Lorentz Transformations

How do the coordinates from different frames of reference moving with respect to each other relate? In this section, the relation between different coordinates systems will be studied.

 In the drawing at the left, two frames of reference move with respect to each other with the velocity v.  These two frames are O and O' . The coordinates of the vertex of a triangle is at position x' with respect to the reference frame O' at this instant of time, as shown in the diagram. The observer O wants to determine that coordinate, x'O. For observer O, the coordinates of the vertex of the triangle are at the coordinate x. Thus, the following relation between the time, velocity, and the coordinates x and x'O can be established: . This relation between the coordinates is conceptually equivalent to the classical result or Galileo Galilei transformation.  From the special relativity point of view, this coordinate, x'O, can not be measured with a ruler; thus, this is an improper length. On the other side, the coordinate x' belongs to the frame of reference O' implying that this coordinate is a proper length. Therefore, the relation between these coordinates is .

Putting together the two boxed relations, the following result is obtained

When the motion between the different frames of reference is only along one of the axis, the other coordinates remain unchanged; that is, y' = y and z' = z.

The time measured by different frames of reference are different. The following discussion will help to understand the relation between the times measured by the two frames of reference.

 The two frames of reference O and O' represented in the drawing on the left moves with respect to each other with the velocity v along their horizontals axis. When their origins coincide, a pulse of light is emitted at the origin of coordinates. Precisely, at that instance, the two clocks, at the origin of both systems, are synchronized to read zero time. In the drawing, the top clock is fixed to the origin of coordinates of the coordinates O' while the bottom clock is fixed to the coordinates represented by O.

After a time t, with respect to frame of reference O, and t', with respect to frame of reference O', the pulse is used for synchronizing secondary clocks in both frames of reference. The yellow circle correspond to the pulse of light as observed by the observer O while the orange circle correspond to the view of the same pulse from the point of view of the observer O'. The coordinates x and x' correspond to the distance traveled by the pulse in the two frames of reference. From these coordinates, the times t and t' can be calculated considering that the light travels at the same speed c for both frames of reference,

and

The relation between x' and x was obtained above for the Lorentz transformation for spatial dimensions.

This result is completely different than the classical mechanic result that where the time is always the same independent of the motion of the frame of reference. In this case, the times run different for different frames of reference in motion with respect to each other.

The following table resume the result encounter for the Lorentz Transformations.

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