field arising from special relativity
The velocities and of two bodies moving along a lineobey, by the special theory of relativity, the addition rule
(1) |
where is the velocity of light. As is unreachable for any material body, itplays for the velocities of the bodies the role of the infinity. These velocities thus satisfy always
By (1) we get
for ; so behaves like the infinity.
One can define the mapping (http://planetmath.org/mapping) by setting
(2) |
which is easily seen to be a bijection.
Define also the binary operation (http://planetmath.org/binaryoperation) for the numbers (http://planetmath.org/number) of theopen interval
(http://planetmath.org/interval) by
(3) |
Then the system may be checked to be a ringand the bijective mapping (2) to behomomorphic (http://planetmath.org/structurehomomorphism):
Consequently, the system , as thehomomorphic image (http://planetmath.org/homomorphicimageofgroup) ofthe field , also itself is a field.
Baker [1] calls the numbers of the set , i.e. ,the Einstein numbers.
References
- 1 G. A. Baker, Jr.: “Einstein numbers”. –Amer. Math. Monthly 61(1954), 39–41.
- 2 H. T. Davis: College algebra
. Prentice-Hall, N.Y. (1940), 351.
- 3 T. Gregor & J. Haluška: Two-dimensional Einsteinnumbers and associativity.http://arxiv.org/abs/1309.0660arXiv (2013)