Velocity Transformation

Considering the Lorentz transformations for the spatial coordinates and time, the velocity transformation from one frame of reference to another can be derived.

Let us consider a spaceship moving along a direction diagonal to the horizontal and vertical axis of the frames of reference of two observers, O and O' as shown in the diagrams below. For the purpose of the analysis, the velocity of the spaceship is constant for the two frames of reference. These frames of reference are moving at the velocity v along the horizontal axis with respect to each other.

 The frames of reference cross their origin of coordinate at a given instant of time, at which the two observer agree to synchronize their clocks to read . For the purpose of the illustrations, the horizontal axis are not in top of each other at this time but it is assumed that they are coincident at the origin of coordinates at this time. The spaceship is moving diagonally along the direction indicated by the arrow. The velocity of the spaceship is different for the two different observers, what formula does relate their velocities? The relation between the velocities can be obtained by analyzing the position and the time later on the motion of the spaceship. Later on the motion, the spaceship is at the position represented at the drawing on the left with corresponding coordinates for the two frames of reference. The times indicated by the clocks for the two frames of reference is also shown on this drawing. For the two frames of reference the constant velocities can be calculated accordingly with the usual definition of speed. Since the spaceship is moving diagonally with respect to the horizontal and vertical axis, the vector velocity must be broken down into component along the vertical and horizontal axis. Thus, for the horizontal coordinates,       and     while for the vertical coordinates the relations are the following,       and

Since the velocities are constant, the derivative are not necessaries. To find the relation among the different coordinates, the Lorentz transformations must be used.

For the horizontal components the following derivation results in the final relation,

and the final relation between these two velocities is

The previous result is the addition of velocities for the component of the spaceship velocity along the direction parallel to the relative motion between the two frames of reference.

A similar derivation relates the velocities in the vertical direction,

resulting in

This result relates the component of the velocities perpendicular to the direction of relative motion  between the two frames of reference. Notice that if the velocity is zero along a direction in one frame of reference, the velocity is also zero along any other frame of reference moving parallel to each other. For example in the case presented above the velocity along the Z direction is zero for both frames of reference.

when uz = 0.

 by Luis F. Sáez, Ph. D. Comments and Suggestions: LSaez@dallaswinwin.com