Proper Velocity - Introduction

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 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. This makes it 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 space-time 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.