# 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

 $\displaystyle u\oplus v\;:=\;\frac{u+v}{1+\frac{uv}{c^{2}}},$ (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

 $|v|\;<\;c.$

By (1) we get

 $c\oplus c\;=\;c,\quad c\oplus v\;=\;c$

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

One can define the mapping (http://planetmath.org/mapping)  $f:\;\mathbb{R}\to(-c,\,c)=S$  by setting

 $\displaystyle f(x)\;:=\;c\;\tanh{x}$ (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

 $\displaystyle u\odot v\;=\;c\;\tanh\left[\left(\mbox{artanh}\frac{u}{c}\right)% \left(\mbox{artanh}\frac{v}{c}\right)\right].$ (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):

 $f(x+y)\;=\;f(x)\oplus f(y),\quad f(xy)\;=\;f(x)\odot f(y)$

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 FieldArisingFromSpecialRelativity 2016-04-20 13:42:53 2016-04-20 13:42:53 pahio (2872) pahio (2872) 8 pahio (2872) Topic