New Transformation Equations and the Electric Field Four-vector

1. Transformation of Velocity

It might be helpful to elaborate on the transformation, or addition, of velocity four-vectors as defined in Eqs. (11) of The Velocity Four-vector in Original Article: May 7, 1999

on this website. The magnitude of any velocity four-vector is always the same. It is always, c, the speed of light. This result can be obtained by dividing Eq. (9) of Additions: June 1, 1999

by , yielding

where

and

If , then , and if , as in the case of light, then .

Using the transformation matrix, , in Eq. (1) of Additions: June, 3, 1999

as the preferred transformation matrix in place of and substituting the velocity four-vector, V, for U in Eqs. (1) above in order to distinguish its components from the components of , we get the equations for the transformation of velocity,

Now, for velocities less than c along the x-axis we have and . The components of the combined velocities in the primed frame are and

At spatial velocities much less than c, we have , and we get the Galilean result and . The signs in the previous sentence are actually "approximately equal to" not "equal to". We see that, for low velocities, most of the velocity is still in the time direction in both primed and unprimed frames of reference. Incidentally, any time the magnitude of the spatial components of a velocity four-vector is less than c, we will also have a time component of the velocity.

For velocities in transverse directions, for example and , we have

For combined velocities along the x-axis at the speed of light and we get , but

In the primed frame, there are no spatial components in the transformed velocity, but there is a component directed along the negative t'-axis. The magnitude of V', however, is still

For spatial velocities at the speed of light in transverse directions, for example and , we have and

The magnitude of V' is still c, but its direction is rotated 90 degrees due to the rotation of the primed coordinates relative to the unprimed coordinates. For spatial velocities at the speed of light in the same direction as in Eq. (11) above, the coordinates associated with V are rotated 90 degrees relative to the unprimed frame, and 180 degrees relative to the primed frame.

E-mail: drutherford@softcom.net