TABLE OF CONTENTS We have seen that the quantity dx ?/ds transforms as a 4-vector under a general Lorentz transformation [see Equation (10.67)]. Since ds ? d ? it follows that
also transforms as a 4-vector. This quantity is known as the 4-velocity. Likewise, the quantity
is a 4-vector, and is called the 4-acceleration.
For events along the world-line of a particle traveling with 3-velocity u, we have
where use has been made of Equation (10.100). This gives the relationship between a particle's 3-velocity and its 4-velocity. The relationship between the 3-acceleration and the 4-acceleration is less straightforward. We have
where a = d u/dt is the 3-acceleration. In the rest frame of the particle, 
and 
. It follows that
(note that 
is an invariant quantity). In other words, the 4-acceleration of a particle is always orthogonal to its 4-velocity.
