New Transformation Equations and the Electric Field Four-vector

1. Times in the Transverse Direction

Using the transformation matrix, ,

from "Additions: July 29, 1999" on this website, we will compare the time components of the spacetime vectors in two frames of reference between two events where there is a separation in time as well as a separation in space transverse to the direction of motion. Let a primed frame of reference move with uniform four-velocity along the x-axis of an unprimed frame. Two events in the unprimed frame with coordinates and are separated by the spacetime vector where

and c is the speed of light. The same two events in the primed frame have coordinates and , respectively. The spacetime vector between these two events in the primed frame is where

For our purposes in this case, we will define the spacetime vector in the unprimed frame as . We wish now to find the spacetime vector between the two events as measured by an observer at rest in the primed frame. To do this, we use the transformation equations

where

and

Expanding Eqs. (4) and inserting the components of and for this case, we get

so the spacetime vector in the primed frame is . Since we have so we can write the last equality in Eqs. (8) as

We see from Eq. (9) that there is a dilation of time in the primed frame transverse to the direction of motion just as there is a dilation in the direction of motion, as we have shown previously. It is interesting to note that, in the direction of motion, there may be a time component of the spacetime vector in one frame while there is none in the other, as shown in "Additions: August 17, 1999" on this website. But in the transverse direction, if there is a time component in one frame there will be a time component in both, and there will be none if there is none in either, so long as there is a time component in the four-velocity. Also we see that none of the components of the spacetime vector in either frame in this case is proper since each spacetime vector contains both space and time components.

E-mail: drutherford@softcom.net