History of Topics in Special Relativity/Four-current
History of 4-Vectors ( | )|||
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Overview edit
The w:Four-current is the four-dimensional analogue of the w:electric current density
where c is the w:speed of light, the four-velocity, ρ is the w:charge density, the rest charge density , and j the conventional w:current density. Alternatively, it can be defined in terms of the inhomogeneous Maxwell equations as the negative product of the D'Alembert operator and the electromagnetic potential , or the four-divergence of the electromagnetic tensor :
and the generally covariant form
The Lorentz transformation of the four-potential components was given by #Poincaré (1905/6) and #Marcolongo (1906). It was explicitly formulated in modern form by #Minkowski (1907/15) and reformulated in different notations by #Born (1909), #Bateman (1909/10), #Ignatowski (1910), #Sommerfeld (1910), #Lewis (1910), Wilson/Lewis (1912), #Von Laue (1911), #Silberstein (1911). The generally covariant form was first given by #Kottler (1912) and #Einstein (1913).
Historical notation edit
Poincaré (1905/6) edit
w:Henri Poincaré (June 1905[R 1]; July 1905, published 1906[R 2]) showed that the four quantities related to charge density are connected by a Lorentz transformation:
and in his July paper he further stated the continuity equation and the invariance of Jacobian D:[R 3]
Even though Poincaré didn't directly use four-vector notation in those cases, his quantities are the components of four-current (a).
Marcolongo (1906) edit
Following Poincaré, w:Roberto Marcolongo defined the general Lorentz transformation of the components of the four independent variables and its continuity equation:[R 4]
equivalent to the components of four-current (a), and pointed out its relation to the components of the four-potential
equivalent to the components of Maxwell's equations (b).
Minkowski (1907/15) edit
w:Hermann Minkowski from the outset employed vector and matrix representation of four-vectors and six-vectors (i.e. antisymmetric second rank tensors) and their product. In a lecture held in November 1907, published 1915, Minkowski defined the four-current in vacuum with as charge density and as velocity:[R 5]
equivalent to (a), and the electric four-current in matter with as current and as charge density:[R 6]
In another lecture from December 1907, Minkowski defined the “space-time vector current” and its Lorentz transformation[R 7]
equivalent to (a). In moving media and dielectrics, Minkowski more generally used the current density vector “electric current” which becomes in isotropic media:[R 8]
Born (1909) edit
Following Minkowski, w:Max Born (1909) defined the “space-time vector of first kind” (four-vector) and its continuity equation[R 9]
equivalent to (a), and pointed out its relation to Maxwell's equations as the product of the D'Alembert operator with the electromagnetic potential :
equivalent to (c). He also expressed the four-current in terms of rest charge density and four-velocity
equivalent to (b).
Bateman (1909/10) edit
A discussion of four-current in terms of integral forms (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, who defined the Lorentz transformations of its components [R 10]
forming the following invariant relations together with the differential four-position and four-potential:[R 11]
with in relativity.
Ignatowski (1910) edit
w:Wladimir Ignatowski (1910) defined the “vector of first kind” using charge density and three-velocity :[R 12]
equivalent to four-current (a).
Sommerfeld (1910) edit
In influential papers on 4D vector calculus in relativity, w:Arnold Sommerfeld defined the four-current P, which he called four-density (Viererdichte):[R 13]
equivalent to (a). In the second paper he pointed out its relation to four-potential and the electromagnetic tensor (six-vector) f together with the continuity condition:[R 14]
equivalent to Maxwell's equations (c). The scalar product with the four-potential[R 15]
he called “electro-kinetic potential” whereas the vector product with the electromagnetic tensor[R 16]
he called the electrodynamic force (four-force density).
Lewis (1910), Wilson/Lewis (1912) edit
w:Gilbert Newton Lewis (1910) devised an alternative 4D vector calculus based on w:Dyadics which, however, never gained widespread support. The four-current is a “1-vector”:[R 17]
equivalent to (a) and its relation to the four-potential and electromagnetic tensor :
equivalent to (c,d).
In 1912, Lewis and w:Edwin Bidwell Wilson used only real coordinates, writing the above expressions as[R 18]
equivalent to (c,d).
Von Laue (1911) edit
In the first textbook on relativity in 1911, w:Max von Laue elaborated on Sommerfeld's methods and explicitly used the term “four-current” (Viererstrom) of density in relation to four-potential and electromagnetic tensor :[R 19]
equivalent to (a,c,d). He went on to define four-force density F as vector-product with , four-convection K and four-conduction using four-velocity Y:[R 20]
- ,
Silberstein (1911) edit
w:Ludwik Silberstein devised an alternative 4D calculus based on w:Biquaternions which, however, never gained widespread support. He defined the “current-quaternion” (i.e. four-current) C and its relation to the “electromagnetic bivector” (i.e. field tensor) and “potential-quaternion” (i.e. four-potential) [R 21]
Kottler (1912) edit
w:Friedrich Kottler defined the four-current and its relation to four-velocity , four-potential , four-force , electromagnetic field-tensor , stress-energy tensor :[R 22]
equivalent to (a,b,c,d) and subsequently was the first to give the generally covariant formulation of Maxwell's equations using metric tensor [R 23]
equivalent to (e).
Einstein (1913) edit
Independently of Kottler (1912), w:Albert Einstein defined the general covariant four-current in the context of his Entwurf theory (a precursor of general relativity):[R 24]
equivalent to (a), and the generally covariant formulation of Maxwell's equations
equivalent to (e) in the case of being the Minkowski tensor.
Historical sources edit
- ↑ Poincaré (1905a), p. 1505
- ↑ Poincaré (1905b), p. 133–134
- ↑ Poincaré (1905b), p. 134
- ↑ Marcolongo (1906), p. 348-349
- ↑ Minkowski (1907/15), p. 929
- ↑ Minkowski (1907/15), p. 933
- ↑ Minkowski (1907/8), p. 57, 60, 67
- ↑ Minkowski (1907/8), p. 71
- ↑ Born (1909), p. 573-574, 576-577
- ↑ Bateman (1910), p. 241
- ↑ Bateman (1910), p. 252
- ↑ Ignatowsky (1910), p. 24-25
- ↑ Sommerfeld (1910a), p. 751
- ↑ Sommerfeld (1910b), p. 651
- ↑ Sommerfeld (1910a), p. 764
- ↑ Sommerfeld (1910a), p. 770
- ↑ Lewis (1910), p. 176ff
- ↑ Wilson & Lewis (1910), p. 488ff
- ↑ Laue (1911), p. 77f, 100f
- ↑ Laue (1911), p. 80, 102, 119
- ↑ Silberstein (1911), p. 796, 801f
- ↑ Kottler (1912), p. 1661, 1686ff
- ↑ Kottler (1912), p. 1688-1689
- ↑ Einstein & Grossmann (1913), p. 241
- 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
- Born, M. (1909), "Die träge Masse und das Relativitätsprinzip", Annalen der Physik, 333 (3): 571–584
- Einstein, A. & Grossmann, M. (1913), "Entwurf einer verallgemeinerten Relativitätstheorie und eine Theorie der Gravitation", Zeitschrift für Mathematik und Physik, 62: 225–261
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- 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
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- Laue, M. v. (1911), Das Relativitätsprinzip, Braunschweig: Vieweg
- Lewis, G. N. (1910), "On Four-Dimensional Vector Analysis, and Its Application in Electrical Theory", Proceedings of the American Academy of Arts and Sciences, 43 (7): 165–181
- 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
{{citation}}
: CS1 maint: multiple names: authors list (link) - Marcolongo, R. (1906), "Sugli integrali delle equazioni dell'elettrodinamica" (PDF), Atti della Reale Accademia dei Lincei Rendiconti, 15: 344–349
- Minkowski, H. (1915) [1907], "Das Relativitätsprinzip", Annalen der Physik, 352 (15): 927–938
- Das Relativitätsprinzip on German Wikisource
- 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
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- Poincaré, H. (1905), "Sur la dynamique de l'électron", Comptes Rendus, 140: 1504–1508
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- Sur la dynamique de l’électron on French Wikisource
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- Silberstein, L. (1912) [1911], "Quaternionic form of relativity", The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 23 (137): 790–809, doi:10.1080/14786440508637276
- Sommerfeld, A. (1910a), "Zur Relativitätstheorie I: Vierdimensionale Vektoralgebra", Annalen der Physik, 337 (9): 749–776
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- Sommerfeld, A. (1910b), "Zur Relativitätstheorie II: Vierdimensionale Vector Analysis", Annalen der Physik, 338 (14): 649–689
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