Copyright © 1999 David E. Rutherford
All Rights Reserved

The Velocity Four-vector

The components of the velocity four-vector, , of a body in motion are

where is the proper time and

We will represent the velocity four-vector by . The magnitude of the four-velocity of any body, as we will now show, is invariant.

Let two events occur at the same place, but at different times in an inertial frame. Since an observer at rest in the frame measures no space interval between the events, his time measurement is the proper time, , and his four-velocity is the four-velocity of the frame. The element of interval in this case is then

Dividing both sides of Eq. (12) by , we get

or

We note that the right-hand side of Eq. (14) is the square of the magnitude of the four-velocity, , so we have

Since represents an arbitrary four-velocity, we conclude that the magnitude of the four-velocity of any body is always equal to . Therefore, every body is perpetually travelling at the speed of light through spacetime. The magnitude of can never be changed; only its direction can be altered. Therefore, any acceleration is at right angles to the four-velocity. A body which appears to be stationary has four-velocity, , and a body moving so that has four-velocity, , where .

The velocity, , can be derived from the four-velocity, , by first dividing Eq. (12) by , then rearranging terms to get

Using Eqs. (10) and (16), we can write

or

We see that is always greater than , and that as approaches , the magnitude of approaches infinity.

Copyright © 1999 David E. Rutherford
All Rights Reserved