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
$u\oplus v:={\displaystyle \frac{u+v}{1+\frac{uv}{{c}^{2}}}},$ | (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 infinity^{}. These velocities $v$ thus satisfy always
$$ |
By (1) we get
$$c\oplus c=c,c\oplus v=c$$ |
for $$; so $c$ behaves like the infinity.
One can define the mapping (http://planetmath.org/mapping) $f:\mathbb{R}\to (-c,c)=S$ by setting
$f(x):=c\mathrm{tanh}x$ | (2) |
which is easily seen to be a bijection^{}.
Define also the binary operation^{} (http://planetmath.org/binaryoperation) $\odot $ for the numbers (http://planetmath.org/number) $u,v$ of the open interval^{} (http://planetmath.org/interval) $(-c,c)$ by
$u\odot v=c\mathrm{tanh}\left[\left(\text{artanh}{\displaystyle \frac{u}{c}}\right)\left(\text{artanh}{\displaystyle \frac{v}{c}}\right)\right].$ | (3) |
Then the system $(S,\oplus ,\odot )$ may be checked to be a ring and the bijective^{} mapping (2) to be homomorphic (http://planetmath.org/structurehomomorphism):
$$f(x+y)=f(x)\oplus f(y),f(xy)=f(x)\odot f(y)$$ |
Consequently, the system $(S,\oplus ,\odot )$, as the homomorphic image^{} (http://planetmath.org/homomorphicimageofgroup) of the field $(\mathbb{R},+,\cdot )$, also itself is a field.
Baker [1] calls the numbers of the set $S$, i.e. $(-c,c)$,
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 Einstein numbers and associativity. http://arxiv.org/abs/1309.0660arXiv (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 |