Proper velocity

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 == Proper velocity addition formula ==

The proper velocity addition formula: : \mathbf{u} \oplus \mathbf{v}=\mathbf{u}+\mathbf{v}+\left\{ {\frac{\beta_\mathbf{u}}{1+\beta_\mathbf{u}}}{\frac{\mathbf{u}\cdot\mathbf{v}}{c^2}} + {\frac{1 - \beta_\mathbf{v}}{\beta_\mathbf{v}}} \right\} \mathbf{u} where \beta_\mathbf{w} is the beta factor given by \beta_\mathbf{w}=\frac{1}{\sqrt{1+\frac{|\mathbf{w}|^2}{c^2}}}. 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 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. 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 (or unidirectional velocity) that flow from Minkowski's flat-space metric equation: : (c \delta \tau)^2 = (c \delta t)^2 - (\delta x)^2.\, . Lorentz factor γ: energy over mc2 ≥ 1 : \gamma \equiv \frac{dt}{d\tau}= \sqrt{1+\left(\frac{w}{c}\right)^2} = \frac{1}{\sqrt{1-(\frac{v}{c})^2}} = \cosh(\eta) \equiv \frac{e^{\eta} + e^{-\eta}}{2} Proper velocity w: momentum per unit mass : \frac{w}{c} \equiv \frac{1}{c} \frac{dx}{d\tau} = \frac{v}{c} \frac{1}{\sqrt{1-(\frac{v}{c})^2}} = \sinh(\eta)\equiv \frac{e^{\eta} - e^{-\eta}}{2} = \pm \sqrt{\gamma^2 - 1} Coordinate velocity: v ≤ c : \frac{v}{c} \equiv \frac{1}{c} \frac{dx}{dt} = \frac{w}{c} \frac{1}{\sqrt{1 + (\frac{w}{c})^2}} = \tanh(\eta) \equiv \frac{e^{2\eta} - 1} {e^{2\eta} + 1}= \pm \sqrt{1 - \left(\frac{1}{\gamma}\right)^2} Hyperbolic velocity angle or rapidity : \eta = \sinh^{-1}\left(\frac{w}{c}\right) = \tanh^{-1}\left(\frac{v}{c}\right) = \pm \cosh^{-1} \left(\gamma \right) or in terms of logarithms: : \eta = \ln\left(\frac{w}{c} + \sqrt{\left(\frac{w}{c}\right)^2 + 1}\right) = \frac{1}{2} \ln\left(\frac{1+\frac{v}{c}}{1-\frac{v}{c}}\right) = \pm \ln\left(\gamma + \sqrt{\gamma^2 - 1}\right). == Applications ==

Comparing velocities at high speed Proper velocity is useful for comparing the speed of objects with momentum per unit rest mass (w) greater than lightspeed c. The coordinate speed of such objects is generally near lightspeed, whereas proper velocity indicates 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 had a Lorentz factor γ of about 88,000 (45 GeV divided by the electron rest mass of 511 keV). Its coordinate speed v was about sixty-four trillionths shy of lightspeed c at 1 light-second per map second. On the other hand, its proper speed was w = γv ~ 88,000 light-seconds per traveler second. By comparison the coordinate speed of a 250 GeV electron in the proposed International Linear Collider (ILC) will be only minutely closer to 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 : \frac{p_{AC}}{m_1}=w_{AC}= \gamma_{AC} v_{AC} =\gamma_{AB} \gamma_{BC} \left( v_{AB}+v_{BC} \right) = \gamma_{AB} w_{BC}+w_{AB} \gamma_{BC}\, 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 (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 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 "(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 the speed of light, the Planck 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. Unidirectional acceleration via proper velocity Proper acceleration 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 (or spacetime slice) 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: : \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}\left(\frac{w}{c}\right) = \tanh^{-1}\left(\frac{v}{c}\right) = \pm \cosh^{-1}\left(\gamma\right). These equations describe some consequences of accelerated travel at high speed. For example, imagine a spaceship that can accelerate its passengers at 1 g (or 1.03 lightyears/year2) 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 (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. 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, while rocket accelerations of 1 g can easily be achieved, they cannot be sustained over long periods of time. == See also ==