History of Topics in Special Relativity/Four-force (electromagnetism)

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

Overview[edit | edit source]

The electromagnetic w:four-force or covariant w:Lorentz force can be expressed as

(a) the rate of change in the w:four-momentum of a particle with respect to the particle's w:proper time ,
(b) function of three-force
(c) assuming constant mass as the product of invariant mass m and four-acceleration
(d) using the Lorentz force per unit charge
(e) the product of the electromagnetic tensor with the four-velocity and charge q
(f) by integrating the four-force density with respect to rest unit volume

The corresponding four-force density is defined as

(a1) the rate of change of four-momentum density with rest mass density and four-velocity ,
(b1) function of three-force density
(c1) assuming constant mass the product of rest mass density and four-acceleration
(d1) using the Lorentz force density with as charge density,
(e1) as the product of electromagnetic tensor with four-current or with four-velocity using rest charge density ,
(f1) as the negative four-divergence of the electromagnetic energy-momentum tensor (compare with History of Topics in Special Relativity/Stress-energy tensor (electromagnetic)):

Historical notation[edit | edit source]

Poincaré (1905/6)[edit | edit source]

w:Henri Poincaré (July 1905, published January 1906) defined the Lorentz force and its Lorentz transformation:[R 1]

with and .

in which (X,Y,Z,T) represent the components of four-force density (b1, d1) because


Additionally, he explicitly obtained the four-force per unit charge by setting :[R 2]

with and

equivalent to (b, d) and obtained its Lorentz transformation by multiplying the transformation of (X,Y,Z,T) by


Minkowski (1907/8)[edit | edit source]

w:Hermann Minkowski employed matrix representation of four-vectors and six-vectors (i.e. antisymmetric second rank tensors) and their product from the outset. In a lecture held in November 1907, published 1915, Minkowski defined the four-force density X,Y,Z,iA as the product of the electromagnetic tensor and the four-current with as charge density and as velocity:[R 3]

equivalent to (b1, d1, e1) because .

In another lecture from December 1907, he used the symbols (i.e. ) and (i.e. ) which he represented in the form of “vectors of second kind”, i.e. the electromagnetic tensor f and its dual , from which he derived the electric rest force and magnetric rest force as the product with four-velocity w, which in turn can be used to express F and f and four-conductivity:[R 4]

which becomes the covariant Lorentz force (d,e) when multiplied with the charge. He gave the general expression of four-force density as

equivalent to (f).

In his lecture “space and time” (1908, published 1909), Minkowski defined the “moving force vector” as[R 5]

equivalent to (b) because is the four-velocity and .

Born (1909)[edit | edit source]

w:Max Born (1909) summarized Minkowski's work using index notation, defining the equations of motion in terms of rest mass density and rest charge density [R 6]

equivalent to (c1, d1, e1).

Frank (1909)[edit | edit source]

w:Philipp Frank (1909) discussed “electromagnetic mechanics” by defining four-force (X,Y,Z,T) as follows:[R 7]

corresponding to (b, c).

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

While Minkowski used the four-force density as with , w:Max Abraham (1909) directly used as four-force density, and expressed it in terms of “momentum equations” and an “energy equation” using momentum density , energy density , Poynting vector , Joule heat Q[R 8]

or alternatively by introducing “relative stresses” and the “relative energy flux”:

all equivalent to (f1).

In a subsequent paper (1909) he formulated these relations as follows[R 9]

In addition, Abraham gave arguments in favor of his choice to use directly as four-force density: He argued that Minkowski's definition of force as the product of constant rest mass with four-acceleration together with a complementary force is disadvantageous at non-adiabatic motion, because mass-energy equivalence according to which mass depends on its energy content would suggest a variable rest mass , and showed that is compatible with the general definition of force density over volume dv as the rate of momentum change without the need of complementary components[R 10]


As he showed in 1910, this implies that the equations of motion in terms of four-force density and rest mass density and four-velocity assume the form:[R 11]

This is equivalent to (a1) defining four-force density as the rate of change of four-momentum density.

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

The first discussion in an English language paper of four-force 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 following invariants (with in relativity):[R 12]

equivalent to (e) because it can be seen as the product of charge, four-velocity and the electromagnetic tensor, which is the definition of the covariant Lorentz force.

Ignatowski (1910)[edit | edit source]

w:Wladimir Ignatowski (1910) formulated the four-force in terms of the Lorentz force as well as the equation of motion, and following Abraham (1909) he assumed variable rest mass in the case of non-adiabatic motion[R 13]

equivalent to (b, d). He also derived the four-force density in terms of the density of three-force or Lorentz force by integrating three-force with respect to volume:[R 14]

equivalent to (b1, d1).

Sommerfeld (1910)[edit | edit source]

In an influential paper, w:Arnold Sommerfeld translated Minkowski's matrix formalism into a four-dimensional vector formalism involving four-vectors and six-vectors. He defined the four-force density as the covariant product of a six-vector denoted as “field vector” f (now known as electromagnetic tensor) and the four-current P, which he related to Lorentz force density :[R 15]

equivalent to (b1, d1, e1).

Laue (1911-13)[edit | edit source]

In the first textbook on relativity, w:Max von Laue (1911) followed Abraham in defining the four-force density (Viererkraft) F as the product of a six-vector denoted as “field vector” (now known as electromagnetic tensor) and the four-current related to Lorentz force density , and alternatively as the divergence of the electromagnetic stress-energy tensor T,[R 16] which included some printing errors corrected in the 1913 edition[R 17]

equivalent to (b1, d1, e1, f1).

In 1913, Laue also showed that four-force density F and three force density can be used to derive the “Minkowskian force vector” (i.e. four-force) K and three-force per unit charge by defining the volume :[R 18]

Silberstein (1911-14)[edit | edit source]

w:Ludwik Silberstein in 1911 and published 1912, devised an alternative 4D calculus based on w:Biquaternions which, however, never gained widespread support. He defined the "force-quaternion" (i.e. four-force density) as the electrical part of P, which he related to the “current-quaternion” (i.e. four-current) C and the “electromagnetic bivector” (i.e. w:Weber vector) }[R 19]

equivalent to (d1,e1,f1).

In his textbook on quaternionic special relativity written 1914, he defined four-force density using stress-energy tensor like Abraham and Laue, as well as in terms of Minkowski's alternative force definition using four-velocity Y:[R 20]

equivalent to (d1,e1,f1).

Lewis and Wilson (1912)[edit | edit source]

w:Gilbert Newton Lewis and w:Edwin Bidwell Wilson explicitly defined a four-vector called “extended momentum” (i.e. four-momentum) , deriving the “extended force” (i.e. four force) using as four-acceleration:[R 21]

equivalent to (a, c).

Kottler (1912)[edit | edit source]

While formulating electrodynamics in a generally covariant way, w:Friedrich Kottler expressed the “Minkowski force” in terms of the electromagnetic field-tensor , four-current , stress-energy tensor :[R 22]

equivalent to (e, f).

In 1914 Kottler derived the equations of motion from the “Nordström tensor” (i.e. dust solution of the stress energy tensor) in terms of rest mass density , which he equated to the action of a constant external electromagnetic field in terms of electromagnetic tensor and charge density :[R 23]

equivalent to (f).

Einstein (1912-14)[edit | edit source]

In an unpublished manuscript on special relativity, written between 1912-1914, w:Albert Einstein wrote the four-force density in terms of electromagnetic tensor , four-current , stress-energy tensor :[R 24]

equivalent to (d1,e1,f1).

References[edit | edit source]

  1. Poincaré (1905/06), p. 169
  2. Poincaré (1905/06), p. 173
  3. Minkowski (1907/15), p. 930.
  4. Minkowski (1907/8), p. 84f.
  5. Minkowski (1907/8), p. 85
  6. Born (1909), p. 582
  7. Frank (1909), p. 437, 441
  8. Abraham (1909a), p. 5-6, 17
  9. Abraham (1909b), p. 737-738
  10. Abraham (1909b), p. 739
  11. Abraham (1910), p. 45
  12. Bateman (1910), p. 241
  13. Ignatowsky (1910), p. 33-34
  14. Ignatowsky (1910), p. 27
  15. Sommerfeld (1910a), p. 770
  16. Laue (1911), p. 79–80, 82
  17. Laue (1913), p. 92, 94
  18. Laue (1913), p. 176–177
  19. Silberstein (1911), p. 807
  20. Silberstein (1914), p. 219f, 240
  21. Lewis & Wilson (1912), p. 420, 445
  22. Kottler (1912), p. 1703
  23. Kottler (1914a), p. 718
  24. Einstein (1912), p. 82f
  • Kottler, Friedrich (1914a). "Relativitätsprinzip und beschleunigte Bewegung". Annalen der Physik 349 (13): 701–748. doi:10.1002/andp.19143491303. http://gallica.bnf.fr/ark:/12148/bpt6k15347v.image.f737. 
  • Laue, Max von (1911). Das Relativitätsprinzip. Braunschweig: Vieweg. https://archive.org/details/dasrelativittsp00lauegoog. 
  • Laue, Max von (1913). Das Relativitätsprinzip (2. Edition). Braunschweig: Vieweg. http://digitale.beic.it/primo_library/libweb/action/dlDisplay.do?vid=BEIC&docId=39bei_digitool6467296. 
  • 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. https://www.jstor.org/stable/20022840. 
  • Minkowski, Hermann (1915) [1907], "Das Relativitätsprinzip", Annalen der Physik, 352 (15): 927–938