field arising from special relativity

The velocities u and v of two bodies moving along a line obey, by the special theory of relativity, the addition rule

uv:=u+v1+uvc2, (1)

where c is the velocity of light.  As c is unreachable for any material body, it plays for the velocities of the bodies the role of the infinityMathworldPlanetmathPlanetmath.  These velocities v thus satisfy always


By (1) we get


for |v|<c;  so c behaves like the infinity.

One can define the mapping (  f:(-c,c)=S  by setting

f(x):=ctanhx (2)

which is easily seen to be a bijectionMathworldPlanetmath.

Define also the binary operationMathworldPlanetmath ( for the numbers ( u,v of the open intervalPlanetmathPlanetmath (  (-c,c) by

uv=ctanh[(artanhuc)(artanhvc)]. (3)

Then the system (S,,) may be checked to be a ring and the bijectiveMathworldPlanetmath mapping (2) to be homomorphic (


Consequently, the system (S,,), as the homomorphic imagePlanetmathPlanetmathPlanetmath ( of the field (,+,), also itself is a field.

Baker [1] calls the numbers of the set S, i.e. (-c,c), the Einstein numbers.


  • 1 G. A. Baker, Jr.: “Einstein numbers”. –Amer. Math. Monthly 61 (1954), 39–41.
  • 2 H. T. Davis: College algebraMathworldPlanetmathPlanetmath. Prentice-Hall, N.Y. (1940), 351.
  • 3 T. Gregor & J. Haluška: Two-dimensional Einstein numbers and associativity. (2013)

Title field arising from special relativity
Canonical name FieldArisingFromSpecialRelativity
Date of creation 2016-04-20 13:42:53
Last modified on 2016-04-20 13:42:53
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Topic