# gyr: Gyr function In RobinHankin/gyrogroup: The Lorentz Transform in Relativistic Physics

## Usage

 ```1 2 3``` ```gyr(u, v, x) gyr.a(u, v, x) gyrfun(u, v) ```

## Arguments

 `u,v,x` Three-velocities, objects of class `3vel`

## Details

Function `gyr(u,v,x)` returns the three-vector gyr[u,v]x.

Function `gyrfun(u,v)` returns a function that returns a three-vector; see examples.

The speed of light (1 by default) is not used directly by these functions; set it with `sol()`.

## Note

Function `gyr()` is slightly faster than `gyr.a()`, which is included for pedagogical reasons.

Function `gyr()` is simply

`add3(neg3(add3(u,v)),add3(u,add3(v,x)))`

while function `gyr.a()` uses the slower but more transparent idiom

` -(u+v) + (u+(v+x)) `

## Author(s)

Robin K. S. Hankin

## References

• Ungar 2006. “Thomas precession: a kinematic effect...”. European Journal of Physics, 27:L17-L20.

• Sbitneva 2001. “Nonassociative geomery of special relativity”. International Journal of Theoretical Physics, volume 40, number 1, pages 359–362

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34``` ```u <- r3vel(10) v <- r3vel(10) w <- r3vel(10) x <- as.3vel(c(0.4,0.1,-0.5)) y <- as.3vel(c(0.1,0.2,-0.7)) z <- as.3vel(c(0.2,0.3,-0.1)) gyr(u,v,x) # gyr[u,v]x f <- gyrfun(u,v) g <- gyrfun(v,u) f(x) f(r3vel(10)) f(g(x)) - x # zero, by eqn 9 g(f(x)) - x # zero, by eqn 9 (x+y) - f(y+x) # zero by eqn 10 (u+(v+w)) - ((u+v)+f(w)) # zero by eqn 11 # Following taken from Sbitneva 2001: rbind(x+(y+(x+z)) , (x+(y+x))+z) # left Bol property rbind((x+y)+(x+y) , x+(y+(y+x))) # left Bruck property sol(299792458) # speed of light in SI as.3vel(c(1000,3000,1000)) + as.3vel(c(1000,3000,1000)) ## should be close to Galilean result sol(1) # revert to default c=1 ```

RobinHankin/gyrogroup documentation built on Feb. 19, 2019, 3:20 a.m.