) Moving rods are contracted by a factor of γ along their direction of motion.
(iii) Simultaneity is relative.
(iv) The length of moving rods remains unchanged perpendicular to their direction of motion.
(v) Moving clocks run slowly by a factor of γ.
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Relativity