History of Topics in Special Relativity/Four-momentum

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

Overview edit

The w:four-momentum   is defined as the product of mass and w:four-velocity   or alternatively can be obtained by integrating the four-momentum density   with respect to volume V (the four-momentum density corresponds to components   of the stress energy tensor combining energy density W and momentum density  ). In addition, replacing rest mass with rest mass density   in terms of rest volume   produces the mass four-current   in analogy to the electric four-current:


Without explicitly defining the four-momentum vector, the Lorentz transformation of all components of (a) was given by #Planck (1907), while the Lorentz transformation of all components of (c) were given by #Laue (1911-13). The first explicit definition of (a) was given by #Minkowski (1908), followed by #Lewis and Wilson (1912), #Einstein (1912-14), #Cunningham (1914), #Weyl (1918-19). Four-momentum density (c) played a role in the papers of #Einstein (1912-14) and #Lewis and Wilson (1912). The material four-current (d) was given by #Laue (1913) and #Weyl (1918-19).

Historical notation edit

Planck (1907) edit

After w:Albert Einstein gave the energy transformation into the rest frame in 1905 and the general energy transformation in May 1907, w:Max Planck in June 1907 defined the transformation of both momentum   and energy E as follows[R 1]


or simplifying in terms of enthalpy R=E+pV:[R 2]


and the transformations into the rest frame[R 3]


Even though Planck wasn't using four-vectors, his formulas correspond to the Lorentz transformation of four-vector  , becoming the ordinary four-momentum   by setting the pressure p=0.

Minkowski (1907-09) edit

In 1907 (published 1908) w:Hermann Minkowski defined the following continuity equation with   as rest mass density and w as four-velocity:[R 4]


which implies the mass four-current equivalent to (d).

The first mention of four-momentum (a) was given by Minkowski in his lecture “space and time” from 1908 (published 1909), calling it "momentum-vector" (“Impulsvektor”) as the product of mass m with the motion-vector (i.e. four-velocity) at a point P[R 5]. He further noted that if the time component of four-momentum is multiplied by   it becomes the kinetic energy:


Laue (1911-13) edit

w:Max von Laue (1911) in his influential first textbook on relativity, gave the Lorentz transformation of the components of the symmetric “world tensor” T (i.e. stress energy tensor), with the l=ict components being energy flux  , momentum density  , energy density W, and pointed out that the divergence of those l-components represents the energy conservation theorem (with A as power of the force density):[R 6]


which components correspond to four-momentum density (c) in case of vanishing pressure p, even though Laue didn't directly denoted it as a four-vector.

In the second edition (1912, published 1913), Laue discussed hydrodynamics in special relativity, defining the four-current of a material volume element in terms of rest mass density   and four-velocity Y, and its continuity equation:[R 7]


equivalent to material four-current (d).

Lewis and Wilson (1912) edit

w:Edwin Bidwell Wilson and w:Gilbert Newton Lewis (1912) devised an alternative 4D vector calculus based on w:Dyadics which, however, never gained widespread support. They explicitly defined “extended momentum” (i.e. four-momentum)   and used it to derive the “extended force” (i.e. four force) together with   as four-acceleration:[R 8]


equivalent to (a). Using rest mass density  , they also defined the extended vector[R 9]


equivalent to the material four-current (d). Then they defined the four-momentum of radiant energy representing total momentum and energy per volume   by integrating electromagnetic energy density   and the Poynting vector  :[R 10]


equivalent to (b). They added, however, that the corresponding energy density vector   is not a four-vector because it is not independent of the chose axis.

Einstein (1912-14) edit

In an unpublished manuscript on special relativity (written around 1912/14), w:Albert Einstein showed how to derive the components of the momentum-energy four-vector from the components   (four-momentum density) of the stress-energy tensor (overline indicates integration over volume,   is four-velocity):[R 11]


equivalent to (a,b,c).

In the context of his Entwurf theory (a precursor of general relativity), Einstein (1913) formulated the following equations for momentum J and energy E using the metric tensor  , from which he concluded that momentum and energy of a material point form a “covariant vector” (i.e. covariant four-momentum), and also showed that the corresponding volume densities are equal to certain components of the stress-energy tensor   (i.e. w:dust solution):[R 12]


equivalent to (a,b,c) in the case of   being the Minkowski tensor.

In 1914 Einstein summarized his previous arguments using the covariant four-vector   (i.e. covariant four-momentum) and explicitly showed that in the case of   being the Minkowski tensor it becomes the ordinary four-momentum of special relativity. He also argued in a footnote why (in terms of his theory of gravitation) this covariant four-momentum   is preferable over the contravariant four-momentum  :[R 13]


equivalent to (a).

Cunningham (1914) edit

Like Wilson and Lewis, w:Ebenezer Cunningham used the expression “extended momentum”   (i.e. four-momentum), and derived the four-force from it:[R 14]


equivalent to (a, b).

Weyl (1918-19) edit

In the first edition of his book “space time matter”, w:Hermann Weyl (1918) defined the “material current” in terms of rest mass density and four-velocity, together with its continuity equation:[R 15]


equivalent to (d).

In 1919, in the framework of general relativity, he expressed the pseudotensor density of total energy as  , with the integral   (i.e. four-momentum) of   (i.e. four-momentum density) in space   = const. representing energy (i=0) and momentum (i=1,2,3). For an arbitrary coordinate system he defined is as the product of mass and four-velocity[R 16]


equivalent to (a,b,c).

In the third edition of his book (1919), the description of the material current remained the same as in the first edition,[R 17] but this time he also included a description of four-momentum   in terms of four-momentum density  :[R 18]


equivalent to (a,b,c).

References edit

  1. Planck (1907), eq. 28,29,33
  2. Planck (1907), eq. 28,31,34
  3. Planck (1907), eq. 43,45,46
  4. Minkowski (1907), p. 102
  5. Minkowski (1908), p. 85
  6. Laue (1911), p. 82ff, 184f
  7. Laue (1913), p. 230
  8. Lewis/Wilson (1911), p. 420, 444
  9. Lewis/Wilson (1911), p. 467
  10. Lewis/Wilson (1911), p. 478f
  11. Einstein (1912/14), p. 96f
  12. Einstein/Grossmann (1913), p. 227, 229, 232
  13. Einstein (1914), p. 1060, 1082
  14. Cunningham (1914), p. 162f
  15. Weyl (1918), p. 148
  16. Weyl (1918), p. 125-128
  17. Weyl (1919), p. 158
  18. Weyl (1919), p. 233-234, 257-259
  • Cunningham, E. (1914), The principle of relativity, Cambridge: University Press
  • 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)
  • Einstein, A. & Grossmann, M. (1913), "Entwurf einer verallgemeinerten Relativitätstheorie und eine Theorie der Gravitation", Zeitschrift für Mathematik und Physik, 62: 225–261{{citation}}: CS1 maint: multiple names: authors list (link)
  • Einstein, A. (1914), "Die Formale Grundlage der allgemeinen Relativitätstheorie", Berliner Sitzungsberichte, 1914 (2): 1030–1085
  • 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{{citation}}: 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
  • Minkowski, H. (1909) [1908], "Raum und Zeit", Physikalische Zeitschrift, 10: 75–88
  • Planck, M. (1907), "Zur Dynamik bewegter Systeme", Berliner Sitzungsberichte, Erster Halbband, 29: 542–570
    Reprinted in: Planck, M. (1908) [1907], "Zur Dynamik bewegter Systeme", Annalen der Physik, 331 (6): 1–34