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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 is the product of two other derivatives in special relativity that describe an object's rate of travel: coordinate velocity and the Lorentz factor . 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 (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〔William Shurcliff (1996) ''Special relativity: the central ideas'' (19 Appleton St, Cambridge MA 02138)〕 mentioned its early use in the Sears and Brehme text.〔Francis W. Sears & Robert W. Brehme (1968) ''Introduction to the theory of relativity'' (Addison-Wesley, NY) (LCCN 680019344 ), section 7–3〕 Fraundorf has explored its pedagogical value〔P. Fraundorf (1996) "A one-map two-clock approach to teaching relativity in introductory physics" ((arXiv:physics/9611011 ))〕 while Ungar,〔A. A. Ungar (2006) "(The relativistic proper velocity transformation group )", ''Progress in Electromagnetics Research'' 60, 85–94.〕 Baylis〔W. E. Baylis (1996) ''Clifford (geometric) algebras with applications to physics'' (Springer, NY) ISBN 0-8176-3868-7〕 and Hestenes〔D. Hestenes (2003) "(Spacetime physics with geometric algebra )", ''Am. J. Phys.'' 71, 691–714〕 have examined its relevance from group theory and geometric algebra perspectives. Proper velocity is sometimes referred to as celerity.〔Bernard Jancewicz (1988) ''Multivectors and Clifford algebra in electrodynamics'' (World Scientific, NY) ISBN 9971-5-0290-9〕 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 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.〔G. Oas (2005) "On the use of relativistic mass in various published works" ((arXiv:physics/0504111 ))〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「proper velocity」の詳細全文を読む スポンサード リンク
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