History of Topics in Special Relativity/Four-acceleration

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

Overview edit

The w:four-acceleration follows by differentiation of the four-velocity   of a particle with respect to the particle's w:proper time  . It can be represented as a function of three-velocity   and three-acceleration  :


and its inner product is equal to the proper acceleration


Historical notation edit

Killing (1884/5) edit

w:Wilhelm Killing discussed Newtonian mechanics in non-Euclidean spaces by expressing coordinates (p,x,y,z), velocity v=(p',x',y',z'), acceleration (p”,x”,y”,z”) in terms of four components, obtaining the following relations:[M 1]


If the Gaussian curvature   (with k as radius of curvature) is negative the acceleration becomes related to the hyperboloid model of hyperbolic space, which at first sight becomes similar to the relativistic four-acceleration 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. Also the dot product of acceleration and velocity differs from the relativistic result.

Minkowski (1907/08) edit

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 his lecture on December 1907, he didn't directly define a four-acceleraton vector, but used it implicitly in the definition of four-force and its density in terms of mass density  , mass m, four-velocity w:[R 1]


corresponding to (a).

In 1908, he denoted the derivative of the motion vector (four-velocity) with respect to proper time as "acceleration vector":[R 2]


corresponding to (a).

Frank (1909) edit

w:Philipp Frank (1909) didn't explicitly mentions four-acceleration as vector, though he used its components while defining four-force (X,Y,Z,T):[R 3]


corresponding to (a, b).

Bateman (1909/10) edit

The first discussion in an English language paper of four-acceleration (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. He first defined four-velocity[R 4]


from which he derived four-acceleration


equivalent to (a, b) as well as its inner product


equivalent to (c). He also defined the four-jerk


Wilson/Lewis (1912) edit

w:Gilbert Newton Lewis and w:Edwin Bidwell Wilson devised an alternative 4D vector calculus based on w:Dyadics which, however, never gained widespread support. They defined the “extended acceleration” as a “1-vector”, its norm, and its relation to the “extended force”:[R 5]


equivalent to (a,b).

Kottler (1912) edit

w:Friedrich Kottler defined four-acceleration in terms of four-velocity V as:[R 6]


equivalent to (a,b). He related its inner product to curvature   (in terms of Frenet-Serret formulas) and the “Minkowski acceleration” b:[R 7]


equivalent to (c) and defined the four-jerk


Von Laue (1912/13) edit

w:Max von Laue explicitly used the term “four-acceleration” (Viererbeschleunigung) for   and defined its inner product, and its relation to four-force K as well:[R 8]


corresponding to (a, c).

Silberstein (1914) edit

While w:Ludwik Silberstein used Biquaternions already in 1911, his first mention of the “acceleration-quaternion” Z was given in 1914. He also defined its conjugate, its Lorentz transformation, the relation of four-velocity Y, and its relation to four-force X:[R 9]


equivalent to (a,b).

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. Minkowski (1907/8), p. 107-108
  2. Minkowski (1908), p. 84
  3. Frank (1909), p. 437
  4. Bateman (1909/10), p. 253f
  5. Lewis/Wilson (1912), p. 444f
  6. Kottler (1912), p. 1663
  7. Kottler (1912), p. 1663, 1707
  8. Laue (1913), p. 69–70, 176
  9. Silberstein (1914), p. 183ff
  • Minkowski, H. (1909) [1908], "Raum und Zeit", Physikalische Zeitschrift, 10: 75–88