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)  $f:ℝ\to (-c,c)=S$  by setting

$f(x):=c\tanh x$ (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

$u\odot v=c\tanh \left[\left(\text{artanh}{\displaystyle \frac{u}{c}}\right)\left(\text{artanh}{\displaystyle \frac{v}{c}}\right)\right].$ (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 $(ℝ,+,\cdot )$, also itself is a field.

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

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