(1721) |

also transforms as a 4-vector. This quantity is known as the

is a 4-vector, and is called the

For events along the world-line of a particle traveling with 3-velocity , we have

(1723) |

where use has been made of Equation (1719). 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 is the 3-acceleration. In the rest frame of the particle, and . It follows that

(1725) |

(note that is an invariant quantity). In other words, the 4-acceleration of a particle is always orthogonal to its 4-velocity.