History of Topics in Special Relativity/Four-velocity
History of 4-Vectors ( | )|||
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Overview
editThe w:four-velocity is defined as the rate of change in the w:four-position of a particle with respect to the particle's w:proper time , while the derivative of four-velocity with respect to proper time is four-acceleration), and the product of four-velocity and mass is four-momentum. It can be represented as a function of three-velocity :
- .
Historical notation
editKilling (1884/5)
editw:Wilhelm Killing discussed Newtonian mechanics in non-Euclidean spaces by expressing coordinates (p,x,y,z), velocity v=(p',x',y',z'), acceleration (p”,x”,y”,z”) in terms of four components, obtaining the following relations:[M 1]
If the Gaussian curvature (with k as radius of curvature) is negative the velocity becomes related to the hyperboloid model of hyperbolic space, which at first sight becomes similar to the relativistic four-velocity in Minkowski space by setting with c as speed of light. However, Killing obtained his results by differentiation with respect to Newtonian time t, not relativistic proper time, so his expressions aren't relativistic four-vectors in the first place, in particular they don't involve a limiting speed. Also the dot product of acceleration and velocity differs from the relativistic result.
Poincaré (1905/6)
editw:Henri Poincaré explicitly defined the four-velocity as:[R 1]
- with
which is equivalent to (b) because
Minkowski (1907/8)
editw:Hermann Minkowski employed matrix representation of four-vectors and six-vectors (i.e. antisymmetric second rank tensors) and their product from the outset. He defined the “velocity vector” (Geschwindigkeitsvektor or Raum-Zeit-Vektor Geschwindigkeit):[R 2]
which is equivalent to (b), which he used to form further four-vectors using electromagnetic quantities etc.:
In 1908, he denoted the derivative of the position vector[R 3]
corresponding to (a). He went on to define the derivative of four-velocity with respect to proper time as "acceleration vector" (i.e. four-acceleration), and the product of four-velocity and mass as “momentum vector” (i.e. four-momentum).
Bateman (1909/10)
editThe first discussion of four-velocity in an English language paper (even though in the broader context of w:spherical wave transformations), was given by w:Harry Bateman in a paper read 1909 and published 1910:[R 4]
equivalent to (b), from which he derived four-acceleration and four-jerk:
Ignatowski (1910)
editw:Wladimir Ignatowski defined the “vector of first kind”:[R 5]
equivalent to (b).
Laue (1911)
editIn the first textbook on relativity, w:Max von Laue explicitly used the term “four-velocity” (Vierergeschwindigkeit) for Y:[R 6]
which is equivalent to (b), which he used to formulate the four-potential of an arbitrarily moving point charge de, as well as the four-convection and four-conduction using four-current P:[R 7]
- ,
and the vector products with electromagnetic tensor and displacement tensor and their duals:[R 8]
In the second edition (1913), Laue used four-velocty Y in order to define the four-acceleration , four force K in terms of four-acceleration, and material four-current M (i.e. four-momentum density) using rest mass density :[R 9]
- ,
De Sitter (1911)
editw:Willem De Sitter defined the four-velocity in terms of both velocity and w:Proper velocity as:[R 10]
equivalent to (b).
Wilson/Lewis (1912)
editw:Gilbert Newton Lewis and w:Edwin Bidwell Wilson devised an alternative 4D vector calculus based on w:Dyadics which, however, never gained widespread support. They defined the “extended velocity” as a “1-vector”:[R 11]
equivalent to (a,b).
Kottler (1912)
editw:Friedrich Kottler defined four-velocity as:[R 12]
equivalent to (a,b) from which he derived four-acceleration and four-jerk, and demonstrated its relation to the tangent c_{1}^{(\alpha)} in terms of Frenet-Serret formulas, its derivative with respect to proper time, and its relation to four-acceleration and curvature :[R 13]
and derived four-acceleration and four-jerk:[R 14]
Einstein (1912-14)
editIn an unpublished manuscript on special relativity, written between 1912-1914, w:Albert Einstein defined four-velocity as:[R 15]
equivalent to (b) and used it to define four-momentum by multiplication with rest energy:[R 16]
Silberstein (1914)
editWhile w:Ludwik Silberstein used Biquaternions already in 1911, his first mention of the “velocity-quaternion” was given in 1914. He also defined its conjugate, its Lorentz transformation, the relation of four-acceleration Z, and its relation to the equation of motion as follows:[R 17]
equivalent to (a,b).
References
edit- Mathematical
- ↑ Killing (1884/5), p. 5
- Killing, W. (1885) [1884], "Die Mechanik in den Nicht-Euklidischen Raumformen", Journal für die Reine und Angewandte Mathematik, 98: 1–48
- Relativity
- ↑ Poincaré (1905/06), p. 173
- ↑ Minkowski (1907/8), p. 64–65,73
- ↑ Minkowski (1908), p. 84
- ↑ Bateman (1909/10), p. 253f
- ↑ Ignatowski (1910), p. 23
- ↑ Laue (1911), p. 63
- ↑ Laue (1911), p. 102, 119
- ↑ Laue (1911), p. 124
- ↑ Laue (1913), p. 69–70, 80, 230
- ↑ De Sitter (1911), p. 392
- ↑ Lewis/Wilson (1912), p. 443f
- ↑ Kottler (1912), p. 1663
- ↑ Kottler (1912), p. 1707
- ↑ Kottler (1912), p. 1663
- ↑ Einstein (1912), p. 84
- ↑ Einstein (1912), p. 97
- ↑ Silberstein (1914), p. 183
- Bateman, H. (1910) [1909], "The Transformation of the Electrodynamical Equations", Proceedings of the London Mathematical Society, 8: 223–264
- The Transformation of the Electrodynamical Equations on English Wikisource
- deSitter, W. (1911), "On the bearing of the Principle of Relativity on Gravitational Astronomy", Monthly Notices of the Royal Astronomical Society, 71: 388–415
- Einstein, A. (1912–14), "Einstein's manuscript on the special theory of relativity", The collected papers of Albert Einstein, vol. 4, pp. 3–108
{{citation}}
: CS1 maint: date format (link) - Ignatowsky, W. v. (1910), "Das Relativitätsprinzip", Archiv der Mathematik und Physik 17: 1-24, 18: 17-40
- Das Relativitätsprinzip on German Wikisource
- Kottler, F. (1912), "Über die Raumzeitlinien der Minkowski'schen Welt", Wiener Sitzungsberichte 2a, 121: 1659–1759
- On the spacetime lines of a Minkowski world on English Wikisource
- Laue, M. v. (1911), Das Relativitätsprinzip, Braunschweig: Vieweg
- Laue, M. v. (1913), Das Relativitätsprinzip (2. Edition), Braunschweig: Vieweg
- Lewis, G. N. & Wilson, E. B. (1912), "The Space-time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics", Proceedings of the American Academy of Arts and Sciences, 48: 387–507
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: CS1 maint: multiple names: authors list (link) - Minkowski, H. (1908) [1907], "Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern", Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse: 53–111
- Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern on German Wikisource
- The Fundamental Equations for Electromagnetic Processes in Moving Bodies on English Wikisource
- Minkowski, H. (1909) [1908], "Raum und Zeit", Physikalische Zeitschrift, 10: 75–88
- Raum und Zeit on German Wikisource
- Space and Time on English Wikisource
- Silberstein, L. (1914), The Theory of Relativity, London: Macmillan