History of Topics in Special Relativity/Four-potential

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History of Topics in Special Relativity: History of 4-Vectors (edit)

Overview[edit | edit source]

The w:Electromagnetic four-potential combines both an w:electric scalar potential and a w:magnetic vector potential into a single four-vector satisfying the w:Lorenz gauge condition:

Its product with the w:D'Alembert operator can be related to the four-current or the four-divergence of the w:electromagnetic tensor representing w:Maxwell equations:

and by forming the w:exterior derivative (four-curl) it produces the electromagnetic tensor:

The four-potential has the solution:

The electric part of (b) was given by Riemann in 1858, while the complete equation (b) together with the Lorenz gauge condition of the four-potential was given by L. Lorenz in 1867, all of which was popularized by H. A. Lorentz in 1892. Solution (e) of the four-potential was given by #Herglotz (1904), while the Lorentz transformation of all components of (a) was given by #Poincaré (1905/6) and #Marcolongo (1906). The modern treatment of the four-potential was given by #Minkowski (1907/15) and was elaborated by #Born (1909), #Bateman (1909/10), #Sommerfeld (1910), #Lewis (1910), Wilson/Lewis (1912), #Von Laue (1911/13), #Silberstein (1911), and embedded in a generally covariant treatment of electromagnetism by #Kottler (1912) and #Einstein (1916).

Historical notation[edit | edit source]

Kirchhoff, Riemann, Lorenz, Lorentz (1857-1892)[edit | edit source]

w:Gustav Kirchhoff (1857) defined the w:continuity equation for electric density and the "Kirchhoff gauge condition” for potentials in which ”u,v,w” depend on U,V,W.[R 1] This is similar to the Lorentz gauge condition (L) with a sign change, yet Kirchhoff was still thinking in terms of Weber's electrodynamics involving actions at a distance.

In a lecture given in 1858, published 1867, w:Bernhard Riemann defined the retarded electric potential ”U” satisfying[R 2]

This corresponds to the electrical part of the inhomogeneous electromagnetic wave equation (b1), but Riemann didn't discuss the corresponding magnetic part (b2) in his model.

w:Ludvig Lorenz (1867) gave the first gave a complete formulation of the electromagnetic potential. Elaborating on Kirchhoff's work, Lorenz redefined u, v, w by replacing U, V, W with the retarded magnetic potentials and by the retarded electric potential , independently derived Maxwell's equations, and formulated the following conditions[R 3]

Equations (1) is the continuity condition for the electromagnetic four-current, (2) the Lorenz gauge condition of the four-potential (L), while (3) is used to define the inhomogeneous electromagnetic wave equation.

These methods were popularized by w:Hendrik Lorentz (1892) who incorporated them into his theory of electrons and immobile aether, in which fields and electrons were strictly separated[R 4]

Herglotz (1904)[edit | edit source]

w:Gustav Herglotz – similar to w:Arthur W. Conway in 1903[R 5] – showed that the wave equation in terms of potential [R 6]

has the solution in terms of the complex variable

from which he obtained the retarded potential in terms of charge density and four coordinates as follows

Using these results, Herglotz went on to determine the force exerted by an electron on another one by defining the potential , which he interpreted as the "ordinary four-dimensional mutual potential of two three-dimensional spherical disks located in four-dimensional space", in which is the radius and T the distance of their centers.

#Sommerfeld (1910) called Herglotz's equations "the most natural representation of electrodynamic potential in the sense of relativity theory", which is notable because this solution was given by Herglotz already in 1904 before the spacetime representations of Poincaré and Minkowski. Sommerfeld also remarked that Minkowski privately told him that the four-dimensional symmetry of electrodynamics is latently contained and mathematically applied in Herglotz's paper.[R 7]

Poincaré (1905/6)[edit | edit source]

w:Henri Poincaré in July 1905, published 1906, showed that the four quantities related to the electromagnetic potential (defined in relation to the components of the four-current using the D'Alembert operator) in different frames are related to each other by Lorentz transformations[R 8]

satisfying the Lorenz gauge condition[R 9]

Even though Poincaré didn't directly use four-vector notation in this case, his quantities are the components of the four-potential in arbitrary inertial frames.

Marcolongo (1906)[edit | edit source]

w:Roberto Marcolongo, citing Poincaré, defined the general Lorentz transformation of the components of the four-potential :[R 10]

equivalent to the components of (a) and the relation to the components of the four-current

equivalent to the components of (b).

Minkowski (1907/15)[edit | edit source]

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-potential with as the components of the vector potential, and as scalar potential:[R 11]

[R 12]

equivalent to (a), pointed out the relation to the four-current using the D'Alembert operator

equivalent to (b), as well as the relation to the electromagnetic tensor (which he called "Traktor”) by setting the exterior derivative (four-curl):

equivalent to (d).

Born (1909)[edit | edit source]

w:Max Born (1909) defined the four-potential and the Lorenz gauge condition[R 13]

equivalent to (a), and pointed out that its product with the D'Alembert operator corresponds to the four-current

equivalent to (b), and its exterior derivative (four-curl) forming the electromagnetic tensor

equivalent to (d).

Bateman (1909/10)[edit | edit source]

A discussion of four-potential 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 14]

forming the following invarant relations using differential four-position and four-current:[R 15]

Sommerfeld (1910)[edit | edit source]

In influential papers on 4D vector calculus in relativity, w:Arnold Sommerfeld simplified Minkowski's spacetime formalism and defined the four-potential in relation to the four-current P and the electromagnetic tensor (six-vector) f together with the Lorenz gauge condition:[R 16]

equivalent to (a,b,c,d), with the Herglotz solution[R 17]

equivalent to (e). He also formulated the "electro-kinetic potential” as the scalar product with the four-current[R 18]

Lewis (1910), Wilson/Lewis (1912)[edit | edit source]

w:Gilbert Newton Lewis (1910) devised an alternative 4D vector calculus based on w:Dyadics which, however, never gained widespread support. The four-potential is a “1-vector”[R 19]

equivalent to (a), and its relation to the four-current and electromagnetic tensor :

equivalent to (b,c,d).

In 1912, Lewis and w:Edwin Bidwell Wilson used only real coordinates, writing the above operators as[R 20]

equivalent to (b,c,d).

Von Laue (1911/13)[edit | edit source]

In the first textbook on relativity in 1911, w:Max von Laue elaborated on Sommerfeld's methods and explicitly introduced the term four-potential (Viererpotential) in terms of vector potential and scalar potential , showing its showed its relation to the four-current P and the electromagnetic tensor (six-vector) together with the Lorenz gauge condition[R 21]

equivalent to (a,b,c,d).

In the second edition (preface dated 1912, published 1913), von Laue also formulated the Herglotz solution:[R 22]

equivalent to (e).

Silberstein (1911)[edit | edit source]

w:Ludwik Silberstein devised an alternative 4D calculus based on w:Biquaternions which, however, never gained widespread support. He defined the “potential-quaternion” (i.e. four-potential) in relation to the “current-quaternion” (i.e. four-current) C and the “electromagnetic bivector” (i.e. field tensor) [R 23]

equivalent to (a,b,c,d).

Kottler (1912)[edit | edit source]

w:Friedrich Kottler defined the four potential and its relation to four-current , electromagnetic field-tensor , and the Herglotz solution[R 24]

equivalent to (a,b,c,d,e) and subsequently was the first to give the generally covariant formulation of the inhomogeneous Maxwell's equations using metric tensor [R 25]

Einstein (1916)[edit | edit source]

In 1916, after finishing his general relativity, w:Albert Einstein also used the electromagnetic potential as a covariant four-vector, relating it to the covariant six-vector of the electromagnetic field (i.e. electromagnetic field tensor):[R 26]

equivalent to (a,d).

Historical sources[edit | edit source]

  1. Kirchhoff (1857)
  2. Riemann (1858)
  3. Lorenz (1867), pp. 292ff, 299
  4. Lorentz (1892), pp. 114–121
  5. Conway (1903), p. 154f
  6. Herglotz (1904), p. 550f
  7. Sommerfeld (1910a)
  8. Poincaré (1905b), p. 134
  9. Poincaré (1905b), p. 135
  10. Marcolongo (1906), p. 349
  11. Minkowski (1907/15), p. 929
  12. Minkowski (1907/15), p. 930
  13. Born (1909), p. 573-574
  14. Bateman (1910), p. 241
  15. Bateman (1910), p. 252
  16. Sommerfeld (1910b), p. 654, 665
  17. Sommerfeld (1910b), p. 667
  18. Sommerfeld (1910a), p. 764
  19. Lewis (1910), p. 176ff
  20. Wilson & Lewis (1910), p. 488ff
  21. Laue (1911), p. 100, 104
  22. Laue (1913), p. 125
  23. Silberstein (1911), p. 805f
  24. Kottler (1912), p. 1687ff
  25. Kottler (1912), p. 1688-1689
  26. Einstein (1916), p. 812
  • Born, M. (1909). "Die träge Masse und das Relativitätsprinzip". Annalen der Physik 333 (3): 571-584. http://hdl.handle.net/2027/uc1.b5622478. 
  • Conway, A. W. (1903), "The field of force due to a moving electron", Proc. London Math. Soc, 2(1): 154–65
  • Einstein, A. (1916). "Die Grundlage der allgemeinen Relativitätstheorie". Annalen der Physik 354 (7): 769-822. https://einsteinpapers.press.princeton.edu/vol6-doc/311. 
  • Herglotz, Gustav (1904). "Über die Berechnung retardierter Potentiale". Gött. Nachr. (6): 549–556. http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN252457811_1904&DMDID=DMDLOG_0056. 
  • Kirchhoff, G. (1857). "Ueber die Bewegung der Elektricität in Leitern". Annalen der Physik 178 (12): 529-544. 
  • Kottler, Friedrich (1912). "Über die Raumzeitlinien der Minkowski'schen Welt". Wiener Sitzungsberichte 2a 121: 1659–1759. https://hdl.handle.net/2027/mdp.39015051107277.