In relativity, proper velocity (also known as celerity) w of an object relative to an observer is the ratio between observer-measured displacement vector x {\displaystyle {\textbf {x}}} and proper time τ elapsed on the clocks of the traveling object:
w = d x d τ {\displaystyle {\textbf {w}}={\frac {d{\textbf {x}}}{d\tau }}}
It is an alternative to ordinary velocity, the distance per unit time where both distance and time are measured by the observer. The two types of velocity, ordinary and proper, are very nearly equal at low speeds. However, at high speeds proper velocity retains many of the properties that velocity loses in relativity compared with Newtonian theory. For example, proper velocity equals momentum per unit mass at any speed, and therefore has no upper limit. At high speeds, as shown in the figure at right, it is proportional to an object's energy as well. Proper velocity w can be related to the ordinary velocity v via the Lorentz factor γ:
w = d x d t d t d τ = v γ ( v ) {\displaystyle {\textbf {w}}={\frac {d{\textbf {x}}}{dt}}{\frac {dt}{d\tau }}={\textbf {v}}\,\gamma (v)}
where t is coordinate time or "map time". For unidirectional motion, each of these is also simply related to a traveling object's hyperbolic velocity angle or rapidity η by
η = sinh − 1 w c = tanh − 1 v c = ± cosh − 1 γ {\displaystyle \eta =\sinh ^{-1}{\frac {w}{c}}=\tanh ^{-1}{\frac {v}{c}}=\pm \cosh ^{-1}\gamma } .
Introduction In flat spacetime, proper velocity is the ratio between distance traveled relative to a reference map frame (used to define simultaneity) and proper time τ elapsed on the clocks of 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 and unlike four-vector velocity, it resides in the three-dimensional slice of spacetime defined by the map frame. As shown below and in the example figure at right, proper-velocities even add as three vectors with rescaling of the out-of-frame component. This makes them more useful for map-based (e.g. engineering) applications, and less useful for gaining coordinate-free insight. Proper speed divided by lightspeed c is the hyperbolic sine of rapidity η, just as the Lorentz factor γ is rapidity's hyperbolic cosine, and coordinate speed v over lightspeed is rapidity's hyperbolic tangent. Imagine an object traveling through a region of spacetime locally described by Hermann Minkowski's flat-space metric equation (cdτ)2 = (cdt)2 − (dx)2. Here a reference map frame of yardsticks and synchronized clocks define map position x and map time t respectively, and the d preceding a coordinate means infinitesimal change. A bit of manipulation allows one to show that proper velocity w = dx/dτ = γv where as usual coordinate velocity v = dx/dt. Thus finite w ensures that v is less than lightspeed c. By grouping γ with v in the expression for relativistic momentum p, proper velocity also extends the Newtonian form of momentum as mass times velocity to high speeds without a need for relativistic mass.
Proper velocity addition formula The proper velocity addition formula:
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