Proper velocity

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Log-log plot of γ (blue), v/c (cyan), and η (yellow) versus proper velocity w/c (i.e. momentum p/mc). Note that w/c is tracked by v/c at low speeds and by γ at high speeds. The dashed red curve is γ-1 (kinetic energy K/mc2), while the dashed magenta curve is the relativistic Doppler factor.
Log-log plot of γ (blue), v/c (cyan), and η (yellow) versus proper velocity w/c (i.e. momentum p/mc). Note that w/c is tracked by v/c at low speeds and by γ at high speeds. The dashed red curve is γ-1 (kinetic energy K/mc2), while the dashed magenta curve is the relativistic Doppler factor.

Proper-velocity, the distance traveled per unit time elapsed on the clocks of a traveling object, equals coordinate velocity at low speeds. At any speed it equals momentum per unit mass, and it therefore has no upper limit. It is one of three related derivatives in special relativity (coordinate velocity v=dx/dt, proper-velocity w=dx/dτ, and Lorentz factor γ=dt/dτ) that describe an object's rate of travel. Each of these is also simply related to a traveling object's hyperbolic velocity angle or rapidity η.

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[1] mentioned its early use in the Sears and Brehme text[2]. Fraundorf has explored its pedagogical value[3] while Ungar[4], Baylis[5] and Hestenes[6] have examined its relevance from group theory and geometric algebra perspectives. Proper-velocity is sometimes referred to as celerity[7].

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[8].

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[edit] Applications

[edit] Comparing proper velocities at high speed

Proper-velocity is useful for comparing the speed of objects with momentum per unit mass (w) 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 (LEP) at Cern in 1989 would have had a Lorentz factor γ of about 88,000 (90 GeV divided by the electron rest mass of 511 keV). 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[9] (ILC) 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 wAC = γABγBC(vAB+vBC) where A, B and C refer to different objects or frames of reference[10]. 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 (A and C) in a head-on collision at 45 GeV in the lab frame (B) would see the other coming toward them at vAC ~c and wAC = 88,0002(1+1) ~1.55×1010 lightseconds per traveler second. Thus colliders can explore higher-speed collisions than can fixed-target accelerators.

[edit] Proper-velocity-based dispersion relations

Plots of (γ-1)c2 × mass, versus proper velocity × mass, for a range of mass values along both axes.
Plots of (γ-1)c2 × mass, versus proper velocity × mass, for a range of mass values along both axes.

Plotting "(γ-1) 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 (horizontally related) that carry different amounts of momentum, as well as how the speed of a low-mass object compares (by vertical extrapolation) to the speed after perfectly inelastic collision with a large object at rest. Highly sloped lines (rise/run=2) 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.

[edit] Unidirectional acceleration via proper velocity

In flat spacetime, proper acceleration is the three-vector acceleration experienced in the instantaneously-varying frame of an accelerated object[11]. Its magnitude α is the frame-invariant magnitude of that object's four-acceleration. Proper-acceleration is also useful from the vantage point (or spacetime slice) of an observer. 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. Δw=αΔt for constant α. At low speeds this reduces to the well-known relation between coordinate velocity and coordinate acceleration times map-time, i.e. Δv=aΔt. 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:

\alpha=\frac{\Delta w}{\Delta t}=c \frac{\Delta \eta}{\Delta \tau}=c^2 \frac{\Delta \gamma}{\Delta x},

where as noted above the various velocity parameters are related by

\eta = \sinh^{-1}[\frac{w}{c}] = \tanh^{-1}[\frac{v}{c}] = \pm \cosh^{-1}[\gamma] .

These equations describe some consequences of accelerated travel at high speed. For example, imagine a spaceship that can accelerate its passengers at "1-gee" (or 1.03 lightyears/year2) halfway to their destination, and then decelerate them at "1-gee" 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 mid-point Lorentz factor (up from its unit rest value) of γmid=1+α(ΔxAB/2)/c2. Hence the round-trip time on traveler clocks will be Δτ = 4(c/α)cosh-1[γmid], during which the time elapsed on map clocks will be Δt = 4(c/α)sinh[cosh-1[γmid]].

Plot of velocity parameters and times on the horizontal axis, versus position on the vertical axis, for an accelerated twin roundtrip to a destination with ΔxAB=10c2/α ~10 lightyears away if α~9.8 m/s2.
Plot of velocity parameters and times on the horizontal axis, versus position on the vertical axis, for an accelerated twin roundtrip to a destination with ΔxAB=10c2/α ~10 lightyears away if α~9.8 m/s2.

This imagined spaceship could offer round trips to Proxima Centauri lasting about 7.1 traveler years (~12 years on earth clocks), round trips to the Milky Way's central black hole of about 40 years (~54,000 years elapsed on earth clocks), and round trips to Andromeda Galaxy lasting around 57 years (over 5 million years on earth clocks). Unfortunately, sustaining 1-gee acceleration for years is easier said than done.

[edit] See Also

[edit] Notes and References

  1. ^ W. A. Shurcliff (1996) Special relativity: the central ideas (19 Appleton St, Cambridge MA 02138)
  2. ^ Francis W. Sears & Robert W. Brehme (1968) Introduction to the theory of relativity (Addison-Wesley, NY) LCCN 680019344, section 7-3
  3. ^ P. Fraundorf (1996) "A one-map two-clock approach to teaching relativity in introductory physics" (arXiv:physics/9611011)
  4. ^ A. A. Ungar (2006) "The relativistic proper-velocity transformation group", Progress in Electromagnetics Research 60, 85-94.
  5. ^ W. E. Baylis (1996) Clifford (geometric) algebras with applications to physics (Springer, NY) ISBN 0-8176-3868-7
  6. ^ D. Hestenes (2003) "Spacetime physics with geometric algebra", Am. J. Phys. 71, 691-714
  7. ^ Bernard Jancewicz (1988) Multivectors and Clifford algebra in electrodynamics (World Scientific, NY) ISBN 9971502909
  8. ^ G. Oas (2005) "On the use of relativistic mass in various published works" (arXiv:physics/0504111)
  9. ^ B. Barish, N. Walker and H. Yamamoto, "Building the next generation collider" Scientific American (Feb 2008) 54-59
  10. ^ This velocity-addition rule is easily derived from rapidities α and β, since Sinh[α+β]=Cosh[α]Cosh[β](Tanh[α]+Tanh[β]).
  11. ^ Edwin F. Taylor & John Archibald Wheeler (1966 1st ed. only) Spacetime Physics (W.H. Freeman, San Francisco) ISBN 0-7167-0336-X

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