Proper velocity
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In relativity, proper velocity (also known as celerity) w of an object relative to an observer is the ratio between observer-measured
displacement vector In geometry and mechanics, a displacement is a vector whose length is the shortest distance from the initial to the final position of a point P undergoing motion. It quantifies both the distance and direction of the net or total motion along a ...
\textbf and
proper time In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The proper time interval b ...
elapsed on the clocks of the traveling object: :\textbf = \frac It is an alternative to ordinary
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
, 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 Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical mec ...
. For example, proper velocity equals
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
per unit
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
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 The Lorentz factor or Lorentz term is a quantity expressing how much the measurements of time, length, and other physical properties change for an object while that object is moving. The expression appears in several equations in special relativit ...
''γ'': :\textbf = \frac\frac=\textbf \,\gamma(v) where ''t'' is
coordinate time In the theory of relativity, it is convenient to express results in terms of a spacetime coordinate system relative to an implied observer. In many (but not all) coordinate systems, an event is specified by one time coordinate and three spatial ...
or "map time". For unidirectional motion, each of these is also simply related to a traveling object's hyperbolic velocity angle or
rapidity In relativity, rapidity is commonly used as a measure for relativistic velocity. Mathematically, rapidity can be defined as the hyperbolic angle that differentiates two frames of reference in relative motion, each frame being associated with di ...
''η'' by :\eta = \sinh^\frac = \tanh^\frac = \pm \cosh^\gamma .


Introduction

In
flat spacetime In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inerti ...
, proper velocity is the ratio between distance traveled relative to a reference map frame (used to define simultaneity) and
proper time In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The proper time interval b ...
Ï„ 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 William Asahel Shurcliff (March 27, 1909 – June 20, 2006) was an American physicist. Biography He received his BA cum laude in 1930, a PhD in Physics in 1934, and a degree in Business Administration in 1935, all from Harvard University. In the 1 ...
's monograph
William Shurcliff William Asahel Shurcliff (March 27, 1909 – June 20, 2006) was an American physicist. Biography He received his BA cum laude in 1930, a PhD in Physics in 1934, and a degree in Business Administration in 1935, all from Harvard University. In the 1 ...
(1996) ''Special relativity: the central ideas'' (19 Appleton St, Cambridge MA 02138)
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 In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
and
geometric algebra In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the ge ...
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 In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the u ...
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 Hermann Minkowski (; ; 22 June 1864 – 12 January 1909) was a German mathematician and professor at Königsberg, Zürich and Göttingen. He created and developed the geometry of numbers and used geometrical methods to solve problems in number t ...
'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 The word "mass" has two meanings in special relativity: ''invariant mass'' (also called rest mass) is an invariant quantity which is the same for all observers in all reference frames, while the relativistic mass is dependent on the velocity of ...
.


Proper velocity addition formula

The proper velocity addition formula:Analytic hyperbolic geometry and Albert Einstein's special theory of relativity
Abraham A. Ungar, World Scientific, 2008,
:\mathbf \oplus \mathbf=\mathbf+\mathbf+\left\ \mathbf where \beta_\mathbf is the beta factor given by \beta_\mathbf=\frac. This formula provides a proper velocity
gyrovector space A gyrovector space is a mathematical concept proposed by Abraham A. Ungar for studying hyperbolic geometry in analogy to the way vector spaces are used in Euclidean geometry.Abraham A. Ungar (2005), "Analytic Hyperbolic Geometry: Mathematical Fo ...
model of
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...
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= \sqrt = \frac = \cosh(\eta) \equiv \frac


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

:\frac \equiv \frac \frac = \frac \frac = \sinh(\eta)\equiv \frac = \pm \sqrt


Coordinate velocity: v ≤ c

:\frac \equiv \frac \frac = \frac \frac = \tanh(\eta) \equiv \frac = \pm \sqrt


Hyperbolic velocity angle or rapidity

:\eta = \sinh^\left(\frac\right) = \tanh^\left(\frac\right) = \pm \cosh^ \left(\gamma \right) or in terms of logarithms: :\eta = \ln\left(\frac + \sqrt\right) = \frac \ln\left(\frac\right) = \pm \ln\left(\gamma + \sqrt\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 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 The Large Electron–Positron Collider (LEP) was one of the largest particle accelerators ever constructed. It was built at CERN, a multi-national centre for research in nuclear and particle physics near Geneva, Switzerland. LEP collided electr ...
(LEP) at Cern in 1989 would have had a Lorentz factor γ of about 88,000 (45 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 The International Linear Collider (ILC) is a proposed linear particle accelerator. It is planned to have a collision energy of 500 GeV initially, with the possibility for a later upgrade to 1000 GeV (1 TeV). Although early proposed ...
(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 :\frac=w_= \gamma_ v_ =\gamma_ \gamma_ \left( v_+v_ \right) = \gamma_ w_+w_ \gamma_\, where A, B and C refer to different objects or frames of reference. For example, ''w''AC 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 ''v''AC ~ ''c'' and ''w''AC = 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 ''mc''2 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 Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
, a spaceship in orbit around the sun, molecules at room temperature, a fighter jet at Mach 3, one radio wave
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they always ...
, a person moving at one lightyear per traveler year, the pulse of a 1.8 MegaJoule
laser A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The fir ...
, 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 In relativity theory, proper acceleration is the physical acceleration (i.e., measurable acceleration as by an accelerometer) experienced by an object. It is thus acceleration relative to a free-fall, or inertial, observer who is momentarily at ...
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 In the theory of relativity, four-acceleration is a four-vector (vector in four-dimensional spacetime) that is analogous to classical acceleration (a three-dimensional vector, see three-acceleration in special relativity). Four-acceleration has ap ...
. 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 In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...
times map time, i.e. . For constant unidirectional proper acceleration, similar relationships exist between rapidity ''η'' and elapsed proper time Δ''Ï„'', as well as between Lorentz factor ''γ'' and distance traveled Δ''x''. To be specific: :\alpha=\frac=c \frac=c^2 \frac, where as noted above the various velocity parameters are related by :\eta = \sinh^\left(\frac\right) = \tanh^\left(\frac\right) = \pm \cosh^\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 ³mid during which the time elapsed on map clocks will be Δt = 4(c/α)sinh osh−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 The Andromeda Galaxy (IPA: ), also known as Messier 31, M31, or NGC 224 and originally the Andromeda Nebula, is a barred spiral galaxy with the diameter of about approximately from Earth and the nearest large galaxy to the Milky Way. The gala ...
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.Extract of page 365
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See also

*
Kinematics Kinematics is a subfield of physics, developed in classical mechanics, that describes the Motion (physics), motion of points, Physical object, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause ...
: for studying ways that position changes with time *
Lorentz factor The Lorentz factor or Lorentz term is a quantity expressing how much the measurements of time, length, and other physical properties change for an object while that object is moving. The expression appears in several equations in special relativit ...
: ''γ'' = ''dt''/''d''τ or kinetic energy over ''mc''2 *
Rapidity In relativity, rapidity is commonly used as a measure for relativistic velocity. Mathematically, rapidity can be defined as the hyperbolic angle that differentiates two frames of reference in relative motion, each frame being associated with di ...
: hyperbolic velocity angle in imaginary radians *
Four-velocity In physics, in particular in special relativity and general relativity, a four-velocity is a four-vector in four-dimensional spacetimeTechnically, the four-vector should be thought of as residing in the tangent space of a point in spacetime, spacet ...
: combining travel through time and space * Uniform acceleration: holding coordinate acceleration fixed *
Gullstrand–Painlevé coordinates Gullstrand–Painlevé coordinates are a particular set of coordinates for the Schwarzschild metric – a solution to the Einstein field equations which describes a black hole. The ingoing coordinates are such that the time coordinate follows the ...
: free-float frames in curved spacetime.


Notes and references

{{reflist


External links


''Spacetime Physics'' by Edwin F. Taylor and John Archibald Wheeler
Minkowski spacetime