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).
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.
In 1907 (published 1908) 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:
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).
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.
In an unpublished manuscript on special relativity (written around 1912/14), w:Albert Einstein showed how to derive the components 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 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).
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).
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).
- ↑ Planck (1907), eq. 28,29,33
- ↑ Planck (1907), eq. 28,31,34
- ↑ Planck (1907), eq. 43,45,46
- ↑ Minkowski (1907), p. 102
- ↑ Minkowski (1908), p. 85
- ↑ Laue (1911), p. 82ff, 184f
- ↑ Laue (1913), p. 230
- ↑ Lewis/Wilson (1911), p. 420, 444
- ↑ Lewis/Wilson (1911), p. 467
- ↑ Lewis/Wilson (1911), p. 478f
- ↑ Einstein (1912/14), p. 96f
- ↑ Einstein/Grossmann (1913), p. 227, 229, 232
- ↑ Einstein (1914), p. 1060, 1082
- ↑ Cunningham (1914), p. 162f
- ↑ Weyl (1918), p. 148
- ↑ Weyl (1918), p. 125-128
- ↑ Weyl (1919), p. 158
- ↑ Weyl (1919), p. 233-234, 257-259
- Cunningham, E. (1914). The principle of relativity. Cambridge: University Press. https://archive.org/details/principleofrelat00cunniala.
- Einstein, A. (1912-14). "Einstein's manuscript on the special theory of relativity". The collected papers of Albert Einstein. 4. pp. 3-108. https://einsteinpapers.press.princeton.edu/vol4-doc/25.
- Einstein, A. & Grossmann, M. (1913). "Entwurf einer verallgemeinerten Relativitätstheorie und eine Theorie der Gravitation". Zeitschrift für Mathematik und Physik 62: 225-261. https://einsteinpapers.press.princeton.edu/vol4-doc/324.
- Einstein, A. (1914). "Die Formale Grundlage der allgemeinen Relativitätstheorie". Berliner Sitzungsberichte 1914 (2): 1030–1085. https://einsteinpapers.press.princeton.edu/vol6-doc/100.
- 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 (1908) . "Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern". Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse: 53–111. https://archive.org/details/nachrichten09klasgoog.
- Minkowski, Hermann (1909) . "Raum und Zeit". Physikalische Zeitschrift 10: 75–88.