Proper velocity the traveling object. It equals the object's momentum p divided by its rest mass "m", and is made up of the space-like components of the object's four-vector velocity. William Shurcliff's monograph mentioned its early use in the Sears and Brehme text. Fraundorf has explored its pedagogical value while Ungar, Baylis and Hestenes have examined its relevance from group theory and geometric algebra perspectives. Proper velocity is sometimes referred to as celerity. Unlike the more familiar coordinate velocity v, proper velocity is synchrony-free (does not require synchronized clocks) and is useful for describing both super-relativistic and sub-relativistic motion. Like coordinate velocity
Velocity-addition formula v \times(\mathbf v \times \mathbf u')\right],\end{align}</math> where the last expression is by the standard vector analysis formula . The first expression extends to any number of spatial dimensions, but the cross product is defined in three dimensions only. The objects with having velocity relative to and having velocity relative to can be anything. In particular, they can be three frames, or they could be the laboratory, a decaying particle and one of the decay products of the decaying particle. The relativistic addition of 3-velocities is non-linear for any real numbers and , although it is true that Also, due to