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

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

Overview edit

The w:four-force   is not only applicable to electromagnetic phenomena (compare with History of Topics in Special Relativity/Four-force (electromagnetism)), but also applies to mechanics in general, thus it can be used in relation to fluids, dust, Lorentz invariant gravity models etc.. It is defined as

(a) the rate of change in the four-momentum   with respect to proper time  ,
(b) function of three-force  
(c) assuming constant mass as the product of invariant mass m and four-acceleration  .
(d) 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  
(b1) function of three-force density  
(c1) assuming constant mass the product of rest mass density   and four-acceleration  
(d1) the four-divergence of the energy-momentum tensor   (such as for fluids or dust). In case  , the four corresponding equations represent the energy and momentum conservation laws.


Examples are the four-force density using the perfect fluid stress energy tensor (compare with History of Topics in Special Relativity/Stress-energy tensor (matter)):


or using the dust solution in case of vanishing pressure:


Historical notation edit

Killing (1884/5) edit

w:Wilhelm Killing discussed Newtonian mechanics in non-Euclidean spaces by expressing position, velocity, acceleration and force in terms of four components. He expressed the four components of force R=(P,X,Y,Z), its norm, its inner product with coordinates (p,x,y,z), and the equations of motion as the product of mass with acceleration as follows:[M 1]


If the Gaussian curvature   (with k as radius of curvature) is negative the force becomes related to the hyperboloid model of hyperbolic space, which at first sight becomes similar to the relativistic four-force in Minkowski space by setting   with c as speed of light. However, Killing obtained his results by differentiation with respect to Newtonian time t, not relativistic proper time, so his expressions aren't relativistic four-vectors in the first place, in particular they don't involve a limiting speed.

Poincaré (1905/6) edit

w:Henri Poincaré (July 1905, published January 1906) argued, that the Lorentz transformation not only applies to electrodynamics, but to all other phenomena as well including mechanics. For instance, he explicitly defined gravitation as non-electromagnetic in origin and applied the following expression of four-force to his Lorentz invariant model of gravitation:[R 1]

  with   and  

equivalent to (b) because


Minkowski (1907) edit

In an appendix to his lecture from December 1907 (published 1908), w:Hermann Minkowski extended the postulate of relativity to mechanics in general, defining four-force density   with w as four-velocity and S as stress energy tensor[R 2]


equivalent to (d1). He went on to show that these relations can be also used to define the equations of motion of mechanics in terms of constant rest mass density  :[R 3]


equivalent to (b1, c1, d1) as well as (f) since the first line includes the dust stress-energy tensor. Eventually he defined a “moving force” as the product of constant rest mass and four-acceleration[R 4]


equivalent to (c). Minkowski's assumption of constant rest mass was later challenged by Abraham (see next section).

Abraham (1909-12) edit

In 1909, w:Max Abraham pointed out that the relativity principle requires that the mechanical forces must transform like the electromagnetic ones, so there must be a four-dimensional tensor for mechanics (i.e. mechanical stress energy tensor) in analogy to the electromagnetic one, and that the relation   can alternatively be interpreted as relation between mechanical momentum and energy density:[R 5]


equivalent to (a) when interpreted as mechanical force. While Minkowski (1907) assumed constant rest mass density, Abraham (1909) held that mass-energy equivalence, according to which mass depends on its energy content, would suggest a variable rest mass  .

In 1912, Abraham introduced the expression “world tensor of motion”   (equivalent to the dust tensor) while formulating his first theory of gravitation. It has ten components representing kinetic stresses, energy flux   and momentum   of matter in terms of rest mass density  , which he combined with the world tensor   (representing the electromagnetic-, gravitational-, and stress field) in order to formulate the momentum and energy conservation theorems:[R 6]


equivalent to (f) when only the dust tensor   is considered.

Nordström (1910–13) edit

The force definitions of both Abraham (variable rest mass) and Minkowski (constant rest mass) were elaborated by w:Gunnar Nordström (1910), who defined two variants of four-force density using a “four-dimensional tensor” (i.e. dust solution)   consisting of rest mass density   and four-velocity  . The first formulation was based on Abraham's assumption of variable rest mass density:[R 7]


equivalent to (f), and the second one on Minkowski's assumption of constant rest mass density:[R 8]


equivalent to (f). In 1911, Nordström only used the first variant with variable rest mass density and considered pressure in a “material fluid” as well.[R 9]

In 1913, he added an “elastic stress tensor” p in order to reformulated Laue's symmetrical four dimensional tensor T representing spatial stresses and mechanical momentum and energy density, which he used to add an elastic component   to the four-force-density   to give the equation of motion:[R 10]


equivalent to (f). He went on to employ this notion in his theory of gravitation.

Ignatowski (1911) edit

w:Wladimir Ignatowski derived the hydrodynamic four-force density in terms of mass density   and pressure   in case of perfect fluids:[R 11]


which corresponds to the four-force of perfect fluid (e).

Von Laue (1911-20) edit

In the first textbook on relativity (1911), w:Max von Laue defined the mechanical ponderomotive force F based on world tensor T (i.e. mechanical stress-energy tensor), implying the complete reduction of mechanical inertia to energy and stresses:[R 12]


equivalent to (c).

In the second edition (1912, published 1913), he followed #Herglotz (1911) and #Lamla (1911/12) in defining the four-force K in order to produce the Lagrangian (1) and Eulerian (2) fundamental equations of hydrodynamics, as well as for the case of least compressibility (3), where   is the normal rest mass, p the pressure, Y the four-velocity:[R 13]


equivalent to (c).

In the fourth edition (1921), he defined the four-force density using the kinetic stress energy tensor, with rest energy density  , rest energy  , rest volume  :[R 14]


equivalent to (b,c) and the dust solution (f).

Herglotz (1911) edit

w:Gustav Herglotz gave a complete theory of elasticity in special relativity, including equations of motion in different forms, which he defined using coordinates   after deformation,   and   before deformation, from which he derived the deformation quantities   and  , together with the kinetic potential  . He gave the Lagrangian equation of motion in terms of four-force density:[R 15]


and the Euler equations of motion by defining stress-energy tensor  , whose components can be related to momentum density  , energy density  , velocity u,v,w:[R 16]


Then he formulated a third kind of equations of motion by introducing “relative” stresses   into the Euler equations:[R 17]


He finally showed how to modify   using mass density m and pressure p, so that previous equations become the equations of motion of a perfect fluid:[R 18]


which corresponds to the four-force of perfect fluid (e).

Lamla (1911/12) edit

Ernst Lamla (1911, published 1912) derived the equation of motion of hydrodynamics independently of Herglotz. Using pressure p and rest density g, he gave the Lagrangian form:[R 19]


which corresponds to the four-force of perfect fluid (e), as well as Euler's equation of motion using four-velocity w:[R 20]


and also gave the Euler equations of motion in the case of substances of least compressibility[R 21]


Lewis & Wilson (1912) edit

w:Edwin Bidwell Wilson and w:Gilbert Newton Lewis devised an alternative 4D vector calculus based on w:Dyadics which, however, never gained widespread support. They defined the dyadic   using four-velocity   and rest mass density   in order to formulate the fundamental equation of hydrodynamics:[R 22]


equivalent to the (f).

Einstein (1913-16) edit

In 1913, in the context of his Entwurf theory (a precursor of general relativity), w:Albert Einstein defined the equation for incoherent matter using the dust stress energy tensor  :[R 23]


equivalent to (f), and in 1916 he used the perfect fluid tensor  :[R 24]


equivalent to (e) in the case of   being the Minkowski tensor.

Kottler (1914) edit

w:Friedrich Kottler derived the equations of motion from the “Nordström tensor” (i.e. dust tensor) in terms of rest mass density  :[R 25]


equivalent to (f), which he then related to the action of a constant external electromagnetic field.

References edit

  1. Killing (1884/5), p. 5
  • Killing, W. (1885) [1884], "Die Mechanik in den Nicht-Euklidischen Raumformen", Journal für die Reine und Angewandte Mathematik, 98: 1–48
  1. Poincaré (1905/06), p. 173
  2. Minkowski (1907/8), p. 97
  3. Minkowski (1907/8), p. 107
  4. Minkowski (1907/8), p. 107
  5. Abraham (1909b), p. 737f
  6. Abraham (1912), p. 737f
  7. Nordström (1910), eq. 4'
  8. Nordström (1910), eq. 4
  9. Nordström (1911)
  10. Nordström (1913), eq. 6
  11. Ignatowsky (1911), p. 442
  12. Laue (1911), p. 149, 184f
  13. Laue (1913), p. 232, 238, 244
  14. Laue (1921), p. 207-209, 237
  15. Herglotz (1911), pp. 505-506
  16. Herglotz (1911), pp. 507-508
  17. Herglotz (1911), p. 510
  18. Herglotz (1911), p. 514
  19. Lamla (1912), p. 27-29
  20. Lamla (1912), p. 39-40
  21. Lamla (1912), p. 64-65
  22. Lewis & Wilson (1912), p. 494ff
  23. Einstein (1913), p. 261
  24. Einstein (1916), p. 809
  25. Kottler (1914a), p. 718
  • Poincaré, H. (1906) [1905], "Sur la dynamique de l'électron", Rendiconti del Circolo Matematico di Palermo, 21: 129–176