Proper velocity


In relativity, proper velocity, also known as celerity, is an alternative to velocity for measuring motion. Whereas velocity relative to an observer is distance per unit time where both distance and time are measured by the observer, proper velocity relative to an observer divides observer-measured distance by the time elapsed on the clocks of the traveling object. Proper velocity equals velocity at low speeds. Proper velocity at high speeds, moreover, 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 defined via two derivatives, from the coordinate velocity v and the Lorentz factor γ:
Linking with the naming scheme for proper velocity, is referred to as proper time and t as 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

Introduction

In flat spacetime, proper velocity is the ratio between distance traveled relative to a reference map frame 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 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 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. 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 where as usual coordinate velocity. 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:
where is the beta factor given by.
This formula provides a proper velocity gyrovector space model of hyperbolic geometry that uses a whole space compared to other models of hyperbolic geometry which use discs or half-planes.
In physics notation, therefore, local proper-velocities w ≡ dx/dτ add as 3-vectors much like coordinate-velocities at low speed, provided that we rescale the magnitude of the "out-of-frame" vector. In other words:
where Lorentz-factor γ = 1/β, and the magnitude of wAB is rescaled into frame C according to:
In the unidirectional case this becomes commutative and simplifies to a Lorentz factor product times a coordinate velocity sum, e.g. to, as discussed in the application section below.

Relation to other velocity parameters

Speed table

The table below illustrates how the proper velocity of or "one map-lightyear per traveler-year" is a natural benchmark for the transition from sub-relativistic to super-relativistic motion.

Condition/ParameterCoordinate velocity v
dx/dt in units of c		Velocity angle η
in i-radians		Proper velocity w
dx/dτ in units of c		Lorentz factor γ
dt/dτ = E/mc2
Traveler stopped in map-frame ⇔
1 map-year/traveler-year		0001
Momentum = ½mc ⇔
0.5 map-lightyear/traveler-year		1/ ≅ 0.447ln ≅ 0.481½/2 ≅ 1.118
Rapidity of 0.5 hyperbolic radian/ ≅ 0.462½½ ≅ 0.521½ ≅ 1.128
Coordinate velocity = ½c ⇔
0.5 map-lightyear/map-year		½½ln ≅ 0.5491/ ≅ 0.5772/ ≅ 1.155
Momentum = mc ⇔
1 map-lightyear/traveler-year		1/ ≅ 0.707ln ≅ 0.8811 ≅ 1.414
Rapidity of 1 hyperbolic radian/ ≅ 0.7611½ ≅ 1.175½ ≅ 1.543
Kinetic energy = mc2 ⇔
2 map-years/traveler-year		/2 ≅ 0.866ln ≅ 1.317 ≅ 1.7322
Momentum = 2mc ⇔
2 map-lightyears/traveler-year		2/ ≅ 0.894ln ≅ 1.4442 ≅ 2.236
Rapidity of 2 hyperbolic radians/ ≅ 0.9642½ ≅ 3.627½ ≅ 3.762
Coordinate velocity = c ⇔
1 map-lightyear/map-year		1∞∞∞


Note from above that velocity angle η and proper-velocity w run from 0 to infinity and track coordinate-velocity when. On the other hand, when, proper velocity tracks Lorentz factor while velocity angle is logarithmic and hence increases much more slowly.

Interconversion equations

The following equations convert between four alternate measures of speed that flow from Minkowski's flat-space metric equation:

Lorentz factor γ: energy over mc2 ≥ 1

Proper velocity ''w'': momentum per unit mass

Coordinate velocity: v ≤ c

Hyperbolic velocity angle or rapidity

or in terms of logarithms:

Applications

Comparing velocities at high speed

Proper velocity is useful for comparing the speed of objects with momentum per unit rest mass greater than lightspeed c. The coordinate speed of such objects is generally near lightspeed, whereas proper velocity tells us how rapidly they are covering ground on traveling-object clocks. This is important for example if, like some cosmic ray particles, the traveling objects have a finite lifetime. Proper velocity also clues us in to the object's momentum, which has no upper bound.
For example, a 45 GeV electron accelerated by the Large Electron–Positron Collider at Cern in 1989 would have had a Lorentz factor γ of about 88,000. Its coordinate speed v would have been about sixty four trillionths shy of lightspeed c at 1 lightsecond per map second. On the other hand, its proper speed would have been w = γv ~ 88,000 lightseconds per traveler second. By comparison the coordinate speed of a 250 GeV electron in the proposed International Linear Collider will remain near c, while its proper speed will significantly increase to ~489,000 lightseconds per traveler second.
Proper velocity is also useful for comparing relative velocities along a line at high speed. In this case
where A, B and C refer to different objects or frames of reference. For example, wAC refers to the proper speed of object A with respect to object C. Thus in calculating the relative proper speed, Lorentz factors multiply when coordinate speeds add.
Hence each of two electrons in a head-on collision at 45 GeV in the lab frame would see the other coming toward them at vAC ~ c and wAC = 88,0002 ~ 1.55×1010 lightseconds per traveler second. Thus from the target's point of view, colliders can explore collisions with much higher projectile energy and momentum per unit mass.

Proper velocity-based dispersion relations

Plotting " versus proper velocity" after multiplying the former by mc2 and the latter by mass m, for various values of m yields a family of kinetic energy versus momentum curves that includes most of the moving objects encountered in everyday life. Such plots can for example be used to show where lightspeed, Planck's constant, and Boltzmann energy kT figure in.
To illustrate, the figure at right with log-log axes shows objects with the same kinetic energy that carry different amounts of momentum, as well as how the speed of a low-mass object compares to the speed after perfectly inelastic collision with a large object at rest. Highly sloped lines mark contours of constant mass, while lines of unit slope mark contours of constant speed.
Objects that fit nicely on this plot are humans driving cars, dust particles in Brownian motion, a spaceship in orbit around the sun, molecules at room temperature, a fighter jet at Mach 3, one radio wave photon, a person moving at one lightyear per traveler year, the pulse of a 1.8 MegaJoule laser, a 250 GeV electron, and our observable universe with the blackbody kinetic energy expected of a single particle at 3 kelvin.

Unidirectional acceleration via proper velocity

at any speed is the physical acceleration experienced locally by an object. In spacetime it is a three-vector acceleration with respect to the object's instantaneously varying free-float frame. Its magnitude α is the frame-invariant magnitude of that object's four-acceleration. Proper acceleration is also useful from the vantage point of external observers. Not only may observers in all frames agree on its magnitude, but it also measures the extent to which an accelerating rocket "has its pedal to the metal".
In the unidirectional case i.e. when the object's acceleration is parallel or anti-parallel to its velocity in the spacetime slice of the observer, the change in proper velocity is the integral of proper acceleration over map time i.e. for constant α. At low speeds this reduces to the well-known relation between coordinate velocity and coordinate acceleration times map time, i.e.. For constant unidirectional proper acceleration, similar relationships exist between rapidity η and elapsed proper time Δτ, as well as between Lorentz factor γ and distance traveled Δx. To be specific:
where as noted above the various velocity parameters are related by
These equations describe some consequences of accelerated travel at high speed. For example, imagine a spaceship that can accelerate its passengers at 1 g halfway to their destination, and then decelerate them at 1 g for the remaining half so as to provide earth-like artificial gravity from point A to point B over the shortest possible time. For a map distance of ΔxAB, the first equation above predicts a midpoint Lorentz factor of γmid=1+α/c2. Hence the round-trip time on traveler clocks will be Δτ = 4cosh−1, during which the time elapsed on map clocks will be Δt = 4sinhcosh−1[γmid.
This imagined spaceship could offer round trips to [Proxima Centauri lasting about 7.1 traveler years, round trips to the Milky Way's central black hole of about 40 years, and round trips to Andromeda Galaxy lasting around 57 years. Unfortunately, while rocket accelerations of 1 g can easily be achieved, they cannot be sustained over long periods of time.